quantum field theory and the standard model

Quantum Field Theory and the Standard Model

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

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Quantum Field Theoryand the Standard Model

MATTHEW D. SCHWARTZHarvard University






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Library of Congress Cataloging in Publication dataSchwartz, Matthew Dean, 1976–Quantum field theory and the standard model / Matthew D. Schwartz.pages cmISBN 978-1-107-03473-0 (hardback)1. Quantum field theory – Textbooks. 2. Particles (Nuclear physics) – Textbooks. I. Title.QC174.45.S329 2014530.14′3–dc23

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4th printing 2015






To my mother,and to Carolyn, Eve and Alec






Contents

Preface page xv

Part I Field theory 1

1 Microscopic theory of radiation 31.1 Blackbody radiation 31.2 Einstein coefficients 51.3 Quantum field theory 7

2 Lorentz invariance and second quantization 102.1 Lorentz invariance 102.2 Classical plane waves as oscillators 172.3 Second quantization 20

Problems 27

3 Classical field theory 293.1 Hamiltonians and Lagrangians 293.2 The Euler–Lagrange equations 313.3 Noether’s theorem 323.4 Coulomb’s law 373.5 Green’s functions 39

Problems 42

4 Old-fashioned perturbation theory 464.1 Lippmann–Schwinger equation 474.2 Early infinities 52

Problems 55

5 Cross sections and decay rates 565.1 Cross sections 575.2 Non-relativistic limit 635.3 e+e− → μ+μ− with spin 65

Problems 67

6 The S-matrix and time-ordered products 696.1 The LSZ reduction formula 706.2 The Feynman propagator 75

Problems 77vii






viii Contents

7 Feynman rules 787.1 Lagrangian derivation 797.2 Hamiltonian derivation 847.3 Momentum-space Feynman rules 937.4 Examples 977.A Normal ordering and Wick’s theorem 100

Problems 103

Part II Quantum electrodynamics 107

8 Spin 1 and gauge invariance 1098.1 Unitary representations of the Poincare group 1098.2 Embedding particles into fields 1138.3 Covariant derivatives 1208.4 Quantization and the Ward identity 1238.5 The photon propagator 1288.6 Is gauge invariance real? 1308.7 Higher-spin fields 132

Problems 138

9 Scalar quantum electrodynamics 1409.1 Quantizing complex scalar fields 1409.2 Feynman rules for scalar QED 1429.3 Scattering in scalar QED 1469.4 Ward identity and gauge invariance 1479.5 Lorentz invariance and charge conservation 150

Problems 155

10 Spinors 15710.1 Representations of the Lorentz group 15810.2 Spinor representations 16310.3 Dirac matrices 16810.4 Coupling to the photon 17310.5 What does spin 1

2 mean? 17410.6 Majorana and Weyl fermions 178

Problems 181

11 Spinor solutions and CPT 18411.1 Chirality, helicity and spin 18511.2 Solving the Dirac equation 18811.3 Majorana spinors 19211.4 Charge conjugation 19311.5 Parity 195






Contents ix

11.6 Time reversal 198Problems 201

12 Spin and statistics 20512.1 Identical particles 20612.2 Spin-statistics from path dependence 20812.3 Quantizing spinors 21112.4 Lorentz invariance of the S -matrix 21212.5 Stability 21512.6 Causality 219

Problems 223

13 Quantum electrodynamics 22413.1 QED Feynman rules 22513.2 γ-matrix identities 22913.3 e+e− → μ+μ− 23013.4 Rutherford scattering e−p+ → e−p+ 23413.5 Compton scattering 23813.6 Historical note 246

Problems 248

14 Path integrals 25114.1 Introduction 25114.2 The path integral 25414.3 Generating functionals 26114.4 Where is the iε? 26414.5 Gauge invariance 26714.6 Fermionic path integral 26914.7 Schwinger–Dyson equations 27214.8 Ward–Takahashi identity 277

Problems 283

Part III Renormalization 285

15 The Casimir effect 28715.1 Casimir effect 28715.2 Hard cutoff 28915.3 Regulator independence 29115.4 Scalar field theory example 296

