Contents
1 Particles and Interactions: An Overview . . . . . . 1
1.1 A Preview . . . . 1
1.2 Particles . . . . . . . . . . . 3
1.2.1 Leptons .......................... 4
1.2. 2 Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Hadrons . . . . . 6
1.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1. 5 Physical Units . . . . . . . . . . . . . . 13
Problems . . . . . . . . . . . . . . 15
Suggestions for Further Reading . . .. .... 16
2 Boson Fields . . . . . . . . . . . . . . 17
2.1 Lorentz Symmetry . . . . . . . .. . 18
2.1.1 Lorentz Transformations . . . . . 18
2.1.2 Tensor Algebra . 23
2.1.3 Tensor Fields . . . . . . . . . . . . . 24
2.2 Scalar Fields . . . . . . . . . . . . . . . 25
2.2.1 Space-Time Translation of a Scalar Field 25
2.2.2 Lorentz Transformation of a Scalar Field . . . . . . . . . 28
2. 3 Vector Fields . . . . . . . . . . . . . . . . .. . 30
2.4
The Klein-Gordon Equation . . . . . . . . . . . . . . . . . . 31
2.4.1 Free-Particle Solutions . . . . . . . . . . . . . . . . . . 31
2.4.2 Particle Probability . . . . . . . . . . . . . . . . . . . . 32
2.4.3 Second Quantization . . . . . . . . . . . . . 34
2.4.4 Operator Algebra . . . . . . . . . . . . . . . . . . . . . 35
2.4.5 Physical Significance of the Fock Operators . . . . . . . 37
2.5 Quantized Vector Fields . . . . . . . . . . . . . . . . . . . . 39
2.5.1 Massive Vector Fields . . . . . , . . . . . . . . 39
2.5.2 The Maxwell Equations . . . . . . . . . : . 40
2.5.3 Quantization of the Electromagnetic Field . 42
2.5.4 Field Energy and Momentum . . . . . . . . . . . . 46
2.6 The Action . . . . . . . . . . . .. ......... . 47
2.6.1 The Euler-IJagrange Equation 47
2.6.2 Conserved Current . . . . . . . . .. . 50
Problems . . . . . . . . . . . . . 55
Suggestions for Further Reading . 56
3 Fermion Fields . . . . . . . . . . . . . . . . . . . . . . . . .. . 57
3.1 The Dirac Equation .......... . . . .. 57
3.2 Lorentz Symmetry . . . . . . . . . . . . . . . . . . 60
3.2.1 Covariance of the Dirac Equation .. 60
3.2.2 Spin of the Dirac Field . . . . : 63
3.2.3 Bilinear Covariants . . 64
3.3 Free-Particle Solutions . . . . . . . . 65
3.3.1 Normalized Spinors . . . . 66
3.3.2 Completeness Relations . . . . . . . . . . . . . . . . . . 68
3.3.3 Helicities . . . . . . . . . . . . . . . 71
3.4 The Lagrangian for a Free Dirac Particle . . . 73
3.5 Quantization of the Dirac Field .. . . . . . 76
3.5.1 Spins and Statistics . . . . . . . . . . . . . . . . . . . . 77
3.5.2 Dirac Field Observables . . . . . . 79
3.5.3 Fock Space . . . . 80
3.6 Zero-Mass Fermions ..... . . . . 82
Pro blems . . . . . . . . . . . . . . . . . . . 86
Suggestions for Further Reading . . . . . . . . . . . . . 88
4 Collisions and Decays . . . . . . . 89
4.1 Interaction Representation . . . . . . . . . . . . . . 90
4.1.1 The Three Pictures . . . . 90
4.1.2 Time Evolution in the Interaction Picture . 92
4.1.3 The S-matrix . . . . . . . . . . . . 95
4.2 Cross-Sections and Decay Rates . . . . . . . . . 96
4.2.1 General Formulas . . . . . . . . . . . . . . . . 96
