Elementary Particle Physics
Quantum Field Theory and Particles V1

Meeting the need for a coherently written and comprehensive compendium combining field theory and particle physics for advanced students and researchers, this volume directly links the theory to the experiments. It is clearly divided into two sections covering approaches to field theory and the Standard Model, and rounded off with numerous useful appendices. A timely work for high energy and theoretical physicists, as well as astronomers, graduate students and lecturers in physics.

From the contents:

• Particles and Fields
• Lorentz Invariance
• Dirac Equation
• Field Quantization
• Scattering Matrix
• QED: Quantum Electrodynamics
• Radiative Corrections and Tests of Qed
• Symmetries
• Path Integral : Basics
• Path Integral Approach to Field Theory
• Accelerator and Detector Technology
• Spectroscopy
• The Quark Model
• Weak Interaction
• Neutral Kaons and CP Violation
• Hadron Structure
• Gauge Theories
• Appendices

Volume 2 (2013, ISBN 3-527-40966-1) will concentrate on the main aspects of the Standard Model by addressing its recent developments and future prospects. Furthermore, it will give some thought to intriguing ideas beyond the Standard Model, including the Higgs boson, the neutrino, the concepts of the Grand Unified Theory and supersymmetry, axions, and cosmological developments.

Foreword V

Preface XVII

Acknowledgements XXI

Part One a Field Theoretical Approach 1

1 Introduction 3

1.1 An Overview of the Standard Model 3

1.1.1 What is an Elementary Particle? 3

1.1.2 The Four Fundamental Forces and Their Unification 4

1.1.3 The Standard Model 7

1.2 The Accelerator as a Microscope 11

2 Particles and Fields 13

2.1 What is a Particle? 13

2.2 What is a Field? 21

2.2.1 Force Field 21

2.2.2 Relativistic Wave Equation 25

2.2.3 Matter Field 27

2.2.4 Intuitive Picture of a Field and Its Quantum 28

2.2.5 Mechanical Model of a Classical Field 29

2.3 Summary 32

2.4 Natural Units 33

3 Lorentz Invariance 37

3.1 Rotation Group 37

3.2 Lorentz Transformation 41

3.2.1 General Formalism 41

3.2.2 Lorentz Vectors and Scalars 43

3.3 Space Inversion and Time Reversal 45

3.4 Covariant Formalism 47

3.4.1 Tensors 47

3.4.2 Covariance 48

3.4.3 Supplementing the Time Component 49

3.4.4 Rapidity 51

3.5 Lorentz Operator 53

3.6 Poincaré Group* 56

4 Dirac Equation 59

4.1 Relativistic Schrödinger Equation 59

4.1.1 Dirac Matrix 59

4.1.2 Weyl Spinor 61

4.1.3 Interpretation of the Negative Energy 64

4.1.4 Lorentz-Covariant Dirac Equation 69

4.2 Plane-Wave Solution 71

4.3 Properties of the Dirac Particle 75

4.3.1 Magnetic Moment of the Electron 75

4.3.2 Parity 77

4.3.3 Bilinear Form of the Dirac Spinor 78

4.3.4 Charge Conjugation 79

4.3.5 Chiral Eigenstates 82

4.4 Majorana Particle 84

5 Field Quantization 89

5.1 Action Principle 89

5.1.1 Equations of Motion 89

5.1.2 Hamiltonian Formalism 90

5.1.3 Equation of a Field 91

5.1.4 Noether’s Theorem 95

5.2 Quantization Scheme 100

5.2.1 Heisenberg Equation of Motion 100

5.2.2 Quantization of the Harmonic Oscillator 102

5.3 Quantization of Fields 105

