Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information

Cambridge University Press 978-1-108-49399-4 — Introduction to Quantum Field Theory Horatiu Nastase Frontmatter More Information

Introduction to Quantum Field Theory

Quantum field theory provides a theoretical framework for understanding fields and the particles associated with them, and is the basis of particle physics and condensed matter research. This graduate-level textbook provides a comprehensive introduction to quantum field theory, giving equal emphasis to operator and path-integral formalisms. It covers modern research such as helicity spinors, BCFW construction, and generalized unitarity cuts, as well as treating advanced topics including BRST quantization, loop equations, and finite-temperature field theory. Various quantum fields are described, including scalar and fermionic fields, abelian vector fields and quantum electrodynamics (QED), and finally non-abelian vector fields and quantum chromodynamics (QCD). Applications to scattering cross-sections in QED and QCD are also described. Each chapter ends with exercises and an important concepts section, allowing students to identify the key aspects of the chapter and test their understanding.

Hora¸tiu Nastase˘ is a Researcher at the Institute for Theoretical Physics at the State University of São Paulo, Brazil. To date, his career has spanned four continents. As an undergraduate he studied at the University of Bucharest and Copenhagen University. He later completed his Ph.D. at the State University of New York, Stony Brook, before moving to the Insti- tute for Advanced Study, Princeton University, New Jersey, where his collaboration with David Berenstein and Juan Maldacena deﬁned the pp-wave correspondence. He has also held research and teaching positions at Brown University, Rhode Island and the Tokyo Institute of Technology. He has published three other books with Cambridge University Press: Introduction to the AdS/CFT Correspondence (2015), String Theory Methods for Condensed Matter Physics (2017), and Classical Field Theory (2019).

Introduction to Quantum Field Theory

HORA¸TIU NASTASE˘ State University of São Paulo, Brazil

With material from unpublished notes by Jan Ambjorn and Jens Lyng Petersen

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www.cambridge.org Information on this title: www.cambridge.org/9781108493994 DOI: 10.1017/9781108624992 © Cambridge University Press 2020 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 2020 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Nastase, Horatiu, 1972– author. Title: Introduction to quantum field theory / Horatiu Nastase (Universidade Estadual Paulista, Sao Paulo). Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2020. | Includes bibliographical references and index. Identifiers: LCCN 2019006491| ISBN 9781108493994 (alk. paper) | ISBN 1108493998 (alk. paper) Subjects: LCSH: Quantum field theory. Classification: LCC QC174.45 .N353 2020 | DDC 530.14/3–dc23 LC record available at https://lccn.loc.gov/2019006491 ISBN 978-1-108-49399-4 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 the memory of my mother, who inspired me to become a physicist

Contents

Preface page xxv Acknowledgments xxvi Introduction xxvii

I Quantum Fields, General Formalism, and Tree Processes 1

1 Review of Classical Field Theory: Lagrangians, Lorentz Group and its Representations, Noether Theorem 3 1.1 What is and Why Do We Need Quantum Field Theory? 3 1.2 Classical Mechanics 6 1.3 Classical Field Theory 6 1.4 Noether Theorem 8 1.5 Fields and Lorentz Representations 9 Further Reading 11 Exercises 12

2 Quantum Mechanics: Harmonic Oscillator and Quantum Mechanics in Terms of Path Integrals 13 2.1 The Harmonic Oscillator and its Canonical Quantization 13 2.2 The Feynman Path Integral in Quantum Mechanics in Phase Space 15 2.3 Gaussian Integration 18 2.4 Path Integral in Conﬁguration Space 19 2.5 Correlation Functions 19 Further Reading 21 Exercises 22

3 Canonical Quantization of Scalar Fields 23 3.1 Quantizing Scalar Fields: Kinematics 23 3.2 Quantizing Scalar Fields: Dynamics and Time Evolution 25 3.3 Discretization 27 3.4 Fock Space and Normal Ordering for Bosons 28 3.4.1 Fock Space 28 3.4.2 Normal Ordering 29 3.4.3 Bose–Einstein Statistics 30 Further Reading 30 Exercises 31 vii

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4 Propagators for Free Scalar Fields 32 4.1 Relativistic Invariant Canonical Quantization 32 4.2 Canonical Quantization of the Complex Scalar Field 33 4.3 Two-Point Functions and Propagators 35 4.4 Propagators: Retarded and Feynman 37 4.4.1 Klein–Gordon Propagators 37 4.4.2 Retarded Propagator 38 4.4.3 Feynman Propagator 38 Further Reading 39 Exercises 40

