The Quantum Theory of Fields. Volume II Modern Applications Steven Weinberg

Transcription

1 The Quantum Theory of Fields Volume II Modern Applications Steven Weinberg

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3 PREFACE TO VOLUME II NOTATION xvii xx 15 NON-ABELIAN GAUGE THEORIES Gauge Invariance 2 Gauge transformations q Structure constants q Jacobi identity q Adjoint representation q Yang-Mills theory q Covariant derivatives q Field strength tensor q Finite gauge transformations q Analogy with general relativit y 15.2 Gauge Theory Lagrangians and Simple Lie Groups 7 Gauge field Lagrangian q Metric q Antisymmetric structure constants q Simple, semisimple, and U(1) Lie algebras q Structure of gauge algebra q Compac t algebras q Coupling constant s 15.3 Field Equations and Conservation Laws 1 2 Conserved currents q Covariantly conserved currents q Inhomogeneous field equations q Homogeneous field equations q Analogy with energy-momentum tensor q Symmetry generators 15.4 Quantization 14 Primary and secondary first-class constraints q Axial gauge q Gribov ambiguity q Canonical variables q Hamiltonian q Reintroduction of A q Covariant actio n q Gauge invariance of the measure 15.5 The De Witt-Faddeev-Popov Method 19 Generalization of axial gauge results q Independence of gauge fixing functionals q Generalized Feynman gauge q Form of vertices 15.6 Ghosts 24 Determinant as path integral q Ghost and antighost fields q Feynman rules for ghosts q Modified action q Power counting and renormalizability

4 15.7 BRST Symmetry 27 Auxiliary field h a q BRST transformation q Nilpotence q Invariance of new action q BRST-cohomology q Independence of gauge fixing q Application to electrodynamics q BRST-quantization q Geometric interpretatio n 15.8 Generalizations of BRST Symmetry * 36 De Witt notation q General Faddeev-Popov--De Witt theorem q BRST transformations q New action q Slavnov operator q Field-dependent structure constants q Generalized Jacobi identity q Invariance of new action q Independence of gauge fixing q Beyond quadratic ghost actions q BRST quantization q BRST cohomology q Anti-BRST symmetry 15.9 The Batalin-Vilkovisky Formalism * 42 Open gauge algebras q Antifields q Master equation q Minimal fields and trivial pairs q BRST-transformations with antifields q Antibrackets q Anticanonical transformations q Gauge fixing q Quantum master equatio n Appendix A A Theorem Regarding Lie Algebras 50 Appendix B The Cartan Catalog 54 Problems 5 8 References EXTERNAL FIELD METHODS The Quantum Effective Action 6 3 Currents q Generating functional for all graphs q Generating functional for connected graphs q Legendre transformation q Generating functional for oneparticle-irreducible graphs q Quantum-corrected field equations q Summing tree graphs 16.2 Calculation of the Effective Potential 6 8 Effective potential for constant fields q One loop calculation q Divergences q Renormalization q Fermion loop s 16.3 Energy Interpretation 7 2 Adiabatic perturbation q Effective potential as minimum energy q Convexity q Instability between local minima q Linear interpolation 16.4 Symmetries of the Effective Action 7 5 Symmetry and renormalization q Slavnov-Taylor identities q Linearly realized symmetries q Fermionic fields and current s Problems 7 8 References 78

5 17 RENORMALIZATION OF GAUGE THEORIES The Zinn-Justin Equation 80 Slavnov-Taylor identities for BRST symmetry q External fields K (x) q Antibracket s 17.2 Renormalization : Direct Analysis 8 2 Recursive argument q BRST-symmetry condition on infinities q Linearity in K (x) q New BRST symmetry q Cancellation of infinities q Renormalization constants q Nonlinear gauge condition s 17.3 Renormalization : General Gauge Theories * 9 1 Are `non-renormalizable' gauge theories renormalizable? q Structural constraint s q Anticanonical change of variables q Recursive argument q Cohomology theorem s 17.4 Background Field Gauge 9 5 New gauge fixing functions q True and formal gauge invariance q Renormalization constants 17.5 A One-Loop Calculation in Background Field Gauge 100 One-loop effective action q Determinants q Algebraic calculation for constant background fields q Renormalization of gauge fields and couplings q Interpretation of infinitie s Problems 109 References RENORMALIZATION GROUP METHODS Where do the Large Logarithms Come From? 112 Singularities at zero mass q `Infrared safe' amplitudes and rates q Jets q Zer o mass singularities from renormalization q Renormalized operators 18.2 The Sliding Scale 11 9 Gell-Mann-Low renormalization q Renormalization group equation q One - loop calculations q Application to 0 4 theory q Field renormalization factors q Application to quantum electrodynamics q Effective fine structure constant q Field-dependent renormalized couplings q Vacuum instabilit y 18.3 Varieties of Asymptotic Behavior 130 Singularities at finite energy q Continued growth q Fixed point at finite coupling q Asymptotic freedom q Lattice quantization q Triviality q Universal coefficients in the beta function