4.2.2 Two-Body Reaction to Two-Body Final States ..... 99
4.2.3 Decay Rates . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Interaction Models ............... . . 104
4.4 Decay Modes of Scalar Particles ............... 105
4.4.1 Neutral Decay Mode . . . . . . . . . . . . . . . . . . . 105
4.4.2 Charged Decay Mode . . . . . . 108
4.5 Pion Scattering . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5.1 The Scalar Boson Propagator . . . . . . 110
4.5.2 Scattering Processes . . . . . . . . . . . 112
4.5.3 Summary and Generalization . . . . . . . . 116
4.6 Electron-Proton Scattering . . . . . . . : . . . . . . . . . . 118
4.6.1 The Electromagnetic Interaction . . . . . . . . .. 119
4.6.2 Electron-Proton Scattering Cross-Section. ..... 120
4.7 Electron-Positron Annihilation . . . . . 127
4.8 Compton Scattering . . . . . . 133
Problems . . . . . . . . . . . . . . 141
Suggestions for Further Reading . . . . . . . 142
5 Discrete Symmetries . . . . . . . . . . . . 143
5.1 Parity . . . . . . . . . . . . . . . . . . . 144
5.1.1 Parity in Quantum Mechanics . . . . . . . . . . . . . . 144
5.1.2 Parity in Field Theories . . . . . 146
5.1.3 Parity and Interactions . . . . . . . . . . 150
5.2 Time Inversion ........... . . . . . . . . . 155
5.2.1 Time Inversion in Quantum Mechanics 156
5.2.2 Time Inversion in Field Theories . . . . 158
5.2.3 T and Interactions . . . . . . . . . . . 162
5.3 Charge Conjugation . . . . . . . . . . . . . . . . . .. . . . . 163
5.3.1 Additive Quantum Numbers ......... . . . . . 164
5.3.2 Charge Conjugation in Field Theories . . . . . . . . . . 169
5.3.3 Interactions . . . . . . . . . . 174
5.4 The CPT Theorem . . . . . . . .. .... 178
5.4.1 Implications of CPT Invariance 180
5. 4. 2 C, P, T, and CPT . . . . . . . . . . . . . . . 181
Problems . . . . . . . . . . . . . . 182
Suggestions for Further Reading . . . . . . . 184
6 Hadrons and Isospin . . . . . . . . . . . . . 185
6.1 Charge Symmetry and Charge Independence . . . . . . . . . 185
6.2 Nucleon Field in Isospin Space . . . . . . . . . 187
6.3 Pion Field in Isospin Space . . . . . . . . . . . 193
6.4 G-Parity . . . . . . . . . . . . . 198
6.4.1 Nucleon and Pion Fields . . . 199
6.4.2 Other Unflavored Hadrons .204
6.5 Isospin of Strange Particles . . . .. .... . . 205
6.6 Isospin Violations . . . . . . . . . . . . . 207
6.6.1 Electromagnetic Interactions . . . . 207
6.6.2 Weak Interactions . . . . . . . . . 208
Problems . . . . . . . . . . . . . . . . . 213
Suggestions for Further Reading . . . . . . .. .. 214
7 Quarks and SU(;3) Symmetry ......... . . 215
7.1 Isospin: SU(2) Symmetry . . . . . 216
7.2 Hypercharge: SU(3) Symmetry . . 222
7.2.1 The Fundamental Representation . . 222
7.2.2 Higher-Dimensional Representations ... . . . . . . . 224
7.2.3 Physical Significance of F3 and F8 ........... 228
7.2.4 3 x 3* Equal Mesons . . . . . . . . . . . . . . . . . . . 230
7.2.53x3x3EquaIBaryons................ .233
7.3 Mass Splitting of the Hadron Multiplets . . . . . . . . . . . 236
7.3.1 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.3.2 Mesons . . . . . . . . 239
