Verfügbarkeit: Gauge field theories

Gauge field theories:

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1. Verfasser:	Pokorski, Stefan (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Cambridge Cambridge Univ. Press 2001
Ausgabe:	2. ed., transferred to digital print.
Schriftenreihe:	Cambridge monographs on mathematical physics
Schlagworte:	Eichtheorie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XIX, 609 S. graph. Darst.
ISBN:	9780521478168 0521472458

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adam_text	Contents Preface to the First Edition page xvii Preface to the Second Edition xviii 0 Introduction I 0.1 Gauge invariance 1 0.2 Reasons for gauge theories of strong and electroweak interactions 3 QCD 3 Electroweak theory 5 1 Classical Reids, symmetries and their breaking 11 1.1 The action, equations of motion, symmetries and conservation laws 12 Equations of motion 12 Global symmetries 13 Space-time transformations 16 Examples 1 8 1.2 Classical field equations 20 Scalar field theory and spontaneous breaking of global symmetries 20 Spinor fields 22 1.3 Gauge field theories 29 U (I) gauge symmetry 29 Non-abelian gauge symmetry 31 1.4 From classical to quantum fields (canonical quantization) 35 Scalar fields 36 The Feynman propagator 39 Spinor fields 40 Symmetry transformations for quantum fields 45 1.5 Discrete symmetries 48 Space reflection 48 Time reversal 53 Charge conjugation 56 Summary and the CPT transformation 62 CP violation in the neutral Ko-R° -system 64 Problems 68 ix x Contents 2 Path integral formulation of quantum field theory 71 2.1 Path integrals in quantum mechanics 71 Transition matrix elements as path integrals 71 Matrix elements of position operators 75 2.2 Vacuum-to-vacuum transitions and the imaginary time formalism 76 General discussion 76 Harmonic oscillator 78 Euclidean Green s functions 82 2.3 Path integral formulation of quantum field theory 83 Green s functions as path integrals 83 Action quadratic in fields 87 Gaussian integration 88 2.4 Introduction to perturbation theory 90 Perturbation theory and the generating functional 90 Wick s theorem 92 An example: four-point Green s function in λΦ4 93 Momentum space 97 2.5 Path integrals for fermions; Grassmann algebra 100 Anticommuting c-numbers 100 Dirac propagator 102 2.6 Generating functionals for Green s functions and proper vertices; effective potential 105 Classification of Green s functions and generating functionals 105 Effective action 107 Spontaneous symmetry breaking and effective action 109 Effective potential 111 2.7 Green s functions and the scattering operator 113 Problems 120 3 Feynman rules for Yang-Mills theories 124 3.1 The Faddeev-Popov determinant 124 Gauge invariance and the path integral 124 Faddeev-Popov determinant 126 Examples 129 Non-covariant gauges 132 3.2 Feynman rules for QCD 133 Calculation of the Faddeev-Popov determinant 133 Feynman rules 135 3.3 Unitarity, ghosts, Becchi-Rouet-Stora transformation 140 Unitarity and ghosts 140 BRS and anti-BRS symmetry 143 Problems 147 4 Introduction to the theory of renormalization 148 4.1 Physical sense of renormalization and its arbitrariness 148 Bare and physical quantities 148 Counterterms and the renormalization conditions 152 Contents xi Arbitrariness of renormalization 153 Final remarks 156 4.2 Classification of the divergent diagrams 157 Structure of the UV divergences by momentum power counting 157 Classification of divergent diagrams 159 Necessary counterterms 161 4.3 λΦ4: low order renormalization 164 Feynman rules including counterterms 164 Calculation of Fig. 4.8(fc) 166 Comments on analytic continuation to η φ 4 dimensions 168 Lowest order renormalization 170 4.4 Effective field theories 173 Problems 175 5 Quantum electrodynamics 177 5.1 Ward-Takahashi identities 179 General derivation by the functional technique 179 Examples 181 5.2 Lowest order QED radiative corrections by the dimensional regularization technique 184 General introduction 184 Vacuum polarization 185 Electron self-energy correction 187 Electron self-energy: IR singularities