Quantum Field Theory and Particle Physics Badis Ydri Département de Physique, Faculté des Sciences Université d’Annaba Annaba, Algerie 2 YDRI QFT Contents 1 Introduction and References 1 I Free Fields, Canonical Quantization and Feynman Diagrams 5 2 Relativistic Quantum Mechanics 7 2.1 SpecialRelativity............................... ..... 7 2.1.1 Postulates.................................... 7 2.1.2 RelativisticEffects . ... 8 2.1.3 LorentzTransformations: Boosts . ...... 10 2.1.4 Spacetime.................................... 11 2.1.5 Metric...................................... 13 2.2 Klein-GordonEquation .. .. .. .. .. .. .. .. .. .. .. .. .... 14 2.3 DiracEquation ................................... .. 17 2.4 FreeSolutionsofTheDiracEquation. ........ 19 2.5 LorentzCovariance............................... .... 21 2.6 ExercisesandProblems .. .. .. .. .. .. .. .. .. .. .. .. .... 24 3 Canonical Quantization of Free Fields 29 3.1 ClassicalMechanics. .. .. .. .. .. .. .. .. .. .. .. .. ..... 29 3.1.1 D’AlembertPrinciple . .. 29 3.1.2 Lagrange’sEquations . .. 32 3.1.3 Hamilton’s Principle: The Principle of Least Action . ........... 33 3.1.4 TheHamiltonEquationsofMotion. .... 35 3.2 ClassicalFreeFieldTheories . ....... 37 3.2.1 TheKlein-GordonLagrangianDensity . ...... 37 3.2.2 TheDiracLagrangianDensity . ... 39 3.3 Canonical Quantization of a Real Scalar Field . ........... 40 3.4 Canonical Quantization of Free Spinor Field . ........... 44 3.5 Propagators..................................... .. 47 3.5.1 ScalarPropagator .............................. 47 3.5.2 DiracPropagator............................... 50 3.6 DiscreteSymmetries .............................. .... 51 3.6.1 Parity ...................................... 51 3.6.2 TimeReversal ................................. 52 3.6.3 ChargeConjugation .. .. .. .. .. .. .. .. .. .. .. .. 54 3.7 ExercisesandProblems .. .. .. .. .. .. .. .. .. .. .. .. .... 55 4 YDRI QFT 4 The S Matrix and Feynman Diagrams For Phi-Four Theory 61 − 4.1 ForcedScalarField............................... .... 61 4.1.1 AsymptoticSolutions . .. 61 4.1.2 The Schrodinger, Heisenberg and Dirac Pictures . ......... 63 4.1.3 The S Matrix ................................. 65 − 4.1.4 Wick’s TheoremForForcedScalarField . ...... 67 4.2 The Φ FourTheory .................................. 69 − 4.2.1 TheLagrangianDensity . .. 69 4.2.2 The S Matrix ................................. 70 − 4.2.3 TheGell-MannLowFormula . 72 4.2.4 LSZ Reduction Formulae and Green’s Functions . ....... 74 4.3 Feynman Diagrams For φ FourTheory ....................... 76 − 4.3.1 PerturbationTheory . .. 76 4.3.2 Wick’s Theorem For Green’s Functions . ..... 77 4.3.3 The 2 PointFunction............................. 79 − 4.3.4 ConnectednessandVacuumEnergy . ... 83 4.3.5 Feynman Rules For Φ FourTheory ..................... 86 − 4.4 ExercisesandProblems .. .. .. .. .. .. .. .. .. .. .. .. .... 88 II Quantum Electrodynamics 93 5 The Electromagnetic Field 95 5.1 Covariant Formulation of Classical Electrodynamics . .............. 95 5.2 Gauge Potentials and Gauge Transformations . .......... 98 5.3 Maxwell’sLagrangianDensity. ....... 99 5.4 PolarizationVectors .. .. .. .. .. .. .. .. .. .. .. .. ..... 101 5.5 Quantization of The Electromagnetic Gauge Field . ........... 103 5.6 Gupta-BleulerMethod . .. .. .. .. .. .. .. .. .. .. .. .. .... 107 5.7 Propagator ...................................... 110 5.8 ExercisesandProblems .. .. .. .. .. .. .. .. .. .. .. .. .... 111 6 Quantum Electrodynamics 113 6.1 LagrangianDensity............................... .... 113 6.2 Review of φ4 Theory.................................. 114 6.3 Wick’sTheoremforForcedSpinorField . ........ 115 6.3.1 GeneratingFunction . 115 6.3.2 Wick’sTheorem ................................ 120 6.4 Wick’s Theorem for Forced Electromagnetic Field . ............ 121 6.5 The LSZ Reduction fromulas and The S Matrix .................. 122 − 6.5.1 TheLSZReductionfromulas . 122 6.5.2 The Gell-Mann Low Formula and the S Matrix .............. 126 − 6.5.3 PerturbationTheory:TreeLevel. ...... 129 6.5.4 Perturbation Theory: One-Loop Corrections . ........ 131 6.6 LSZReductionformulasforPhotons . ....... 137 6.6.1 Example II: e− + γ e− + γ ........................ 137 −→ 6.6.2 PerturbationTheory . 138 6.7 FeynmanRulesforQED.............................. 140 6.8 CrossSections................................... .. 142 YDRI QFT 5 6.9 TreeLevelCrossSections: AnExample . ....... 146 6.10 ExercisesandProblems . ..... 151 7 Renormalization of QED 155 7.1 Example III: e− + µ− e− + µ− ......................... 155 −→ 7.2 Example IV : Scattering From External Electromagnetic Fields .......... 156 7.3 One-loop Calculation I: Vertex Correction . ........... 159 7.3.1 FeynmanParametersandWickRotation . .... 