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Graduate Texts in Physics Edouard B. Manoukian Quantum Field Theory I Foundations and Abelian and Non-Abelian Gauge Theories
Graduate Texts in Physics Series editors Kurt H. Becker, Polytechnic School of Engineering, Brooklyn, USA Sadri Hassani, Illinois State University, Normal, USA Bill Munro, NTT Basic Research Laboratories, Atsugi, JapanRichard Needs, University of Cambridge, Cambridge, UKJean-Marc Di Meglio, Université Paris Diderot, Paris, France William T. Rhodes, Florida Atlantic University, Boca Raton, USA Susan Scott, Australian National University, Acton, AustraliaH. Eugene Stanley, Boston University, Boston, USA Martin Stutzmann, TU München, Garching, Germany Andreas Wipf, Friedrich-Schiller-Univ Jena, Jena, Germany
Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging ﬁeldswithin physics, both pure and applied. These textbooks serve students at theMS- or PhD-level and their instructors as comprehensive sources of principles,deﬁnitions, derivations, experiments and app lications (as relevant) for their mastery and teaching, respectively. Internati onal in scope and relevance, the textbooks correspond to course syllabi sufﬁciently to serve as required reading. Their didacticstyle, comprehensiveness and coverage of fundamental material also make themsuitable as introductions or references for scientists entering, or requiring timelyknowledge of, a research ﬁeld. More information about this series at http://www.springer.com/series/8431
Edouard B. Manoukian Quantum Field Theory I Foundations and Abelian and Non-Abelian Gauge Theories 123
Edouard B. Manoukian The Institute for Fundamental StudyNaresuan UniversityPhitsanulok, Thailand ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-319-30938-5 ISBN 978-3-319-30939-2 (eBook)DOI 10.1007/978-3-319-30939-2 Library of Congress Control Number: 2016935720 © Springer International P ublishing Switzerland 2016 This work is subject to copyright. All rights are rese rved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of tra nslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any oth er physical way, and transmission or information storage and retrieval, electronic adaptation, com puter software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations an d therefore free for general use. The publisher, the authors and the editors are safe t o assume that the advice and information in this book are believed to be true and accurate at the date of pub lication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. Printed on acid-free paperThis Springer imprint is published by Springer Nature The registered company is Springer Int ernational Publishing AG Switzerland
Preface to Volume I This textbook is based on lectures given in quantum ﬁeld theory (QFT) over the years to graduate students in theoretical and experimental physics. The writing of the book spread over three continents: North America (Canada), Europe (Ireland),and Asia (Thailand). QFT was born about 90 years ago, when quantum mechanicsmet relativity, and is still going strong. The book covers, pedagogically, the wide spectrum of developments in QFT emphasizing, however, those parts which arereasonably well understood and for which satisfactory theoretical descriptions havebeen given. The legendary Richard Feynman inhis 1958 Cornell, 1959–1960 Cal Tech lectures on QFT of fundamental processes, the ﬁrst statement he makes, the veryﬁrst one, is that the lectures cover all of physics . 1One quickly understands what Feynman meant by covering all of physics. The role of fundamental physics is todescribe the basic interactions of Nature and QFT, par excellence, is supposed to
do just that . Feynman’s statement is obviously more relevant today than it was then, since the recent common goal is t o provide a uniﬁed description of allthe fundamental interactions in nature. The book requires as background a good knowledge of quantum mechanics, including rudiments of the Dirac equation, as well as elements of the Klein-Gordonequation, and the reader would beneﬁt m uch by reading relevant sections of my earlier book : Quantum Theory: A Wide Spectrum (2006), Springer in this respect. This book differs from QFT books that have appeared in recent years 2in several respects and, in particular, it offers something new in its approach to the subject, andthe reader has plenty of opportunity to be exposed to many topics not covered, or 1R. P. Feynman, The Theory of Fundamental Processes, The Benjamin/Cummings Publishing Co., Menlo Park, California. 6th Printing (1982), page 1. 2Some of the ﬁne books that I am f amiliar with are: L. H. Ryder, Quantum Field Theory;
S. Weinberg, The Quantum Theory of Fields I (1995) & II (1996), Cambridge: Cambridge University Press; M. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, NewYork: Westview Press (1995); B. D eWitt, The Globa l Approach to Quantum Field Theory, Oxford: Oxford University Press (2014). v
vi Preface to Volume I just touched upon, in standard references. Some notable differences are seen, partly, from unique features in the following material included in ours: The very elegant functional differential approach of Schwinger, referred to as the quantum dynamical (action) principle, and its underlying theory are used systematically in generating the so-called vacuum-to-vacuum transitionamplitude of both abelian and non-abelian g auge theories, in addition to the well- known functional integral approach of Feynman, referred to as the path-integralapproach, which are simply related by functional Fourier transforms and deltafunctionals. Transition amplitudes are readily extracted by a direct expansion of the vacuum- to-vacuum transition amplitude in terms of a unitarity sum, which is most closelyrelated to actual experimental setups with particles emitted and detected prior andafter a given process and thus represent the underlying physics in the clearestpossible way.
Particular emphasis is put on the concept of a quantum ﬁeld and its particle content, both physically and technically, as providing an appropriate descriptionof physical processes at sufﬁciently high energies, for which relativity becomesthe indispensable language to do physics and explains the exchange that takesplace between energy and matter, allowing the creation of an unlimited numberof particles such that the number of particles need not be conserved, and forwhich a variable number of particles may be created or destroyed. Moreover,quantum mechanics implies that a wavefunction renormalization arises in QFTﬁeld independent of any perturbation theory – a point not sufﬁciently emphasizedin the literature. The rationale of the stationary action principle and emergence of ﬁeld equations,
via ﬁeld variations of transformation functions and generators of ﬁeld variations.The introduction of such generators lead, self consistently, to the ﬁeld equations.Such questions are addressed as: “Why is the variation of the action, within the boundaries of transformation functions, set equal to zero which eventually leads to the Euler-Lagrange equations?”, “How does the Lagrangian density appear inthe formalism?” “What is the signiﬁcance in commuting/anti-commuting ﬁeldcomponents within the interaction Lagrangian density in a theory involving ﬁeldoperators?” These are some of the questions many students seem to worry about. A panorama of all the ﬁelds encountered in present high-energy physics, together with the details of the underlying derivations are given. Schwinger’s point splitting method of currents is developed systematically in
studying abelian and non-abelian gauge theories anomalies. Moreover, an explicitexperimental test of the presence of an anomaly is shown by an example. Derivation of the Spin & Statistics connection and CPT symmetry, emphasizing for the latter that the invariance of the action under CPT transformation is notsufﬁcient for CPT symmetry, but one has also to consider the roles of incomingand outgoing particles.
Preface to V olume I vii The ﬁne-structure effective coupling ˛'1=128 at high energy corresponding to the mass of the neutral Z0vector boson based on all the charged leptons and all those contributing quarks of the three generations. Emphasis is put on renormalization theory, including its underlying general subtractions scheme, often neglected in treatments of QFT. Elementary derivation of Faddeev-Popov factors directly from the functional differential formalism, with constraints, and their modiﬁcations , and how they may even arise in some abelian gauge theories. A fairly detailed presentation is given of “deep inelastic” experiments as a fundamental application of quantum chromodynamics. Schwinger line integrals, origin of Wilson loops, lattices, and quark conﬁnement. Neutrino oscillations,3neutrino masses, neutrino mass differences, and the “seesaw mechanism.” QCD jets and parton splitting, in cluding gluon sp litting to gluons.
Equal importance is put on both abelian and non-abelian gauge theories, witnessing the wealth of information also stored in the abelian case.4 A most important, fairly detailed, and semi-technical introductory chapter is given which traces the development of QFT since its birth in 1926 without tears, in abelian and non-abelian gauge theories, including aspects of quantumgravity, as well as examining the impact of supersymmetry, string theory, and thedevelopment of the theory of renormalization, as a pedagogical strategy for the reader to be able to master the basic ideas of the subject at the outset before they are encountered in glorious technical details later. Solutions to all the problems are given right at the end of the book. With the mathematical rigor that renormalization has met over the years and
the reasonable agreement between gauge theories and experiments, the underlyingtheories are in pretty good shape. This volume is organized as follows. The ﬁrstintroductory chapter traces the subject of QFT since its birth, elaborating on many of its important developments which are conveniently described in a fairlysimple language and will be quite useful for understanding the underlying technicaldetails of the theory covered in later chapters including those in V olume II. Apreliminary chapter follows which includes the study of symmetry transformationsin the quantum world, as well as of intricacies of functional differentiation andfunctional integration which are of great importance in ﬁeld theory. Chapter 3 deals with quantum ﬁeld theory methods of spin 1/2 culminating in the study ofanomalies in the quantum world. The latter refers to the fact that a conservation law 3It is rather interesting to point out that the theory of neutrino oscillations was written up in this
book much earlier tha n the 2015 Nobel Prize in Physics was announced on neutri no oscilla tions. 4With the development of non-abelian gauge theories, unfortunately, it seems that some students are not even exposed to such derivations as of the “Lamb shift” and of the “anomalous magnetic moment of the electron” in QED.
viii Preface to V olume I in classical physics does not necessarily hold in the quantum world. Chapter 4,a critical one, deals with the concept of a quantum ﬁeld, the Poincaré algebra, and particle states. Particular attention is given to the stationary action principle aswell as in developing the solutions of QFT via the quantum dynamical principle.This chapter includes the two celebrated theorems dealing with CPT symmetryand of the Spin & Statistics connection. A detailed section is involved with thebasic quantum ﬁelds one encounters in present day high-energy/elementary-particlephysics and should provide a useful reference source for the reader. Chapter 5treats abelian gauge theories (QED, scalar boson electrodynamics) in quite details andincludes, in particular, the derivations of two of the celebrated results of QEDwhich are the anomalous magnetic moment of the electron and the Lamb shift.Chapter 6is involved with non-abelian gauge theories (electroweak, QCD, Grand uniﬁcation).