regularized by photon mass 190 On-shell vertex correction 191 5.3 Massless QED 194 5.4 Dispersion calculation of О (a) virtual corrections in massless QED, in (4 =f ε) dimensions 196 Self-energy calculation 197 Vertex calculation 198 5.5 Coulomb scattering and the IR problem 200 Corrections of order a 200 IR problem to all orders in a 205 Problems 208 6 Renormalization group 209 6.1 Renormalization group equation (RGE) 209 Derivation of the RGE 209 Solving the RGE 212 Green s functions for rescaled momenta 214 RGE in QED 215 6.2 Calculation of the renormalization group functions β, γ, ym 216 6.3 Fixed points; effective coupling constant 219 Fixed points 219 Effective coupling constant 222 6.4 Renormalization scheme and gauge dependence of the RGE parameters 224 xii Contents Renormalization scheme dependence 224 Effective α in QED 226 Gauge dependence of the ^-function 227 Problems 228 7 Scale invariance and operator product expansion 230 7.1 Scale invariance 230 Scale transformations 230 Dilatation current 233 Conformai transformations 235 7.2 Broken scale invariance 237 General discussion 237 Anomalous breaking of scale invariance 238 7.3 Dimensional transmutation 242 7.4 Operator product expansion (OPE) 243 Short distance expansion 243 Light-cone expansion 247 7.5 The relevance of the light-cone 249 Electron-positron annihilation 249 Deep inelastic hadron leptoproduction 250 Wilson coefficients and moments of the structure function 254 7.6 Renormalization group and OPE 256 Renormalization of local composite operators 256 RGE for Wilson coefficients 259 OPE beyond perturbation theory 261 7.7 OPE and effective field theories 262 Problems 269 8 Quantum chromodynamics 272 8.1 General introduction 272 Renormalization and BRS invariance; counterterms 272 Asymptotic freedom of QCD 274 The Slavnov-Taylor identities 277 8.2 The background field method 279 8.3 The structure of the vacuum in non-abelian gauge theories 282 Homotopy classes and topological vacua 282 Physical vacuum 284 Θ -vacuum and the functional integral formalism 287 8.4 Perturbative QCD and hard collisions 290 Parton picture 290 Factorization theorem 291 8.5 Deep inelastic electron-nucleon scattering in first order QCD (Feynman gauge) 293 Structure functions and Born approximation 293 Deep inelastic quark structure functions in the first order in the strong coupling constant 298 Final result for the quark structure functions 302 Contents xiii Hadron structure functions; probabilistic interpretation 304 8.6 Light-cone variables, light-like gauge 306 8.7 Beyond the one-loop approximation 312 Comments on the IR problem in QCD 314 Problems 315 9 Chiral symmetry; spontaneous symmetry breaking 317 9.1 Chiral symmetry of the QCD lagrangian 317 9.2 Hypothesis of spontaneous chiral symmetry breaking in strong interactions 320 9.3 Phenomenological chirally symmetric model of the strong interactions (σ -model) 324 9.4 Goldstone bosons as eigenvectors of the mass matrix and poles of Green s functions in theories with elementary scalars 327 Goldstone bosons as eigenvectors of the mass matrix 327 General proof of Goldstone s theorem 330 9.5 Patterns of spontaneous symmetry breaking 333 9.6 Goldstone bosons in QCD 337 10 Spontaneous and explicit global symmetry breaking 342 10.1 Internal symmetries and Ward identities 342 Preliminaries 342 Ward identities from the path integral 344 Comparison with the operator language 347 Ward identities and short-distance singularities of the operator products 348 Renormalization of currents 351 10.2 Quark masses and chiral perturbation theory 353 Simple approach 353 Approach based on use of the Ward identity 354 10.3 Dashen s theorems 356 Formulation of Dashen s theorems 356 Dashen s conditions and global symmetry broken by weak gauge interactions 358 10.4 Electromagnetic π+-π° mass difference and spectral function sum rules 362 Electromagnetic π+-π° mass difference from Dashen s formula 362 Spectral function sum rules 363 Results 366 11 Higgs mechanism in gauge theories 369 11.1 Higgs mechanism 369 11.2 Spontaneous gauge symmetry breaking