5 Interaction Picture and Wick Theorem for λφ4 in Operator Formalism 41 5.1 Quantum Mechanics Pictures 41 5.1.1 Schrödinger Picture (Usual) 42 5.1.2 Heisenberg Picture 42 5.1.3 Dirac (Interaction Picture) 42 5.2 Physical Scattering Set-up and Interaction Picture 43 5.2.1 λφ4 Theory 43 5.3 Evolution Operator and the Feynman Theorem 44 5.4 Wick’s Theorem 47 Further Reading 49 Exercises 49

6 Feynman Rules for λφ4 from the Operator Formalism 50 6.1 Diagrammatic Representation of Free Four-Point Function 50 6.2 Interacting Four-Point Function: First-Order Result and its Diagrammatic Representation 51 6.3 Other Contractions and Diagrams 52 6.4 x-Space Feynman Rules for λφ4 53 6.5 p-Space Feynman Rules and Vacuum Bubbles 54 6.5.1 Canceling of the Vacuum Bubbles in Numerator vs. Denominator in Feynman Theorem 55 Further Reading 57 Exercises 57

7 The Driven (Forced) Harmonic Oscillator 59 7.1 Set-up 59 7.2 Sloppy Treatment 60 7.3 Correct Treatment: Harmonic Phase Space 64 Further Reading 66 Exercises 67

8 Euclidean Formulation and Finite-Temperature Field Theory 68 8.1 Phase-Space and Conﬁguration-Space Path Integrals and Boundary Conditions 68

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8.2 Wick Rotation to Euclidean Time and Connection with Statistical Mechanics Partition Function 70 8.3 Quantum-Mechanical Statistical Partition Function and Correlation Functions 72 8.4 Example: Driven Harmonic Oscillator 73 Further Reading 75 Exercises 75

9 The Feynman Path Integral for a Scalar Field 77 9.1 Euclidean Formulation 77 9.2 Perturbation Theory 78 9.3 Dyson Formula for Perturbation Theory 79 9.4 Solution of the Free Field Theory 80 9.5 Wick’s Theorem 81 Further Reading 82 Exercises 82

10 Wick’s Theorem for Path Integrals and Feynman Rules Part I 83 10.1 Examples 83 10.2 Wick’s Theorem: Second Form 86 10.3 Feynman Rules in x Space 87 Further Reading 89 Exercises 89

11 Feynman Rules in x Space and p Space 91 11.1 Proof of the Feynman Rules 91 11.2 Statistical Weight Factor (Symmetry Factor) 92 11.3 Feynman Rules in p Space 93 11.4 Most General Bosonic Field Theory 95 Further Reading 97 Exercises 97

12 Quantization of the Dirac Field and Fermionic Path Integral 98 12.1 The Dirac Equation 98 12.2 Weyl Spinors 100 12.3 Solutions of the Free Dirac Equation 101 12.4 Quantization of the Dirac Field 103 12.5 The Fermionic Path Integral 105 12.5.1 Deﬁnitions 105 Further Reading 108 Exercises 109

13 Wick Theorem, Gaussian Integration, and Feynman Rules for Fermions 110 13.1 Gaussian Integration for Fermions 110 13.1.1 Gaussian Integration – The Real Case 110 13.1.2 Real vs. Complex Integration 112

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13.2 The Fermionic Harmonic Oscillator and Generalization to Field Theory 112 13.3 Wick Theorem for Fermions 115 13.4 Feynman Rules for Yukawa Interaction 116 Further Reading 118 Exercises 118

14 Spin Sums, Dirac Field Bilinears, and C, P, T Symmetries for Fermions 120 14.1 Spin Sums 120 14.2 Dirac Field Bilinears 121 14.3 C, P, T Symmetries for Fermions 122 14.3.1 Parity 123 14.3.2 Time Reversal 125 14.3.3 Charge Conjugation 126 Further Reading 127 Exercises 128

15 Dirac Quantization of Constrained Systems 129 15.1 Set-up and Hamiltonian Formalism 129 15.2 System with Constraints in Hamiltonian Formalism: Primary/Secondary and First/Second-Class Constraints 130 15.3 Quantization and Dirac Brackets 133 15.4 Example: Electromagnetic Field 135 Further Reading 137 Exercises 137

16 Quantization of Gauge Fields, their Path Integral, and the Photon Propagator 139 16.1 Physical Gauge 139 16.2 Quantization in Physical Gauge 140 16.3 Lorenz Gauge (Covariant) Quantization 142 16.4 Fadeev–Popov Path-Integral Quantization 144 16.5 Photon Propagator 147 Further Reading 148 Exercises 149

