Beyond the Standard Model A.N. Schellekens [Word cloud by www.worldle.net] Last modified 12 June 2017 1 Contents 1 Introduction 9 1.1 A Complete Theory? . .9 1.2 Gravity and Cosmology . 10 1.3 The Energy Balance of the Universe . 11 1.4 Environmental Issues . 16 1.5 Baryogenesis . 18 1.6 Beyond the Standard Model . 18 2 Gauge Theories 19 2.1 Classical Electrodynamics . 19 2.2 Gauge Invariance . 20 2.3 Noether's Theorem . 22 2.4 Covariant Derivatives . 22 2.5 Non-Abelian Gauge Theories . 23 2.6 Coupling to Fermions . 24 2.7 Gauge Kinetic Terms . 25 2.8 Feynman Rules . 26 2.9 Other Gauge Groups . 27 3 The Higgs Mechanism 28 3.1 Vacuum Expectation Values . 28 3.2 The Goldstone Theorem . 29 3.3 Higgs Mechanism for Abelian Gauge Symmetry . 31 3.4 The Mexican Hat Potential . 33 4 The Standard Model 34 4.1 QED and QCD . 34 4.1.1 Chiral Symmetry Breaking . 36 4.1.2 The θ-parameter . 37 4.2 The Weak Interactions . 40 4.2.1 Fermion Representations . 40 4.2.2 The Higgs Field . 41 4.2.3 Vector Boson Masses . 42 4.2.4 Electromagnetism . 43 4.2.5 The Low-energy Spectrum . 43 4.2.6 Parameters . 44 4.2.7 The Higgs Boson . 44 4.3 Masses and Mixing angles . 45 4.3.1 Yukawa Couplings . 45 4.3.2 Mass Matrix Diagonalization . 46 4.3.3 The CKM matrix . 47 4.3.4 Counting Free Parameters in the CKM Matrix . 47 2 4.3.5 Flavor Changing Neutral Currents and the GIM Mechanism . 49 5 A First Look Beyond 50 5.1 The Left-handed Representation . 50 5.1.1 Replacing Particles by Anti-Particles . 50 5.1.2 The Standard Model in Left-handed Representation . 51 5.1.3 Fermion Masses in the Left-handed Representation . 52 5.1.4 Yukawa Couplings in the Left-handed Representation . 53 5.1.5 Real Representations . 53 5.1.6 Mirror Fermions . 54 5.2 Neutrino Masses . 54 5.2.1 Modifications of the Standard Model . 56 5.2.2 Adding a Dimension 5 Operator . 56 5.2.3 Neutrino-less Double-beta Decay . 57 5.2.4 Adding Right-handed Neutrinos . 58 5.2.5 The See-Saw Mechanism . 58 5.2.6 Neutrino Oscillations . 60 5.2.7 Neutrino Experiments . 64 5.3 C,P and CP . 67 5.4 Continuous Global Symmetries . 68 5.5 Anomalies . 69 5.5.1 Feynman Diagram Computation . 70 5.5.2 Anomalous Local Symmetries . 74 5.5.3 Anomalous Global Symmetries . 76 5.5.4 Global Anomalies in Field-Theoretic Form . 77 5.5.5 Global Anomalies in QCD QED . 77 5.5.6 The π0 γγ Decay Width× . 78 5.5.7 The Axial! U(1) Symmetry . 79 5.5.8 Baryon and Lepton Number Anomalies . 79 5.5.9 Proton decay by Instantons and Sphalerons . 79 5.5.10 Anomaly-free Global Symmetries . 80 5.5.11 Mixed Gauge and Gravitational Anomalies . 80 5.5.12 Other Anomalous Diagrams . 81 5.5.13 Symplectic Anomalies . 81 5.6 Axions . 81 5.6.1 Phases in Quark Masses . 82 5.6.2 The Peccei-Quinn Mechanism . 84 5.6.3 General Axion Models . 86 5.6.4 Axions in the Standard Model . 88 5.6.5 The Mass of the Original QCD Axion . 91 5.6.6 Invisible Axions . 93 5.6.7 Two-photon coupling . 94 5.6.8 Axion-electron coupling . 94 3 5.6.9 Generic Axions . 95 5.6.10 Multiple gauge group factors . 99 6 Loop Corrections of the Standard Model 101 6.1 Divergences and Renormalization . 101 6.1.1 Ultraviolet Divergences . 101 6.1.2 Regularization . 102 6.1.3 The Origin of Ultraviolet Divergences . 103 6.1.4 Renormalization . 103 6.1.5 Renormalizability . 105 6.1.6 Dimensional Analysis . 106 6.1.7 The Meaning of Renormalizability . 108 6.2 Running Coupling Constants . 109 6.2.1 Example: Scalar Field Theories . 110 6.2.2 The Renormalization Group Equation . 113 6.2.3 Summing Leading Logarithms . 114 6.2.4 Asymptotic Freedom . 116 6.2.5 Abelian gauge theories . 117 6.2.6 Yukawa Couplings . 118 6.2.7 The Higgs Self-coupling . 120 7 Intermezzo: Standard Model problems 123 7.1 The Hierarchy Problem . 123 7.2 The Strong CP problem . 124 7.3 The Multiverse and Anthropic Reasoning . 125 7.4 Cosmological Problems . 126 8 Grand Unification 127 8.1 Convergence of Standard Model Couplings . 127 8.1.1 Coupling Constant Unification: Generalities . 128 8.2 Electric Charge Quantization . 132 8.3 Gauge Unification in SU(5) GUTs. 134 8.4 Embedding the Standard Model Gauge Group. 135 8.4.1 Decomposition of SU(5) Representations . 136 8.4.2 Normalization of Generators. 136 8.5 Fermion Representations . 137 8.5.1 Intuition from Anomaly Cancellation. 137 8.5.2 Matter in the Five-Dimensional Representation. 137 8.5.3 Particle Content of the Ten-dimensional Representation . 138 8.5.4 Detailed Particle Decompositions . 138 8.5.5 Distributing Family Members. 139 8.6 The Standard Model Higgs Field. 139 8.7 Choosing the GUT-breaking Higgs Field . 139 4 8.8 Baryon Number Violation . 140 8.9 Fermion Masses . 142 8.10 Proton Decay . 146 8.10.1 B-L . 146 8.10.2 The Proton Lifetime . 147 8.10.3 Historical Remarks . 147 8.11 The Higgs System . 148 8.12 Magnetic Monopoles . 150 8.13 Other GUTs . ..