Modern chapter of theoretical particle physics Background material An introduction to the foundations of modern physics SS 2021 at the KFU Graz Axel Maas 2 Contents 1 Fundamentals of physics 1 1.1 Introduction................................... 1 1.2 Naturalunits .................................. 2 1.3 Particlephysics ................................. 3 1.3.1 The purview of particle physics . 3 1.3.2 Elementaryparticles .......................... 4 1.3.3 Fermionsandbosons .......................... 5 1.3.4 Particles and anti-particles . 6 1.3.5 Identicalparticles............................ 7 1.4 Interactions ................................... 8 1.5 Gravityandquantumgravity ......................... 9 1.6 Bottom-up, top-down, and effective theories . .... 10 2 Scattering experiments 12 2.1 Transitions.................................... 12 2.2 Non-relativistic ................................. 15 2.2.1 Elasticscattering ............................ 15 2.2.2 Crosssection .............................. 16 2.2.3 Luminosity ............................... 17 2.3 Relativistic ................................... 18 2.3.1 Repetition: Relativistic notation . 18 2.3.2 Inelasticscattering ........................... 20 2.3.3 Partialwavedecomposition . 22 2.4 Formfactors ................................... 24 2.5 Decays...................................... 25 2.6 Cross-sectionsandparticles . 26 2.7 Feynmandiagrams ............................... 28 i ii Contents 3 A role model: Quantum electrodynamics 30 3.1 Electronsandphotons ............................. 30 3.2 TheDiracequation............................... 33 3.3 Formulationasafieldtheory.......................... 35 3.4 Gaugesymmetry ................................ 36 3.5 Boundstatesandthepositronium. 37 3.6 MuonsandTaus ................................ 38 3.7 Radiative corrections and renormalization . 38 3.8 Fixing the inputs and running quantities . 40 3.9 Landaupoles .................................. 42 4 Invariances and quantum numbers 44 4.1 Symmetries,groups,andalgebras . 44 4.2 Noether’stheorem ............................... 46 4.3 Electriccharge ................................. 48 4.4 Implications of space-time symmetries . 49 4.5 Parity ...................................... 50 4.6 Timereversal .................................. 51 4.7 Chargeparity .................................. 52 4.8 CPT....................................... 52 4.9 Symmetrybreaking............................... 53 4.10Fermionnumber................................. 55 4.11Flavor ...................................... 55 4.12Chiralsymmetry ................................ 56 4.13Generators.................................... 57 5 Strong interactions 58 5.1 Nucleiandthenuclearforce .......................... 59 5.2 Mesons...................................... 60 5.3 Nucleons,isospin,andbaryons. 61 5.4 Thequarkmodel ................................ 63 5.5 Colorandgluons ................................ 66 5.6 Chiralsymmetrybreaking ........................... 68 5.7 TheGoldstonetheorem............................. 69 5.8 Confinementandasymptoticfreedom . 74 5.9 Hadronicresonances .............................. 76 5.10 Glueballs, hybrids, tetraquarks, and pentaquarks . ....... 77 Contents iii 5.11 Flavorsymmetryandstrangeness . .. 78 5.12 Charm,bottomandquarkonia. 81 5.13Top........................................ 84 5.14 Partonsandscatteringexperiments . ... 84 5.14.1 Fragmentationfunctionsandjets . 85 5.14.2 Parton distribution functions . 86 5.14.3 Monte-Carlogenerators. 87 5.15 Somealgebraandgrouptheory . 88 5.16 Yang-MillstheoryandQCD . .. .. 92 5.17Cosmicradiation ................................ 97 5.18 TheQCDphasediagram............................ 98 6 Weak interactions 103 6.1 β decayandparityviolation . 104 6.2 Neutrinos .................................... 106 6.3 Flavor-changingcurrents . 107 6.4 WandZbosons................................. 108 6.5 The Brout-Englert-Higgs effect and the Higgs boson . 111 6.6 Parity violation and fermion masses . 115 6.7 Weak isospin, hypercharge, and electroweak unification . ..... 116 6.7.1 Constructingthegaugegroup . 117 6.7.2 Hidingtheelectroweaksymmetry . 119 6.8 CPviolationandtheCKMmatrix . 124 6.9 Neutrino oscillations and the PMNS matrix . 127 6.10 Theweakinteractionsasagaugetheory . 130 6.11Anomalies .................................... 134 6.12 Landau-poles, triviality, and naturalness . 136 6.13 Baryon-numberviolation . 137 6.14Theearlyuniverse ............................... 140 6.14.1 Globalsymmetry ............................ 141 6.14.2 Abeliancase............................... 145 6.14.3 Theelectroweakcase . 146 7 Gravity 149 7.1 Space-timeandthemetric ........................... 149 7.2 Dynamics and the Einstein equations . 