5.3.1 Complex Fields 106

5.3.2 Real Field 111

5.3.3 Dirac Field 112

5.3.4 Electromagnetic Field 114

5.4 Spin and Statistics 119

5.5 Vacuum Fluctuation 121

5.5.1 The Casimir Effect* 122

6 Scattering Matrix 127

6.1 Interaction Picture 127

6.2 Asymptotic Field Condition 131

6.3 Explicit Form of the S-Matrix 133

6.3.1 Rutherford Scattering 135

6.4 Relativistic Kinematics 136

6.4.1 Center of Mass Frame and Laboratory Frame 136

6.4.2 Crossing Symmetry 139

6.5 Relativistic Cross Section 141

6.5.1 Transition Rate 141

6.5.2 Relativistic Normalization 142

6.5.3 Incoming Flux and Final State Density 144

6.5.4 Lorentz-Invariant Phase Space 145

6.5.5 Cross Section in the Center of Mass Frame 145

6.6 Vertex Functions and the Feynman Propagator 147

6.6.1 eeγ Vertex Function 147

6.6.2 Feynman Propagator 151

6.7 Mott Scattering 157

6.7.1 Cross Section 157

6.7.2 Coulomb Scattering and Magnetic Scattering 161

6.7.3 Helicity Conservation 161

6.7.4 A Method to Rotate Spin 161

6.8 Yukawa Interaction 162

7 Qed: Quantum Electrodynamics 167

7.1 e–μ Scattering 167

7.1.1 Cross Section 167

7.1.2 Elastic Scattering of Polarized e–μ 171

7.1.3 e^_ e⁺ + μ^_ μ⁺Reaction 174

7.2 Compton Scattering 176

7.3 Bremsstrahlung 181

7.3.1 Soft Bremsstrahlung 183

7.4 Feynman Rules 186

8 Radiative Corrections and Tests of Qed* 191

8.1 Radiative Corrections and Renormalization* 191

8.1.1 Vertex Correction 191

8.1.2 Ultraviolet Divergence 193

8.1.3 Infrared Divergence 197

8.1.4 Infrared Compensation to All Orders* 199

8.1.5 Running Coupling Constant 204

8.1.6 Mass Renormalization 208

8.1.7 Ward–Takahashi Identity 210

8.1.8 Renormalization of the Scattering Amplitude 211

8.2 Tests of QED 213

8.2.1 Lamb Shift 213

8.2.2g - 2 214

8.2.3 Limit of QED Applicability 216

8.2.4 E821 BNL Experiment 216

9 Symmetries 221

9.1 Continuous Symmetries 222

9.1.1 Space and Time Translation 223

9.1.2 Rotational Invariance in the Two-Body System 227

9.2 Discrete Symmetries 233

9.2.1 Parity Transformation 233

9.2.2 Time Reversal 240

9.3 Internal Symmetries 251

9.3.1 U(1) Gauge Symmetry 251

9.3.2 Charge Conjugation 252

9.3.3 CPT Theorem 258

9.3.4 SU(2) (Isospin) Symmetry 260

10 Path Integral: Basics 267

10.1 Introduction 267

10.1.1 Bra and Ket 267

10.1.2 Translational Operator 268

10.2 Quantum Mechanical Equations 271

10.2.1 Schrödinger Equation 271

10.2.2 Propagators 272

10.3 Feynman’s Path Integral 274

10.3.1 Sum over History 274

10.3.2 Equivalence with the Schrödinger Equation 278

10.3.3 Functional Calculus 279

10.4 Propagators: Simple Examples 282

10.4.1 Free-Particle Propagator 282

10.4.2 Harmonic Oscillator 285

10.5 Scattering Matrix 294

10.5.1 Perturbation Expansion 295

10.5.2 S-Matrix in the Path Integral 297

10.6 Generating Functional 300

10.6.1 Correlation Functions 300

10.6.2 Note on Imaginary Time 302

10.6.3 Correlation Functions as Functional Derivatives 304

10.7 Connection with Statistical Mechanics 306

11 Path Integral Approach to Field Theory 311

11.1 From Particles to Fields 311

11.2 Real Scalar Field 312

11.2.1 Generating Functional 312

