An Introduction to Particle Physics and the Standard Model

602 Pages·2009·29.895 MB·English

by  Robert Mann

AN INTRODUCTION TO
PARTICLE PHYSICS
AND THE
STANDARD MODEL
82981_FM.indd 1 10/23/09 11:36:32 AM

ROBERT MANN
UNIVERSITY OF WATERLOO
ONTARIO, CANADA
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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Contents
Preface xiii
Acknowledgments xvii
Further Reading xix
1 Introduction and Overview 1
1.1 Methods of Study . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Large Accelerators . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Precision Measurements . . . . . . . . . . . . . . . . . 3
1.1.4 Cosmological Data . . . . . . . . . . . . . . . . . . . . 4
1.1.5 Empirical Analysis . . . . . . . . . . . . . . . . . . . . 6
1.1.6 Model Building . . . . . . . . . . . . . . . . . . . . . . 6
1.1.7 Numerical Computation . . . . . . . . . . . . . . . . . 7
1.1.8 Mathematical Foundations. . . . . . . . . . . . . . . . 7
1.1.9 Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Bosons and Fermions . . . . . . . . . . . . . . . . . . . 9
1.2.2 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.3 Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 A Review of Special Relativity 23
2.1 Basic Review of Relativity . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Relativity of Simultaneity . . . . . . . . . . . . . . . . 24
2.1.2 Length Contraction. . . . . . . . . . . . . . . . . . . . 25
2.1.3 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.4 Velocity Addition . . . . . . . . . . . . . . . . . . . . . 28
2.2 4-Vector Notation . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Spacetime Structure . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Momentum and Energy . . . . . . . . . . . . . . . . . . . . . 33
2.5 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.1 Broadside Collision . . . . . . . . . . . . . . . . . . . . 37
2.5.2 Compton Scattering . . . . . . . . . . . . . . . . . . . 38
2.5.3 3-Body Decay . . . . . . . . . . . . . . . . . . . . . . . 39
v
vi
2.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Symmetries 45
3.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Axioms of a Group . . . . . . . . . . . . . . . . . . . . 46
3.1.2 Representations . . . . . . . . . . . . . . . . . . . . . . 48
3.1.3 Irreducible Representations . . . . . . . . . . . . . . . 50
3.1.4 Multiplication Tables . . . . . . . . . . . . . . . . . . . 51
3.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Lie Algebras. . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 The Rotation Group SO(3) . . . . . . . . . . . . . . . . . . . 55
3.5 Appendix: Lie Algebras from Lie Groups . . . . . . . . . . . 58
3.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Conservation Laws 63
4.1 The Action Principle . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Spacetime Symmetries and Their Noether Currents . . . . . . 68
4.3.1 Spatial Translations . . . . . . . . . . . . . . . . . . . 68
4.3.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.3 Time Translations . . . . . . . . . . . . . . . . . . . . 69
4.4 Symmetries and Quantum Mechanics . . . . . . . . . . . . . . 70
4.4.1 Spatial Translations . . . . . . . . . . . . . . . . . . . 71
4.4.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.3 Time Translations . . . . . . . . . . . . . . . . . . . . 73
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Particle Classification 77
5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Basic Classification . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Spectroscopic Notation . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Adding Angular Momenta . . . . . . . . . . . . . . . . . . . . 84
5.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Appendix: Tools for Angular Momenta . . . . . . . . . . . . . 87
5.5.1 Pauli-Matrices . . . . . . . . . . . . . . . . . . . . . . 87
5.5.2 Clebsch-Gordon Tables . . . . . . . . . . . . . . . . . . 88
5.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Discrete Symmetries 93
6.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1.1 Parity of the Photon . . . . . . . . . . . . . . . . . . . 96
6.1.2 Parity Conservation . . . . . . . . . . . . . . . . . . . 96
6.2 Time-Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
vii
6.2.1 Detailed Balance . . . . . . . . . . . . . . . . . . . . . 98
6.3 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . . 100
6.3.1 Charge Conjugation of the Pion . . . . . . . . . . . . . 101
6.3.2 Charge Conjugation of Fermions . . . . . . . . . . . . 101
6.4 Positronium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4.1 A Puzzle with ORTHO . . . . . . . . . . . . . . . . . 104
6.4.2 Testing Fermion-Antifermion Parity . . . . . . . . . . 105
6.5 The CPT Theorem . . . . . . . . . . . . . . . . . . . . . . . 107
6.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Accelerators 111
7.1 DC Voltage Machines . . . . . . . . . . . . . . . . . . . . . . 112
7.2 Linacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2.1 Focusing the Beam . . . . . . . . . . . . . . . . . . . . 116
7.3 Synchrotrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3.1 Focusing Beams at Synchrotrons . . . . . . . . . . . . 120
7.4 Colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.5 The Future of Accelerators . . . . . . . . . . . . . . . . . . . 132
7.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8 Detectors 135
8.1 Energy Transfer and Deposition . . . . . . . . . . . . . . . . . 135
8.1.1 Charged Particles . . . . . . . . . . . . . . . . . . . . . 135
8.1.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.2 Detector Types . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.2.1 Scintillation Counters . . . . . . . . . . . . . . . . . . 144
8.2.2 Cloud Chambers . . . . . . . . . . . . . . . . . . . . . 146