by radiative corrections 373 11.3 Dynamical breaking of gauge symmetries and vacuum alignment 379 Dynamical breaking of gauge symmetry 379 Examples 382 Problems 388 12 Standard electroweak theory 389 12.1 The lagrangian 391 xiv Contents 12.2 Electroweak currents and physical gauge boson fields 394 12.3 Fermion masses and mixing 398 12.4 Phenomenology of the tree level lagrangian 402 Effective four-fermion interactions 403 Z° couplings 406 12.5 Beyond tree level 407 Renormalization and counterterms 407 Corrections to gauge boson propagators 411 Fermion self-energies 418 Running α (μ) in the electroweak theory 419 Muon decay in the one-loop approximation 422 Corrections to the Z° partial decay widths 430 12.6 Effective low energy theory for electroweak processes 435 QED as the effective low energy theory 438 12.7 Flavour changing neutral-current processes 441 QCD corrections to CP violation in the neutral kaon system 445 Problems 456 13 Chiral anomalies 457 13.1 Triangle diagram and different renormalization conditions 457 Introduction 457 Calculation of the triangle amplitude 459 Different renormalization constraints for the triangle amplitude 464 Important comments 465 13.2 Some physical consequences of the chiral anomalies 469 Chiral invariance in spinor electrodynamics 469 л° -► 2γ 471 Chiral anomaly for the axial (7(1) current in QCD; U (i) problem 473 Anomaly cancellation in the SU(2) χ U( ) electroweak theory 475 Anomaly-free models 478 13.3 Anomalies and the path integral 478 Introduction 478 Abelian anomaly 480 Non-abelian anomaly and gauge invariance 481 Consistent and covariant anomaly 484 13.4 Anomalies from the path integral in Euclidean space 486 Introduction 486 Abelian anomaly with Dirac fermions 488 Non-abelian anomaly and chiral fermions 491 Problems 492 14 Effective lagrangians 495 14.1 Non-linear realization of the symmetry group 495 Non-linear σ -model 495 Effective lagrangian in the ђа(х) basis 500 Matrix representation for Goldstone boson fields 502 14.2 Effective lagrangians and anomalies 504 Contents xv Abelian anomaly 505 The Wess-Zumino term 506 Problems 508 15 Introduction to supersynimetry 509 15.1 Introduction 509 15.2 The supersymmetry algebra 511 15.3 Simple consequences of the supersymmetry algebra 513 15.4 Superspace and superfields for N = 1 supersymmetry 515 Superspace 515 Superfields 519 15.5 Supersymmetric lagrangian; Wess-Zumino model 521 15.6 Supersymmetry breaking 524 15.7 Supergraphs and the non-renormalization theorem 531 Appendix A: Spinors and their properties 539 Lorentz transformations and two-dimensional representations of the group SL(2, C) 539 Solutions of the free Weyl and Dirac equations and their properties 546 Parity 550 Time reversal 551 Charge conjugation 552 Appendix B: Feynman rules for QED and QCD and Feynman integrals 555 Feynman rules for the λΦ4 theory 555 Feynman rules for QED 556 Feynman rules for QCD 557 Dirac algebra in η dimensions 558 Feynman parameters 559 Feynman integrals in η dimensions 559 Gaussian integrals 560 λ -parameter integrals 560 Feynman integrals in light-like gauge η ■ A = 0, n2 = 0 561 Convention for the logarithm 561 Spence functions 562 Appendix C: Feynman rules for the Standard Model 563 Propagators of fermions 563 Propagators of the gauge bosons 564 Propagators of the Higgs and Goldstone bosons 565 Propagators of the ghost fields 566 Mixed propagators (only counterterms exist) 567 Gauge interactions of fermions 567 Yukawa interactions of fermions 570 Gauge interactions of the gauge bosons 571 Self-interactions of the Higgs and Goldstone bosons 573 Gauge interactions of the Higgs and Goldstone bosons 574 Gauge interactions of the ghost fields 578 Interactions of ghosts with Higgs and Goldstone bosons 579 xvi Contents Appendix D: One-loop Feynman integrals 583 Two-point functions 583 Three- and four-point functions 585 General expressions for the one-loop vector boson self-energies 586 Appendix E: Elements of group theory 591 Definitions 591 Transformation of operators 593 Complex and real representations 593 Traces 594 σ -model 596 References 599 Index 605
