Quantum Field Theory 2 nd Edition

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1 Quantum Field Theory 2 nd Edition FRANZ MANDL and GRAHAM SHAW School of Physics & Astromony, The University of Manchester, Manchester, UK WILEY A John Wiley and Sons, Ltd., Publication

2 Contents Preface Notes xi xiii 1 Photons and the Electromagnetic Field Particles and Fields The Electromagnetic Field in the Absence of Charges The classical field Harmonic oscillator The quantized radiation field The Electric Dipole Interaction The Electromagnetic Field in the Presence of Charges Classical electrodynamics Quantum electrodynamics Radiative transitions in atoms Thomson scattering Appendix: The Schrödinger, Heisenberg and Interaction Pictures 20 Problems 22 2 Lagrangian Field Theory Relativistic Notation Classical Lagrangian Field Theory Quantized Lagrangian Field Theory Symmetries and Conservation Laws 31 Problems 37 3 The Klein Gordon Field The Real Klein Gordon Field The Complex Klein Gordon Field Covariant Commutation Relations The Meson Propagator 48 Problems 53

3 vi Contents 4 The Dirac Field The Number Representation for Fermions The Dirac Equation Second Quantization The spin-statistics theorem The Fermion Propagator The Electromagnetic Interaction and Gauge Invariance 70 Problems 71 5 Photons: Covariant Theory The Classical Fields Covariant Quantization The Photon Propagator 81 Problems 84 6 The S-Matrix Expansion Natural Dimensions and Units The S-Matrix Expansion Wick's Theorem 94 7 Feynman Diagrams and Rules in QED Feynman Diagrams in Configuration Space Feynman Diagrams in Momentum Space The first-order terms S (1) Compton scattering Electron electron scattering Closed loops Feynman Rules for QED Leptons 121 Problems QED Processes in Lowest Order The Cross-Section Spin Sums Photon Polarization Sums Lepton Pair Production in (e + e ) Collisions Bhabha Scattering Compton Scattering Scattering by an External Field Bremsstrahlung The Infrared Divergence 155 Problems Radiative Corrections The Second-Order Radiative Corrections of QED The Photon Self-Energy 167

4 Contents vii 9.3 The Electron Self-Energy External Line Renormalization The Vertex Modification Applications The anomalous magnetic moments The Lamb shift The Infrared Divergence Higher-Order Radiative Corrections Renormalizability 198 Problems Regularization Mathematical Preliminaries Some standard integrals Feynman parameterization Cut-Off Regularization: The Electron Mass Shift Dimensional Regularization Introduction General results Vacuum Polarization The Anomalous Magnetic Moment 214 Problems Gauge Theories The Simplest Gauge Theory: QED Quantum Chromodynamics Colour and confinement Global phase invariance and colour conservation SU(3) gauge invariance Quantum chromodynamics Alternative Interactions? Non-minimal interactions Renormalizability Appendix: Two Gauge Transformation Results The transformation law (11.26b) The SU(3) gauge invariance of Eq. (11.34) 237 Problems Field Theory Methods Green Functions Feynman Diagrams and Feynman Rules The perturbation expansion The vacuum amplitude The photon propagator Connected Green functions 252

5 viii Contents 12.3 Relation to S-Matrix Elements Crossing Functionals and Grassmann Fields Functionals Grassmann algebras and Grassmann fields The Generating Functional The free-field case The perturbation expansion 270 Problems Path Integrals Functional Integration Classical fields Grassmann generators Grassmann fields Path Integrals The generating functional Free and interacting fields The free electromagnetic field The free spinor fields Perturbation Theory Wick's theorem Interactions Gauge Independent Quantization? 297 Problems Quantum Chromodynamics Gluon Fields The generating functional A mathematical analogy The Faddeev Popov Method Gauge fixing and ghosts The electromagnetic field revisited Including Quarks The QCD Lagrangian The generating functional Free fields Perturbation Theory Wick's theorem and propagators The perturbation expansion The vertex factors Feynman Rules for QCD Renormalizability of QCD 321 Problems 323

6 Contents ix 15 Asymptotic Freedom Electron Positron Annihilation Two-jet events Three-jet events The Renormalization Scheme The electron propagator The photon propagator Charge renormalization The Renormalization Group The renormalization group equations Scale transformations The running charge The Strong Coupling Constant Colour factors Null diagrams Renormalization of the coupling constant The running coupling Applications Appendix: Some Loop Diagrams in QCD The gluon self-energy graphs The quark gluon vertex corrections 360 Problems Weak Interactions Introduction Leptonic Weak Interactions The Free Vector Boson Field The Feynman Rules for the IVB Theory Decay Rates Applications of the IVB Theory Muon decay Neutrino scattering The leptonic decay of the W boson Neutrino Masses Neutrino oscillations Dirac or Majorana neutrinos? Difficulties with the IVB Theory 385 Problems A Gauge Theory of Weak Interactions QED Revisited Global Phase Transformations and Conserved Weak Currents The Gauge-Invariant Electroweak Interaction Properties of the Gauge Bosons Lepton and Gauge Boson Masses 401

7 x Contents 18 Spontaneous Symmetry Breaking The Goldstone Model The Higgs Model The Standard Electroweak Theory The Standard Electroweak Theory The Lagrangian Density in the Unitary Gauge Feynman Rules Elastic Neutrino Electron Scattering Electron Positron Annihilation The Higgs Boson Higgs boson decays Higgs boson searches 446 Problems 448 Appendix A The Dirac Equation 451 A.1 The Dirac Equation 451 A.2 Contraction Identities 453 A.3 Traces 453 A.4 Plane Wave Solutions 455 A.5 Energy Projection Operators 456 A.6 Helicity and Spin Projection Operators 456 A.7 Relativistic Properties 458 A.8 Particular Representations of the -y-matrices 460 Problems 462 Appendix B Feynman Rules and Formulae for Perturbation Theory 463 Index 473

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