151 7.3 Blackholes ................................... 152 iv Contents 7.4 Compactstellarobjects............................. 154 7.5 Gravitationalwaveastronomy . 156 8 Why there needs to be something beyond 158 8.1 Inconsistencies of the standard model . 159 8.2 Gravity ..................................... 161 8.2.1 Problems with quantization . 161 8.2.2 Asymptoticsafety............................ 162 8.3 Observations from particle physics experiments . 163 8.4 Astronomicalobservations . 164 8.4.1 Thecosmologicalstandardmodel . 164 8.4.2 Directobservations . .. .. 169 8.4.3 Darkmatter............................... 169 8.4.4 Inflation................................. 171 8.4.5 Curvature, cosmic expansion, and dark energy . 171 8.4.6 Matter-antimatterasymmetry . 172 8.5 How can new physics be discovered? . 173 8.5.1 Resonances ............................... 174 8.5.2 Missingenergy ............................. 175 8.5.3 Precisionobservables . 175 8.5.4 Right-handedneutrinos. 176 8.6 WhyoneTeV? ................................. 177 9 Candidate theories beyond the standard model 179 9.1 Supersymmetry ................................. 179 9.1.1 A first encounter with supersymmetry . 179 9.1.2 Supersymmetricconcepts. 182 9.1.2.1 Generatorsofsupersymmetry . 182 9.1.2.2 Spectrum of supersymmetric Hamilton operators . 184 9.1.2.3 Breaking supersymmetry and the Witten index . 185 9.1.2.4 Interactionsandthesuperpotential . 186 9.1.3 The simplest supersymmetric field theory . 188 9.1.3.1 Setup ............................. 188 9.1.3.2 Supermultiplets . 191 9.1.3.3 Off-shellsupersymmetry . 192 9.1.4 Supersymmetric Abelian gauge theory . 193 9.1.5 Supersymmetric Yang-Mills theory . 195 Contents v 9.1.6 SupersymmetricQED . .. .. 196 9.1.7 SupersymmetricQCD . .. .. 197 9.1.8 Supersymmetrybreaking. 198 9.1.9 The minimal supersymmetric standard model . 200 9.1.10 Supergravity............................... 205 9.2 Technicolor ................................... 207 9.2.1 Simple technicolor . 208 9.2.2 Extendingtechnicolor . .. .. 210 9.2.3 Conformalfieldtheories . 213 9.2.4 Walking technicolor . 215 9.3 Grandunifiedtheories ............................. 218 9.3.1 Setup .................................. 218 9.3.2 Aspecificexample ........................... 219 9.3.3 Runningcoupling............................ 221 9.3.4 Baryonnumberviolation. 223 9.4 Other low-scale extensions of the standard model . 224 9.4.1 Little Higgs . 224 9.4.2 Hiddensectors ............................. 225 9.4.3 Flavons ................................. 226 9.4.4 See-saw ................................. 227 9.4.5 Axions and other light particles . 228 9.4.6 2-Higgsdoubletmodels. 228 9.5 AtthePlanckscale............................... 229 9.5.1 Largeextradimensions. 229 9.5.1.1 Separable gravity-exclusive extra dimensions . 230 9.5.1.2 Universal extra dimensions . 232 9.5.1.3 Warped extra dimensions . 233 9.5.1.4 Symmetry breaking by orbifolding . 235 9.5.1.5 Deconstructedextradimensions . 235 9.5.1.6 Black holes . 236 9.5.2 Quantumgravity ............................ 237 9.5.2.1 Quantization . 237 9.5.2.2 Loopquantumgravity . 238 9.5.2.3 Non-commutativegeometry . 239 9.5.3 Stringtheory .............................. 239 9.5.3.1 Point particle . 242 vi Contents 9.5.3.2 Classical strings . 245 9.5.3.3 Lightconegauge ....................... 250 9.5.3.4 Point particle: Quantization . 251 9.5.3.5 Openstring.......................... 253 9.5.3.6 Closed string spectrum . 264 9.5.3.7 Dualities . 272 Chapter 1 Fundamentals of physics 1.1 Introduction Our current understanding of the fundamental laws of physics consists out of two parts. The first is particle physics. Its theory and details are quite involved, and it is a genuine quantum theory. In its current form it has evolved in the 1970ies. The second part is the non-quantum theory of general relativity. It is much older, from the 1910ies, and, in a sense, much more simple, as it does not require too much about the matter content. While both theories have been extremely successful, explaining many physical phenomena, partly to extreme precision, the situation is far from satisfactory. The most obvious problem is that these are two disjoint theories. However, entities like electrons exist in both, but behave fundamentally different, and their description cannot be joined. This will be elaborated more later, but the main problem is that one is a quantum theory and one is not. Beyond that, there are also several other hints, mostly from astrophysics and mathe- matical consistency, that even a joint version of these theories may not yet be the answer. Among these are questions pertaining to dark matter and the origin of the universe or why there was a big bang. However, it has become more and more clear that any