17 Generating Functional for Connected Green’s Functions and the Eﬀective Action (1PI Diagrams) 150 17.1 Generating Functional of Connected Green’s Functions 150 17.2 Effective Action and 1PI Green’s Functions 152 17.2.1 Example: Free Scalar Field Theory in the Discretized Version 152 17.2.2 1PI Green’s Functions 153 17.3 The Connected Two-Point Function 155 17.4 Classical Action as Generating Functional of Tree Diagrams 156 Further Reading 158 Exercises 158

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18 Dyson–Schwinger Equations and Ward Identities 160 18.1 Dyson–Schwinger Equations 160 18.1.1 Speciﬁc Interaction 161 18.2 Iterating the Dyson–Schwinger Equation 162 18.2.1 Example 162 18.3 Noether’s Theorem 165 18.4 Ward Identities 166 Further Reading 168 Exercises 168

19 Cross-Sections and the S-Matrix 170 19.1 Cross-Sections and Decay Rates 170 19.1.1 Decay Rate 172 19.2 In and Out States, the S-Matrix, and Wavefunctions 172 19.2.1 Wavefunctions 173 19.3 The Reduction Formula (Lehmann, Symanzik, Zimmermann) 174 19.4 Cross-Sections from Amplitudes M 175 19.4.1 Particle Decay 178 Further Reading 179 Exercises 179

20 The S-Matrix and Feynman Diagrams 180 20.1 Perturbation Theory for S-Matrices: Feynman and Wick 180 20.2 Example: φ4 Theory in Perturbation Theory and First-Order Differential Cross-Section 181 20.3 Second-Order Perturbation Theory and Amputation 184 20.4 Feynman Rules for S-Matrices 185 Further Reading 186 Exercises 186

21 The Optical Theorem and the Cutting Rules 188 21.1 The Optical Theorem: Formulation 188 21.2 Unitarity: Optical Theorem at One Loop in λφ4 Theory 190 21.3 General Case and the Cutkovsky Cutting Rules 193 Further Reading 194 Exercises 194 ∗ 22 Unitarity and the Largest Time Equation 197 22.1 The Largest Time Equation for Scalars: Propagators 197 22.2 Cut Diagrams 199 22.3 The Largest Time Equation for Scalars: Derivation 200 22.4 General Case 201 Further Reading 203 Exercises 203

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23 QED: Deﬁnition and Feynman Rules; Ward–Takahashi Identities 204 23.1 QED: Deﬁnition 204 23.2 QED Path Integral 205 23.3 QED Feynman Rules 206 23.3.1 Feynman Rules for Green’s Functions in Euclidean Momentum Space 206 23.3.2 Feynman Rules for S-Matrices in Minkowski Space 206 23.4 Ward–Takahashi Identities 207 23.4.1 Example 1: Photon Propagator 208 23.4.2 Example 2: n-Photon Vertex Function for n ≥ 3 209 23.4.3 Example 3: Original Ward–Takahashi Identity 209 Further Reading 210 Exercises 211

24 Nonrelativistic Processes: Yukawa Potential, Coulomb Potential, and Rutherford Scattering 212 24.1 Yukawa Potential 212 24.2 Coulomb Potential 214 24.3 Particle–Antiparticle Scattering 215 24.3.1 Yukawa Potential 215 24.3.2 Coulomb Potential 216 24.4 Rutherford Scattering 217 Further Reading 220 Exercises 220

25 e+e− → ll¯Unpolarized Cross-Section 221 25.1 e+e− → ll¯ Unpolarized Cross-Section: Set-up 221 25.2 Gamma Matrix Identities 223 25.3 Cross-Section for Unpolarized Scattering 226 25.4 Center of Mass Frame Cross-Section 227 Further Reading 229 Exercises 229

26 e+e− → ll¯Polarized Cross-Section; Crossing Symmetry 230 26.1 e+e− → ll¯ Polarized Cross-Section 230 26.2 Crossing Symmetry 233 26.3 Mandelstam Variables 235 Further Reading 239 Exercises 239

27 (Unpolarized) Compton Scattering 241 27.1 Compton Scattering: Set-up 241 27.2 Photon Polarization Sums 242 27.3 Cross-Section for Compton Scattering 243