11.2.2 Calculation of det A 315

11.2.3 n-Point Functions and the Feynman Propagator 318

11.2.4 Wick’s Theorem 319

11.2.5 Generating Functional of Interacting Fields 320

11.3 Electromagnetic Field 321

11.3.1 Gauge Fixing and the Photon Propagator 321

11.3.2 Generating Functional of the Electromagnetic Field 323

11.4 Dirac Field 324

11.4.1 Grassmann Variables 324

11.4.2 Dirac Propagator 331

11.4.3 Generating Functional of the Dirac Field 332

11.5 Reduction Formula 333

11.5.1 Scalar Fields 333

11.5.2 Electromagnetic Field 337

11.5.3 Dirac Field 337

11.6 QED 340

11.6.1 Formalism 340

11.6.2 Perturbative Expansion 342

11.6.3 First-Order Interaction 343

11.6.4 Mott Scattering 345

11.6.5 Second-Order Interaction 346

11.6.6 Scattering Matrix 351

11.6.7 Connected Diagrams 353

11.7 Faddeev–Popov’s Ansatz* 354

11.7.1 A Simple Example* 355

11.7.2 Gauge Fixing Revisited* 356

11.7.3 Faddeev–Popov Ghost* 359

12 Accelerator and Detector Technology 363

12.1 Accelerators 363

12.2 Basic Parameters of Accelerators 364

12.2.1 Particle Species 364

12.2.2 Energy 366

12.2.3 Luminosity 367

12.3 Various Types of Accelerators 369

12.3.1 Low-Energy Accelerators 369

12.3.2 Synchrotron 373

12.3.3 Linear Collider 377

12.4 Particle Interactions with Matter 378

12.4.1 Some Basic Concepts 378

12.4.2 Ionization Loss 381

12.4.3 Multiple Scattering 389

12.4.4 Cherenkov and Transition Radiation 390

12.4.5 Interactions of Electrons and Photons with Matter 394

12.4.6 Hadronic Shower 401

12.5 Particle Detectors 403

12.5.1 Overview of Radioisotope Detectors 403

12.5.2 Detectors that Use Light 404

12.5.3 Detectors that Use Electric Signals 410

12.5.4 Functional Usage of Detectors 415

12.6 Collider Detectors 422

12.7 Statistics and Errors 428

12.7.1 Basics of Statistics 428

12.7.2 Maximum Likelihood and Goodness of Fit 433

12.7.3 Least Squares Method 438

13 Spectroscopy 443

13.1 Pre-accelerator Age (1897–1947) 444

13.2 Pions 449

13.3 πN Interaction 454

13.3.1 Isospin Conservation 454

13.3.2 Partial Wave Analysis 462

13.3.3 Resonance Extraction 466

13.3.4 Argand Diagram: Digging Resonances 472

13.4 ƿ (770) 475

13.5 Final State Interaction 478

13.5.1 Dalitz Plot 478

13.5.2 K Meson 481

13.5.3 Angular Momentum Barrier 484

13.5.4 ω Meson 485

13.6 Low-Energy Nuclear Force 487

13.6.1 Spin–Isospin Exchange Force 487

13.6.2 Effective Range 490

13.7 High-Energy Scattering 491

13.7.1 Black Sphere Model 491

13.7.2 Regge Trajectory* 494

14 The Quark Model 501

14.1 SU(3) Symmetry 501

14.1.1 The Discovery of Strange Particles 502

14.1.2 The Sakata Model 505

14.1.3 Meson Nonets 507

14.1.4 The Quark Model 509

14.1.5 Baryon Multiplets 510

14.1.6 General Rules for Composing Multiplets 511

14.2 Predictions of SU(3) 513

14.2.1 Gell-Mann–Okubo Mass Formula 513

14.2.2 Prediction of Ω 514

14.2.3 Meson Mixing 516

14.3 Color Degrees of Freedom 519

14.4 SU(6) Symmetry 522

14.4.1 Spin and Flavor Combined 522

14.4.2 SU(6) _ O(3) 525

14.5 Charm Quark 525

14.5.1 J/ψ 525

14.5.2 Mass and Quantum Number of J/ψ 527

14.5.3 Charmonium 527

14.5.4 Width of J/ψ 533