5Such important topics are included as “asymptotic freedom,” “deep inelastic” scattering, QCD jets, parton splittings, neutrino oscillations, the “seesawmechanism” and neutrino masses, Schwinge r-line integrals, Wilson loops, lattices, and quark conﬁnement. Uniﬁcation of c oupling parameters of the electroweak theory and of QCD are also studied, as well as of spontaneous symmetry breakingin both abelian and non-abelian gauge theo ries, and of renormalizability aspects of both gauge theories, emphasizing the so-called BRS transformations for the latter.We make it a point, pedagogically, to derive things in detail, and some of suchdetails are relegated to appendices at the end of the respective chapters with the main results given in the sections in question. Five general appendices, at the end
of this volume, cover some additional important topics and/or technical details.In particular, I have included an appendix covering some aspects of the generaltheory of renormalization and its underlying subtractions scheme itself which isoften neglected in books on QFT. Fortunately, my earlier book, with proofs not just words, devoted completely to renormalization theory – Renormalization (1983), Academic Press – may be consulted for more details. The problems given at the end of the chapters form an integral part of the book, and many developments inthe text depend on the problems and may include, in turn, additional material. Theyshould be attempted by every serious student. Solutions to all the problems are given right at the end of the book for the convenience of the reader. The introductorychapter together with the introductions to each chapter provide the motivation and thepedagogical means to handle the technicalities that follow them in the texts.
I hope this book will be useful for a wide range of readers. In particular, I hope that physics graduate students, not only in quantum ﬁeld theory and high-energy physics, but also in other areas of specializations will also beneﬁt from itas, according to my experience, they seem to have been left out of this fundamental area of physics, as well as instructors and researchers in theoretical physics. Thecontent of this volume may be covered in one-year (two semesters) quantum ﬁeldtheory courses. 5QED and QCD stand, respectively, for quantum electrodynamics and quantum chromodynamics.
Preface to Volume I ix In V olume II, the reader is introduced to quantum gravity, supersymmetry, and string theory,6which although may, to some extent, be independently read by a reader with a good background in ﬁeld theory, the present volume sets up the language, the notation, provid es additional background for i ntroducing these topics, and will certainly make it much easier for the reader to follow. In this two-volumeset, aiming for completeness in covering the basics of the subject, I have includedtopics from the so-called conventional ﬁeld theory (the classics) to ones from themodern or the new physics which I believe that every serious graduate studentstudying quantum ﬁeld theory should be exposed to. Without further ado, and with all due respect to the legendary song writer Cole Porter, let us ﬁnd out “what is this thing called QFT?” Edouard B. Manoukian 6Entitled: Quantum Field Theory II: Introductions to Quantum Gravity, Supersymmetry, and String Theory” (2016), Springer.
Acknowledgements In the beginning of it all, I was introduced to the theoretical aspects of quantum ﬁeld theory by Theodore Morris and Harry C. S. Lam, both from McGill and toits mathematical intricacies by Eduard Prugove ˇcki from the University of Toronto. I am eternally grateful to them. Over the years, I was fortunate enough to attenda few lectures by Julian Schwinger and beneﬁted much from his writings as well.Attending a lecture by Schwinger was quite an event. His unique elegant, incisive,physically clear approach and, to top it off, short derivations were impressive. WhenI was a graduate student, I would constantly hear that Schwinger “does no mistakes.”It took me years and years to understand what that meant. My understanding of thisis because he had developed such a powerful formalism to do ﬁeld theory that,unlike some other approaches, everythin g in the theory came out automatically and readily without the need to worry about multip licative factors in computations, such
as2/EM’s and other numerical factors, and, on top of this, is relatively easy to apply. Needless to say this has much inﬂuen ced my own approach to the subject. He had one of the greatest minds in theoretical physics of our time. I want to take this opportunity as well to thank Steven Weinberg, the late Abdus Salam, Raymond Streater, and Eberhard Zeidler for the keen interest they haveshown in my work on renormalization theory. I acknowledge with thanks the support I received from several colleagues, while visiting their research establishments for extended periods, for doing my own thingand writing up the initial notes on this project. These include Yasushi Takahashiand Anton Z. Capri from the University of Alberta, Lochlainn O’Raifeartaigh
and John Lewis from the Dublin Institute for Advanced Studies, and Jiri Pateraand Pavel Winternitz from the University of Montreal. For the ﬁnal developmentsof the project, I would like to thank Sujin Jinahyon, the President of Nare-suan University, Burin Gumjudpai, Seck son Sukhasena, Suchittra Sa-Nguansin, and Jiraphorn Chomdaeng of the university’s Institute for Fundamental Studyfor encouragement, as well as Ahpisit Ungkitchanukit and Chai-Hok Eab fromChulalongkorn University. I am also indebted to many of my former graduate students, who are now established physicists in their own rights, particularly to Chaiyapoj Muthaporn, xi
xii Acknowledgements Nattapong Yongram, Siri Sirininlakul, Tukkamon Vijaktanawudhi (aka Kanchana Limboonsong), Prasopchai Viriyasrisuwattana, and Seckson Sukhasena, whothrough their many questions, several discussions, and collaborations have beenvery helpful in my way of analyzing this subject. Although I have typed the entire manuscripts myself, and drew the ﬁgures as well, Chaiyapoj Muthaporn prepared the L ATEX input ﬁles. Without his constant help in L ATEX, this work would never have been completed. I applaud him, thank him and will always remember how helpful he was. My special thanks also goto Nattapong Yongram for downloading the endless number of papers I needed tocomplete the project. I was fortunate and proud to have been associated with the wonderful editorial
team of Maria Bellantone and Mieke van der Fluit of Springer. I would like toexpress my deepest gratitude to them for their excellent guidance, caring, patience,and hard work in making this project possible and move forward toward itscompletion. I have exchanged more emails with Mieke than with anybody else onthe globe. This has led to such an enjoyable association that I will always cherish. This project would not have been possible without the patience, encouragement, and understanding of my wife Tuenjai. To my parents, who are both gone, this workis affectionately dedicated.