adam_txt	Contents Preface to the First Edition page xvii Preface to the Second Edition xviii 0 Introduction I 0.1 Gauge invariance 1 0.2 Reasons for gauge theories of strong and electroweak interactions 3 QCD 3 Electroweak theory 5 1 Classical Reids, symmetries and their breaking 11 1.1 The action, equations of motion, symmetries and conservation laws 12 Equations of motion 12 Global symmetries 13 Space-time transformations 16 Examples 1 8 1.2 Classical field equations 20 Scalar field theory and spontaneous breaking of global symmetries 20 Spinor fields 22 1.3 Gauge field theories 29 U (I) gauge symmetry 29 Non-abelian gauge symmetry 31 1.4 From classical to quantum fields (canonical quantization) 35 Scalar fields 36 The Feynman propagator 39 Spinor fields 40 Symmetry transformations for quantum fields 45 1.5 Discrete symmetries 48 Space reflection 48 Time reversal 53 Charge conjugation 56 Summary and the CPT transformation 62 CP violation in the neutral Ko-R° -system 64 Problems 68 ix x Contents 2 Path integral formulation of quantum field theory 71 2.1 Path integrals in quantum mechanics 71 Transition matrix elements as path integrals 71 Matrix elements of position operators 75 2.2 Vacuum-to-vacuum transitions and the imaginary time formalism 76 General discussion 76 Harmonic oscillator 78 Euclidean Green's functions 82 2.3 Path integral formulation of quantum field theory 83 Green's functions as path integrals 83 Action quadratic in fields 87 Gaussian integration 88 2.4 Introduction to perturbation theory 90 Perturbation theory and the generating functional 90 Wick's theorem 92 An example: four-point Green's function in λΦ4 93 Momentum space 97 2.5 Path integrals for fermions; Grassmann algebra 100 Anticommuting c-numbers 100 Dirac propagator 102 2.6 Generating functionals for Green's functions and proper vertices; effective potential 105 Classification of Green's functions and generating functionals 105 Effective action 107 Spontaneous symmetry breaking and effective action 109 Effective potential 111 2.7 Green's functions and the scattering operator 113 Problems 120 3 Feynman rules for Yang-Mills theories 124 3.1 The Faddeev-Popov determinant 124 Gauge invariance and the path integral 124 Faddeev-Popov determinant 126 Examples 129 Non-covariant gauges 132 3.2 Feynman rules for QCD 133 Calculation of the Faddeev-Popov determinant 133 Feynman rules 135 3.3 Unitarity, ghosts, Becchi-Rouet-Stora transformation 140 Unitarity and ghosts 140 BRS and anti-BRS symmetry 143 Problems 147 4 Introduction to the theory of renormalization 148 4.1 Physical sense of renormalization and its arbitrariness 148 Bare and 'physical' quantities 148 Counterterms and the renormalization conditions 152 Contents xi Arbitrariness of renormalization 153 Final remarks 156 4.2 Classification of the divergent diagrams 157 Structure of the UV divergences by momentum power counting 157 Classification of divergent diagrams 159 Necessary counterterms 161 4.3 λΦ4: low order renormalization 164 Feynman rules including counterterms 164 Calculation of Fig. 4.8(fc) 166 Comments on analytic continuation to η φ 4 dimensions 168 Lowest order renormalization 170 4.4 Effective field theories 173 Problems 175 5 Quantum electrodynamics 177 5.1 Ward-Takahashi identities 179 General derivation by the functional technique 179 Examples 181 5.2 Lowest order QED radiative corrections by the dimensional regularization technique 184 General introduction 184 Vacuum polarization 185 Electron self-energy correction 187 Electron self-energy: IR singularities regularized by photon mass 190 On-shell vertex correction 191 5.3 Massless QED 194 5.4 Dispersion calculation of О (a) virtual corrections in massless QED, in (4 =f ε) dimensions 196 Self-energy calculation 197 Vertex calculation 198 5.5 Coulomb scattering and the IR problem 200 Corrections of order a 200 IR problem to all orders in a 205 Problems 208 6 Renormalization group 209 6.1 Renormalization group equation (RGE) 209 Derivation of the RGE 209 Solving the RGE 212 Green's functions for rescaled