Quantum Field Theory and the Standard Model Matthew D

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Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information

Quantum Field Theory and the Standard Model

Providing a comprehensive introduction to quantum field theory, this textbook covers the development of particle physics from its foundations to the discovery of the Higgs boson. Its combination of clear physical explanations, with direct connections to experimental data, and mathematical rigor make the subject accessible to students with a wide vari- ety of backgrounds and interests. Assuming only an undergraduate-level understanding of quantum mechanics, the book steadily develops the Standard Model and state-of-the art calculation techniques. It includes multiple derivations of many important results, with modern methods such as effective field theory and the renormalization group playing a prominent role. Numerous worked examples and end-of-chapter problems enable students to reproduce classic results and to master quantum field theory as it is used today. Based on a course taught by the author over many years, this book is ideal for an introductory to advanced quantum field theory sequence or for independent study.

MATTHEW D. SCHWARTZ is an Associate Professor of Physics at Harvard University. He is one of the world’s leading experts on quantum ﬁeld theory and its applications to the Standard Model.

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03473-0 - Quantum Field Theory and the Standard Model Matthew D. Schwartz Frontmatter More information

Quantum Field Theory and the Standard Model

MATTHEW D. SCHWARTZ Harvard University

University Printing House, Cambridge CB2 8BS, United Kingdom

Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence.

www.cambridge.org Information on this title: www.cambridge.org/9781107034730 c M. Schwartz 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 4th printing 2015 Printed in the United States of America by Sheridan Books, Inc. A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Schwartz, Matthew Dean, 1976– Quantum ﬁeld theory and the standard model / Matthew D. Schwartz. pages cm ISBN 978-1-107-03473-0 (hardback) 1. Quantum ﬁeld theory – Textbooks. 2. Particles (Nuclear physics) – Textbooks. I. Title. QC174.45.S329 2014 530.143–dc23 2013016195 ISBN 978-1-107-03473-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To my mother, and to Carolyn, Eve and Alec

Contents

Preface page xv

Part I Field theory 1

1 Microscopic theory of radiation 3 1.1 Blackbody radiation 3 1.2 Einstein coefﬁcients 5 1.3 Quantum ﬁeld theory 7

2 Lorentz invariance and second quantization 10 2.1 Lorentz invariance 10 2.2 Classical plane waves as oscillators 17 2.3 Second quantization 20 Problems 27

3 Classical ﬁeld theory 29 3.1 Hamiltonians and Lagrangians 29 3.2 The Euler–Lagrange equations 31 3.3 Noether’s theorem 32 3.4 Coulomb’s law 37 3.5 Green’s functions 39 Problems 42

4 Old-fashioned perturbation theory 46 4.1 Lippmann–Schwinger equation 47 4.2 Early inﬁnities 52 Problems 55

5 Cross sections and decay rates 56 5.1 Cross sections 57 5.2 Non-relativistic limit 63 5.3 e+e− → μ+μ− with spin 65 Problems 67

6 The S-matrix and time-ordered products 69 6.1 The LSZ reduction formula 70 6.2 The Feynman propagator 75 Problems 77 vii

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7 Feynman rules 78 7.1 Lagrangian derivation 79 7.2 Hamiltonian derivation 84 7.3 Momentum-space Feynman rules 93 7.4 Examples 97 7.A Normal ordering and Wick’s theorem 100 Problems 103

Part II Quantum electrodynamics 107

8 Spin 1 and gauge invariance 109 8.1 Unitary representations of the Poincare´ group 109 8.2 Embedding particles into ﬁelds 113 8.3 Covariant derivatives 120 8.4 Quantization and the Ward identity 123 8.5 The photon propagator 128 8.6 Is gauge invariance real? 130 8.7 Higher-spin ﬁelds 132 Problems 138

9 Scalar quantum electrodynamics 140 9.1 Quantizing complex scalar ﬁelds 140 9.2 Feynman rules for scalar QED 142 9.3 Scattering in scalar QED 146 9.4 Ward identity and gauge invariance 147 9.5 Lorentz invariance and charge conservation 150 Problems 155

10 Spinors 157 10.1 Representations of the Lorentz group 158 10.2 Spinor representations 163 10.3 Dirac matrices 168 10.4 Coupling to the photon 173 1 10.5 What does spin 2 mean? 174 10.6 Majorana and Weyl fermions 178 Problems 181