Contents 1 Introduction Donkey Electron, Bare Electron, Electroweak Frog, God Particle, “Colored” Quarks and Gluons, AsymptoticFreedom, Beyond Resonances into the Deep Inelastic Region, Partons, QCD Jets, Conﬁned Quarks, Bekenstein– Hawking Entropy of a Black Hole, Sparticles, Strings,Branes, Various Dimensions and even Quanta of Geometry,AdS/CFT Correspondence and Holographic Principle,CPT, and Spin & Statistics .................................................. 1 References ..................................................................... 33 Recommended Reading ....................................................... 42 2 Preliminaries ................................................................. 45 2.1 Wigner’s Symmetry Transformations in the Quantum World ....... 46 2.1.1 Wigner’s Symmetry Transformations ....................... 47 2.2 Minkowski Spacetime: Common Arena of Elementary Particles ... 50 2.3 Representations of the Dirac Gamma Matrices;
Majorana Spinors ...................................................... 54 2.4 Differentiation and Integration with Respect to Grassmann Variables ............................................... 56 2.5 Fourier Transforms Involving Grassmann Variables ................. 60 2.6 Functional Differentiation a nd Integration; Functional Fourier Transforms .................................................... 63 2.7 Delta Functionals ...................................................... 68 Problems ....................................................................... 70 References ..................................................................... 71 Recommended Reading ....................................................... 71 3 Quantum Field Theory Methods of Spin 1=2............................. 73 3.1 Dirac Quantum Field, Propagator and Energy-Momentum Transfer: Schwinger-FeynmanBoundary Condition ................................................... 74 xiii
xiv Contents 3.2 The Dirac Quantum Field Concept, Particle Content, and C, P, T Tansformations ............................................ 78 3.2.1 Charge Conjugation (C), Parity Transformation (T), and Time Reversal (T) of the Dirac Quantum Field ............................. 83 3.3 Re-Discovering the Positron and Eventual Discovery of Anti-Matter .......................................................... 85 3.3.1h0Cj0/NULifor the Dirac Equation ........................... 89 3.4 Coulomb Scattering of Relativistic Electrons ........................ 93 3.5 Spin & Statistics and the Dirac Quantum Field; Anti-Commutativity Properties Derived .............................. 96 3.6 Electromagnetic Current, Gauge Invariance andh0Cj0/NULiwith External Electromagnetic Field ................. 98 3.7h0Cj0/NULi.e/in the Presence of a Constant F/SYN/ETBField and Effective Action ................................................... 103
3.8 Pair Creation by a Constant Electric Field ........................... 109 3.9 Fermions and Anomalies in Field Theory: Abelian Case ............ 111 3.9.1 Derivation of the Anomaly .................................. 112 3.9.2 Experimental Veriﬁcation of the Anomaly: /EM0!/CR/CRDecay ............................................. 117 3.10 Fermions and Anomalies in Field Theory: Non-Abelian Case ...... 120 Appendix A: Evaluation of L/SYN1/SYN2............................................ 125 Appendix B: Inﬁnitesimal Variation of the Exponential of a Matrix ....... 128 Problems ....................................................................... 129 References ..................................................................... 131 Recommended Reading ....................................................... 132 4 Fundamental Aspects of Quantum Field Theory ......................... 133 4.1 The Field Concept, Partic le Aspect and Wavefunction
Renormalization ....................................................... 135 4.2 PoincarKe Algebra and Particle States ................................. 139 4.3 Principle of Stationary Action of Quantum Field Theory: The Rationale Of ............................................. 146 4.3.1 A Priori Imposed Variations of Dynamical Variables and Generators of Field Variations:Field Equations ............................................... 148 4.3.2 Commutation/Anti-commutation Relations ................ 155 4.3.3 Generators for Quantum Responses to Field Variations: Internal Symmetry Groups ...................... 156 4.3.4 Variations of Boundary Surfaces ............................ 157 4.4 Inhomogeneous Lorentz Transformations and Energy-Momentum Tensor ....................................... 159 4.5 Spin and Statistics Connection ........................................ 162 4.5.1 Summary ..................................................... 166
4.5.2 The Hamiltonian ............................................. 166 4.5.3 Constraints ................................................... 167
Contents xv 4.6 Quantum Dynamical Principle (QDP) of Field Theory .............. 168 4.6.1 Summary ..................................................... 175 4.7 A Panorama of Fields .................................................. 176 4.7.1 Summary of Salient Features of Some Basic Fields ........ 177 4.7.2 Spin 0 ......................................................... 184 4.7.3 Spin 1 ......................................................... 188 4.7.4 Spin 3=2...................................................... 192 4.7.5 Spin 2 ......................................................... 199 4.8 Further Illustrations and Applications of the QDP ................... 204 4.9 Time-Ordered Products, How to Write Down Lagrangians and Setting Up the Solution of Field Theory ........................................................ 209 4.10 CPT ..................................................................... 213
Appendix A: Basic Equalities Involving the CPT Operator ................. 217 Problems ....................................................................... 219 References ..................................................................... 221 Recommended Reading ....................................................... 222 5 Abelian Gauge Theories ..................................................... 223 5.1 Spin One and the General Vector Field ............................... 224 5.2 Polarization States of Photons ........................................ 226 5.3 Covariant Formulation of the Propagator ............................. 228 5.4 Casimir Effect .......................................................... 231 5.5 Emission and Detection of Photons ................................... 235 5.6 Photons in a Medium .................................................. 238 5.7 Quantum Electr odynamics, Covariant Gauges: Setting
Up the Solution ........................................................ 241 5.7.1 The Differential Formalism (QDP) and Solution of QED in Covariant Gauges ...................... 241 5.7.2 From the Differential Formalism to the Path Integral Expression for h0Cj0/NULi........................... 246 5.8 Low Order Contributions to lnh 0Cj0/NULi............................. 248 5.9 Basic Processes ........................................................ 253 5.9.1 e/NULe/NUL!e/NULe/NUL,eCe/NUL!eCe/NUL........................... 257 5.9.2 e/NUL/CR!e/NUL/CR;eCe/NUL!/CR/CR and Polarizations Correlations ................................... 264 5.9.3 e/NUL/SYN/NUL!e/NUL/SYN/NUL............................................... 269 5.10 Modiﬁed Propagators .................................................. 271 5.10.1 Electron Self-Energy and Its Interpretation ................. 273 5.10.2 Photon Self-Energy and Its Interpretation; Coulomb Potential ........................................... 278
5.11 Vertex Part ............................................................. 283 5.11.1 Charge renormalization and External lines ................. 287 5.11.2 Anomalous Magnetic Moment of the Electron ............. 290
xvi Contents 5.12 Radiative Correction to Coulomb Scattering and Soft Photon Contribution ................................................... 290 5.13 Lamb Shift ............................................................. 293 5.14 Coulomb Gauge Formulation ......................................... 304 5.14.1h0Cj0/NULiin the Coulomb Gauge in the Functional Differential Form ................................ 304 5.14.2 From the QDP to the Path Integral of h0Cj0/NULiin Coulomb Gauge ............................... 308 5.15 Gauge Transformations of the Full Theory ........................... 310 5.16 Vertex Function and Ward-Takahashi Identity; Full Propagators ... 316 5.16.1 Ward-Takahashi Identity ..................................... 316 5.16.2 Equations for the Electron Propagator and the Vertex Function .............................................. 318 5.16.3 Spectral Representation of the Photon Propagator, Charge Renormalization, Coulomb
Potential in Full QED, Unordered Products of Currents ... 323 5.16.4 Integral Equation for the Vacuum Polarization Tensor ..... 333 5.17 The Full Renormalized Theory ....................................... 335 5.18 Finiteness of the Renormalized Theory; Renormalized Vertex Function and Renormalized Propagators ..................... 338 5.18.1 Finiteness of the Renormalized Theory ..................... 339 5.18.2 The Renormalized Vertex and the Renormalized Electron Propagator .......................... 342 5.18.3 The Renormalized Photon Propagator ...................... 343 5.19 Effective Charge and the Renormalization Group ................... 347 5.19.1 Renormalization Group Analysis ........................... 347 5.19.2 The Fine-Structure Effective Coupling at High Energy Corresponding to the Mass of theNeutral Z 0Vector Boson ................................... 354 5.20 Scalar Boson Electrodynamics, Effective Action
and Spontaneous Symmetry Breaking ................................ 355 5.20.1 Change of Real Field Variables of Integration in a Path Integral ............................................. 356 5.20.2 Goldstone Bosons and Spontaneous Symmetry Breaking ...................................................... 357 Problems ....................................................................... 361 References ..................................................................... 364 Recommended Reading ....................................................... 367 6 Non-Abelian Gauge Theories ............................................... 369 6.1 Concept of Gauge Fields and Internal Degrees of Freedom: From Geometry to Dynamics ........................... 371 6.1.1 Generators of SU( N)......................................... 381 6.2 Quantization of Non-Abelian Gauge Fields in the Coulomb Gauge ................................................. 382
Contents xvii 6.3 Functional Fourier Transform and Transition to Covariant Gauges; BRS Transformations and Renormalization of Gauge Theories ............................. 388 6.3.1 Functional Fourier Transform and The Coulomb Gauge .............................................. 388 6.3.2 Trasformation Law from the Coulomb Gauge to Covariant Gauges in Non-abelian GaugeField Theories ................................................ 395 6.3.3 BRS Trasformations and Renormalization of Non-abelian Gauge Field Theories ....................... 395 6.4 Quantum Chromodynamics ........................................... 403 6.5 e Ce/NULAnnihilation ..................................................... 411 6.6 Self-Energies and Vertex Functions in QCD ......................... 413 6.6.1 Fermion Inverse Propagator ................................. 414 6.6.2 Inverse Gluon Propagator .................................... 415 6.6.3 Fermion-Gluon vertex ....................................... 417