7.4 Including Spin: SU(6) . . . . . . . . . . . . . . . . . . . . . 241
7.4.1 Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.4.2 Baryons . . . . . . . . . . . . . . . . . . . . . 245
7.4.3 Application: Magnetic Moments of Hadrons . . . . . . 246
7.5 T he Co lor 0 f Quarks ..................... 248
7.6 The New Particles . . . . . . . . . . . . . . .. .. 250
7.6.1 J /1/J and Charm . . . . . . . . . . . . . . . . . . . . . 250
7.6.2 The Tau Lepton . . . . . . . . . . . . . . . . . . . . . 258
7.6.3 From Bottom to Top . . . . . . . . . . . . . . . . . . 260
Problems . . . . . . . . . . . . . . . . . . 263
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . 265
8 Gauge Field Theories ..................... 267
8.1 Symrnetries and Interactions . . . . . . . . . . . . . . . . . 267
8. 2 Abelian Gauge Invariance . . . . . . . . . . . . . . . . . . .. 269
8.3 Non-Abelian Gauge Invariance . . . . . . . . . . . . . 271
8.4 Quantum Chromodynamics . . . . . . . . . . . . . .. . 277
8.5 Spontaneous Breaking of Global Symmetries . . . . .. . 283
8.5.1 The Basic Idea . . . . . . . . . . . . . . . . . .. . 284
8.5.2 Breakdown of Discrete Symmetry . . . . . . . .. . 286
8.5.3 Breakdown of Abelian Symmetry . . . . . . . . . . . . 287
8.5.4 Breakdown of Non-Abelian Symmetry . . . . . . . . . 289
8.6 Spontaneous Breaking of Local Symmetries . . . . . . . . . . 293
8.6.1 Abelian Symmetry . . . . . . . . . . . . . . . . . . . . 293
8.6.2 Non-Abelian Symmetry . . . . . . . . . . . . . . . . . 298
Problems . . . . . . . . . . .. .... . . . . . 301
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . 303
9 The Standard Model of the Electroweak Interaction 305
9.1 The Weak Interaction Before the Gauge Theories 305
9.2 Gauge-Invariant Model of One-Lepton Family 307
9.2.1 Global Symmetry 308
9.2.2 Gauge Invariance ...... . . . . . . . . . . . . 312
9.2.3 Spontaneous Symmetry Breaking . . . . . . 313
9.2.4 Feynman Rul
s for One-Lepton Family . 322
9.3 Including u and d Quarks . . . . . . . . . . . . . . . 326
9.4 Multigeneration Model . . . . . . . . . . 330
9.4.1 The GIM Mechanism . . . . . . . . . . 330
9.4.2 Classification Scheme for Fermions . . . . 333
9.4.3 Fermion Families and the CKM Matrix . . 333
9.4.4 Summary and Extensions . . . . . . . . . . 338
Problems . . . . . . . . . . .. ......... . . 341
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . 342
1 0 Electron-Nucleon Scattering .... . . . . . . . . . . . . 343
10.1 Electromagnetic and Weak Form Factors . . . . . . . . . . 343
10.2 Analyticity and Dispersion Relation .. . . 352
10.3 Exclusive Reaction: Elastic Scattering. ........ 355
10.4 Inclusive Reaction: Deep Inelastic Scattering . . . . . . . . 361
10.4.1 Structure Functions . . . . . . . . . . . . . .. . 362
10.4.2 Bjorken Scaling and the Feynman Quark Parton . . . 366
Pro blems . . . . . . . . . . . . . . . . . . . . 373
Suggestions for Further Reading . . . . . . 375
11 Neutral K Mesons and CP Violation . . . . . . . . 377
11.1 The Two Neutral K Mesons . . . . . . 378