159 7.3.2 Pauli-VillarsRegularization . ....... 164 7.3.3 Renormalization (Minimal Subtraction) and Anomalous Magnetic Moment 166 7.4 Exact Fermion 2 PointFunction ........................... 169 − 7.5 One-loop Calculation II: Electron Self-Energy . ............. 171 7.5.1 ElectronMassatOne-Loop . 171 7.5.2 The Wave-Function Renormalization Z2 ................... 174 7.5.3 The Renormalization Constant Z1 ...................... 175 7.6 Ward-TakahashiIdentities . ....... 177 7.7 One-Loop Calculation III: Vacuum Polarization . ............ 180 7.7.1 The Renormalization Constant Z3 and Renormalization of the Electric Charge ..................................... 180 7.7.2 DimensionalRegularization . ..... 182 7.8 RenormalizationofQED. .... 186 7.9 ExercisesandProblems .. .. .. .. .. .. .. .. .. .. .. .. .... 187 III Path Integrals, Gauge Fields and Renormalization Group 191 8 Path Integral Quantization of Scalar Fields 193 8.1 FeynmanPathIntegral. .. .. .. .. .. .. .. .. .. .. .. .. .... 193 8.2 ScalarFieldTheory ............................... .. 197 8.2.1 PathIntegral .................................. 197 8.2.2 The Free 2 PointFunction .......................... 199 − 8.2.3 LatticeRegularization . .... 200 8.3 TheEffectiveAction ............................... .. 203 8.3.1 Formalism.................................... 203 8.3.2 PerturbationTheory . 206 8.3.3 Analogy with Statistical Mechanics . ....... 208 8.4 The O(N) Model.................................... 209 8.4.1 The 2 Point and 4 PointProperVertices . 210 − − 8.4.2 MomentumSpaceFeynmanGraphs . 211 8.4.3 Cut-offRegularization . .. 212 8.4.4 Renormalization at 1 Loop.......................... 215 − 8.5 Two-LoopCalculations. ..... 216 8.5.1 The Effective Action at 2 Loop ....................... 216 − 8.5.2 The Linear Sigma Model at 2 Loop ..................... 217 − 8.5.3 The 2 Loop Renormalization of the 2 PointProperVertex. 219 − − 8.5.4 The 2 Loop Renormalization of the 4 PointProperVertex. 224 − − 8.6 RenormalizedPerturbationTheory . ........ 226 8.7 Effective Potential and Dimensional Regularization . .............. 228 8.8 SpontaneousSymmetryBreaking . ...... 232 6 YDRI QFT 8.8.1 Example: The O(N) Model.......................... 232 8.8.2 Glodstone’sTheorem. 238 9 Path Integral Quantization of Dirac and Vector Fields 241 9.1 FreeDiracField .................................. 241 9.1.1 CanonicalQuantization . .. 241 9.1.2 Fermionic Path Integral and Grassmann Numbers . ....... 242 9.1.3 TheElectronPropagator . 247 9.2 FreeAbelianVectorField. ..... 248 9.2.1 Maxwell’sAction............................... 248 9.2.2 Gauge Invariance and Canonical Quantization . ........ 250 9.2.3 Path Integral Quantization and the Faddeev-Popov Method........ 252 9.2.4 ThePhotonPropagator . 254 9.3 GaugeInteractions............................... .... 255 9.3.1 Spinor and Scalar Electrodynamics: Minimal Coupling ........... 255 9.3.2 The Geometry of U(1) GaugeInvariance. 257 9.3.3 Generalization: SU(N) Yang-MillsTheory . 260 9.4 Quantization and Renormalization at 1 Loop.................... 266 − 9.4.1 The Fadeev-Popov Gauge Fixing and Ghost Fields . ....... 266 9.4.2 Perturbative Renormalization and Feynman Rules . ......... 270 9.4.3 The Gluon Field Self-Energy at 1 Loop................... 273 − 9.4.4 The Quark Field Self-Energy at 1 Loop................... 283 − 9.4.5 The Vertex at 1 Loop............................. 285 − 10 The Renormalization Group 293 10.1 Critical Phenomena and The φ4 Theory ....................... 293 10.1.1 Critical Line and Continuum Limit . ...... 293 10.1.2 MeanFieldTheory. .. .. .. .. .. .. .. .. .. .. .. .. 298 10.1.3 CriticalExponentsinMeanField. ...... 302 10.2 The Callan-Symanzik Renormalization Group Equation . ............. 307 10.2.1 PowerCountingTheorems. .. 307 10.2.2 Renormalization Constants and Renormalization Conditions. 311 10.2.3 Renormalization Group Functions and Minimal Subtraction ....... 314 10.2.4 CS Renormalization Group Equation in φ4 Theory ............. 317 10.2.5 Summary .................................... 323 10.3 Renormalization Constants and Renormalization Functions atTwo-Loop . 325 10.3.1 The Divergent Part of the Effective Action . ....... 325 10.3.2 RenormalizationConstants . ..... 330 10.3.3 RenormalizationFunctions . ..... 332 10.4 CriticalExponents . .. .. .. .. .. .. .. .. .. .. .. .. ..... 333 10.4.1 CriticalTheoryandFixedPoints . ...... 333 10.4.2 Scaling Domain (T >Tc) ........................... 339 10.4.3 Scaling Below Tc ................................ 342 10.4.4 Critical Exponents from 2 Loop and Comparison with Experiment . 345 − 10.5 The Wilson Approximate Recursion Formulas . .......... 348 10.5.1 Kadanoff-WilsonPhase Space Analysis. ....... 348 10.5.2 RecursionFormulas . 350 10.5.3 TheWilson-FisherFixedPoint