6 18.4 Multiple Couplings and Mass Effects 139 Behavior near a fixed point q Invariant eigenvalues q Nonrenormalizable theories q Finite dimensional critical surfaces q Mass renormalization at zero mass q Renormalization group equations for masse s 18.5 Critical Phenomena* 14 5 Low wave numbers q Relevant, irrelevant, and marginal couplings q Phase transitions and critical surfaces q Critical temperature q Behavior of correlation length q Critical exponent q 4 - e dimensions q Wilson-Fisher fixed point q Comparison with experiment q Universality classes 18.6 Minimal Subtraction 14 8 Definition of renormalized coupling q Calculation of beta function q Application to electrodynamics q Modified minimal subtraction q Non-renormalizable interaction s 18.7 Quantum Chromodynamics 15 2 Quark colors and flavors q Calculation of beta function q Asymptotic freedom q Quark and gluon trapping q Jets q e+-e- annihilation into hadrons q Accidenta l symmetries q Non-renormalizable corrections q Behavior of gauge coupling q Experimental results for g s and A 18.8 Improved Perturbation Theory * 15 7 Leading logarithms q Coefficients of logarithm s Problems 158 References SPONTANEOUSLY BROKEN GLOBAL SYMMETRIES Degenerate Vacua 163 Degenerate minima of effective potential q Broken symmetry or symmetric superpositions? q Large systems q Factorization at large distances q Diagonalizatio n of vacuum expectation values q Cluster decomposition 19.2 Goldstone Bosons 167 Broken global symmetries imply massless bosons q Proof using effective potential q Proof using current algebra q F factors and vacuum expectation values q Interactions of soft Goldstone boson s 19.3 Spontaneously Broken Approximate Symmetries 177 Pseudo-Goldstone bosons q Tadpoles q Vacuum alignment q Mass matrix q Positivity

7 19.4 Pions as Goldstone Bosons 18 2 S U(2) x S U(2) chiral symmetry of quantum chromodynamics q Breakdown to isospin q Vector and axial-vector weak currents q Pion decay amplitude q Axia l form factors of nucleon q Goldberger-Treiman relation q Vacuum alignment q Quark and pion masses q Soft pion interactions q Historical note 19.5 Effective Field Theories : Pions and Nucleons 192 Current algebra for two soft pions q Current algebra justification for effectiv e Lagrangian q a-model q Transformation to derivative coupling q Nonlinear realization of SU(2) x SU(2) q Effective Lagrangian for soft pions q Direct justification of effective Lagrangian q General effective Lagrangian for pions q Power counting q Pion-pion scattering for massless pions q Identification o f F-factor q Pion mass terms in effective Lagrangian q Pion-pion scattering for real pions q Pion-pion scattering lengths q Pion-nucleon effective Lagrangian q Covariant derivatives q ga 1 q Power counting with nucleons q Pion-nucleon scattering lengths q a-terms q Isospin violation q Adler-Weisberger sum rule 19.6 Effective Field Theories : General Broken Symmetries 21 1 Transformation to derivative coupling q Goldstone bosons and right cosets q Symmetric spaces q Cartan decomposition q Nonlinear transformation rules q Uniqueness q Covariant derivatives q Symmetry breaking terms q Application to quark mass terms q Power counting q Order parameter s 19.7 Effective Field Theories : SU(3) x SU(3) 22 5 SU(3) multiplets and matrices q Goldstone bosons of broken SU(3) x SU(3) q Quark mass terms q Pseudoscalar meson masses q Electromagnetic correction s q Quark mass ratios q Higher terms in Lagrangian q Nucleon mass shifts 19.8 Anomalous Terms in Effective Field Theories * 23 4 Wess-Zumino-Witten term q Five-dimensional form q Integer coupling q Uniqueness and de Rham cohomolog y 19.9 Unbroken Symmetries 23 8 Persistent mass conjecture q Vafa-Witten proof q Small non-degenerate quark masses The U(1) Problem 24 3 Chiral U(1) symmetry q Implications for pseudoscalar masses Problems 246 References 247