6.7 Renormalization Constants, Effective Coupling, Asymptotic Freedom, and What is Responsible for the Latter? ..... 419 6.7.1 What Part of the Dynamics is Responsible for Asymptotic Freedom? ....................................... 423 6.8 Renormalization Group and QCD Corrections to eCe/NUL Annihilation ............................................................ 423 6.9 Deep Inelastic Scattering: Differential Cross Section and Structure Functions ............................................... 429 6.10 Deep Inelastic Scattering, The Parton Model, Parton Splitting; QCD Jets .................................................... 434 6.10.1 The Parton Model ............................................ 434 6.10.2 Parton Splitting ............................................... 437 6.10.3 QCD Jets ..................................................... 444 6.11 Deep Inelastic Scattering: QCD Corrections ......................... 447 6.12 From the Schwinger Line-Integral to the Wilson Loop
and How the latter Emerges ........................................... 457 6.13 Lattices and Quark Conﬁnement ...................................... 463 6.14 The Electroweak Theory I ............................................. 469 6.14.1 Development of the Theory: From the Fermi Theory to the Electroweak Theory .......................... 470 6.14.2 Experimental Determination of sin2/DC2W................... 481 6.14.3 Masses of the Neutrinos and the “Seesaw Mechanism” ................................................. 482 6.14.4 Neutrino Oscillations: An Interlude ......................... 485 6.15 Electroweak Theory II: Incorporation of Quarks; Anomalies and Renormalizability .................................... 488 6.15.1 Quarks and the Electroweak Theory ........................ 489 6.15.2 Anomalies and Renormalizability .......................... 491 6.16 Grand Uniﬁcation ...................................................... 495
xviii Contents Problems ....................................................................... 504 References ..................................................................... 506 Recommended Reading ....................................................... 511 General Appendices .............................................................. 513 Appendix I: The Dirac Formalism ........................................... 515 Appendix II: Doing Integrals in Field Theory ............................... 521 Appendix III: Analytic Continuation in Spacetime Dimension and Dimensional Regularization ........................................... 529 Appendix IV: Schwinger’s Point Splitting Method of Currents: Arbitrary Orders ............................................................. 533 Appendix V: Renormalization and the Underlying Subtractions ......... 541 Recommended Reading ....................................................... 549
Solutions to the Problems ....................................................... 551 Index ............................................................................... 583
Notation and Data ıLatin indices i;j;k;::: are generally taken to run over 1,2,3, while the Greek indices/SYN;/ETB;::: over0;1;2;3 in 4D. Variations do occur when there are many different types of indices to be used, and the meanings should be evident from the presentations. ıThe Minkowski metric /DC1/SYN/ETBis deﬁned by Œ/DC1/SYN/ETB/c141DdiagŒ/NUL1;1;1;1/c141DŒ/DC1/SYN/ETB/c141in 4D. ıUnless otherwise stated, the fundamental constants „;c are set equal to one. ıThe gamma matrices satisfy the anti-commutation relations f/CR/SYN;/CR/ETBgD/NUL2/DC1/SYN/ETB. ıThe Dirac, the Majorana, and the chiral representations of the /CR/SYNmatrices are deﬁned in Appendix Iat the end of the book. ıThe charge conjugation matrix is deﬁned by CDi/CR2/CR0. ı D /c142/CR0,uDu/c142/CR0,vDv/c142/CR0. A Hermitian conjugate of a matrix Mis denoted by M/c142, while its complex conjugate is denoted by M/ETX. ıThe step function is denoted by ™.x/which is equal to 1 for x>0, and 0 for x<0.
ıThe symbol "is used in dimensional regularization (see Appendix III)./SIis used in deﬁning the boundary condition in the denominator of a propagator.Q 2Cm2/NULi/SI/and should not be confused with "used in dimensional regularization. We may also use either one when dealing with an inﬁnitesimalquantity, in general, with /SImore frequently, and this should be self-evident from the underlying context. ıFor units and experimental data, see the compilation of the “Particle DataGroup”: Beringer et al. [1 ]a n dO l i v ee ta l .[ 2]. The following (some obviously approximate) numerical values should, however, be noted: 1MeVD10 6eV 1GeVD103MeV 103GeVD1TeV 1ergD10/NUL7J xix
xx Notation and Data 1JD6:242/STX109GeV cD2:99792458/STX1010cm/s (exact) „D 1:055/STX10/NUL34Js „cD197:33 MeV fm 1fmD 10/NUL13cm (Masses) MpD938:3 MeV=c2,MnD939:6 MeV=c2, MWD80:4 GeV=c2,MZD91:2 GeV=c2, meD0:511 MeV=c2,m/SYND105:66 MeV=c2,m/FSD1777 MeV=c2. Mass of/ETBe<2eV=c2,M a s s o f/ETB/SYN<0:19 MeV=c2,M a s s o f/ETB/FS<18:2 MeV=c2, Mass of the neutral Higgs H0/EM125:5 GeV=c2. For approximate mass values of some of the quarks taken, see Table 5.1in Sect. 5.19.2 . For more precise range of values, see Olive et al. [ 2]. (Newton’s gravitational constant) G ND6:709/STX10/NUL39„c5/G e V2. (Fermi weak interaction constant) G FD1:666/STX10/NUL5„3c3/G e V2. Planck massp „c=GN/EM1:221/STX1019GeV=c2, Planck lengthp „GN=c3/EM1:616/STX10/NUL33cm. Fine structure constant ˛D1=137:04 atQ2D0,a n d/EM1=128 atQ2/EMM2 Z. For the weak-mixing angle /DC2W,s i n2/DC2W/EM0:232,a t Q2/EMM2 Z. ˛=sin2/DC2W/EM0:034,a t Q2/EMM2 Z. Strong coupling constant ˛s/EM0:119,a t Q2/EMM2 Z. References
1. Beringer, J., et al. (2012). Particle data group. Physical Review D, 86 , 010001. 2. Olive, K. A., et al. (2014). Particle data group. Chinese Physics C, 38, 090001.
Chapter 1 Introduction Donkey Electron, Bare Electron, Electroweak Frog, God Particle, “Colored” Quarks and Gluons, Asymptotic Freedom, Beyond Resonances into the Deep InelasticRegion, Partons, QCD Jets, Conﬁned Quarks, Bekenstein – Hawking Entropyof a Black Hole, Sparticles, Strings, Branes, Various Dimensions and even Quantaof Geometry, AdS/CFT Correspondence and Holographic Principle, CPT, andSpin & Statistics The major theme of quantum ﬁeld theory is the development of a uniﬁed theory that may be used to describe nature from m icroscopic to cosmological distances. Quantum ﬁeld theory was born 90 years ago, when quantum theory met relativity,and has captured the hearts of the brightest theoretical physicists in the world. It isstill going strong. It has gone through various stages, met various obstacles on the
way, and has been struggling to provide us with a coherent description of naturein spite of the “patchwork” of seemingly different approaches that have appearedduring the last 40 years or so, but still all, with the common goal of uniﬁcation. As mentioned in our Preface, Feynm an, in his 1958 Cornell, 1959–1960 Cal Tech, lectures on the quantum ﬁeld theory of fundamental processes, the ﬁrststatement he makes, the very ﬁrst one, is that the lectures will cover allof physics [76, p. 1]. One quickly understands what Feynman meant by covering all of physics. After all, the role of fundamental physics is to describe the basic interactions wehave in nature and quantum ﬁeld theory is supposed to do just that. Feynman’sstatement is obviously more relevant today than it was then, since the recent common goal is to provide a uniﬁed description of allthe fundamental forces in
nature. With this in mind, let us trace the development of this very rich subject fromthe past to the present, and see what the theory has been telling us all these years. When the energy and momentum of a quantum particle are large enough, one is
confronted with the requirement of developing a formalism, as imposed by nature,which extends quantum theory to the relativistic regime. A relativistic theory, asa result of the exchange that takes place between energy and matter, allows thecreation of an unlimited number of particles and the number of particles in a givenphysical process need not be conserved. An appropriate description of such physicalprocesses for which a variable number of particles may be created or destroyed, inthe quantum world, is provided by the very rich concept of a quantum ﬁeld. Forexample, photon emissions and absorptions, in a given process, are explained by theintroduction of the electromagnetic quantum ﬁeld. The theory which emerges fromextending quantum physics to the relativistic regime is called “Relativistic Quantum © Springer International Publishing Switzerland 2016 E.B. Manoukian, Quantum Field Theory I , Graduate Texts in Physics, DOI 10.1007/978-3-319-30939-2_11
2 1 Introduction Field Theory” or just “Quantum Field Theory” . Quantum Electrodynamics is an example of a quantum ﬁeld theory and is the most precise theory devised by man when confronted with experiments. The essence of special relativity is that allinertial frames are completely equivalent in explaining a physical theory as oneinertial frame cannot be distinguished from another. This invariance property ofphysical theories in all inertial frames, as required by special relativity, as well as bythe many symmetries one may impose on such theories, are readily implemented inthe theory of quantum ﬁelds. The implementation of symmetries and describingtheir roles in the explanation of obser ved phenomena has played a key role in elementary particle physics. Of course it took years before the appropriate language of quantum ﬁeld theory,
described concisely above, by marrying quantum theory and relativity, was spelledout and applied consistently to physical processes in the quantum world in therelativistic regime. An appropriate place to start in history is when Dirac [47 –49] developed his relativistic equation of spin 1/ 2, from which one learns quite a bitabout the subsequent development of the subject as a multi-particle theory. We willthen step back a year or two, and then move again forward in time to connectthe dots between the various stages of th e underlying exciting developments. His relativistic equation, which incorporated the spin of the electron, predicted theexistence of negative energy states with negative mass, with energies going downto/NUL1, implying the instability of the corresponding systems. For example, an electron in the ground-state energy of a n atom would spontaneously decay to such
lower and lower negative energy states emitting radiation of arbitrary large energiesleading eventually to the collapse of the atom with the release of an inﬁnite amountof energy. Historically, a relativistic equation for spin 0, was developed earlier byKlein and Gordon in 1926, 1referred to as the Klein-Gordon equation, which also shared this problem, but unlike Dirac’s theory it led to negative probabilities as well.Dirac being aware of the negative probab ilities encountered in the theory of the latter authors, was able to remedy this problem in his equation. To resolve the dilemmaof negative energies, Dirac, in 1930, 2assumed that a priori all the negative energy states are ﬁlled with electrons in accord to the P auli exclusion principle, giving rise to the so-called Dirac sea or the Dirac vacuum, so that no transitions to such statesare possible, thus ensuring the stability of the atom. The consequences of the assumption made by Dirac above were many. A negative
energy electron in the Dirac sea, may absorb radiation of sufﬁcient energy so asto overcome an energy gap arising from the level /NULmc 2toCmc2,w h e r e mis the mass of an electron, thus making such a negative energy electron jump to apositive energy state, leaving behind a su rplus of positive energy and a surplus of 1Klein [ 128] and Gordon [ 101]. 2Dirac [ 50,51].