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Further Reading 247 Exercises 247 ∗ 28 The Helicity Spinor Formalism 248 28.1 Helicity Spinor Formalism for Spin 1/2 248 28.2 Helicity Spinor Formalism for Spin 1 253 28.3 Amplitudes with External Spinors 253 Further Reading 254 Exercises 254 29∗ Gluon Amplitudes, the Parke–Taylor Formula, and the BCFW Construction 256 29.1 Amplitudes with External Gluons and Color-Ordered Amplitudes 256 29.2 Amplitudes of Given Helicity and Parke–Taylor Formula 258 29.3 Kleiss–Kluijf and BCJ Relations 260 29.4 The BCFW Construction 262 29.5 Application of BCFW: Proof of the Parke–Taylor Formula 264 Further Reading 265 Exercises 265 30 Review of Path Integral and Operator Formalism and the Feynman Diagram Expansion 266 30.1 Path Integrals, Partition Functions, and Green’s Functions 266 30.1.1 Path Integrals 266 30.1.2 Scalar Field 266 30.2 Canonical Quantization, Operator Formalism, and Propagators 268 30.3 Wick Theorem, Dyson Formula, and Free Energy in Path-Integral Formalism 270 30.4 Feynman Rules, Quantum Effective Action, and S-Matrix 271 30.4.1 Feynman Rules in x Space (Euclidean) 271 30.4.2 Simpliﬁed Rules 271 30.4.3 Feynman Rules in p Space 271 30.4.4 Simpliﬁed Momentum-Space Rules 272 30.4.5 Classical Field 272 30.4.6 Quantum Effective Action 272 30.4.7 S-Matrix 273 30.4.8 Reduction Formula (LSZ) 273 30.5 Fermions 274 30.6 Gauge Fields 275 30.7 Quantum Electrodynamics 276 30.7.1 QED S-Matrix Feynman Rules 276 Further Reading 277 Exercises 277

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II Loops, Renormalization, Quantum Chromodynamics, and Special Topics 279

31 One-Loop Determinants, Vacuum Energy, and Zeta Function Regularization 281 31.1 Vacuum Energy and Casimir Force 281 31.2 General Vacuum Energy and Regularization with Riemann Zeta ζ (−1) 283 31.3 Zeta Function and Heat Kernel Regularization 285 31.3.1 Heat Kernel Regularization 286 31.4 Saddle Point Evaluation and One-Loop Determinants 286 31.4.1 Path Integral Formulation 287 31.4.2 Fermions 288 Further Reading 289 Exercises 289

32 One-Loop Divergences for Scalars; Power Counting 290 32.1 One-Loop UV and IR Divergences 290 32.2 Analytical Continuation of Integrals with Poles 293 32.3 Power Counting and UV Divergences 294 32.4 Power-Counting Renormalizable Theories 296 32.4.1 Examples 298 32.4.2 Divergent φ4 1PI Diagrams in Various Dimensions 298 Further Reading 299 Exercises 299

33 Regularization, Definitions: Cut-off, Pauli–Villars, Dimensional Regularization, and General Feynman Parametrization 301 33.1 Cut-off Regularization and Regularizations of Infinite Sums 301 33.1.1 Infinite Sums 302 33.2 Pauli–Villars Regularization 303 33.3 Derivative Regularization 305 33.4 Dimensional Regularization 305 33.5 Feynman Parametrization 307 33.5.1 Feynman Parametrization with Two Propagators 307 33.5.2 General One-Loop Integrals and Feynman Parametrization 309 33.5.3 Alternative Version of the Feynman Parametrization 310 33.6 Dimensionally Continuing Lagrangians 311 Further Reading 312 Exercises 313

34 One-Loop Renormalization for Scalars and Counterterms in Dimensional Regularization 314 34.1 Divergent Diagrams in φ4 Theory in D = 4 and its Divergences 315 34.1.1 Divergent Parts 316 34.2 Divergent Diagrams in φ3 Theory in D = 6 and its Divergences 317 34.2.1 Divergent Parts 319 34.3 Counterterms in φ4 and φ3 Theories 319

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34.4 Renormalization 321 34.4.1 Examples 323 Further Reading 324 Exercises 324

35 Renormalization Conditions and the Renormalization Group 325 35.1 Renormalization of n-Point Functions 325 35.2 Subtraction Schemes and Normalization Conditions 327 35.2.1 Subtraction Schemes 327 35.2.2 Normalization Conditions 327 35.3 Renormalization Group Equations and Anomalous Dimensions 328 35.3.1 Renormalization Group in MS Scheme 328 35.3.2 φ4 in Four Dimensions 328 35.4 Beta Function and Running Coupling Constant 331 35.4.1 Possible Behaviors for β(λ) 331 35.5 Perturbative Beta Function in Dimensional Regularization in MS Scheme 334 35.5.1 Examples 335 35.6 Perturbative Calculation of γm and γd in Dimensional Regularization in the MS Scheme 335 Further Reading 337 Exercises 337

