Elementary Particles and Their Interactions - Concepts and Phenomena

Vietsciences- Hồ Kim Quang - Phạm Xuân Yêm          27/03/2009         Sửa chữa ngày 20/09/2013

 

Những bài cùng tác giả

 

 

 

 

 

 

 

 

 

  • 1 Particles and Interactions: An Overview

    2 Boson Fields

    3 Fermion Fields

    4 Collisions and Decays

    5 Discrete Symmetries

    6 Hadrons and Isospin

    7 Quarks and SU(3) Symmetry

    8 Gauge Field Theories

    9 The Standard Model of the Electroweak Interaction

    10 Electron–Nucleon Scattering

    11 Neutral K Mesons and CP Violation

    12 The Neutrinos

    13 Muon and Tau Lepton Decays

    14 One-Loop QCD Corrections

    15 Asymptotic Freedom in QCD

    16 Heavy Flavors

    17 Status and Perspectives of the Standard Model

    Index

    Selected Solutions

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

               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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