Problems 299

16 Vacuum polarization 30016.1 Scalar φ3 theory 30216.2 Vacuum polarization in QED 304






x Contents

16.3 Physics of vacuum polarization 309Problems 314

17 The anomalous magnetic moment 31517.1 Extracting the moment 31517.2 Evaluating the graphs 318

Problems 321

18 Mass renormalization 32218.1 Vacuum expectation values 32318.2 Electron self-energy 32418.3 Pole mass 33018.4 Minimal subtraction 33418.5 Summary and discussion 336

Problems 338

19 Renormalized perturbation theory 33919.1 Counterterms 33919.2 Two-point functions 34219.3 Three-point functions 34519.4 Renormalization conditions in QED 34919.5 Z1 = Z2: implications and proof 350

Problems 354

20 Infrared divergences 35520.1 e+e− → μ+μ− (+γ) 35620.2 Jets 36420.3 Other loops 36620.A Dimensional regularization 373

Problems 380

21 Renormalizability 38121.1 Renormalizability of QED 38221.2 Non-renormalizable field theories 386

Problems 393

22 Non-renormalizable theories 39422.1 The Schrodinger equation 39522.2 The 4-Fermi theory 39622.3 Theory of mesons 40022.4 Quantum gravity 40322.5 Summary of non-renormalizable theories 40722.6 Mass terms and naturalness 40722.7 Super-renormalizable theories 414

Problems 416






Contents xi

23 The renormalization group 41723.1 Running couplings 41923.2 Renormalization group from counterterms 42323.3 Renormalization group equation for the 4-Fermi theory 42623.4 Renormalization group equation for general interactions 42923.5 Scalar masses and renormalization group flows 43523.6 Wilsonian renormalization group equation 442

Problems 450

24 Implications of unitarity 45224.1 The optical theorem 45324.2 Spectral decomposition 46624.3 Polology 47124.4 Locality 475

Problems 477

Part IV The Standard Model 479

25 Yang–Mills theory 48125.1 Lie groups 48225.2 Gauge invariance and Wilson lines 48825.3 Conserved currents 49325.4 Gluon propagator 49525.5 Lattice gauge theories 503

Problems 506

26 Quantum Yang–Mills theory 50826.1 Feynman rules 50926.2 Attractive and repulsive potentials 51226.3 e+e− → hadrons and αs 51326.4 Vacuum polarization 51726.5 Renormalization at 1-loop 52126.6 Running coupling 52626.7 Defining the charge 529

Problems 533

27 Gluon scattering and the spinor-helicity formalism 53427.1 Spinor-helicity formalism 53527.2 Gluon scattering amplitudes 54227.3 gg → gg 54527.4 Color ordering 54827.5 Complex momenta 55127.6 On-shell recursion 55527.7 Outlook 558

Problems 559






xii Contents

28 Spontaneous symmetry breaking 56128.1 Spontaneous breaking of discrete symmetries 56228.2 Spontaneous breaking of continuous global symmetries 56328.3 The Higgs mechanism 57528.4 Quantization of spontaneously broken gauge theories 580

Problems 583

29 Weak interactions 58429.1 Electroweak symmetry breaking 58429.2 Unitarity and gauge boson scattering 58829.3 Fermion sector 59229.4 The 4-Fermi theory 60229.5 CP violation 605

Problems 614

30 Anomalies 61630.1 Pseudoscalars decaying to photons 61730.2 Triangle diagrams with massless fermions 62230.3 Chiral anomaly from the integral measure 62830.4 Gauge anomalies in the Standard Model 63130.5 Global anomalies in the Standard Model 63430.6 Anomaly matching 638

Problems 640

31 Precision tests of the Standard Model 64131.1 Electroweak precision tests 64231.2 Custodial SU(2), ρ, S, T and U 65331.3 Large logarithms in flavor physics 657

Problems 666

32 Quantum chromodynamics and the parton model 66732.1 Electron–proton scattering 66832.2 DGLAP equations 67732.3 Parton showers 68232.4 Factorization and the parton model from QCD 68532.5 Lightcone coordinates 695

Problems 698

Part V Advanced topics 701

33 Effective actions and Schwinger proper time 70333.1 Effective actions from matching 70433.2 Effective actions from Schwinger proper time 70533.3 Effective actions from Feynman path integrals 71133.4 Euler–Heisenberg Lagrangian 713