36 One-Loop Renormalizability in QED 338 36.1 QED Feynman Rules and Power-Counting Renormalizability 338 36.2 Dimensional Regularization of Gamma Matrices 340 36.3 Case 1: Photon Polarization μν(p) 340 36.4 Case 2: Fermion Self-energy (p) 343 36.5 Case 3: Fermions–Photon Vertex Ŵμαβ 345 Further Reading 347 Exercises 347

37 Physical Applications of One-Loop Results I: Vacuum Polarization 348 37.1 Systematics of QED Renormalization 348 37.2 Vacuum Polarization 349 37.3 Pair Creation Rate 353 Further Reading 354 Exercises 355

38 Physical Applications of One-Loop Results II: Anomalous Magnetic Moment and Lamb Shift 356 38.1 Anomalous Magnetic Moment 356 38.2 Lamb Shift 358 Further Reading 362 Exercises 362

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39 Two-Loop Example and Multiloop Generalization 363 39.1 Types of Divergences at Two Loops and Higher 363 39.2 Two Loops in φ4 in Four Dimensions: Set-up 365 39.3 One-Loop Renormalization 367 39.4 Calculation of Two-Loop Divergences in φ4 in Four Dimensions and their Renormalization 368 Further Reading 373 Exercises 373

40 The LSZ Reduction Formula 374 40.1 The LSZ Reduction Formula and Wavefunction Renormalization 374 40.2 Adding Wavepackets 377 40.3 Diagrammatic Interpretation 378 Further Reading 379 Exercises 379 ∗ 41 The Coleman–Weinberg Mechanism for One-Loop Potential 380 41.1 One-Loop Effective Potential in λφ4 Theory 380 41.2 Renormalization and Coleman–Weinberg Mechanism 381 41.3 Coleman–Weinberg Mechanism in Scalar-QED 382 Further Reading 384 Exercises 384

42 Quantization of Gauge Theories I: Path Integrals and Fadeev–Popov 385 42.1 Review of Yang–Mills Theory and its Coupling to Matter Fields 385 42.2 Fadeev–Popov Procedure in Path Integrals 387 42.2.1 Correlation Functions 387 42.3 Ghost Action 391 Further Reading 393 Exercises 393

43 Quantization of Gauge Theories II: Propagators and Feynman Rules 394 43.1 Propagators and Effective Action 394 43.1.1 Propagators 394 43.1.2 Interactions 394 43.2 Vertices 395 43.3 Feynman Rules 396 43.4 Example of Feynman Diagram Calculation 398 Further Reading 401 Exercises 402

44 One-Loop Renormalizability of Gauge Theories 403 44.1 Divergent Diagrams of Pure Gauge Theory 403 44.2 Counterterms in MS Scheme 407 44.3 Renormalization and Consistency Conditions 408 44.4 Gauge Theory with Fermions 409

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Further Reading 498 Exercises 498

53 Factorization and the Kinoshita–Lee–Nauenberg Theorem 500 53.1 The KLN Theorem 500 53.2 Statement and Proof of Lemma 502 53.3 Factorization and Evolution 504 53.3.1 Factorization Theorem 504 Further Reading 506 Exercises 506

54 Perturbative Anomalies: Chiral and Gauge 508 54.1 Chiral Invariance in Classical and Quantum Theory 508 54.2 Chiral Anomaly 509 54.2.1 Anomaly in d = 2 Euclidean Dimensions 510 54.2.2 Anomaly in d = 4 Dimensions 512 54.3 Properties of the Anomaly 513 54.4 Chiral Anomaly in Nonabelian Gauge Theories 514 54.5 Gauge Anomalies 515 Further Reading 516 Exercises 516 55 Anomalies in Path Integrals: The Fujikawa Method, Consistent vs. Covariant Anomalies, and Descent Equations 517 55.1 Chiral Basis vs. V–A Basis 517 55.2 Anomaly in the Path Integral: Fujikawa Method 518 55.3 Consistent vs. Covariant Anomaly 521 55.4 Descent Equations 522 Further Reading 524 Exercises 524 56 Physical Applications of Anomalies, ’t Hooft’s UV–IR Anomaly Matching Conditions, and Anomaly Cancellation 526 56.1 π 0 → γγ Decay 526 56.2 Nonconservation of Baryon Number in Electroweak Theory 529 56.3 The U(1) Problem 531 56.4 ’t Hooft’s UV–IR Anomaly Matching Conditions 531 56.5 Anomaly Cancellation in General and in the Standard Model 532 56.5.1 The Standard Model 533 Further Reading 535 Exercises 535 ∗ 57 The Froissart Unitarity Bound and the Heisenberg Model 536 57.1 The S-Matrix Program, Analyticity, and Partial Wave Expansions 536 57.2 The Froissart Unitarity Bound 537 57.2.1 Application to Strong Interactions 538

