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The Standard Model of Particle Physics

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The Standard Model of Particle Physics Uwe-Jens Wiese Institute for Theoretical Physics University of Bern April 11, 2021 Contents 1 Introduction 7 I FUNDAMENTAL CONCEPTS 11 2 Concepts of Quantum Field Theory and the Standard Model 13 2.1 Point Particles versus Fields at the Classical Level . 14 2.2 Particles versus Waves in Quantum Theory . 15 2.3 Classical and Quantum Gauge Fields . 18 2.4 Ultraviolet Divergences, Regularization, and Renormalization 19 2.5 The Standard Model: Renormalizability, Triviality, and Incorporation of Gravity . 20 2.6 Fundamental Standard Model Parameters . 22 2.7 Hierarchies of Scales and Approximate Global Symmetries . 23 2.8 Local and Global Symmetries . 25 2.9 Explicit versus Spontaneous Symmetry Breaking . 27 2.10 Anomalies in Local and Global Symmetries . 28 1 2 CONTENTS 2.11 Euclidean Quantum Field Theory versus Classical Statistical Mechanics . 29 II CONSTRUCTION OF THE STANDARD MODEL 31 3 From Cooper Pairs to Higgs Bosons 33 3.1 A Charged Scalar Field for Cooper Pairs . 34 3.2 The Higgs Doublet . 38 3.3 The Goldstone Theorem . 41 3.4 The Mermin-Wagner Theorem . 44 3.5 Low-Energy Effective Field Theory . 45 3.6 The Hierarchy Problem . 49 3.7 Triviality of the Standard Model . 53 3.8 Electroweak Symmetry Restauration at High Temperature . 55 3.9 Extended Model with Two Higgs Doublets . 56 4 From Superconductivity to Electroweak Gauge Bosons 61 4.1 Scalar Quantum Electrodynamics . 62 4.2 The Higgs Mechanism in the Electroweak Theory . 66 4.3 Accidental Custodial Symmetry . 72 4.4 Lattice Gauge-Higgs Models . 74 4.5 From Electroweak to Grand Unification . 79 5 One Generation of Leptons and Quarks 87 CONTENTS 3 5.1 Weyl and Dirac Spinors . 88 5.2 Parity, Charge Conjugation, and Time-Reversal . 90 5.3 Electrons and Left-handed Neutrinos . 94 5.4 CP and T Invariance of Gauge Interactions . 97 5.5 Fixing the Lepton Weak Hypercharges . 100 5.6 Gauge Anomalies in the Lepton Sector . 103 5.7 Up and Down Quarks . 106 5.8 Anomaly Cancellation . 109 5.9 Electric Charges of Quarks and Baryons . 111 5.10 Anomaly Matching . 113 5.11 Right-handed Neutrinos . 115 5.12 Lepton and Baryon Number Anomalies . 116 5.13 An Anomaly-Free Technicolor Model . 119 6 Fermion Masses 123 6.1 Electron and Down Quark Masses . 124 6.2 Up Quark Mass . 127 6.3 Massive Neutrinos . 129 6.4 The Majorana mass term . 130 7 Three Generations of Quarks and Leptons 133 7.1 The CKM Quark Mixing Matrix . 133 8 The Structure of the Strong Interactions 141 4 CONTENTS 8.1 Quantum Chromodynamics . 143 8.2 Chiral Symmetry . 147 III SPECIAL TOPICS 153 9 Phenomenology of the Strong Interactions 155 9.1 Isospin Symmetry . 157 9.2 Nucleon and ∆-Isobar in the Constituent Quark Model . 159 9.3 Anti-Quarks and Mesons . 163 9.4 Strange Hadrons . 164 9.5 The Gellman-Okubo Baryon Mass Formula . 167 9.6 Meson Mixing . 170 10 Topology of Gauge Fields 175 10.1 The Anomaly . 175 10.2 Topological Charge . 177 10.3 Topology of a Gauge Field on a Compact Manifold . 182 10.4 The Instanton in SU(2) . 185 10.5 θ-Vacua . 187 10.6 The U(1)-Problem . 191 10.7 Baryon Number Violation in the Standard Model . 196 11 The Strong CP-Problem 199 11.1 Rotating θ into the Mass Matrix . 202 CONTENTS 5 11.2 The θ-Angle in Chiral Perturbation Theory . 203 11.3 The θ-Angle at Large Nc . 204 11.4 The Peccei-Quinn Symmetry . 205 11.5 U(1)PQ Breaking and the Axion . 208 IV APPENDICIES 213 A Units, Scales, and Hierarchies in Particle Physics 215 A.1 Man-Made versus Natural Units . 215 A.2 Energy Scales and Particle Masses . 219 A.3 Fundamental Standard Model Parameters . 224 B Basics of Quantum Field Theory 229 B.1 From Point Particle Mechanics to Classical Field Theory . 230 B.2 The Quantum Mechanical Path Integral . 233 B.3 The Path Integral in Euclidean Time . 241 B.4 Spin Models in Classical Statistical Mechanics . 243 B.5 Analogies between Quantum Mechanics and Classical Statistical Mechanics . 247 B.6 Lattice Field Theory . 250 C Group Theory of SN and SU(n) 265 C.1 The Permutation Group SN . 265 C.2 The Group SU(n) . 268 6 CONTENTS D Canonical Quantization of Free Weyl, Dirac, and Majorana Fermions 273 D.1 Massless Weyl Fermions . 273 D.2 Momentum, Angular Momentum, and Helicity of Weyl Fermions277 D.3 Fermion Number, Parity, and Charge Conjugation . 279 D.4 Massive Dirac Fermions . 284 D.5 Massive Majorana Fermions . 289 D.6 Massive Weyl Fermions . 292 E Fermionic Functional Integrals 295 E.1 Grassmann Algebra, Pfaffian, and Fermion Determinant . 295 E.2 The Dirac Equation . 305 E.3 The Weyl and Majorana Equations . 308 E.4 Euclidean Fermionic Functional Integral . 310 E.5 Euclidean Lorentz Group . 313 E.6 Charge Conjugation, Parity, and Time-Reversal for Weyl Fermions . 