8 20 OPERATOR PRODUCT EXPANSIONS The Expansion : Description and Derivation 25 3 Statement of expansion q Dominance of simple operators q Path-integral derivation 20.2 Momentum Flow" contribution for two large momenta q Renormalized operators q Integra l equation for coefficient function q 02 contribution for many large moment a 20.3 Renormalization Group Equations for Coefficient Functions 26 3 Derivation and solution q Behavior for fixed points q Behavior for asymptotic freedo m 20.4 Symmetry Properties of Coefficient Functions 26 5 Invariance under spontaneously broken symmetrie s 20.5 Spectral Function Sum Rules 26 6 Spectral functions defined q First, second, and third sum rules q Application to chiral S U(N) x S U(N) q Comparison with experiment 20.6 Deep Inelastic Scattering 27 2 Form factors WI and W2 q Deep inelastic differential cross section q Bjorken scaling q Parton model q Callan-Gross relation q Sum rules q Form factors Tl and T2 q Relation between Tr and Wr q Symmetric tensor operators q Twist q Operators of minimum twist q Calculation of coefficient functions q Sum rules for parton distribution functions q Altarelli-Parisi differential equations q Logarithmic corrections to Bjorken scalin g 20.7 Renormalons* 28 3 Borel summation of perturbation theory q Instanton and renormalon obstructions q Instantons in massless 04 theory q Renormalons in quantum chromodynamic s Appendix Momentum Flow: The General Case 28 8 Problems 29 2 References SPONTANEOUSLY BROKEN GAUGE SYMMETRIES Unitarity Gauge 29 5 Elimination of Goldstone bosons q Vector boson masses q Unbroken symmetrie s and massless vector bosons q Complex representations q Vector field propagator q Continuity for vanishing gauge couplings

9 21.2 Renormalizable c-gauges 300 Gauge fixing function q Gauge-fixed Lagrangian q Propagators 21.3 The Electroweak Theory 305 Lepton-number preserving symmetries q SU(2) x U(1) q W±, Z, and photon s q Mixing angle q Lepton-vector boson couplings q W ± and z masses q Muon decay q Effective fine structure constant q Discovery of neutral currents q Quark currents q Cabibbo angle q c quark q Third generation q Kobayashi-Maskawa matrix q Discovery of W ± and z q Precise experimental tests q Accidental symmetries q Nonrenormalizable corrections q Lepton nonconservation and neutrino masses q Baryon nonconservation and proton decay 21.4 Dynamically Broken Local Symmetries* 31 8 Fictitious gauge fields q Construction of Lagrangian q Power counting q General mass formula q Example : SU(2) x SU(2) q Custodial SU(2) x S U(2) q Technicolo r 21.5 Electroweak-Strong Unification 32 7 Simple gauge groups q Relations among gauge couplings q Renormalization group flow q Mixing angle and unification mass q Baryon and lepton nonconservation 21.6 Superconductivity* 332 U(1) broken to Z 2 q Goldstone mode q Effective Lagrangian q Conservatio n of charge q Meissner effect q Penetration depth q Critical field q Flux quantization q Zero resistance ~ ac Josephson effect q Landau-Ginzburg theory q Correlation length q Vortex lines q U(1) restoration q Stability q Type I an d II superconductors q Critical fields for vortices q Behavior near vortex center q Effective theory for electrons near Fermi surface q Power counting q Introduction of pair field q Effective action q Gap equation q Renormalization group equations q Conditions for superconductivity Appendix General Unitarity Gauge 352 Problems 35 3 References ANOMALIES The it Decay Problem 35 9 Rate for x 2y q Naive estimate q Suppression by chiral symmetry q Comparison with experimen t 22.2 Transformation of the Measure : The Abelian Anomaly 36 2 Chiral and non-chiral transformations q Anomaly function q Chern-Pontryagin density q Nonconservation of current q Conservation of gauge-non-invariant

