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Unformatted text preview: Advanced Quantum Gauge Field Theory P. van Nieuwenhuizen Contents 1 A brief history of quantum gauge field theory 11 1 QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Weak interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5 Quantization, unitarity and renormalizability . . . . . . . . . . . . . . 122 A Relativistic corrections to the spectrum of hydrogen. . . . . . . . . . 178 B Anomalous magnetic moment . . . . . . . . . . . . . . . . . . . . . . 197 2 BRST symmetry 208 1 Invariance of the quantum action for gauge fields . . . . . . . . . . . 212 2 Nilpotency and auxiliary field . . . . . . . . . . . . . . . . . . . . . . 221 3 The BRST Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 4 Anti-BRST symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5 Nonrenormalizability of massive gauge theory . . . . . . . . . . . . . 234 6 BRST, Faddeev-Popov and string-like quantization . . . . . . . . . . 244 7 Classical and quantum Yang-Mills theory from the Noether method . 252 8 Gauge invariance from tree unitarity . . . . . . . . . . . . . . . . . . 257 9 Historical and other comments . . . . . . . . . . . . . . . . . . . . . . 261 A Heat kernel regularization of the BRST Jacobian. . . . . . . . . . . . 275 2 3 CONTENTS 3 Renormalization of unbroken gauge theories 280 1 The Ward identities for divergences in proper graphs . . . . . . . . . 287 2 Multiplicative renormalizability of QCD . . . . . . . . . . . . . . . . 306 3 Multiplicative renormalizability of quarks and gluons . . . . . . . . . 316 4 On-shell renormalization in QED . . . . . . . . . . . . . . . . . . . . 323 5 Nonlinear gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 6 Noncovariant algebraic gauges . . . . . . . . . . . . . . . . . . . . . . 332 7 Asymptotic freedom in the Coulomb gauge . . . . . . . . . . . . . . . 337 8 One-loop Z-factors in QCD . . . . . . . . . . . . . . . . . . . . . . . 344 9 The one-loop beta function and running masses . . . . . . . . . . . . 347 10 The two-loop β function . . . . . . . . . . . . . . . . . . . . . . . . . 351 A Proof that Γ = Γren even with external sources . . . . . . . . . . . . . 361 B Functional methods for external sources C Details of the renormalization of the Dirac-Yang-Mills system . . . . 374 . . . . . . . . . . . . . . . . 366 4 Renormalization of Higgs models 377 1 Renormalization of Goldstone models . . . . . . . . . . . . . . . . . . 381 2 The Goldstone theorem at one- and higher-loop level . . . . . . . . . 391 3 The spontaneously broken SU (2) Higgs model . . . . . . . . . . . . . 399 4 Renormalization of the SU(2) Higgs model . . . . . . . . . . . . . . . 409 5 Perturbative unitarity from the cutting rules 425 1 The largest-time equation:unitarity for scalars . . . . . . . . . . . . . 437 2 Unitarity for spin 1/2 fields . . . . . . . . . . . . . . . . . . . . . . . 449 3 Unitarity for massless spin 1 fields . . . . . . . . . . . . . . . . . . . . 455 4 Unitarity for spontaneously broken gauge theories . . . . . . . . . . . 466 5 Unitarity and renormalizability . . . . . . . . . . . . . . . . . . . . . 472 6 Locality of counter terms, causality and statistics . . . . . . . . . . . 480 7 Gauge-choice independence of the S-matrix . . . . . . . . . . . . . . . 490 4 CONTENTS 6 Anomalies 497 1 The V-A basis and the chiral basis . . . . . . . . . . . . . . . . . . . 507 2 Anomalies in triangle, box and pentagon graphs . . . . . . . . . . . . 512 3 Gauge anomalies ruin renormalizability and unitarity . . . . . . . . . 529 4 When do anomalies cancel, and when should they cancel? . . . . . . . 541 5 π 0 → 2γ: a good anomaly . . . . . . . . . . . . . . . . . . . . . . . . 554 6 Consistency conditions and Bardeen anomaly . . . . . . . . . . . . . 563 7 The Wess Zumino term . . . . . . . . . . . . . . . . . . . . . . . . . . 570 8 Consistent and covariant anomalies. Descent equations . . . . . . . . 576 9 The Pauli-Villars method . . . . . . . . . . . . . . . . . . . . . . . . . 593 10 The Fujikawa method . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 7 The background field method 623 1 Background gauge invariant effective actions . . . . . . . . . . . . . . 629 2 The S matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 3 Renormalization of background gauge field theory . . . . . . . . . . . 