14.5.5 Lifetime of Charmed Particles 536

14.5.6 Charm Spectroscopy: SU(4) 537

14.5.7 The Fifth Quark b (Bottom) 539

14.6 Color Charge 539

14.6.1 Color Independence 542

14.6.2 Color Exchange Force 544

14.6.3 Spin Exchange Force 545

14.6.4 Mass Formulae of Hadrons 547

15 Weak Interaction 553

15.1 Ingredients of the Weak Force 553

15.2 Fermi Theory 555

15.2.1 Beta Decay 555

15.2.2 Parity Violation 562

15.2.3 π Meson Decay 564

15.3 Chirality of the Leptons 567

15.3.1 Helicity and Angular Correlation 567

15.3.2 Electron Helicity 569

15.4 The Neutrino 571

15.4.1 Detection of the Neutrino 571

15.4.2 Mass of the Neutrino 572

15.4.3 Helicity of the Electron Neutrino 576

15.4.4 The Second Neutrino ν_μ 578

15.5 The Universal V–A Interaction 579

15.5.1 Muon Decay 579

15.5.2 CVC Hypothesis 584

15.6 Strange Particle Decays 589

15.6.1 ΔS = ΔQ Rule 589

15.6.2 ΔI = 1/2 Rule 591

15.6.3 K_l3 : K⁺→ π⁰ + l⁺ + ν 592

15.6.4 Cabibbo Rotation 596

15.7 Flavor Conservation 598

15.7.1 GIM Mechanism 598

15.7.2 Kobayashi–Maskawa Matrix 600

15.7.3 Tau Lepton 601

15.7.4 The Generation Puzzle 605

15.8 A Step Toward a Unified Theory 608

15.8.1 Organizing the Weak Phenomena 608

15.8.2 Limitations of the Fermi Theory 610

15.8.3 Introduction of SU(2) 614

16 Neutral Kaons and CP Violation* 617

16.1 Introduction 618

16.1.1 Strangeness Eigenstates and CP Eigenstates 618

16.1.2 Schrödinger Equation for K⁰ - K⁰ States 619

16.1.3 Strangeness Oscillation 622

16.1.4 Regeneration of K1 626

16.1.5 Discovery of CP Violation 630

16.2 Formalism of CP and CPT Violation 632

16.2.1 CP, T, CPT Transformation Properties 632

16.2.2 Definition of CP Parameters 635

16.3 CP Violation Parameters 640

16.3.1 Observed Parameters 640

16.3.2 є and є’ 644

16.4 Test of T and CPT Invariance 653

16.4.1 Definition of T- and CPT-Violating Amplitudes 654

16.4.2 T Violation 654

16.4.3 CPT violation 656

16.4.4 Possible Violation of Quantum Mechanics 662

16.5 Experiments on CP Parameters 664

16.5.1 CPLEAR 664

16.5.2 NA48/KTeV 666

16.6 Models of CP Violation 673

17 Hadron Structure 679

17.1 Historical Overview 679

17.2 Form Factor 680

17.3 e–p Elastic Scattering 683

17.4 Electron Proton Deep Inelastic Scattering 687

17.4.1 Cross-Section Formula for Inelastic Scattering 687

17.4.2 Bjorken Scaling 690

17.5 Parton Model 693

17.5.1 Impulse Approximation 693

17.5.2 Electron–Parton Scattering 696

17.6 Scattering with Equivalent Photons 699

17.6.1 Transverse and Longitudinal Photons 699

17.6.2 Spin of the Target 702

17.6.3 Photon Flux 703

17.7 How to Do Neutrino Experiments 705

17.7.1 Neutrino Beams 705

17.7.2 Neutrino Detectors 709

17.8 ν–p Deep Inelastic Scattering 712

17.8.1 Cross Sections and Structure Functions 712

17.8.2 ν, ν–q Scattering 715

17.8.3 Valence Quarks and Sea Quarks 716

17.8.4 Comparisons with Experimental Data 717

17.8.5 Sum Rules 719

17.9 Parton Model in Hadron–Hadron Collisions 721

17.9.1 Drell–Yan Process 721

17.9.2 Other Hadronic Processes 724

17.10 A Glimpse of QCD’s Power 725

18 Gauge Theories 729

18.1 Historical Prelude 729