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57.3 The Heisenberg Model for Saturation of the Froissart Bound 539 Further Reading 543 Exercises 543 58 The Operator Product Expansion, Renormalization of Composite Operators, and Anomalous Dimension Matrices 545 58.1 Renormalization of Composite Operators 545 58.2 Anomalous Dimension Matrix 546 58.3 Anomalous Dimension Calculation 547 58.3.1 Tree Level: O(1) 547 58.3.2 One-Loop Level: O(λ) 548 58.4 The Operator Product Expansion 549 58.5 QCD Example 550 Further Reading 552 Exercises 553 59∗ Manipulating Loop Amplitudes: Passarino–Veltman Reduction and Generalized Unitarity Cut 554 59.1 Passarino–Veltman Reduction of One-Loop Integrals 554 59.2 Box Integrals 557 59.3 Generalized Unitarity Cuts 560 Further Reading 562 Exercises 562 ∗ 60 Analyzing the Result for Amplitudes: Polylogs, Transcendentality, and Symbology 564 60.1 Polylogs in Amplitudes 564 60.2 Maximal and Uniform Transcendentality of Amplitudes 565 60.3 Symbology 568 Further Reading 570 Exercises 570 61∗ Representations and Symmetries for Loop Amplitudes: Amplitudes in Twistor Space, Dual Conformal Invariance, and Polytope Methods 571 61.1 Twistor Space 571 61.2 Amplitudes in Twistor Space 572 61.2.1 Dual Space and Momentum Twistors 573 61.3 Dual Conformal Invariance 574 61.4 Polytopes and Amplitudes 574 61.5 Leading Singularities of Amplitudes and a Conjecture for Them 578 Further Reading 581 Exercises 581

62 The Wilsonian Eﬀective Action, Eﬀective Field Theory, and Applications 582 62.1 The Wilsonian Effective Action 582 62.1.1 φ4 Theory in Euclidean Space 582 62.2 Calculation of c ,i 583

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62.3 Effective Field Theory 586 62.3.1 Nonrenormalizable Theories 587 62.3.2 Removing the Cut-off 587 Further Reading 589 Exercises 589 63 Kadanoﬀ Blocking and the Renormalization Group: Connection with Condensed Matter 590 63.1 Field Theories as Classical Spin Systems 590 63.2 Kadanoff Blocking 593 63.3 Expansion Near a Critical Point 595 63.4 Critical Exponents (Near the Fixed Point) 596 Further Reading 598 Exercises 598

64 Lattice Field Theory 600 64.1 Continuum Limit 600 64.1.1 Gaussian Fixed Point 600 64.2 Beta Function 601 64.3 Lattice Gauge Theory 602 64.4 Lattice Gauge Theory: Continuum Limit 605 64.5 Adding Matter 606 Further Reading 607 Exercises 607

65 The Higgs mechanism 608 65.1 Abelian Case 608 65.2 Abelian Case: Unitary Gauge 610 65.3 Abelian Case: Gauge Symmetry 611 65.4 Nonabelian Case 611 65.4.1 SU(2) Case 613 65.5 Standard Model Higgs: Electroweak SU(2) × U(1) 613 Further Reading 616 Exercises 616 66 Renormalization of Spontaneously Broken Gauge Theories I: The Goldstone Theorem and Rξ Gauges 618 66.1 The Goldstone Theorem 618 66.2 Rξ Gauges: Abelian Case 620 66.3 Rξ Gauges: Nonabelian Case 623 Further Reading 624 Exercises 624

67 Renormalization of Spontaneously Broken Gauge Theories II: The SU(2)-Higgs Model 625 67.1 The SU(2)-Higgs Model 625 67.2 Quantum Theory and LZJ Identities 627

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67.3 Renormalization 628 Further Reading 630 Exercises 630