11 Spinor solutions and CPT 184 11.1 Chirality, helicity and spin 185 11.2 Solving the Dirac equation 188 11.3 Majorana spinors 192 11.4 Charge conjugation 193 11.5 Parity 195

Contents ix

11.6 Time reversal 198 Problems 201

12 Spin and statistics 205 12.1 Identical particles 206 12.2 Spin-statistics from path dependence 208 12.3 Quantizing spinors 211 12.4 Lorentz invariance of the S -matrix 212 12.5 Stability 215 12.6 Causality 219 Problems 223

13 Quantum electrodynamics 224 13.1 QED Feynman rules 225 13.2 γ-matrix identities 229 13.3 e+e− → μ+μ− 230 13.4 Rutherford scattering e−p+ → e−p+ 234 13.5 Compton scattering 238 13.6 Historical note 246 Problems 248

14 Path integrals 251 14.1 Introduction 251 14.2 The path integral 254 14.3 Generating functionals 261 14.4 Where is the iε? 264 14.5 Gauge invariance 267 14.6 Fermionic path integral 269 14.7 Schwinger–Dyson equations 272 14.8 Ward–Takahashi identity 277 Problems 283

Part III Renormalization 285

15 The Casimir effect 287 15.1 Casimir effect 287 15.2 Hard cutoff 289 15.3 Regulator independence 291 15.4 Scalar ﬁeld theory example 296 Problems 299

16 Vacuum polarization 300 16.1 Scalar φ3 theory 302 16.2 Vacuum polarization in QED 304

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16.3 Physics of vacuum polarization 309 Problems 314

17 The anomalous magnetic moment 315 17.1 Extracting the moment 315 17.2 Evaluating the graphs 318 Problems 321

18 Mass renormalization 322 18.1 Vacuum expectation values 323 18.2 Electron self-energy 324 18.3 Pole mass 330 18.4 Minimal subtraction 334 18.5 Summary and discussion 336 Problems 338

19 Renormalized perturbation theory 339 19.1 Counterterms 339 19.2 Two-point functions 342 19.3 Three-point functions 345 19.4 Renormalization conditions in QED 349 19.5 Z1 = Z2: implications and proof 350 Problems 354

20 Infrared divergences 355 20.1 e+e− → μ+μ− (+γ) 356 20.2 Jets 364 20.3 Other loops 366 20.A Dimensional regularization 373 Problems 380

21 Renormalizability 381 21.1 Renormalizability of QED 382 21.2 Non-renormalizable ﬁeld theories 386 Problems 393

22 Non-renormalizable theories 394 22.1 The Schrodinger¨ equation 395 22.2 The 4-Fermi theory 396 22.3 Theory of mesons 400 22.4 Quantum gravity 403 22.5 Summary of non-renormalizable theories 407 22.6 Mass terms and naturalness 407 22.7 Super-renormalizable theories 414 Problems 416

Contents xi

23 The renormalization group 417 23.1 Running couplings 419 23.2 Renormalization group from counterterms 423 23.3 Renormalization group equation for the 4-Fermi theory 426 23.4 Renormalization group equation for general interactions 429 23.5 Scalar masses and renormalization group ﬂows 435 23.6 Wilsonian renormalization group equation 442 Problems 450

24 Implications of unitarity 452 24.1 The optical theorem 453 24.2 Spectral decomposition 466 24.3 Polology 471 24.4 Locality 475 Problems 477

Part IV The Standard Model 479

25 Yang–Mills theory 481 25.1 Lie groups 482 25.2 Gauge invariance and Wilson lines 488 25.3 Conserved currents 493 25.4 Gluon propagator 495 25.5 Lattice gauge theories 503 Problems 506

26 Quantum Yang–Mills theory 508 26.1 Feynman rules 509 26.2 Attractive and repulsive potentials 512 + − 26.3 e e → hadrons and αs 513 26.4 Vacuum polarization 517 26.5 Renormalization at 1-loop 521 26.6 Running coupling 526 26.7 Deﬁning the charge 529 Problems 533

27 Gluon scattering and the spinor-helicity formalism 534 27.1 Spinor-helicity formalism 535 27.2 Gluon scattering amplitudes 542 27.3 gg → gg 545 27.4 Color ordering 548 27.5 Complex momenta 551 27.6 On-shell recursion 555 27.7 Outlook 558 Problems 559

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28 Spontaneous symmetry breaking 561 28.1 Spontaneous breaking of discrete symmetries 562 28.2 Spontaneous breaking of continuous global symmetries 563 28.3 The Higgs mechanism 575 28.4 Quantization of spontaneously broken gauge theories 580 Problems 583