11.2 Strangeness Oscillations . . . . . . . . . .. ... 380
11.3 Regener ation of K. . . . . . . . . . . . . . . . . . . . . 383
11.4 Calculation of m . . . . . . . . . . . . . . . . . . . . . . 385
11.5 CP Violation ............ . . . . .. .... 389
11.5.1 General Formalism . . . . . . . . . . . . . . . . . . 389
11.5.2 Model-Independent Analysis of K L --+ 27r .... 393
11.5.3 The Superweak Scenario . . . . . . . . . . . . . . . 398
11.5.4 Calculations of E and E' in the Standard Model ... 399
11.5.5 The Gluonic Penguin and IE' lEI . . . . . . . . . . . .402
Problems . . . . . . . . . . . . . . . . . . 406
Suggestions for Further Reading . . . . . 406
12 The Neutrinos . . . . . . . . . 407
12.1 On the Neutrino Masses . . . . . . 407
12.1.1 General Properties . . . . . . . . . . . 408
12.1.2 Dirac or Majorana Neutrino? ..... 409
12.1.3 Lepton Mixing . . . . . . . . . . . . . . 411
12.2 Oscillations in the Vacuum . . . . . . . . . . . 412
12.3 Oscillations in Matter . . . . . . . . . . . . . . . . . 415
12.3.1 Index of Refraction, Effective Mass . . . . . . . . . . 416
12.3.2 The MSW Effect . . . . . . . . . . . . . . . . . . . 420
12.3.3 Adiabaticity . . . . . . . . . . . . . . . . . . . . . . 423
12.4 Neutral Currents by Neutrino Scattering 426
12.4.1 Neutral Currents, Why Not? . . . . .. 427
12.4.2 Neutrino-Electron Scattering . . . .. ..... .428
12.5 Neutrino-Nucleon Elastic Scattering . . . . . . . . . 435
12.6 Neutrino-Nucleon Deep Inelastic Collision . . . . . 438
12.6.1 Deep Inelastie Cross-Section 439
12.6.2 Quarks as Partons . 441
Problems . . . . . . . . . . . . . 445
Suggestions for Further Reading 446
13 Muon and Tau Lepton Decays ...... . . . . . . . . . 447
13.1 Weak Decays: Classification and Generalities . . . . . . . . 447
13.2 Leptonic Modes . . . . . . . . . . . . . . . . . . . . . . . 450
13.2.1 Leptonic Branching Ratio ... . . . . . . . . . . . 450
13.2.2 Parity Violation. Energy Spectrum . . . . . . . . . . 451
13.2.3 Angular Distribution. Decay Rate . . . . . . . . . . 456
13.3 Semileptonic Decays . . . . . . . . . . . . . . 460
13.3.1 The One-Pion Mode: T- --+ l/T + 7r- . . 460
13.3.2 The 2n-Pion Mode and CVC . . . . . . . . . . . . . 462
13.4 The Method of Spectral Functions . . . . . . . . . 465
13.4.1 The Three-Pion Mode . . . . . . 467
13.4.2 Spectral Functions of Quark Pairs . . 470
Pro blems . . . . . . . . . . . . . . . . . . . . 473
Suggestions for Further Reading . . . . . . . . . 474
14 One-Loop QCD Corrections
475
14.1 Vertex Function . . . . . .
477
14.2 Quark Self-Energy . . . .. ......
484
14.3 Mass and Field Renormalization . . . . . . . . . . . . .
485
14.3.1 Renormalized Form Factor Pr en ( q2) ... . . . . .
489
14.3.2 Important Consequence of Mass Renormalization
491
14.4 Virtual Gluon Contributions . . . . . . . . . . . . . .
492
14.5 Real Gluon Contributions . . . . . .. ....
496
14.5.1 Infrared Divergence . . . . . . . . . . . .
497
14.5.2 Three-Particle Phase Space
498
14.5.3 Bremsstrahlung Rate
500
14. 6 Fin al Res ul t . . . . . . . . . . . . . . . .
501
Pro blems . . . . . . . . . . . .
