Particle Physics 1 Lecture notes for the first year master course on the electroweak part of the Standard Model Nikhef - autumn 2017 Wouter Hulsbergen, Marcel Merk, Ivo van Vulpen Contents Preliminaries i Introduction 1 i.1 Quantum fields . 1 i.2 The Yukawa interaction . 4 i.3 Feynman diagrams . 7 i.4 The Standard Model . 9 i.5 Units in particle physics . 12 i.6 Four-vector notation . 14 1 Wave Equations and Anti-Particles 17 1.1 Particle-wave duality . 17 1.2 The Schr¨odinger equation . 18 1.3 The Klein-Gordon equation . 20 1.4 Interpretation of negative energy solutions . 23 1.4.1 Dirac’s interpretation . 23 1.4.2 Pauli-Weisskopf interpretation . 24 1.4.3 Feynman-St¨uckelberg interpretation . 24 Exercises....................................... 27 2 Perturbation Theory and Fermi’s Golden Rule 31 2.1 Decay and scattering observables . 31 2.2 Non-relativistic scattering . 33 2.3 Relativistic scattering . 38 2.3.1 Normalisation of the Wave Function . 39 2.3.2 Density of states and phase space factor . 40 2.3.3 The Flux Factor . 42 2.3.4 Golden rules for cross-section and decay . 43 Exercises....................................... 44 3 The Electromagnetic Field 47 3.1 The Maxwell Equations . 47 3.2 Gauge transformations . 49 3.3 The photon . 50 i ii CONTENTS 3.4 Electrodynamics in quantum mechanics . 53 3.5 The Aharanov-Bohm E↵ect ......................... 54 Exercises....................................... 57 4 Electromagnetic Scattering of Spinless Particles 59 4.1 Electromagnetic current . 59 4.2 Coulomb scattering . 61 4.3 Spinless ⇡ K Scattering . 63 − 4.4 Formfactors.................................. 67 4.5 Particles and Anti-Particles . 67 Exercises....................................... 69 5 The Dirac Equation 71 5.1 Spin, spinors and the gyromagnetic ratio . 71 5.2 The Dirac equation . 73 5.3 Covariant form of the Dirac equation . 75 5.4 Dirac algebra . 76 5.5 Adjoint spinors and current density . 77 5.6 Bilinear covariants . 79 5.7 Solutions to the Dirac Equation . 79 5.7.1 Plane waves solutions with p =0 . 79 5.7.2 Plane wave solutions for p =0.................... 80 6 5.8 Antiparticle spinors . 82 5.9 Normalization of the wave function . 83 5.10 The completeness relation . 84 5.11 Helicity . 84 5.12 Charge current and anti-particles . 86 5.13 The charge conjugation operation . 87 Exercises....................................... 88 6 Spin-1/2 Electrodynamics 93 6.1 Feynman rules for fermion scattering . 93 6.2 Electron-muon scattering . 96 + + 6.3 Crossing: the process e−e µ−µ .....................101 ! 6.4 Summary of QED Feynman rules . 104 Exercises....................................... 105 7 The Weak Interaction 107 7.1 Lifetimes and couplings . 107 7.2 The 4-point interaction . 109 7.3 Parity ..................................... 111 7.4 Covariance of the wave equations under parity . 113 7.5 The V A interaction ............................ 114 − 7.6 The propagator of the weak interaction . 116 7.7 Muondecay.................................. 117 CONTENTS iii 7.8 Quark mixing . 120 Exercises....................................... 125 8 Local Gauge Invariance 127 8.1 Symmetries . 127 8.2 The principle of least action . 128 8.3 Lagrangian density for fields . 129 8.4 Global phase invariance and Noether’s theorem . 131 8.5 Local phase invariance . 131 8.6 Application to the Dirac Lagrangian . 133 8.7 Yang-Mills theory . 134 8.8 Historical interlude 1: isospin, QCD and weak isospin . 138 8.9 Historical interlude 2: the origin of the name “gauge theory” . 139 Exercises....................................... 140 9 Electroweak Theory 143 9.1 SU(2) symmetry for left-handed douplets . 144 9.2 The Charged Current . 146 9.3 The Neutral Current . 148 9.4 Couplings for Z ff ............................151 ! 9.5 The mass of the W and Z bosons...................... 152 Exercises....................................... 154 + + 10 The Process e−e µ−µ 157 ! 10.1 Helicity conservation . 157 + + 10.2 The cross section of e−e µ−µ .....................158 ! 10.2.1 Photon contribution . 159 10.2.2 Z0 contribution . 162 10.2.3 Correcting for the finite width of the Z0 ..............163 10.2.4 Total unpolarized cross-section . 164 10.3 Near the resonance . 165 10.4 The forward-backward asymmetry . 167 10.5 The Z0 decay width and the number of light neutrinos . 168 Exercises....................................... 171 11 Symmetry breaking 173 11.1 Problems in the Electroweak Model . 173 11.2 A few basics on Lagrangians . 