Contents xiii

33.5 Coupling to other currents 72233.6 Semi-classical and non-relativistic limits 72533.A Schwinger’s method 728

Problems 732

34 Background fields 73334.1 1PI effective action 73534.2 Background scalar fields 74334.3 Background gauge fields 752

Problems 758

35 Heavy-quark physics 76035.1 Heavy-meson decays 76235.2 Heavy-quark effective theory 76535.3 Loops in HQET 76835.4 Power corrections 772

Problems 775

36 Jets and effective field theory 77636.1 Event shapes 77836.2 Power counting 78036.3 Soft interactions 78236.4 Collinear interactions 79036.5 Soft-Collinear Effective Theory 79536.6 Thrust in SCET 802

Problems 810

Appendices 813

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

Problems 821

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

Problems 833

References 834Index 842






Preface

Quantum field theory (QFT) provides an extremely powerful set of computational methodsthat have yet to find any fundamental limitations. It has led to the most fantastic agreementbetween theoretical predictions and experimental data in the history of science. It providesdeep and profound insights into the nature of our universe, and into the nature of otherpossible self-consistent universes. On the other hand, the subject is a mess. Its foundationsare flimsy, it can be absurdly complicated, and it is most likely incomplete. There are oftenmany 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 introductionto 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 studentsthat, despite their comprehensive understanding of classical and quantum physics, theystill cannot explain what is happening. Where does the light comes from? The emissionand 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. Ithen 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 thelogical generalization of the quantum theory of creation and annihilation of photons to thequantum theory of creation and annihilation of any particle. This book is based on lecturenotes for that class, plus additional material.

The main guiding principle of this book is that QFT is primarily a theory of physics, notof mathematics, and not of philosophy. QFT provides, first and foremost, a set of tools forperforming practical calculations. These calculations take as input measured numbers andpredict, sometimes to absurdly high accuracy, numbers that can be measured in other exper-iments. Whenever possible, I motivate and validate the methods we develop as explainingnatural (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 tounderstand the material. Partly, it is because the connection to data has been critical in thehistorical development of QFT.

The historical connection between theory and experiment weaves through this entirebook. The great sucess of the Dirac equation from 1928 was that it explained the magneticdipole moment of the electron (Chapter 10). Measurements of the Lamb shift in the late1940s helped vindicate the program of renormalization (Chapters 15 to 21). Measurementsof inelastic electron–proton scattering experiments in the 1960s (Chapter 32) showed thatQFT could also address the strong force. Ironically, this last triumph occurred only a few

xv






xvi Preface

years after Geoffrey Chew famously wrote that QFT “is sterile with respect to strong inter-actions 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 QFTwas capable of precision calculations to match the precision experiments that were beingperformed. 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-shellapproaches (Chapters 24 and 27). The agreement of QFT and the Standard Model withdata over the past half century has been truly astounding.

Beyond the connection to experiment, I have tried to present QFT as a set of relatedtools guided by certain symmetry principles. For example, Lorentz invariance, the symme-try group associated with special relativity, plays a essential role. QFT is the theory of thecreation and destruction of particles, which is possible due to the most famous equation ofspecial relativityE = mc2. Lorentz invariance guides the definition of particle (Chapter 8),is critical to the spin-statistics theorem (Chapter 12), and strongly constrains properties ofthe main object of interest in this book: scattering or S-matrix elements (Chapter 6 andonward). On the other hand, QFT is useful in space-times for which Lorentz invarianceis not an exact symmetry (such as our own universe, which since 1998 has been knownto have a positive cosmological constant), and in non-relativistic settings, where Lorentzinvariance is irrelevant. Thus, I am reluctant to present Lorentz invariance as an axiom ofQFT (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 addup to 1. Chapter 24 is entirely dedicated to the implications of unitarity, with reverberationsthroughout Parts IV and V. Unitarity is closely related to other appealing features of ourdescription of fundamental physics, such as causality, locality, analyticity and the clusterdecomposition principle. While unitarity and its avatars are persistent themes within thebook, I am cautious of giving them too much of a primary role. For example, it is not clearhow 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 appliedthrough the hard work of physicists who see QFT from different angles. While I do presentQFT in a more or less linear fashion, I attempt to provide multiple viewpoints wheneverpossible. For example, I derive the Feynman rules in five different ways: in classical fieldtheory (Chapter 3), in old-fashioned perturbation theory (Chapter 4), through a Lagrangianapproach (Chapter 7), through a Hamiltonian approach (also Chapter 7), and through theFeynman 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 readermight already be familiar. The Lagrangian approach illustrates in a transparent way howtree-level diagrams are just classical field theory. The old-fashioned perturbation theoryderivation connects immediately to perturbation theory in quantum mechanics, and moti-vates the distinct advantage of thinking off-shell, so that Lorentz invariance can be keptmanifest at all stages of the calculation. On the other hand, there are some instances wherean 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 Lorentzinvariance 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 Hamiltonianderivation, perturbatively through the Feynman rules, and non-perturbatively through theSchwinger–Dyson equations), and the three ways in which I derive effective actions inChapter 33 (matching, with Schwinger proper time, and with Feynman path integrals).As different students learn in different ways, providing multiple derivations is one way inwhich I have tried to make QFT accessible to a wide audience.

