Beyond the Standard Model A.N. Schellekens [Word cloud by www.worldle.net] Last modified 12 June 2017 1 Contents 1 Introduction9 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................................... 156 8.13.1 SO(10).................................. 156 8.13.2 E6 .................................... 158 8.13.3 Flipped SU(5)............................. 158 8.13.4 Still Larger Groups........................... 159 8.14 Conclusions................................... 159 8.15 References.................................... 160 9 Supersymmetry 160 9.1 The Supersymmetry Algebra.......................... 162 9.2 Multiplets.................................... 163 9.3 Constructing supersymmetric Lagrangians.................. 164 9.4 The Supersymmetrized Standard Model................... 167 9.5 Additional Interactions............................. 168 9.6 Continuous R-symmetries........................... 171 9.7 R-Parity..................................... 172 9.8 Supersymmetry Breaking............................ 172 9.9 Non-renormalization Theorems........................ 174 9.10 Soft Supersymmetry Breaking......................... 175 9.11 Spontaneous Supersymmetry Breaking.................... 175 9.12 The Goldstino.................................. 176 9.13 Mass Sum Rules................................. 177 9.14 The Minimal Supersymmetric Standard Model................ 178 9.15 The Higgs Potential............................... 180 9.15.1 A Weak Symmetry Breaking Minimum................ 183 9.16 Higgs Masses.................................. 185 9.17 Corrections to the Higgs Masses........................ 187 9.18 Neutralino Masses................................ 188 9.19 Rare Processes................................. 188 9.20 Direct Searches................................. 191 9.21 Supersymmetric Unification.......................... 192 9.21.1 MSSM β-functions........................... 193 9.21.2 MSSM versus SM Unification..................... 194 9.21.3 Proton Decay.............................. 195 9.22 Conclusions................................... 195 5 9.23 References.................................... 196 10 Supergravity 196 10.1 Local Supersymmetry.............................. 197 10.2 The Lagrangian................................. 198 10.3 Spontaneous Symmetry