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 variety 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.

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

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

10.5 What does spin 1/2 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

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 ø3 theory 302

16.2 Vacuum polarization in QED 304

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 Z_1 = Z_2: 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

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 α_s 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

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

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

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