317 E.7 C, P, and T for Dirac Fermions . 321 E.8 CPT Invariance in Relativistic Quantum Field Theory . 322 E.9 Connections between Spin and Statistics . 323 E.10 Euclidean Time Transfer Matrix . 325 Chapter 1 Introduction The Standard Model of particle physics summarizes all we know about the fundamental forces of electromagnetism, as well as the weak and strong interactions (but not gravity). It has been tested in great detail up to ener- gies in the hundred GeV range and has passed all these tests very well. The Standard Model is a relativistic quantum field theory that incorporates the basic principles of quantum mechanics and special relativity. Like quantum electrodynamics (QED) the Standard Model is a gauge theory, however, with the non-Abelian gauge group SU(3)c ⊗ SU(2)L ⊗ U(1)Y instead of the simple Abelian U(1)em gauge group of QED. The gauge bosons are the photons mediating the electromagnetic interactions, the W - and Z-bosons mediating the weak interactions, as well as the gluons mediating the strong interactions. Gauge theories can exist in several phases: in the Coulomb phase with massless gauge bosons (like in QED), in the Higgs phase with spontaneously broken gauge symmetry and with massive gauge bosons (e.g. the W - and Z-bosons), and in the confinement phase, in which the gauge bosons do not appear in the spectrum (like the gluons in quantum chromo- dynamics (QCD)). All these different phases are indeed realized in Nature and hence in the Standard Model that describes it. In particle physics symmetries play a central role. One distinguishes global and local symmetries. Global symmetries are usually only approxi- mate. Exact symmetries, on the other hand, are locally realized, and require the existence of a gauge field. Our world is not quite as symmetric as the 7 8 CHAPTER 1. INTRODUCTION theories we use to describe it. This is because many symmetries are broken. The simplest form of symmetry breaking is explicit breaking which is due to non-invariant symmetry breaking terms in the classical Lagrangian of the theory. On the other hand, the quantization of the theory may also lead to explicit symmetry breaking, even if the classical Lagrangian is invariant. In that case one has an anomaly which is due to an explicit symmetry breaking in the measure of the Feynman path integral. Only global symmetries can be explicitly broken (either in the Lagrangian or via an anomaly). Theories with explicitly broken gauge symmetries, on the other hand, are inconsis- tent (perturbatively and even non-perturbatively non-renormalizable). For example, in the Standard Model all gauge anomalies are canceled due to the properly arranged fermion content of each generation. Another interesting form of symmetry breaking is spontaneous sym- metry breaking which is a dynamical effect. When a continuous global symmetry breaks spontaneously, massless Goldstone bosons appear in the spectrum. If there is, in addition, a weak explicit symmetry breaking, the Goldstone bosons pick up a small mass. This is the case for the pions, which arise as a consequence of the spontaneous breaking of the approximate global chiral symmetry in QCD. When a gauge symmetry is spontaneously broken one encounters the so-called Higgs mechanism which gives mass to the gauge bosons. This gives rise to an additional helicity state. This state has the quantum numbers of a Goldstone boson that would arise if the symmetry were global. One says that the gauge boson eats the Goldstone boson and thus becomes massive. The fermions in the Standard Model are either leptons or quarks. Lep- tons participate only in the electromagnetic and weak gauge interactions, while quarks also participate in the strong interactions. Quarks and lep- tons also pick up their masses through the Higgs mechanism. The values of these masses are free parameters of the Standard Model that are presently not understood on the basis of a more fundamental theory. There are six quarks: up, down, strange, charm, bottom, and top, and six leptons: the electron, muon, tau, as well as their corresponding neutrinos. The weak interaction eigenstates are mixed to form the mass eigenstates. The quark mixing Cabbibo-Kobayashi-Maskawa (CKM) matrix contains several more free parameters of the Standard Model. There is convincing experimental evidence for non-zero neutrino masses. This implies that there are not only additional free mass parameters for the electron-, muon-, and tau-neutrinos, 9 but an entire lepton mixing matrix. Altogether, the fermion sector of the Standard Model has so many free parameters that it is hard to believe that there should not be a more fundamental theory that will be able to explain the values of these parameters.
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