1 Introduction 3 positive chargeCjejrelative to the Dirac sea. This has led Dirac eventually,3in 1931 [ 52], to interpret the “hole” left behind by the transition of the negative energy electron to a positive energy state, as a particle that has the same mass as the electron but of opposite charge. It is interesting to note that George Gamow referred4to Dirac’s predicted particle as a “donkey electron” , because it would move in the opposite direction of an appropriate applied force. The physics community foundit difﬁcult to accept Dirac’s prediction until Anderson 5discovered this particle (the positron eC), who apparently was not aware of Dirac’s prediction at the time of the discovery.6With the positron now identiﬁed, the above argument just given has provided an explanation of the so-called pair production /CR!eCe/NULby a photon ( in the vicinity of a nucleus).7Conversely, if a “hole” is created in the vacuum, then
an electron may make a transition to such a state releasing radiation giving rise to thephenomenon of pair annihilation. A Pair created, as described above, in the vicinity of a positively charged nucleus, would lead to a partial screening of the charge ofthe nucleus as the electron within the pair would be attracted by the nucleus andthe positively charged one would be repelled. Accordingly, an electron, in the atom,at sufﬁciently large distances from the nucleus would then see a smaller charge onthe nucleus than an electron nearby (such as one in an s-state). This leads to the concept of vacuum polarization , and also to the concept of charge renormalization as a result of the partial charge screening mentioned above. The Dirac equation is Lorentz covariant, that is, it has the same form in every inertial frame with its variables being simply relabeled reﬂecting the variables usedin the new inertial frame. It predicted, approximately, the gyromagnetic ratio gD
2of the electron, the ﬁne-structure of the atom, and eventually anti-matter was discovered such as antiprotons. 8It was thus tremendously successful. Apparently,9 Dirac himself remarked in one of his talks that his equation was more intelligent than its author .10 Thus the synthesis of relativity and quantum physics, led to the discovery of the antiparticle. The Dirac equation which was initially considered to describe a singleparticle necessarily led to a multi -particle theory, and a single particle description in the relativistic regime turned out to be not complete. A formalism which wouldnaturally describe creation and annihilation of particles and take into account this 3Dirac [ 50,51] assumed that the particle is the proton as the positron was not discovered yet at that time. Apart from the large mass difference between the proton and the electron, there were other inconsistencies with such an assumption. 4Weisskopf [ 242]. 5Anderson [ 5,6]. 6Weisskopf [ 242].
7The presence of the nucleus is to conserve energy and momentum. 8Chamberlain et al. [30 ]. 9Weisskopf [ 242]. 10For a systematic treatment of the intricacies of Dirac’s theory and of the quantum description of relativistic particles, in general, see Manoukian [ 151], Chapter 16.
4 1 Introduction multi-particle aspect became necessary. The so-called “hole” theory although it gave insight into the nature of fundamental processes involving quantum particles in the relativistic regime, and concepts such as vacuum polarization, turned out to be alsonot complete. For example, in the “hole” theory, the number of electrons minus thenumber of positrons, created is conserved by the simultaneous creation of a “hole”for every electron ejected from the Dirac Sea. In nature, there are processes, wherejust an electron or just a positron is created while conserving charge of course.Examples of such processes are ˇ /NULdecay: n!pCe/NULCQ/ETBe, muon decay: /SYN/NUL!e/NULCQ/ETBeC/ETB/SYN,a n dˇCdecay: p!nCeCC/ETBe, for a bound proton
in a nucleus for the latter process. Finally, Dirac’s argument of a sea of negativelycharged bosons did not work with the Klein-Gordon equation because of the verynature of the Bose statistics of the particles. A new description to meet all of theabove challenges including the creation and annihilation of particles, mentionedabove, was necessary. After the conceptual framework of quantum mechanics was developed, Born, Heisenberg, and Jordan in 1926 [26 ], applied quantum mechanical methods to the electromagnetic ﬁeld, now, giving rise to a system with an inﬁnite degrees offreedom, and described as a set of independent harmonic oscillators of variousfrequencies. Then Dirac in 1927 [46 ], prior to the development of his relativistic spin 1/2 equation, also extended quantum mechanical methods to the electromagneticﬁeld now with the latter ﬁeld treated as an operator, and provided a theoretical
description of how photons emerge in the quantization of the electromagnetic ﬁeld.This paper is considered to mark the birthdate of “Quantum Electrodynamics”, aname coined by Dirac himself, and provided a prototype for the introduction ofﬁeld operators for other particles with spin, such as for spin 1/2, where in the lattercase commutators in the theory a re replaced by anti-commutators [ 125,126]f o rt h e fermion ﬁeld. The ﬁrst comprehensive treatment of a general quantum ﬁeld theory, involving Lagrangians, as in modern treatments, was given by Heisenberg and Pauli in 1929,1930 [ 116,117], where canonical quantization procedures were applied directly to the ﬁelds themselves. A classic review of the state of affairs of quantumelectrodynamics in 1932 [ 68] was given by Fermi. The problem of negative energy solutions was resolved and its equivalence to the Dirac “hole” theory wasdemonstrated by Fock in 1933 [ 83], and Furry and Oppenheimer in 1934 [ 90], where
the (Dirac) ﬁeld operator and its adjoint were expanded in terms of appropriatecreation and annihilation operators for the electron and positron, thus providinga uniﬁed description for the particle and its antiparticle. The method had a directgeneralization to bosons. The old “hole” theory became unnecessary and obsolete. 11 The problem of negative energy solutions was also resolved for spin 0 bosons by 11As a young post-doctoral fellow, I remember attending Schwinger’s lecture tracing the Develop- ment of Quantum Electrodynamics in “The Physicist’s Conception of Nature” [ 202], making the statement, regarding the “hole” theory, that it is now best regarded as an historical curiosity, and forgotten.