29 Weak interactions 584 29.1 Electroweak symmetry breaking 584 29.2 Unitarity and gauge boson scattering 588 29.3 Fermion sector 592 29.4 The 4-Fermi theory 602 29.5 CP violation 605 Problems 614

30 Anomalies 616 30.1 Pseudoscalars decaying to photons 617 30.2 Triangle diagrams with massless fermions 622 30.3 Chiral anomaly from the integral measure 628 30.4 Gauge anomalies in the Standard Model 631 30.5 Global anomalies in the Standard Model 634 30.6 Anomaly matching 638 Problems 640

31 Precision tests of the Standard Model 641 31.1 Electroweak precision tests 642 31.2 Custodial SU(2), ρ, S, T and U 653 31.3 Large logarithms in ﬂavor physics 657 Problems 666

32 Quantum chromodynamics and the parton model 667 32.1 Electron–proton scattering 668 32.2 DGLAP equations 677 32.3 Parton showers 682 32.4 Factorization and the parton model from QCD 685 32.5 Lightcone coordinates 695 Problems 698

Part V Advanced topics 701

33 Effective actions and Schwinger proper time 703 33.1 Effective actions from matching 704 33.2 Effective actions from Schwinger proper time 705 33.3 Effective actions from Feynman path integrals 711 33.4 Euler–Heisenberg Lagrangian 713

Contents xiii

33.5 Coupling to other currents 722 33.6 Semi-classical and non-relativistic limits 725 33.A Schwinger’s method 728 Problems 732

34 Background fields 733 34.1 1PI effective action 735 34.2 Background scalar fields 743 34.3 Background gauge fields 752 Problems 758

35 Heavy-quark physics 760 35.1 Heavy-meson decays 762 35.2 Heavy-quark effective theory 765 35.3 Loops in HQET 768 35.4 Power corrections 772 Problems 775

36 Jets and effective ﬁeld theory 776 36.1 Event shapes 778 36.2 Power counting 780 36.3 Soft interactions 782 36.4 Collinear interactions 790 36.5 Soft-Collinear Effective Theory 795 36.6 Thrust in SCET 802 Problems 810 Appendices 813

Appendix A Conventions 815 A.1 Dimensional analysis 815 A.2 Signs 817 A.3 Feynman rules 819 A.4 Dirac algebra 820 Problems 821

Appendix B Regularization 822 B.1 Integration parameters 822 B.2 Wick rotations 823 B.3 Dimensional regularization 825 B.4 Other regularization schemes 830 Problems 833

References 834 Index 842

Preface

Quantum field theory (QFT) provides an extremely powerful set of computational methods that have yet to find any fundamental limitations. It has led to the most fantastic agreement between theoretical predictions and experimental data in the history of science. It provides deep and profound insights into the nature of our universe, and into the nature of other possible self-consistent universes. On the other hand, the subject is a mess. Its foundations are flimsy, it can be absurdly complicated, and it is most likely incomplete. There are often many ways to solve the same problem and sometimes none of them are particularly satisfy- ing. This leaves a formidable challenge for the design and presentation of an introduction to the subject. This book is based on a course I have been teaching at Harvard for a number of years. I like to start my first class by flipping the light switch and pointing out to the students that, despite their comprehensive understanding of classical and quantum physics, they still cannot explain what is happening. Where does the light comes from? The emission and absorption of photons is a quantum process for which particle number is not conserved; it is an everyday phenomenon which cannot be explained without quantum field theory. I then proceed to explain (with fewer theatrics) what is essentially Chapter 1 of this book. As the course progresses, I continue to build up QFT, as it was built up historically, as the logical generalization of the quantum theory of creation and annihilation of photons to the quantum theory of creation and annihilation of any particle. This book is based on lecture notes for that class, plus additional material. The main guiding principle of this book is that QFT is primarily a theory of physics, not of mathematics, and not of philosophy. QFT provides, first and foremost, a set of tools for performing practical calculations. These calculations take as input measured numbers and predict, sometimes to absurdly high accuracy, numbers that can be measured in other experiments. Whenever possible, I motivate and validate the methods we develop as explaining natural (or at least in principle observable) phenomena. Partly, this is because I think hav- ing tangible goals, such as explaining measured numbers, makes it easier for students to understand the material. Partly, it is because the connection to data has been critical in the historical development of QFT. The historical connection between theory and experiment weaves through this entire book. The great sucess of the Dirac equation from 1928 was that it explained the magnetic dipole moment of the electron (Chapter 10). Measurements of the Lamb shift in the late 1940s helped vindicate the program of renormalization (Chapters 15 to 21). Measurements of inelastic electron–proton scattering experiments in the 1960s (Chapter 32) showed that QFT could also address the strong force. Ironically, this last triumph occurred only a few