502
Suggestions for Further Reading 504
15 Asymptotic Freedom in QCD . . . . . . 505
15.1 Running Coupling Constant . . . . . . . . . . . . . 506
15.1.1 Vacuurn Polarization . . . . . . . . . . . . . . . . . 507
15.1.2 Dressed and Renormalized Photon Propagator . 509
15.1.3 Vertex Renormalization . . . . . . . . . . . . .512
15.1.4 Renormalized Vacuum Polarization II ren (q2) . . 515
15.1.5 Physical Effects of II ren(q2) . . . . . . . . . . 517
15.2 The Renormalization Group . . . . . . . . . . .. ... 518
15.2.1 The Callan-Symanzik Equation . . . . . . . . . . . 520
15.2.2 Calculation of the (3- and l'-Functions . . . . . . . . 523
15.2.3 Running Coupling from the Renormalization Group . 525
15.2.4 Solution of the Renormalization Group Equation 526
15.3 One-Loop Computation of the QCD (3-Function . . 529
15.3.1 Quark Self-Energy Counterterm Zq . . . . . . . 529
15.3.2 Quark-Gluon Vertex Counterterm Zl . . . 529
15.3.3 Gluon Self-Energy Counterterm Zglu . . . . . . 531
15.3.4 The Running QCD Coupling . . . . . . . . . . 535
15.4 Ghosts . . . . . . . . . . . . . . . . . . . . . .. ... 538
15.4.1 The Faddeev-Popov Gauge-Fixing Method ..... 538
15.4.2 Ghosts and Unitarity . . . . . 541
Pro blems . . . . . . . . . . . . . . . . . . . 547
Suggestions for Further Reading . . . . . . . . 548
16 Heavy Flavors . . . . . . . . . . . . . . . . . . . 549
16.1 QCD Renormalization of Weak Interactions 550
16.1.1 Corrections to Single Currents . . . . . . . 551
16.1.2 Corrections to Product of Currents . . . . . . . . . . 553
16.1.3 Renormalization Group Improvement . . . . . . . . 557
16.1.4 The
I == 1/2 in Strangeness Hadronic Decays . 560
16.2 Heavy Flavor Symmetry . . . . . . . . . . . . . . . . . . . 562
16.2.1 Basic Physical Pictures . . . . . . . . . . . . . . . . 563
16.2.2 Elements of Heavy Quark Effective Theory (HQET) . 565
16.3 Inclusive Decays . . . . . . . . . . . . . . . . . . . . . . . 567
16.3.1 Gener al Formalism . . . . . . . . . . . . . . . . . . 568
16.3.2 Inchlsive Semileptonic Decay: B --+ e- + V e + Xc 572
16.3.3 Inclusive Nonleptonic Decay: B ---t Hadrons 573
16.4 Exclusive Decays .... . . . . . . . . . 576
16.4.1 Form Factors in B£3 Decays 577
16.4.2 Semileptonic Decay Rates . . . . . . . . . 580
16.4.3 Two-Body Hadronic Decays . . . . . 582
16.5 CP Violation in B Mesons . . . . . . . . . . . . . . . . . . 588
16.5.1 B 0 - B 0 Mixing . . . . . . . . . . . . . . . . . . . . . 588
16.5.2 CP Asymmetries in Neutral B Meson Decays . . 594
Pro b I ems . . . . . . . . . . . . . . . . . . 598
Suggestions for Further Reading . . . . . . . . . . . .. .. 599
1 7 Status and Perspectives of the Standard Model 601
17.1 Production and pecay of the Higgs Boson. .... 602
17.2 Why Go Beyond the Standard Model? . . . . . .. .. 605
17.3 The St
ndard Model as an Effective Theory . . . . . 607
17.3.1 Problems with the Standard Model ......... 608
17.3.2 Renormalization Group Equation Analysis . . . . . . 610
17.3.3 Supersymmetry and Technicolor . . . . . . 611
Pro bl ems . . . . . . . . . . . . . 614
Suggestions for Further Reading . . . . . . 614
Selected Solutions 615
Appendix: Useful Formulas 645
A.l Relativistic Quantum Mechanics 645
A.2 Cross-Sections and Decay Rates 649
A.3 Phase Space and Loop Integrals . . . . . . . .650
A.4 Feynman Rules . . . . . . . . . . . 653
A.5 Parameters of the Standard Model . . . . . . . . . 656
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
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