1 Introduction 5 similar methods by Pauli and Weisskopf in 1934 [170 ]. The ﬁelds thus introduced from these endeavors have become operators for creation and annihilation of particles and antiparticles, rather than probability amplitudes.12 The explanation that interactions are generated by the exchange of quanta was clear in the classic work of Bethe and Fermi in 1932 [18 ]. For example, charged particles, as sources of the electromagnetic ﬁeld, inﬂuence other charged particlesvia these electromagnetic ﬁelds. Fields as operators of creation and destruction ofparticles, and the association of particles w ith forces is a natural consequence of ﬁeld theory. The same idea was used by Yukawa in 1935 [ 249], to infer that a massive
scalar particle is exchanged in describing the strong interaction (as understood inthose days), with the particle necessarily being massive to account for the shortrange nature of the strong force unlike the electromagnetic one which is involvedwith the massless photon describing an interaction of inﬁnite range. The mass /SYN of the particle may be estimated from the expression /SYN/EM„=Rc, obtained formally from the uncertainty principle, where Rdenotes the size of the proton, i.e., RD 1fmD10 /NUL13cm. In natural units, i.e., for „D1,cD1,1fm/EM1=.200 MeV/. This gives/SYN/EM200MeV. Such a particle (the pion) was subsequently discovered by the C. F. Powell group in 1947 [136 ]. As early as 1930s, inﬁnities appeared in explicit computations in quantum electrodynamics by Oppenheimer [ 168], working within an atom, by Waller [233,234] ,a n db yW e i s s k o p f[ 239]. The nature of these divergences, arising
in these computations, came from integrations that one had to carry out overenergies of photons exchanged in describing the interaction of the combinedsystem of electrons and the electromagnetic ﬁeld to arbitrary high-energies. By formally restricting the energies of photons exchanged, as just described, to beless than, say, /DC4, Weisskopf, in his calculations, has shown [ 239,240], within the full quantum electrodynamics, that the divergences encountered in the self-energyacquired by the electron from its interaction with the electromagnetic ﬁeld is ofthe logarithmic 13type/CANln./DC4=mc2/, improving the preliminary calculations done earlier, particularly, by Waller, mentioned above. That such divergences, referredto as “ultraviolet divergence”, 14are encountered in quantum ﬁeld theory should 12It is important to note, however, that the matrix elements of these ﬁeld operators between particle
states and the vacuum naturally lead to amplitudes of particles creation by the ﬁelds and to the concept of wavefunction renormalization (see Sect. 4.1) independently of any perturbation theories. 13The corresponding expression occurs with higher powers of the logarithm for higher orders in the ﬁne-structure constant e2=4/EM„c. 14That is, divergences arising from the high-energy behavior of a theory. Another type of divergence, of different nature occurring in the low energy region, referred to as the “infraredcatastrophe”, was encountered in the evaluatio n of the probability that a photon be em itted in a collision of a charged particle. In computations of the scattering of charged particles, due to thezero mass nature of photons, their simultaneous emissions in arbitrary, actually inﬁnite, in numbermust necessary be taken into account for a complete treatment. By doing so ﬁnite expressions forthe probabilities in question were obtained [ 22].
6 1 Introduction e+ e−e−e−e−γ γ γ) b ( ) a ( Fig. 1.1 Processes leading to an electron self-energy correction, and vacuum polarization, respec- tively be of no surprise as one is assuming that our theories are valid up to inﬁnite energies!15 The2S1=2,2P1=2states of the Hydrogen atom are degenerate in Dirac’s theory. In 1947 [ 134], Lamb and Retherford, however, were able to measure the energy difference between these states, referred to as the “Lamb Shift”, using thennewly developed microwave met hods with great accuracy. Bethe [ 17]t h e nm a d e a successful attempt to compute this energy difference by setting an upper limit forthe energy of photon exchanged in describing the electromagnetic interaction of theorder of the rest energy of the electron mc 2, above which relativistic effects take place, relying on the assumption that the el ectron in the atom is non-relativistic, and, in the process, took into consideration of the mass shift16of the electron.
He obtained a shift of the order of 1000 megacycles which was in pretty good agreement with the Lamb-Retherford experiment. Very accurate computations were then ma de, within the full relativistic quantum electrodynamics, and positron theory. Notably, Schwinger17in 1948 [ 192], com- puted the magnetic moment of the electron modifying the gyromagnetic ratio, gD2 in the Dirac theory, to 2.1C˛=2/EM/ , to lowest order in the ﬁne-structure constant. The computation of the Lamb-Shift was also carried out in a precise manner by Krolland Lamb [ 133], and, for example, by French and Weisskopf [ 85], and Fukuda et al. [88,89]. State of affairs changed quite a bit. It became clear that an electron is accompa- nied by an electromagnetic ﬁeld which in turn tends to alter the nature of the electronthat one was initially aiming to describe. The electron e /NUL, being a charged particle, produces an electromagnetic ﬁeld ./CR/. This ﬁeld, in turn, interacts back with the
electron as shown below in Fig. 1.1a. Similarly, the electromagnetic ﬁeld ( /CR)m a y lead to the creation of an electron-positron pair eCe/NUL, which in turn annihilate each other re-producing an electromagnetic ﬁeld, a process referred to as vacuum- polarization, shown in part (b). Because of these processes, the parameters initially 15See also the discussion in Sect. 5.19. 16See also the important contribution to this by Kramers [ 130]. This reference also includes contributions of his earlier work. 17See also Appendix B of Schwinger [ 193].
1 Introduction 7 e−e− p p E=p2+m2, E=p2+m2, Fig. 1.2 As a result of the self-energy correction in Fig. 1.1a, where an electron emits and re- absorbs a photon, the mass parameter m0, one initially starts with, doe s not represent the physical mass of the electron determined in the lab. Here this is emphasized by the energy dependence on the physical mass mof an electron in a scattering process. The dashed lines represent additional particles participating in the process appearing in the theory, such as mass, say, m0, vis-à-vis Fig. 1.1a, and the electron charge, say, e 0, vis-à-vis Fig. 1.1b, that were associated with the electron one starts with, are not the parameters actually meas ured in the lab. For example, the energy of a scattered electron of momentum p, in a collision process, turned up to be not equal toq p2Cm2 0but rather top p2Cm2, self-consistently,18with midentiﬁed with the actual, i.e., tabulated, mass of the electron, and m¤m0, with a scattering
process shown in Fig. 1.2, where the dashed lines represent other particles (such as /CR;e/NUL;eC), where the total charge as well as the total energy and momentum are conserved in the scattering process. Similarly, the potential energy betw een two widely separated electrons, by a distance r, turned up to be not e20=4/EMrbut rather e2=4/EMr, with e2¤e20,w h e r e e is identiﬁed with the charge, i.e., the tabulated charge, of the electron. As we will see later, the physical parameters are related to the initial ones by scalingfactors, referred to as mass and charge re normalization constants, respectively. An electron parametrized by the couple .m 0;e0/, i sr e f e r r e dt oa sa bare electron as it corresponds to measurements of its properties by going down to “zero”distances all the way into the “core” of the electron – a process that is unattainable experimentally. On the other h and, the physical parameters .m;e/, correspond to
measurements made on the electron from sufﬁciently large distances. One thus, in turn, may generate parameters, corresponding to a wide spectrum of scales running from the very small to the very large. Here one already notices that inquantum ﬁeld theory, one encounters so-called effective parameters which are func-tions of different scales (or energies). Func tions of these effective parameters turn out to satisfy invariance properties under scale transformations, thus introducing aconcept referred to as the renormalization group. Clearly, due to the screening effectvia vacuum polarization of e Ce/NULpairs creation, as discussed earlier, the magnitude of the physical charge is smaller than the magnitude of the bare charge. 18An arbitrary number of photons of vanishingly small energies are understood to be attached to the external electron lines, as discussed in Footnote 14when dealing with infrared divergence problems.
8 1 Introduction A process was, in turn, then carried out , referred to as “renormalization”, to eliminate the initial parameters in the theory in favor of physically observed ones. This procedure related the theory at very small distances to the theory at sufﬁciently large distances at which particles emerge on their way to detectors asit happens in actual experiments. All the difﬁculties associated with the ultravioletdivergences in quantum electrodynamics were isolated in renormalization constants,such as the ones discussed above, and one was then able to eliminate them incarrying out physical applications giving rise to completely ﬁnite results. Thisbasic idea of the renormalization procedure was clearly spelled out in the workof Schwinger, Feynman, and Tomonaga. 19The renormalization group,20mentioned above, describes the connection of renorma lization to scale transformations, and relates, in general, the underlying physics at different energy scales.
In classic papers, Dyson [59 ,60] has shown not only the equivalence of the Schwinger, Feynman, and Tomonaga approaches,21and the ﬁniteness of the so- called renormalized quantum electrodynamics, but also developed a formalism forcomputations that may be readily applied to other interacting quantum ﬁeld theories.Theories that are consistently ﬁnite when all the different parameters appearing initially in the theory are elim inated in favor of the physically observed ones, which are ﬁnite in number, are said to be renormalizable. Dyson’s work, had set up: renormalizab ility as a condition f or generating ﬁeld theory interactions . In units of„D1,cD1,ŒMass/c141DŒLength/c141/NUL1. Roughly speaking, in a renormalizable theory, no coupling constants can have the dimensions of negativepowers of mass. (Because of dimensional r easons, we note, in particular, that one
cannot have too many derivatives of the ﬁelds, describing interactions, as everyderivative necessitates involving a coupling of dimensionality reduced by one inunits of mass.) The photon as the agent for t ransmitting the interaction between charged parti- cles, is described by a vector – the vector potential. In quantum electrodynamics,as a theory of the interaction of photons and electrons, for example, the photon iscoupled locally to the electromagnetic current. The latter is also a vector, and theinteraction is described by their scalar product (in Minkowski space) ensuring the relativistic invariance of the underlying theory. To lowest order in the charge e ofthe electron e, this coupling may be represented by the diagram Fig. 1.3a. On the other hand, for a spin 0 charged boson ', say, one encounters two such diagrams, each shown to lowest order in the charge e in Fig. 1.3b, where we note that in the
second diagram in the latter part, two photons emerge locally from the same point. 19This is well described in their Nobel lectures: Schwinger [ 201], Feynman [ 75], Tomonaga [ 225], as well as in the collection of papers in Schwinger [ 198,201]. 20Stueckelberg and Peterman [ 209], Gell-Mann and Low [ 93], Bogoliubov and Shirkov [ 25], Ovsyannikov [ 169], Callan [ 28,29], Symanzik [213 –215], Weinberg [237 ], and ’t Hooft [ 218]. 21The best sources for these approaches are their Nobel Lectures: Schwinger [ 201], Feynman [ 75], Tomonaga [ 225], as well as Schwinger [ 198].