8.2.3 Bubble Chambers. . . . . . . . . . . . . . . . . . . . . 146
8.2.4 Spark Chambers . . . . . . . . . . . . . . . . . . . . . 148
8.2.5 Wire Chambers . . . . . . . . . . . . . . . . . . . . . . 148
8.2.6 Time Projection Chambers . . . . . . . . . . . . . . . 149
8.2.7 Cerenkov Counters . . . . . . . . . . . . . . . . . . . . 150
8.2.8 Solid State Detectors . . . . . . . . . . . . . . . . . . . 152
8.2.9 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . 155
8.3 Modern Collider Detectors . . . . . . . . . . . . . . . . . . . . 156
8.3.1 Tracking Chambers . . . . . . . . . . . . . . . . . . . . 156
8.3.2 Electromagnetic Shower Detectors . . . . . . . . . . . 156
8.3.3 Hadron Shower Calorimeters . . . . . . . . . . . . . . 157
8.3.4 Muon Chambers . . . . . . . . . . . . . . . . . . . . . 158
8.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
viii
9 Scattering 161
9.1 Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.3 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.4 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.4.1 General Features of Decay Rates and Cross-Sections . 171
9.5 2-Body Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.5.1 2-Body Decay Rate . . . . . . . . . . . . . . . . . . . . 172
9.5.2 2-Body CM Cross-Section . . . . . . . . . . . . . . . . 173
9.5.3 2-Body Lab Cross-Section . . . . . . . . . . . . . . . . 173
9.6 Detailed Balance Revisited . . . . . . . . . . . . . . . . . . . . 174
9.6.1 Pion Spin . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10 A Toy Theory 179
10.1 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 180
10.2 A-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
10.3 Scattering in the Toy Theory . . . . . . . . . . . . . . . . . . 187
10.4 Higher-Order Diagrams . . . . . . . . . . . . . . . . . . . . . 190
10.5 Appendix: n-Dimensional Integration . . . . . . . . . . . . . . 194
10.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11 Wave Equations for Elementary Particles 201
11.1 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . 202
11.2 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.3 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . 206
11.4 Antiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11.5 Appendix: The Lorentz Group and Its Representations . . . . 210
11.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12 Gauge Invariance 219
12.1 Solutions to the Dirac Equation . . . . . . . . . . . . . . . . . 219
12.2 Conserved Current . . . . . . . . . . . . . . . . . . . . . . . . 221
12.3 The Gauge Principle . . . . . . . . . . . . . . . . . . . . . . . 222
12.4 The Maxwell-Dirac Equations . . . . . . . . . . . . . . . . . . 223
12.4.1 Physical Features of the Maxwell-Dirac Equations . . 225
12.5 The Wavefunction of the Photon . . . . . . . . . . . . . . . . 226
12.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
13 Quantum Electrodynamics 231
13.1 Feynman Rules for QED . . . . . . . . . . . . . . . . . . . . . 234
13.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
13.2.1 Electron-Muon Scattering . . . . . . . . . . . . . . . . 236
13.2.2 Bhabha Scattering (electron-positron scattering) . . . 238
13.2.3 Compton Scattering . . . . . . . . . . . . . . . . . . . 240
ix
13.3 Obtaining Cross-Sections . . . . . . . . . . . . . . . . . . . . . 241
13.4 Appendix: Mathematical Tools for QED . . . . . . . . . . . . 246
13.4.1 The Casimir Trick . . . . . . . . . . . . . . . . . . . . 246
13.4.2 Dirac γ−Matrices and Their Traces . . . . . . . . . . 247
13.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
14 Testing QED 251
14.1 Basic Features of QED Scattering . . . . . . . . . . . . . . . . 252
14.1.1 Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . 252
14.1.2 Propagator . . . . . . . . . . . . . . . . . . . . . . . . 253
14.1.3 Matrix element . . . . . . . . . . . . . . . . . . . . . . 253
14.1.4 Dimensionality . . . . . . . . . . . . . . . . . . . . . . 254
14.1.5 Antiparticles . . . . . . . . . . . . . . . . . . . . . . . 254
14.2 Major Tests of QED . . . . . . . . . . . . . . . . . . . . . . . 255
14.2.1 Scattering Processes . . . . . . . . . . . . . . . . . . . 255
14.2.2 Anomalous Magnetic Moments . . . . . . . . . . . . . 256
14.2.3 Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . 262
14.2.4 Running Coupling Constant . . . . . . . . . . . . . . . 264
14.3 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
15 From Nuclei to Quarks 271
15.1 Range of the Nuclear Force . . . . . . . . . . . . . . . . . . . 271
15.2 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
15.3 Strangeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
15.4 Flavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
15.5 Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
15.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
16 The Quark Model 291
16.1 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
16.2 Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
16.3 Mass Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 300
16.4 Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . . . 305
16.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
17 Testing the Quark Model 309
17.1 Vector-Meson Decay . . . . . . . . . . . . . . . . . . . . . . . 309
17.2 Hadron Production . . . . . . . . . . . . . . . . . . . . . . . . 312
17.3 Elastic Scattering of Electrons and Protons . . . . . . . . . . 316
17.3.1 The Photon-Proton Vertex . . . . . . . . . . . . . . . 317
17.3.2 The Rosenbluth Formula. . . . . . . . . . . . . . . . . 318
17.4 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . 320
17.5 Quark Model Predictions . . . . . . . . . . . . . . . . . . . . 323
17.6 Quark Structure Functions . . . . . . . . . . . . . . . . . . . . 326
17.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328