10 current q Calculation of it ---> 2y q Euclidean calculation q Atiyah-Singe r index theore m 223 Direct Calculation of Anomalies : The General Case 370 Fermion non-conserving currents q Triangle graph calculation q Shift vectors q Symmetric anomaly q Bardeen form q Adler-Bardeen theorem q Massive fermions q Another approach q Global anomalie s 22.4 Anomaly-Free Gauge Theories 38 3 Gauge anomalies must vanish q Real and pseudoreal representations q Safe groups q Anomaly cancellation in standard model q Gravitational anomalies q Hypercharge assignments q Another U(1)? 22.5 Massless Bound States * 389 Composite quarks and leptons? q Unbroken chiral symmetries q `t Hooft anomaly matching conditions q Anomaly matching for unbroken chiral S U(n) x SU(n) with SU(N) gauge group q The case N = 3 q q Chiral SU(3) x SU(3 ) must be broken q 't Hooft decoupling condition q Persistent mass condition 22.6 Consistency Conditions 396 Wess-Zumino conditions q BRST cohomology q Derivation of symmetri c anomaly q Descent equations q Solution of equations q Schwinger terms q Anomalies in Zinn-Justin equation q Antibracket cohomology q Algebraic proof of anomaly absence for safe groups 22.7 Anomalies and Goldstone Bosons 408 Anomaly matching q Solution of anomalous Slavnov-Taylor identities q Uniqueness q Anomalous Goldstone boson interactions q The case SU(3) x SU(3) q Derivation of Wess-Zumino-Witten interaction q Evaluation of integer coefficient q Generalization Problems 41 6 References EXTENDED FIELD CONFIGURATIONS The Uses of Topology 422 Topological classifications q Homotopy q Skyrmions q Derrick's theorem q Domain boundaries q Bogomol'nyi inequality q Cosmological problems 0 Instantons q Monopoles and vortex lines q Symmetry restoratio n 23.2 Homotopy Groups 43 0 Multiplication rule for n l (, ) q Associativity q Inverses q n l(sj) q Topological conservation laws q Multiplication rule for 7Ck(./#) q Winding number

11 23.3 Monopoles 43 6 SU(2)/U(1) model q Winding number q Electromagnetic field q Magneti c monopole moment q Kronecker index q `t Hooft-Polyakov monopole q Another Bogomol'nyi inequality q BPS monopole q Dirac gauge q Charge quantization q G/(H' x U(1)) monopoles q Cosmological problems q Monopole-particle interactions q G/H monopoles with G not simply connected q Irrelevance o f field conten t 23.4 The Cartan-Maurer Integral Invariant 445 Definition of the invariant q Independence of coordinate system q Topological invariance q Additivity q Integral invariant for S l H U(1) q Bott's theorem q Integral invariant for S3 1- S U(2) 23.5 Instantons 45 0 Evaluation of Cartan-Maurer invariant q Chern-Pontryagin density q One more Bogomol'nyi inequality q v = 1 solution q General winding number q Solution of U(1) problem q Baryon and lepton non-conservation by electroweak instanton s q Minkowskian approach q Barrier penetration q Thermal fluctuation s 23.6 The Theta Angle 45 5 Cluster decomposition q Superposition of winding numbers q P and CP nonconservation q Complex fermion masses q Suppression of P and CP nonconservation by small quark masses q Neutron electric dipole moment q Peccei- Quinn symmetry q Axions q Axion mass q Axion interaction s 23.7 Quantum Fluctuations around Extended Field Configurations 46 2 Fluctuations in general q Collective parameters q Determinental factor q Coupling constant dependence q Counting collective parameters 23.8 Vacuum Decay 46 4 False and true vacua q Bounce solutions q Four dimensional rotational invariance q Sign of action q Decay rate per volume q Thin wall approximatio n Appendix A Euclidean Path Integrals 46 8 Appendix B A List of Homotopy Groups 47 2 Problems 47 3 References 474 AUTHOR INDEX 47 8 SUBJECT INDEX 484