xvi Preface

years after Geoffrey Chew famously wrote that QFT “is sterile with respect to strong interactions and that, like an old soldier, it is destined not to die but just to fade away.” [Chew, 1961, p. 2]. Once asymptotic freedom (Chapter 26) and the renormalizability of the Stan- dard Model (Chapter 21 and Part IV) were understood in the 1970s, it was clear that QFT was capable of precision calculations to match the precision experiments that were being performed. Our ability to perform such calculations has been steadily improving ever since, for example through increasingly sophisticated effective field theories (Chapters 22, 28, 31, 33, 35 and 36), renormalization group methods (Chapter 23 and onward), and on-shell approaches (Chapters 24 and 27). The agreement of QFT and the Standard Model with data over the past half century has been truly astounding. Beyond the connection to experiment, I have tried to present QFT as a set of related tools guided by certain symmetry principles. For example, Lorentz invariance, the symmetry group associated with special relativity, plays a essential role. QFT is the theory of the creation and destruction of particles, which is possible due to the most famous equation of special relativity E = mc2. Lorentz invariance guides the definition of particle (Chapter 8), is critical to the spin-statistics theorem (Chapter 12), and strongly constrains properties of the main object of interest in this book: scattering or S-matrix elements (Chapter 6 and onward). On the other hand, QFT is useful in space-times for which Lorentz invariance is not an exact symmetry (such as our own universe, which since 1998 has been known to have a positive cosmological constant), and in non-relativistic settings, where Lorentz invariance is irrelevant. Thus, I am reluctant to present Lorentz invariance as an axiom of QFT (I personally feel that as QFT is a work in progress, an axiomatic approach is prema- ture). Another important symmetry is unitarity, which implies that probabilities should add up to 1. Chapter 24 is entirely dedicated to the implications of unitarity, with reverberations throughout Parts IV and V. Unitarity is closely related to other appealing features of our description of fundamental physics, such as causality, locality, analyticity and the cluster decomposition principle. While unitarity and its avatars are persistent themes within the book, I am cautious of giving them too much of a primary role. For example, it is not clear how well cluster decomposition has been tested experimentally. I very much believe that QFT is not a finished product, but rather a work in progress. It has developed historically, it continues to be simplified, clarified, expanded and applied through the hard work of physicists who see QFT from different angles. While I do present QFT in a more or less linear fashion, I attempt to provide multiple viewpoints whenever possible. For example, I derive the Feynman rules in five different ways: in classical field theory (Chapter 3), in old-fashioned perturbation theory (Chapter 4), through a Lagrangian approach (Chapter 7), through a Hamiltonian approach (also Chapter 7), and through the Feynman path integral (Chapter 14). While the path-integral derivation is the quickest, it is also the farthest removed from the type of perturbation theory to which the reader might already be familiar. The Lagrangian approach illustrates in a transparent way how tree-level diagrams are just classical field theory. The old-fashioned perturbation theory derivation connects immediately to perturbation theory in quantum mechanics, and moti- vates the distinct advantage of thinking off-shell, so that Lorentz invariance can be kept manifest at all stages of the calculation. On the other hand, there are some instances where an on-shell approach is advantageous (see Chapters 24 and 27).

Preface xvii

Other examples of multiple derivations include the four explanations of the spin- statistics theorem I give in Chapter 12 (direct calculation, causality, stability and Lorentz invariance of the S-matrix), the three ways I prove the path integral and canonical formula- tions of quantum field theory equivalent in Chapter 14 (through the traditional Hamiltonian derivation, perturbatively through the Feynman rules, and non-perturbatively through the Schwinger–Dyson equations), and the three ways in which I derive effective actions in Chapter 33 (matching, with Schwinger proper time, and with Feynman path integrals). As different students learn in different ways, providing multiple derivations is one way in which I have tried to make QFT accessible to a wide audience. This textbook is written assuming that the reader has a solid understanding of quantum mechanics, such as what would be covered in a year-long undergraduate class. I have found that students coming in generally do not know much classical field theory, and must relearn special relativity, so these topics are covered in Chapters 2 and 3. At Harvard, much of the material in this book is covered in three semesters. The first semester covers Chapters 1 to 22. Including both QED and renormalization in a single semester makes the coursework rather intense. On the other hand, from surveying the students, especially the ones who only have space for a single semester of QFT, I have found that they are universally glad that renormalization is covered. Chapter 22, on non-renormalizable theories, is a great place to end a semester. It provides a qualitative overview of the four forces in the Standard Model through the lens of renormalization and predictivity. The course on which this textbook is based has a venerable history, dominated by the thirty or so years it was taught by the great physicist Sidney Coleman. Sidney provides an evocative description of the period from 1966 to 1979 when theory and experiment collaborated to firmly establish the Standard Model [Coleman, 1985, p. xii]:

