weak - Contents 1 Classical Symmetries 1.1 Noether's...

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Contents 1 — Classical Symmetries 2 1.1 Noether’s Theorem – Classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1a — Quantum Field Theory 11 1a.1 Local Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1a.2 Composite Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1b — Gauge Symmetries 21 1b.1 Noether’s Theorem – Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1b.2 Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1b.3 Global Symmetries of Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 — Weinberg’s Model of Leptons 29 2.1 The electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 SU (2) × U (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Renormalizability? An Interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 Spontaneous Symmetry Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 The Goldstone Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6 The σ -Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.7 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.8 Neutral Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.9 e + e - μ + μ - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 — Quarks and QCD 49 3.1 Color SU (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 A Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Quark Doublets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 GIM and Charm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 The Standard Six–Quark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.6 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 SU (3) and Light Hadron Semileptonic Decays 63 4.1 Weak Decays of Light Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Isospin and the Determination of V ud . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 f π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Strangeness Changing Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5 PT Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 i
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Weak Interactions — Howard Georgi — draft - March 25, 2010 — ii 4.6 Second Class Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.7 The Goldberger-Treiman Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.8 SU (3) - D and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 — Chiral Lagrangians — Goldstone Bosons 76 5.1 SU (3) × SU (3) SU (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Effective Low–Momentum Field Theories . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.4 Symmetry breaking and light quark masses . . . . . . . . . . . . . . . . . . . . . . . 81 5.5 What Happened to the Axial U (1)? . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.6 Light Quark Mass Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.7 The Chiral Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.8 Semileptonic K Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.9 The Chiral Symmetry–Breaking Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.10 Important Loop Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.11 Nonleptonic K Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6 — Chiral Lagrangians — Matter Fields 93 6.1 How Do the Baryons Transform? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 A More Elegant Transformation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 Nonlinear Chiral Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.4 Successes of the Nonrelativistic Quark Model . . . . . . . . . . . . . . . . . . . . . . 98 6.5 Hyperon Nonleptonic Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6a - Anomalies 105 6a.1 Electromagnetic Interactions and π 0 2 γ . . . . . . . . . . . . . . . . . . . . . . . . 105 6a.2 The Steinberger Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6a.3 Spectators, gauge invariance and the anomaly . . . . . . . . . . . . . . . . . . . . . . 111 7 — The Parton Model 123 7.1 Mode Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 Heavy Quark Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.3 Deep Inelastic Lepton-Hadron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.4 Neutrino-Hadron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.5 Neutral Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.6 The SLAC Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8 — Standard Model Precision Tests 134 8.1 Choosing a Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.2 Effective Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.3 The Symmetries of Strong and Electroweak Interactions . . . . . . . . . . . . . . . . 140 8.4 The ρ Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.5 M W and M Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.6 Neutrino-hadron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8.7 Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
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Weak Interactions — Howard Georgi — draft - March 25, 2010 — 1 9 — Nonleptonic Weak Interactions 150 9.1 Why We Can’t Calculate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.2 The Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.3 Charm Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.4 Penguins and the Δ I = 1 2 Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 10 — The Neutral K Mesons and CP Violation 161 10.1 K 0 - K 0 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 10.2 The Box Diagram and the QCD Corrections . . . . . . . . . . . . . . . . . . . . . . . 163 10.3 The Gilman-Wise Δ S = 2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.4 CP Violation and the Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.5 K L ππ and the Parameter 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A Review of Dimensional Regularization 175 A.1 n Dimensional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 A.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B Background Field Gauge 179 B.1 The β function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
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Chapter 1 — Classical Symmetries The concept of symmetry will play a crucial role in nearly all aspects of our discussion of weak interactions. At the level of the dynamics, the fundamental interactions (or at least that subset of
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