68 Pseudo-Goldstone Bosons, Nonlinear Sigma Model, and Chiral Perturbation Theory 632 68.1 QCD, Chiral Symmetry Breaking, and Goldstone Theorem 632 68.2 Pseudo-Goldstone Bosons, Chiral Perturbation Theory, and Nonlinear Sigma Model 634 68.3 The SO(N) Vector Model 636 68.4 Physical Processes and Generalizations 639 68.4.1 Generalization 641 68.4.2 Generalization to SU(3) 641 68.5 Heavy Quark Effective Field Theory 641 68.6 Coupling to Nucleons 642 68.7 Mass Terms 643 Further Reading 644 Exercises 644

∗ 69 The Background Field Method 646 69.1 General Method and Quantum Partition Function 646 69.2 Scalar Field Analysis for Effective Action 648 69.3 Gauge Theory Analysis 650 Further Reading 654 Exercises 654

∗ 70 Finite-Temperature Quantum Field Theory I: Nonrelativistic (“Manybody”) Case 655 70.1 Review of Thermodynamics of Quantum Systems (Quantum Statistical Mechanics) 656 70.2 Nonrelativistic QFT at Finite Temperature: “Manybody” Theory 658 70.3 Paranthesis: Condensed Matter Calculations 659 70.4 Free Green’s Function 661 70.5 Perturbation Theory and Dyson Equations 663 70.5.1 Feynman Rules in x Space 664 70.5.2 Feynman Rules in Momentum Space 664 70.5.3 Dyson Equation 665 70.6 Lehmann Representation and Dispersion Relations 667 70.7 Real-Time Formalism 669 70.7.1 Lehmann Representation and Dispersion Relations 670 70.7.2 Relation with Green–Matsubara Functions 671 70.7.3 Free Green–Zubarev Function 672 70.7.4 Correlation Functions and Scattering 672 Further Reading 674 Exercises 674

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∗ 71 Finite-Temperature Quantum Field Theory II: Imaginary and Real-Time Formalisms 675 71.1 The Imaginary-Time Formalism 675 71.2 Imaginary-Time Formalism: Propagators 677 71.3 KMS (Kubo–Martin–Schwinger) Relation 678 71.4 Real-Time Formalism 679 71.5 Interpretation of Green’s Functions 681 71.6 Propagators and Field Doubling 682 Further Reading 684 Exercises 685 72∗ Finite-Temperature Quantum Field Theory III: Thermoﬁeld Dynamics and Schwinger–Keldysh “In–In” Formalism for Thermal and Nonequilibrium Situations. Applications 686 72.1 Thermoﬁeld Dynamics 686 72.1.1 Thermal Fermionic Harmonic Oscillator 687 72.1.2 Bosonic Harmonic Oscillator 688 72.2 The Schwinger–Keldysh Formalism at T = 0 689 72.3 Schwinger–Keldysh Formalism at Nonzero T 692 72.4 Application of Thermal Field Theory: Finite-Temperature Effective Potential 693 Further Reading 696 Exercises 696

References 697 Index 699

Preface

Quantum field theory is a subject that has been here for a while, and there are many books that teach it. However, I have found several reasons to write a book, based on lecture notes for a two-semester course I gave. One motivation is that, to my knowledge, there are no books that consistently treat together the operator and path-integral formalisms, on an equal footing, and using one or the other as is more convenient for the presentation. People usually have their favorite way of thinking about quantum field theory, either in the operator (usually for more phenomenological reasons) or the path-integral (usually for more theoretical reasons) formalism, and they almost always stick with it. But modern methods use both, and I think it important for students to be proficient in both also. There are many modern topics that don’t make it into a quantum field theory book, but since the subject is constantly evolving, it is worth knowing the most important recent developments. Another reason is that in most physics departments in the USA and Europe, quantum field theory is taught directly after classical mechanics and quantum mechanics, which is oftentimes a tough transition for a graduate student. At our institute, the Institute for Theoretical Physics at UNESP, we teach one semester of classical field theory, followed by two of quantum field theory, which makes the transition smoother, and is easier for the students to follow. For that reason, I have already published a classical field theory book (also with Cambridge University Press), as an extended version of the corresponding course I taught. The present quantum field theory book is conceived as a continuation of that classical field theory course, though I have tried to make it as self-consistent as possible. Notions of classical field theory are thus just reviewed, not treated in great detail. Having decided to include path-integral treatment alongside the operator treatment (more heavily used in standard textbooks) early on, when deciding how to use path integrals in the exposition of most topics, I could think of no better way than the one I learned just before graduate school, in an exchange program at Niels Bohr Institute: a course following unpublished lecture notes by Jan Ambjorn and Jens Lyng Petersen, which were available from the NBI secretariat. So I used a lot of material from those notes, which are unpublished so far, in the building of my exposition, though of course following a different logic of exposition, in the end forming a significant part of the book, maybe a quarter or up to a third. When deciding to publish my notes, I asked Jan Ambjorn and Jens Lyng Petersen for permission to use the material in their notes as I did, with a clarification added in the subtitle. There are many quantum field theory books out there, among which I would highlight perhaps the ones by Peskin and Schroeder, Schwartz, Ramond, Weinberg, Zinn- Justin, Banks, Zee, and Srednicki, but I believe that none exactly match the requirements I had set out to fulfill, as stated above. xxv

