Quantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University

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1 Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS

2 t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation Blackbody radiation Einstein coefficients Quantum field theory 7 2 Lorentz invariance and second quantization Lorentz invariance Classical plane waves as oscillators Second quantization 20 Problems 27 3 Classical field theory Hamiltonians and Lagrangians The Euler-Lagrange equations Noether's theorem Coulomb's law Green's functions 39 Problems 42 4 Old-fashioned perturbation theory Lippmann-Schwinger equation Early infinities 52 Problems 55 5 Cross sections and decay rates Cross sections Non-relativistic limit e+e~ > with spin 65 Problems 67 6 The S-matrix and time-ordered products The LSZ reduction formula The Feynman propagator 75 Problems 77 vii

3 7 Feynman rules Lagrangian derivation Hamiltonian derivation Momentum-space Feynman rules Examples * 97 7.A Normal ordering and Wick's theorem 100 Problems 103 Part II Quantum electrodynamics Spin 1 and gauge invariance Unitary representations of the Poincare group Embedding particles into fields Covariant derivatives Quantization and the Ward identity The photon propagator Is gauge invariance real? Higher-spin fields 132 Problems Scalar quantum electrodynamics Quantizing complex scalar fields Feynman rules for scalar QED Scattering in scalar QED Ward identity and gauge invariance Lorentz invariance and charge conservation 150 Problems Spinors Representations of the Lorentz group Spinor representations Dirac matrices Coupling to the photon What does spin \ mean? Majorana and Weyl fermions 178 Problems Spinor solutions and CPT Chirality, helicity and spin Solving the Dirac equation Majorana spinors Charge conjugation Parity 195

4 11.6 Time reversal 198 Problems Spin and statistics Identical particles Spin-statistics from path dependence Quantizing spinors Lorentz invariance of the S -matrix Stability Causality 219 Problems Quantum electrodynamics QED Feynman rules matrix identities e+e~ n+p Rutherford scattering e p+ e~p Compton scattering Historical note 246 Problems Path integrals Introduction The path integral Generating functionals Where is the iel Gauge invariance Fermionic path integral Schwinger-Dyson equations Ward-Takahashi identity 277 Problems 283 Part III Renormalization The Casimir effect Casimir effect Hard cutoff Regulator independence Scalar field theory example 296 Problems Vacuum polarization Scalar < 3 theory Vacuum polarization in QED 304

5 16.3 Physics of vacuum polarization 309 Problems The anomalous magnetic moment Extracting the moment» Evaluating the graphs 318 Problems Mass renormalization Vacuum expectation values Electron self-energy Pole mass Minimal subtraction Summary and discussion 336 Problems Renormalized perturbation theory Counterterms Two-point functions Three-point functions Renormalization conditions in QED Z Zi\ implications and proof 350 Problems Infrared divergences e+e~ -> n+fi- (+7) Jets Other loops A Dimensional regularization 373 Problems Renormalizability Renormalizability of QED Non-renormalizable field theories 386 Problems Non-renormalizable theories The Schrodinger equation The 4-Fermi theory Theory of mesons Quantum gravity Summary of non-renormalizable theories Mass terms and naturalness Super-renormalizable theories 414 Problems 416

6 23 The renormalization group Running couplings Renormalization group from counterterms Renormalization group equation for the 4-Fermi theory Renormalization group equation for general interactions Scalar masses and renormalization group flows Wilsonian renormalization group equation 442 Problems Implications of unitarity The optical theorem Spectral decomposition Polology Locality 475 Problems 477 Part IV The Standard Model Yang-Mills theory Lie groups Gauge invariance and Wilson lines Conserved currents Gluon propagator Lattice gauge theories 503 Problems Quantum Yang-Mills theory Feynman rules Attractive and repulsive potentials e+e~ > hadrons and as Vacuum polarization Renormalization at 1-loop Running coupling Defining the charge 529 Problems Gluon scattering and the spinor-helicity formalism Spinor-helicity formalism Gluon scattering amplitudes gg -» gg Color ordering Complex momenta On-shell recursion Outlook 558 Problems 559

7 28 Spontaneous symmetry breaking Spontaneous breaking of discrete symmetries Spontaneous breaking of continuous global symmetries The Higgs mechanism Quantization of spontaneously broken gauge theories Problems Weak interactions Electroweak symmetry breaking Unitarity and gauge boson scattering Fermion sector The 4-Fermi theory CP violation 605 Problems Anomalies Pseudoscalars decaying to photons Triangle diagrams with massless fermions Chiral anomaly from the integral measure Gauge anomalies in the Standard Model Global anomalies in the Standard Model Anomaly matching 638 Problems Precision tests of the Standard Model Electroweak precision tests Custodial SU(2), p, S, T and U Large logarithms in flavor physics 657 Problems Quantum chromodynamics and the parton model Electron-proton scattering DGLAP equations Parton showers Factorization and the parton model from QCD Lightcone coordinates 695 Problems 698 Part V Advanced topics Effective actions and Schwinger proper time Effective actions from matching Effective actions from Schwinger proper time Effective actions from Feynman path integrals Euler-Heisenberg Lagrangian 713

8 33.5 Coupling to other currents Semi-classical and non-relativistic limits A Schwinger's method 728 Problems Background fields PI effective action Background scalar fields Background gauge fields 752 Problems Heavy-quark physics Heavy-meson decays Heavy-quark effective theory Loops in HQET Power corrections 772 Problems Jets and effective field theory Event shapes Power counting Soft interactions Collinear interactions Soft-Collinear Effective Theory Thrust in SCET 802 Problems 810 Appendices 813 Appendix A Conventions 815 A.l Dimensional analysis 815 A.2 Signs 817 A.3 Feynman rules 819 A. 4 Dirac algebra 820 Problems 821 Appendix B Regularization 822 B. l Integration parameters 822 B.2 Wick rotations 823 B.3 Dimensional regularization 825 B.4 Other regularization schemes 830 Problems 833 References 834 Index 842