momenta 214 RGE in QED 215 6.2 Calculation of the renormalization group functions β, γ, ym 216 6.3 Fixed points; effective coupling constant 219 Fixed points 219 Effective coupling constant 222 6.4 Renormalization scheme and gauge dependence of the RGE parameters 224 xii Contents Renormalization scheme dependence 224 Effective α in QED 226 Gauge dependence of the ^-function 227 Problems 228 7 Scale invariance and operator product expansion 230 7.1 Scale invariance 230 Scale transformations 230 Dilatation current 233 Conformai transformations 235 7.2 Broken scale invariance 237 General discussion 237 Anomalous breaking of scale invariance 238 7.3 Dimensional transmutation 242 7.4 Operator product expansion (OPE) 243 Short distance expansion 243 Light-cone expansion 247 7.5 The relevance of the light-cone 249 Electron-positron annihilation 249 Deep inelastic hadron leptoproduction 250 Wilson coefficients and moments of the structure function 254 7.6 Renormalization group and OPE 256 Renormalization of local composite operators 256 RGE for Wilson coefficients 259 OPE beyond perturbation theory 261 7.7 OPE and effective field theories 262 Problems 269 8 Quantum chromodynamics 272 8.1 General introduction 272 Renormalization and BRS invariance; counterterms 272 Asymptotic freedom of QCD 274 The Slavnov-Taylor identities 277 8.2 The background field method 279 8.3 The structure of the vacuum in non-abelian gauge theories 282 Homotopy classes and topological vacua 282 Physical vacuum 284 Θ -vacuum and the functional integral formalism 287 8.4 Perturbative QCD and hard collisions 290 Parton picture 290 Factorization theorem 291 8.5 Deep inelastic electron-nucleon scattering in first order QCD (Feynman gauge) 293 Structure functions and Born approximation 293 Deep inelastic quark structure functions in the first order in the strong coupling constant 298 Final result for the quark structure functions 302 Contents xiii Hadron structure functions; probabilistic interpretation 304 8.6 Light-cone variables, light-like gauge 306 8.7 Beyond the one-loop approximation 312 Comments on the IR problem in QCD 314 Problems 315 9 Chiral symmetry; spontaneous symmetry breaking 317 9.1 Chiral symmetry of the QCD lagrangian 317 9.2 Hypothesis of spontaneous chiral symmetry breaking in strong interactions 320 9.3 Phenomenological chirally symmetric model of the strong interactions (σ -model) 324 9.4 Goldstone bosons as eigenvectors of the mass matrix and poles of Green's functions in theories with elementary scalars 327 Goldstone bosons as eigenvectors of the mass matrix 327 General proof of Goldstone 's theorem 330 9.5 Patterns of spontaneous symmetry breaking 333 9.6 Goldstone bosons in QCD 337 10 Spontaneous and explicit global symmetry breaking 342 10.1 Internal symmetries and Ward identities 342 Preliminaries 342 Ward identities from the path integral 344 Comparison with the operator language 347 Ward identities and short-distance singularities of the operator products 348 Renormalization of currents 351 10.2 Quark masses and chiral perturbation theory 353 Simple approach 353 Approach based on use of the Ward identity 354 10.3 Dashen's theorems 356 Formulation of Dashen's theorems 356 Dashen's conditions and global symmetry broken by weak gauge interactions 358 10.4 Electromagnetic π+-π° mass difference and spectral function sum rules 362 Electromagnetic π+-π° mass difference from Dashen's formula 362 Spectral function sum rules 363 Results 366 11 Higgs mechanism in gauge theories 369 11.1 Higgs mechanism 369 11.2 Spontaneous gauge symmetry breaking by radiative corrections 373 11.3 Dynamical breaking of gauge symmetries and vacuum alignment 379 Dynamical breaking of gauge symmetry 379 Examples 382 Problems 388 12 Standard electroweak theory 389 12.1 The lagrangian 391 xiv Contents 12.2 Electroweak currents and physical gauge boson fields 394 12.3 Fermion masses and mixing 398 12.4 Phenomenology of the tree level lagrangian 402 Effective four-fermion interactions 403 Z° couplings 406 12.5 Beyond tree level 407 Renormalization and counterterms 