18.2 Gauge Principle 731

18.2.1 Formal Definition 731

18.2.2 Gravity as a Geometry 733

18.2.3 Parallel Transport and Connection 734

18.2.4 Rotation in Internal Space 737

18.2.5 Curvature of a Space 739

18.2.6 Covariant Derivative 741

18.2.7 Principle of Equivalence 743

18.2.8 General Relativity and Gauge Theory 745

18.3 Aharonov–Bohm Effect 748

18.4 Nonabelian Gauge Theories 754

18.4.1 Isospin Operator 754

18.4.2 Gauge Potential 755

18.4.3 Isospin Force Field and Equation of Motion 757

18.5 QCD 760

18.5.1 Asymptotic Freedom 762

18.5.2 Confinement 767

18.6 Unified Theory of the Electroweak Interaction 770

18.6.1 SU(2) _ U(1) Gauge Theory 770

18.6.2 Spontaneous Symmetry Breaking 774

18.6.3 Higgs Mechanism 778

18.6.4 Glashow–Weinberg–Salam Electroweak Theory 782

18.6.5 Summary of GWS Theory 784

19 Epilogue 787

19.1 Completing the Picture 788

19.2 Beyond the Standard Model 789

19.2.1 Neutrino Oscillation 789

19.2.2 GUTs: Grand Unified Theories 791

19.2.3 Supersymmetry 792

19.2.4 Superstring Model 795

19.2.5 Extra Dimensions 796

19.2.6 Dark Matter 797

19.2.7 Dark Energy 798

Appendix A Spinor Representation 803

A.1 Definition of a Group 803

A.1.1 Lie Group 804

A.2 SU(2) 805

A.3 Lorentz Operator for Spin 1/2 Particle 809

A.3.1 SL(2, C) Group 809

A.3.2 Dirac Equation: Another Derivation 811

Appendix B Coulomb Gauge 813

B.1 Quantization of the Electromagnetic Field in the Coulomb Gauge 814

Appendix C Dirac Matrix and Gamma Matrix Traces 817

C.1 Dirac Plane Wave Solutions 817

C.2 Dirac γ Matrices 817

C.2.1 Traces of the γ Matrices 818

C.2.2 Levi-Civita Antisymmetric Tensor 819

C.3 Spin Sum of |Mfi|² 819

C.3.1 A Frequently Used Example 820

C.3.2 Polarization Sum of the Vector Particle 822

C.4 Other Useful Formulae 823

Appendix D Dimensional Regularization 825

D.1 Photon Self-Energy 825

D.2 Electron Self-Energy 830

Appendix E Rotation Matrix 833

E.1 Angular Momentum Operators 833

E.2 Addition of the Angular Momentum 835

E.3 Rotational Matrix 835

Appendix F C, P, T Transformation 839

Appendix G SU(3), SU(n) and the Quark Model 841

G.1 Generators of the Group 841

G.1.1 Adjoint Representation 842

G.1.2 Direct Product 843

G.2 SU(3) 844

G.2.1 Structure Constants 844

G.2.2 Irreducible Representation of a Direct Product 846

G.2.3 Tensor Analysis 851

G.2.4 Young Diagram 854

Appendix H Mass Matrix and Decaying States 859

H.1 The Decay Formalism 859

Appendix I Answers to the Problems 865

Appendix J Particle Data 915

Appendix K Constants 917

References 919

Index 929 

Yorikiyo Nagashima is Professor Emeritus at the Department of Physics of Osaka University, Japan. An organizer of international conferences, he is also a member of the most important collaboration groups in his field of expertise, including those related to neutrino research. Professor Nagashima was the spokesman of the VENUS group, one of the major detectors of the Japanese first collider accelerator TORISTAN, where he served the first and second term at the start of the project. Professor Nagashima has authored or co-authored 296 papers, some of them cited up to 250 times.