Acknowledgments

I want to thank everybody that shaped who I am as a physicist and helped me along the way, starting first and foremost with my mother Ligia, the first physicist I knew and the per- son who introduced me to the wonderful world of physics. Next, my high-school physics teacher Iosif Sever Georgescu made me realize that I could make a career out of physics, and made me see the path to it. Poul Olesen was my student exchange advisor at the Niels Bohr Institute (NBI), and he introduced me to string theory, which is still my field of study. Also at the NBI I had the wonderful quantum field theory course by Jan Ambjorn, based on the notes by him and Jens Lyng Petersen, which, I mentioned in the Preface, were used a lot in the presentation of the path-integral formalism in this book. Jens Lyng Petersen taught elementary particle physics, also forming my understanding of some of the issues in this book. My Ph.D. advisor, Peter van Nieuwenhuizen, taught me how to be a com- plete theoretical physicist, the beauty of calculations, and the value of rigor, and from him I learned many of the topics presented in this book, in particular various advanced topics in quantum field theory. The book is based on a two-semester course I gave (twice) at the Institute for Theoretical Physics in São Paulo, so I would like to thank all the students in my class for their questions, input, and corrections to the notes that were then expanded into this book. I would like to thank all my collaborators, and especially those with which I worked on various topics that ended up being presented in this book: Howard Schnitzer, Stephen Naculich, Juan Maldacena, David Berenstein, and Jeff Murugan. I want to thank my wife Antonia for her patience when I wrote this book, in the evenings at home, and her encouragement to continue. I want to thank my students and postdocs for accepting the reduced time I spent with them while working on the book. A big thanks to my editor at Cambridge University Press, Simon Capelin, for his encouragement and for helping me get this book, as well as my previous ones, published. To all the staff at Cambridge University Press, thanks for helping me with this book and my previous ones, and for making sure that it is as good as it can be.

xxvi

Introduction

This book is meant as a two-semester course in quantum field theory, skipping some material that can be studied independently. The chapters with an asterisk I have not taught in my class, and they can be skipped in a first reading, or when teaching the material. The book and the corresponding course is supposed to follow a course in classical field theory; however, I have tried to make the book self-contained. This means that only a thorough knowledge of classical mechanics, quantum mechanics, and electromagnetism is really needed, though it is preferable to have first classical field theory. I will only review classical field theory, without going into great detail. Quantum field theory represents the union of quantum mechanics and classical field theory, which itself is but a generalization of classical mechanics to an infinite number of degrees of freedom. As such, one needs to understand both quantum mechanics and classical mechanics (and perhaps its field theory generalization), so I will start by reviewing the necessary concepts. There are two main formalisms for treating quantum field theory, the operator formalism, and the more modern path-integral formalism, a generalization of the path integral for quantum mechanics, unfortunately not often taught in quantum mechanics. I will introduce them in parallel, and then use one or the other as is more convenient for the subject being treated. In Part I, corresponding to the first semester of the course, I will introduce the general formalism and use it in “tree” processes, which are processes in the quantum mechanics form of classical field theory, but with no quantum field theory corrections. I will also give examples of physical processes treated with the quantum field theory formalism in the tree approximation, calculating scattering cross-sections for them. I will also describe briefly the modern formalism of helicity spinors for amplitudes of given helicity, and the BCFW iterative construction of amplitudes. In Part II, corresponding to the second semester of the course, with several added chapters, I will describe “loops,” which are true quantum field theory corrections, and the formalism of renormalization, which is a way to “absorb” the infinities of quantum field theory in the redefinition of parameters of the model, and the basis of the standard treatment. I will also treat nonabelian gauge theories, in particular QCD, IR divergences and anomalies, as well as many advanced topics that are usually less taught, like BRST quantization, the Makeenko–Migdal loop equation and order parameters, the Froissart unitarity bound, renormalization of spontaneously broken gauge theories, background field method, and finite-temperature quantum field theory. I also include more modern topics, like the generalized unitarity cut, polylogs and symbology, amplitudes in twistor space, and dual conformal invariance. xxvii