1 Introduction 9 e e eγ(a) ϕ e ϕγ(b) ϕ e2 ϕγ γ Fig. 1.3 Local couplings for photon emission by an electron, and by a spin 0 charged particle described by the ﬁeld ', respectively Quantum Electrodynamics, was not only the theory of interest. There was also the weak interaction. The preliminary theory of weak interaction dates back to Fermi [69,70] .B a s e do nw e a kp r o c e s s e ss u c ha s ˇ/NULdecay: n!pCe/NULCQ/ETBe, he postulated that the weak interactions may be described by local four-pointinteractions involving a universal coupling parameter G F. The four particles of the process just mentioned, interact locally at a point with a zero range interaction.The Fermi theory was in good agreement in predicting the energy distribution ofthe electron. For dimensional reasons, however, the dimensions of the couplingconstant G Finvolved in the theory has the dimensions of ŒMass/c141/NUL2, giving rise to a non-renormalizable theory.22In analogy to quantum electrodynamics, the situation
with this type of interaction may be somehow improved by introducing, in theprocess, a vector Boson 23W/NULwhich mediates an interaction24between the two pairs (so-called currents), .n;p/and.e/NUL;Q/ETBe/, with both necessarily described by entities carrying (Lorentz) vector indices, t o ensure the invariance of the underlying description. Moreover, in units of „D1,cD1,adimensionless coupling gis introduced. The Fermi interaction and its modiﬁcation are shown, respectively inparts Fig. 1.4a, b. In order that the process in diagram given in part Fig. 1.4b, be consistent with the “short-range” nature of the Fermi interaction, described by the diagram on theleft, the vector particle W /NULmust not only be massive but its mass, MWmust be quite large. This is because the propaga tor of a massive vector particle of mass MW, which mediates an interaction between two spacetime points xandx0,a s w e will discuss below, behaves like /DC1/SYN/ETBı.4/.x/NULx0/=M2 Wfor a large mass, signifying
22It is interesting to point out as one goes to higher and higher orders in the Fermi coupling constant G F, the divergences increase (Sect. 6.14) without any bound and the theory becomes uncontrollable. 23A quantum relativistic treatment of a problem, implies that a theory involving the W/NULparticle, must also include its antiparticle WC, having the same mass as of W/NUL. 24Such a suggestion was made, e.g., by Klein [ 129].
10 1 Introduction (a) (b) (x) (x) (x)GF g gn np p e− e− ˜νe ˜νeW− Fig. 1.4 (a) The old Fermi theory with a coupling G Fis replaced by one in ( b)w h e r et h e interaction is mediated by a vector boson with a dimensionless coupling g necessarily a vanishingly small range of the interaction.25Upon comparison of both diagrams, one may then infer that GF/EMg2 M2 W: (1.1) Evidently, the Fourier transform of the propagator in the energy-momentum descrip- tion, at energies much less than MWis, due to the ı.4/.x/NULx0/function given above, simply/DC1/SYN/ETB=M2 W,a n d ( 1.1) may be obtained from a low-energy limit. With some minimal effort, the reader will understand better the above two limits and some of the difﬁculties encountered with a massive vector boson, in general, if, at this stage, we write down explicitly its propagator between two spacetime points x,x0in describing an interaction carried by the exchange of such a particle which is denoted by26: 4/SYN/ETB C.x/NULx0/DZ.dk/
.2/EM/4eik/ETB.x/ETB/NULx0/ETB/4/SYN/ETBC.k/; .d k/Ddk0dk1dk1dk3; (1.2) 4/SYN/ETBC.k/D1 .k2CM2 W/NULi0//DC2 /DC1/SYN/ETBCk/SYNk/ETB M2W/DC3 ; (1.3) 25Here/DC1/SYN/ETBis the Minkowski metric. 26This expression will be derived in Sect. 4.7. For a so-called virtual particle k2Dk2/NUL.k0/2¤ /NULM2 W.T h e/NULi0in the denominator in ( 1.3) just speciﬁes the bounda ry condition on how the k0 integration is to be carried out. These things will be discussed in detail later on and are not needed here.
1 Introduction 11 where k0is its energy, and kD.k1;k2;k3/its momentum. Formally for M2 W! 1,4/SYN/ETB C.k/!/DC1/SYN/ETB=M2 W, leading from (1.2 )t o 4/SYN/ETB C.x/NULx0/!/DC1/SYN/ETB M2 WZ.dk/ .2/EM/4eik/ETB.x/ETB/NULx0/ETB/D/DC1/SYN/ETB M2Wı4.x/NULx0/; (1.4) signalling, in a limiting sense, a short range interaction for a heavy-mass particle. On the other hand, for jk/ETBj/FS MWfor each component, one has 4/SYN/ETB C.k//EM/DC1/SYN/ETB M2 W; (1.5) in the energy-momentum description. Although the introduction of the intermediate boson Wimproves somehow the divergence problem, it is still problematic. The reason is not difﬁcult to understand. In the energy-momentum description, the propagator of a massive vector particle,as given in (1.3 ), has the following behavior at high energies and momenta 4 /SYN/ETB C.k/!1 k2k/SYNk/ETB M2 W; (1.6) providing no damping in such a limit. Moreover, as one goes to higher orders in
perturbation theory the number of integra tion variables, over energy and momenta arising in the theory, increase, and the divergences in turn increase without bound and the theory becomes uncontrollable.27On the other hand, an inherited property of quantum electrodynamics is gauge symmetry due to the masslessness of the photon. In the present context of ultraviolet divergences, the photon propagator hasa very welcome vanishing property at high energies. This gauge symmetry as wellas the related massless aspect of the photon, which are key ingredients in the self consistency of quantum electrodynamics, turned out to provide a guiding principle in developing the so-called electroweak theory. In 1956 [ 138], an important observation was made by Lee and Yang that parity P is violated in the weak interactions. Her e we recall that, given a process, its parity transformed (mirror) version, is obtained by reversing the directions of the spacevariables.
28This has led Gershtein and Zel’dovich [ 95], Feynman and Gell-Mann [77], Sudarshan and Marshak [ 210], and Sakurai [ 178], to express the currents 27The damping provided by the propagators of a massless vector particle, a spin 1/2 particle, and a spin 0 particle, for example, in the ultraviolet region vanish like 1/energy2, 1/energy, 1/energy2, respectively. 28It was later observed that the product of charge conjugation, where a particle is replaced by its antiparticle, and parity transformation “CP”, is also not conserved in a decay mode of Kmesons at a small level [ 38,39,82]. As the product “CPT”, of charge conjugation, parity transformation, and time reversal “T”, is believed to be conserved, the violation of time reversal also follows. For a test of such a violation see CPLEAR/Collaboration [ 36].
12 1 Introduction constructed out of the pairs of ﬁelds: .n;p/,.e/NUL;Q/ETBe/;::: in the Fermi theory to reﬂect, in particular, this property dicta ted by nature. The various currents were eventually expressed and conveniently parametrized in such a way that the theory was described by the universal coupling parameter G F. The construction of such fundamental currents together with idea of intermediate vector bosons exchanges todescribe the weak interaction led eventually to its modern version. Quantum Electrodynamics may be considered to arise from local gauge invari- ance in which the electron ﬁeld is subjected to a local phase transformation e i#.x/. The underlying group of transformations is denoted by U .1/involving simply the identity as the single generator of transformations with which the photon is associated as the single gauge ﬁeld. In 1954 [ 247], Yang and Mills, and Shaw in 1955 [ 203], generalized the just mentioned abelian gauge group of phase
transformations, encountered in quantum electrodynamics, to a non-abelian29gauge theory, described by the group SU .2/,30and turned out to be a key ingredient in the development of the modern theory of weak interactions. This necessarily requiredthe introduction, in addition to the charged bosons W ˙, a neutral one. What distinguishes a non-abelian gauge theory from an abelian one, is that in the formertheory, direct interactions occur between gauge ﬁelds, carrying speciﬁc quantumnumbers, unlike in the latter, as the gauge ﬁeld – the photon – being uncharged. As early as 1956, Schwinger believed that the weak and electromagnetic interactions should be combined into a gauge theory [ 97,159,199]. Here we may pose to note that both in electrodynamics and in the modiﬁed Fermi theory,interactions are mediated by vector particles. They are both described by universaldimensionless coupling constants e, and by g(see ( 1.1)) in the intermediate vector
boson description, respectively. In a uniﬁed description of electromagnetism and theweak interaction, one expects these couplings to be comparable, i.e., g 2/EMe2D4/EM˛; where˛/EM1 137;GF/EM1:166/STX10/NUL5=.GeV/2:(1.7) in units„D1;cD1.F r o m ( 1.1), we may then estimate the mass of the Wbosons to be MW/EMs 4/EM˛ GF/EM90GeV=c2; (1.8) 29Non-abelian refers to the fact that the generators do not commute. In contrast a U .1/gauge theory, such as quantum electrodynamics, is an abelian one. 30SU.2/consists of2/STX2unitary matrices of determinant one. (The letter S in the group stands for the special property of determinant one.) It involves three generators, with which are associated three gauge ﬁelds. This will be studied in detail in Sect. 6.1.