This textbook is written assuming that the reader has a solid understanding of quantummechanics, such as what would be covered in a year-long undergraduate class. I have foundthat students coming in generally do not know much classical field theory, and must relearnspecial relativity, so these topics are covered in Chapters 2 and 3. At Harvard, much of thematerial in this book is covered in three semesters. The first semester covers Chapters 1 to22. Including both QED and renormalization in a single semester makes the courseworkrather intense. On the other hand, from surveying the students, especially the ones who onlyhave space for a single semester of QFT, I have found that they are universally glad thatrenormalization is covered. Chapter 22, on non-renormalizable theories, is a great place toend a semester. It provides a qualitative overview of the four forces in the Standard Modelthrough the lens of renormalization and predictivity.

The course on which this textbook is based has a venerable history, dominated by thethirty or so years it was taught by the great physicist Sidney Coleman. Sidney providesan evocative description of the period from 1966 to 1979 when theory and experimentcollaborated 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 ofquantum 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 thespectator 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 EriceLectures 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 thesestudents, this book would not have been written. The presentation of most of the materialin 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 questionsthe students posed to me both in and out of class. In ruminating on those questions, anddiscussing them with my colleagues, the notes steadily improved.






xviii Preface

I have to thank in particular the various unbelievable teaching assistants I had for thecourse, particularly David Simmons-Duffin, Clay Cordova, Ilya Feige and Prahar Mitrafor their essential contributions to improving the course material. The material in this bookwas refined and improved due to critical conversations that I had with many people. Inparticular, I would like to thank Frederik Denef, Ami Katz, Aneesh Manohar, YasunoriNomura, Michael Peskin, Logan Ramalingam, Matthew Reece, Subir Sachdev, Iain Stew-art, Matthew Strassler, and Xi Yin for valuable conversations. More generally, my approachto physics, by which this book is organized, has been influenced by three people mostof all: Lisa Randall, Nima Arkani-Hamed and Howard Georgi. From them I learned torespect non-renormalizable field theories and to beware of smoke and mirrors in theoreticalphysics.

I am indebted to Anders Andreassen, David Farhi, William Frost, Andrew Marantan andPrahar Mitra for helping me convert a set of decent lecture notes into a coherent and com-prehensive 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 thebook.

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 existingvast library of fantastic quantum field theory texts. The textbooks by Peskin and Schroederand by Weinberg were especially influential. For example, Peskin and Schroeder’s nearlyperfect Chapter 5 guides my Chapter 13. Weinberg’s comprehensive two volumes of TheQuantum 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 theLSZ 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 Chapters7 and 14), from Halzen and Martin’s very physical book Quarks and Leptons (my Chapter32), Rick Field’s book Applications Perturbative QCD (my Chapter 20). I have alwaysfound Manohar and Wise’s monograph Heavy Quark Physics (on which my Chapter 35is 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 myChapter 15). In addition, a few historical accounts come from Pais’ Inward Bound, whichI 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 thisbook was being written, as well as for some editorial assistance.

Matthew Dean SchwartzCambridge, MassachusettsNovember 2013






quantum field theory and the standard model

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