407 Corrections to gauge boson propagators 411 Fermion self-energies 418 Running α (μ) in the electroweak theory 419 Muon decay in the one-loop approximation 422 Corrections to the Z° partial decay widths 430 12.6 Effective low energy theory for electroweak processes 435 QED as the effective low energy theory 438 12.7 Flavour changing neutral-current processes 441 QCD corrections to CP violation in the neutral kaon system 445 Problems 456 13 Chiral anomalies 457 13.1 Triangle diagram and different renormalization conditions 457 Introduction 457 Calculation of the triangle amplitude 459 Different renormalization constraints for the triangle amplitude 464 Important comments 465 13.2 Some physical consequences of the chiral anomalies 469 Chiral invariance in spinor electrodynamics 469 л° -► 2γ 471 Chiral anomaly for the axial (7(1) current in QCD; U\(i) problem 473 Anomaly cancellation in the SU(2) χ U(\) electroweak theory 475 Anomaly-free models 478 13.3 Anomalies and the path integral 478 Introduction 478 Abelian anomaly 480 Non-abelian anomaly and gauge invariance 481 Consistent and covariant anomaly 484 13.4 Anomalies from the path integral in Euclidean space 486 Introduction 486 Abelian anomaly with Dirac fermions 488 Non-abelian anomaly and chiral fermions 491 Problems 492 14 Effective lagrangians 495 14.1 Non-linear realization of the symmetry group 495 Non-linear σ -model 495 Effective lagrangian in the ђа(х) basis 500 Matrix representation for Goldstone boson fields 502 14.2 Effective lagrangians and anomalies 504 Contents xv Abelian anomaly 505 The Wess-Zumino term 506 Problems 508 15 Introduction to supersynimetry 509 15.1 Introduction 509 15.2 The supersymmetry algebra 511 15.3 Simple consequences of the supersymmetry algebra 513 15.4 Superspace and superfields for N = 1 supersymmetry 515 Superspace 515 Superfields 519 15.5 Supersymmetric lagrangian; Wess-Zumino model 521 15.6 Supersymmetry breaking 524 15.7 Supergraphs and the non-renormalization theorem 531 Appendix A: Spinors and their properties 539 Lorentz transformations and two-dimensional representations of the group SL(2, C) 539 Solutions of the free Weyl and Dirac equations and their properties 546 Parity 550 Time reversal 551 Charge conjugation 552 Appendix B: Feynman rules for QED and QCD and Feynman integrals 555 Feynman rules for the λΦ4 theory 555 Feynman rules for QED 556 Feynman rules for QCD 557 Dirac algebra in η dimensions 558 Feynman parameters 559 Feynman integrals in η dimensions 559 Gaussian integrals 560 λ -parameter integrals 560 Feynman integrals in light-like gauge η ■ A = 0, n2 = 0 561 Convention for the logarithm 561 Spence functions 562 Appendix C: Feynman rules for the Standard Model 563 Propagators of fermions 563 Propagators of the gauge bosons 564 Propagators of the Higgs and Goldstone bosons 565 Propagators of the ghost fields 566 Mixed propagators (only counterterms exist) 567 Gauge interactions of fermions 567 Yukawa interactions of fermions 570 Gauge interactions of the gauge bosons 571 Self-interactions of the Higgs and Goldstone bosons 573 Gauge interactions of the Higgs and Goldstone bosons 574 Gauge interactions of the ghost fields 578 Interactions of ghosts with Higgs and Goldstone bosons 579 xvi Contents Appendix D: One-loop Feynman integrals 583 Two-point functions 583 Three- and four-point functions 585 General expressions for the one-loop vector boson self-energies 586 Appendix E: Elements of group theory 591 Definitions 591 Transformation of operators 593 Complex and real representations 593 Traces 594 σ -model 596 References 599 Index 605
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spellingShingle	Pokorski, Stefan Gauge field theories Eichtheorie (DE-588)4122125-4 gnd
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title	Gauge field theories
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title_full	Gauge field theories Stefan Pokorski
title_fullStr	Gauge field theories Stefan Pokorski
title_full_unstemmed	Gauge field theories Stefan Pokorski
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