This was a great time to be a high-energy theorist, the period of the great triumph of quantum ﬁeld theory. And what a triumph it was, in the old sense of the word: a glori- ous victory parade, full of wonderful things brought back from far places to make the spectator gasp with awe and laugh with joy. Sidney was able to capture some of that awe and joy in his course, and in his famous Erice Lectures from which this quote is taken. Over the past 35 years, the parade has continued. I hope that this book may give you a sense of what all the fuss is about.

Acknowledgements

The book would not have been possible without the perpetual encouragement and enthusi- asm of the many students who took the course on which this book is based. Without these students, this book would not have been written. The presentation of most of the material in this book, particularly the foundational material in Parts I to III, arose from an itera- tive process. These iterations were promoted in no small part by the excellent questions the students posed to me both in and out of class. In ruminating on those questions, and discussing them with my colleagues, the notes steadily improved.

xviii Preface

I have to thank in particular the various unbelievable teaching assistants I had for the course, particularly David Simmons-Duffin, Clay Cordova, Ilya Feige and Prahar Mitra for their essential contributions to improving the course material. The material in this book was refined and improved due to critical conversations that I had with many people. In particular, I would like to thank Frederik Denef, Ami Katz, Aneesh Manohar, Yasunori Nomura, Michael Peskin, Logan Ramalingam, Matthew Reece, Subir Sachdev, Iain Stew- art, Matthew Strassler, and Xi Yin for valuable conversations. More generally, my approach to physics, by which this book is organized, has been influenced by three people most of all: Lisa Randall, Nima Arkani-Hamed and Howard Georgi. From them I learned to respect non-renormalizable field theories and to beware of smoke and mirrors in theoretical physics. I am indebted to Anders Andreassen, David Farhi, William Frost, Andrew Marantan and Prahar Mitra for helping me convert a set of decent lecture notes into a coherent and comprehensive textbook. I also thank Ilya Feige, Yang-Ting Chien, Yale Fan, Thomas Becher, Zoltan Ligeti and Marat Freytsis for critical comments on the advanced chapters of the book. Some of the material in the book is original, and some comes from primary literature. However, the vast majority of what I write is a rephrasing of results presented in the existing vast library of fantastic quantum field theory texts. The textbooks by Peskin and Schroeder and by Weinberg were especially influential. For example, Peskin and Schroeder’s nearly perfect Chapter 5 guides my Chapter 13. Weinberg’s comprehensive two volumes of The Quantum Theory of Fields are unequalled in their rigor and generality. Less general ver- sions of many of Weinberg’s explanations have been incorporated into my Chapters 8, 9, 14 and 24. I have also taken some material from Srednicki’s book (such as the derivation of the LSZ reduction formula in my Chapter 6), from Muta’s book on quantum chromodyanmics (parts of my Chapters 13, 26 and 32), from Banks’ dense and deep Concise Introduction (particularly his emphasis on the Schwinger–Dyson equations which affected my Chapters 7 and 14), from Halzen and Martin’s very physical book Quarks and Leptons (my Chapter 32), Rick Field’s book Applications Perturbative QCD (my Chapter 20). I have always found Manohar and Wise’s monograph Heavy Quark Physics (on which my Chapter 35 is based) to be a valuable reference, in particular its spectacularly efficient first chapter. Zee’s Quantum Field Theory in a Nutshell also had much influence on me (and on my Chapter 15). In addition, a few historical accounts come from Pais’ Inward Bound, which I recommend any serious student of quantum field theory to devour. Finally, I would like to thank my wife Carolyn, for her patience, love and support as this book was being written, as well as for some editorial assistance.

Matthew Dean Schwartz Cambridge, Massachusetts November 2013