175 11.3 Simple example of symmetry breaking . 176 11.3.1 µ2 > 0: Free particle with additional interactions . 176 11.3.2 µ2 < 0: Introducing a particle with imaginary mass ? . 177 11.4 Breaking a global symmetry . 178 11.4.1 µ2 > 0 .................................178 11.4.2 µ2 < 0 .................................179 11.5 Breaking a local gauge invariant symmetry: the Higgs mechanism . 180 iv CONTENTS 11.5.1 Lagrangian under small perturbations . 181 11.5.2 Rewriting the Lagrangian in the unitary gauge . 181 11.5.3 Lagrangian in the unitary gauge: particle spectrum . 182 11.5.4 A few words on expanding the terms with (v + h)2 ........182 Exercises....................................... 183 12 The Higgs mechanism in the Standard Model 185 12.1 Breaking the local gauge invariant SU(2) U(1) symmetry . 185 L ⇥ Y 12.2 Checking which symmetries are broken in a given vacuum . 186 12.3 Scalar part of the Lagrangian: gauge boson mass terms . 187 µ 12.3.1 Rewriting (D φ)† (Dµφ) in terms of physical gauge bosons . 188 12.4 Massesofthegaugebosons ......................... 190 12.4.1 Massive charged and neutral gauge bosons . 190 12.4.2 Massless neutral gauge boson (γ): . 191 12.5 Mass of the Higgs boson . 191 Exercises....................................... 191 13 Fermion masses, Higgs decay and limits on mh 195 13.1 Fermion masses . 195 13.1.1 Lepton masses . 196 13.1.2 Quark masses . 197 13.2 Yukawa couplings and the origin of Quark Mixing . 198 13.3 Higgs boson decay . 202 13.3.1 Higgs boson decay to fermions . 202 13.3.2 Higgs boson decay to gauge bosons . 203 13.3.3 Review Higgs boson couplings to fermions and gauge bosons . 204 13.3.4 Higgs branching fractions . 205 13.4 Theoretical bounds on the mass of the Higgs boson . 205 13.4.1 Unitarity . 205 13.4.2 Triviality and Vacuum stability . 206 13.4.3 Triviality: λ g, g0, h heavy Higgs boson upper limit on m 207 t ! h 13.4.4 Vacuum stability λ g, g0, h light Higgs boson lower limit ⌧ t ! on mh .................................208 13.5 Experimental limits on the mass of the Higgs boson . 210 13.5.1 Indirect measurements . 210 13.5.2 Direct measurements . 211 Exercises....................................... 213 14 Problems with the Higgs mechanism and Higgs searches 215 14.1 Problems with the Higgs boson . 215 14.1.1 Problems with the Higgs boson: Higgs self-energy . 215 14.1.2 Problems with the Higgs boson: the hierarchy problem . 216 14.2 Higgs bosons in models beyond the SM (SUSY) . 217 Exercises....................................... 218 CONTENTS v A Some properties of Dirac matrices ↵i and β 221 B Summary of electroweak theory 223 Preliminaries These are the lecture notes for the Particle Physics 1 (PP1) master course that is taught at Nikhef in the autumn semester of 2014. These notes contain 14 chapters, each corresponding to one lecture session. The topics discussed in this course are: Lecture 1 - 4: Electrodynamics of spinless particles • Lecture 5 - 6: Electrodynamics of spin 1/2 particles • Lecture 7: The weak interaction • Lecture 8 - 10: Gauge symmetries and the electroweak theory • Lecture 11-14: Electroweak symmetry breaking • Each lecture of 2 45 minutes is followed by a 1.5 hour problem solving session. The exercises are included⇥ in these notes, at the end of each chapter. The notes mainly follow the material as discussed in the books of Halzen and Martin. The first ten chapters have been compiled by Marcel Merk in the period 2000-2011, and updated by Wouter Hulsbergen for the PP1 courses of 2012 and 2013. The last four chapters, written by Ivo van Vulpen, were added in 2014. Literature The following is a non-exhaustive list of course books on particle physics. (The comments reflect a personel opinion of your lecturers!) Thomson: “Modern Particle Physics”: This is a new book (2013) that covers practically all the material in these lectures. If you do not have another particle physics book yet, then we recommend that you acquire this book. Halzen & Martin: “Quarks & Leptons: an Introductory Course in Modern Particle Physics ”: This is the book that your lecturers used when they did their university studies. Though most of the theory is timeless, it is a bit outdated when it comes to experimental results. i ii PRELIMINARIES The book builds on earlier work of Aitchison (see below). Most of the course follows this book, but it is no longer in print. Griffiths: “Introduction to Elementary Particle Physics”, second, revised ed. The text is somewhat easier to read than H & M and is.