1 Introduction 13 re-inserting the constant c for convenience, in good agreement with the observed mass. We may also estimate the range of the weak interaction to be RW/EM„c MWc2/EM2:2/STX10/NUL16cm: (1.9) Glashow, a former graduate student of Schwinger, eventually realized [ 96]31the important fact that the larger group SU .2//STXU.1/, is needed to include also electrodynamics within the context of a Ya ng-Mills-Shaw theory. A major problem remained: the local gauge symmetry required that the gauge ﬁelds associated withthe group must, a priori, be massless in the initial formulation of the theory. The problem of the masslessness of the vector bosons was solved by Weinberg [236,238]a n dS a l a m[ 182,183], 32by making use of a process,33referred to as spontaneous symmetry breaking, where a scalar ﬁeld interacting with the vectorbosons, whose expectation value in the vacuum state is non zero, leads to thegeneration of masses to them. 34This is referred to as the Higgs35mechanism, in
which the group SU .2//STXU.1/is spontaneously broken to the group U .1/with the latter associated with the photon, and, in the process, the other bosons, calledW ˙;Z0, acquiring masses, thanks to the Higgs boson, and renormalizability may be achieved. The latter particle has been also called the “God Particle”.36The mere existence of a neutral vector boson Z0implies the existence of a weak interaction component in the theory without a charge transfer, the so-called neutral currents. Atypical process involving the neutral Z 0boson exchange is in Q/ETB/SYNCe/NUL!Q/ETB/SYNCe/NUL shown in Fig. 1.5not involving the muon itself. Neutral currents37have been observed,38and all the vector bosons have been observed39as well. It turned out that the theory with spontaneous symmetry did not spoil the renormalizability of 31See also Salam and Ward [ 189]. 32See also Salam and Ward [ 187–189]a n dS a l a m[ 181].
33Some key papers showing how spontaneous symmetry breaking using spin 0 ﬁeld may generate masses for vector bosons are: Englert and Brout [ 63], Englert et al. [ 64], Guralnik et al. [ 110], and Kibble [ 127]. 34Apparently the Legendary Victor Weisskopf was not impressed by this way of generating masses. In his CERN publication [241 ], on page 7, 11th line from below, he says that this is an awkward way to explain masses and that he believes that Nature should be more inventive, but experiments may prove him wrong. 35Higgs [ 119–121]. This work followed earlier work of Schwinger [ 200], where he shows, by the exactly solvable quantum electrodynamics in two dimensions, that gauge invariance does notprevent the gauge ﬁeld to acquire mass dynamically, as well as of the subsequent work of Anderson[7] in condensed matter physics. 36This name was given by Lederman and Teresi [ 137]. 37Neutral current couplings also appear in Bludman’s [ 23] pioneering work on an SU .2/gauge
theory of weak interactions but did not include electromagnetic interactions. 38Hasert et al. [ 112, 113] and Benvenuti et al. [ 15]. 39See, e.g., C. Rubbia’s Nobel Lecture [ 176].
14 1 Introduction Fig. 1.5 A process involving the exchange of the neutral vector boson Z0 e−e−Z0˜νμ ˜νμ the resulting theory with massive vector bosons. Proofs of renormalizability were given by ’t Hooft [ 216,217].40It seems that Sydney Coleman used to say that ’t Hooft’s proof has turned the Weinberg-Salam frog into an enchanted prince.41 The “Electroweak Theory” turned u p to be quite a successful theory.42 Another interaction which was also developed in the “image” of quantum electrodynamics was quantum chromodynamics, as a theory of strong interactionsbased, however, on the non-abelian gauge symmetry group SU .3/. Here one notes that a typical way to probe the internal structure of the proton is through electron-proton scattering. The composite nature of the proton, as having an underlyingstructure, becomes evident when one compar es the differential cross sections for elastic electron-proton scattering with the proton described as having a ﬁnite
extension to the one described as a point-like particle. With a one photon exchangedescription, the form factors in the differential cross section are seen to vanishrapidly for large momentum transfer (squared) Q 2of the photon imparted to the proton. As Q2is increased further one reaches the so-called resonance region,43 beyond which, one moves into a deep inelastic region, where experimentally thereaction changes “character”, and the corresponding structure functions of thedifferential cross section have approximate scaling properties (Sect. 6.9), instead of the vanishing properties encountered with elastic form factors, the process of whichis depicted in Fig. 1.6. Such properties indicate the presence of approximately free point-like structures within the proton referred as partons, which consist of quarks,gluons together with those emitted 44from their scattering reactions. This led to the development of the so-called parton model,45as a ﬁrst approximation, in which
these point-like particles within the proton are free and the virtual photon interacts 40See also ’t Hooft and Veltman [ 221], Lee and Zinn-Justin [ 139–142], and Becchi et al. [13 ]. 41See Salam [ 183], p. 529. 42The basic idea of the renormalizability of the theory rests on the fact that renormalizability may be established for the theory with completely massless vector bosons, as in QED, one may then invoke gauge symmetry to infer that the theory is also renormalizable for massive vector bosonsvia spontaneous symmetry breaking. 43A typical resonance is /SOHC, of mass 1.232 GeV, consisting of a proton pand a/EM0meson. 44See, e.g., Fig. 6.7c. 45Feynman [ 73,74] and Bjorken and Pachos [ 21].
1 Introduction 15 } nucleon leptonleptonAnything Fig. 1.6 In the process, “Anything” denotes anything that may be created in the process consistent with the underlying conservation laws. The wavy line denotes a neutral particle (/CR ,Z0, ...)o fl a r g e momentum transfer (a) (b) Fig. 1.7 (a) If interactions between quarks may be represented, as an analogy, by people holding hands, then pulling one person would drag everybody else al ong. In the parton model, the situation is represented as in part ( b)r a t h e rt h a ni np a r t( a) with each of its charged constituents independently,46instead of interacting with the proton as a whole. The non-abelian gauge symmetry group SU .3/, is needed to accommodate quarks and gluons, involving eight generators with which the gluons are associated. Here, in particular, a quantum number referred to as “color” (three of them)47is assigned to the quarks. One of the many reasons for this is that the spin 3/2 particle/SOH
CC, which is described in terms of three identical quarks (the so-called u quarks) as a low lying state with no orbital angular momentum between the quarks, behavesas a symmetric state under the exchange of two of its quarks and would violate theSpin & Statistics connection without this additional quantum number. The color degrees of freedom are not observed in the hadronic states themselves and the latterbehave as scalars, that is they are color singlets, under SU .3/transformations. As the group SU .3/involves “color” transforma tions within each quark ﬂavor, it may be denoted by SU .3/ color or just by SU .3/C. The gluons also carry “color” and direct gluon-gluon interactions then n ecessarily occur, unlik e the situation with 46For an analogy to this, see part (b) of Fig. 1.7b. 47Greenberg [ 104], Han and Nambu [ 111], Nambu [163 ], Greenberg and Zwanziger [ 105], Gell- Mann [ 92], and Fritzsch and Gell-Mann [ 87].
16 1 Introduction photons in quantum electrodynamics since photons do not carry a charge. These gluon-gluon interactions turn out to have an anti-screening effect on a source ﬁeldwhich dominate over the screening effect of quark/antiquark interactions leading tothe interesting fact that the effective coupling of quark interactions becomes smallerat high-energies, and eventually vanish 48– a phenomenon referred to as asymptotic freedom. This has far reaching consequences as it allows one to develop perturbation theory at high energies, in the effective coupling, and carry out various applicationswhich were not possible before the development of the theory, and is consistentwith deep-inelastic experiments of leptons with nucleons, with the latter describedby point-like objects which, at high energies, scatter almost like free particles, 49as mentioned above, the process of which is shown Fig. 1.6. A particular experiment which indirectly supports the idea of quarks having

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