643 4 Gauge parameter independence of the beta function . . . . . . . . . . 652 5 Calculation of the β function at two loops . . . . . . . . . . . . . . . 662 6 Further applications of the background field method . . . . . . . . . . 677 A The Slavnov identity with background fields. . . . . . . . . . . . . . . 682 8 Instantons 1 Winding number and embeddings . . . . . . . . . . . . . . . . . . . . 693 1.1 2 3 4 687 Some remarks on nonselfdual instanton solutions . . . . . . . 706 Regular and singular instanton solutions . . . . . . . . . . . . . . . . 707 2.1 Lorentz and spinor algebra . . . . . . . . . . . . . . . . . . . . 708 2.2 Solving the selfduality equations . . . . . . . . . . . . . . . . . 713 Collective coordinates, the index theorem and fermionic zero modes . 717 3.1 Bosonic collective coordinates and the Dirac operator . . . . . 719 3.2 Fermionic moduli and the index theorem . . . . . . . . . . . . 722 Construction of zero modes . . . . . . . . . . . . . . . . . . . . . . . 730 CONTENTS 5 4.1 Bosonic zero modes and their normalization . . . . . . . . . . 730 4.2 Construction of the fermionic zero modes . . . . . . . . . . . . 737 5 6 The measure for zero modes . . . . . . . . . . . . . . . . . . . . . . . 740 5.1 The measure for the bosonic collective coordinates . . . . . . . 741 5.2 The measure for the fermionic collective coordinates . . . . . . 744 One loop determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 747 6.1 7 The exact β function for SYM theories . . . . . . . . . . . . . 754 N = 4 supersymmetric Yang-Mills theory . . . . . . . . . . . . . . . . 759 7.1 7.2 7.3 Minkowskian N = 4 SYM . . . . . . . . . . . . . . . . . . . . 760 Euclidean N = 4 SYM . . . . . . . . . . . . . . . . . . . . . . 761 Involution in Euclidean space . . . . . . . . . . . . . . . . . . 765 8 Large instantons and the Higgs effect . . . . . . . . . . . . . . . . . . 766 9 Instantons as most probable tunnelling paths . . . . . . . . . . . . . . 771 10 False vacua and phase transitions . . . . . . . . . . . . . . . . . . . . 784 11 The strong CP problem 12 The U (1) problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 13 Baryon decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802 14 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 A Winding number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 B ’t Hooft symbols and Euclidean spinors . . . . . . . . . . . . . . . . . 811 C The volume of the gauge orientation moduli space . . . . . . . . . . . 815 D Zero modes and conformal symmetries . . . . . . . . . . . . . . . . . 823 E Instantons at finite temperature . . . . . . . . . . . . . . . . . . . . . 827 . . . . . . . . . . . . . . . . . . . . . . . . . 797 9 The anomalous magnetic moment of the electron and muon 846 A On-shell renormalization of QED . . . . . . . . . . . . . . . . . . . . 870 B The vacuum polarization . . . . . . . . . . . . . . . . . . . . . . . . . 875 C Susy contributions to g − 2 . . . . . . . . . . . . . . . . . . . . . . . . 881 6 CONTENTS 10 The Dirac formalism and Hamiltonian path integrals 892 1 Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 2 The Dirac formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 3 Structure functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 4 Example: nonlinear Lie algebras . . . . . . . . . . . . . . . . . . . . . 925 5 The Hamiltonian BRST charge QH . . . . . . . . . . . . . . . . . . . 930 6 The BRST invariant Hamiltonian . . . . . . . . . . . . . . . . . . . . 933 7 The quantum action . . . . . . . . . . . . . . . . . . . . . . . . . . . 936 8 Boundary conditions and gauge-choice independence . . . . . . . . . 940 11 The antifield formalism 949 1 The antibracket and the quantum action . . . . . . . . . . . . . . . . 952 2 BRST transformations and nilpotency . . . . . . . . . . . . . . . . . 964 3 Examples of irreducible theories . . . . . . . . . . . . . . . . . . . . . 972 3.1 Pure Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . 972 3.2 The point particle . . . . . . . . . . . . . . . . . . . . . . . . . 973 4 Reducible gauge theories and ghosts for ghosts . . . . . . . . . . . . . 976 5 Examples of reducible gauge theories . . . . . . . . . . . . . . . . . . 979 5.1 Antisymmetric tensor gauge fields . . . . . . . . . . . . . . . . 979 5.2 Yang Mills fields coupled to antisymmetric tensors . . . . . . . 984 5.3 Ghosts-for-ghosts without extra ghosts . . . . . . . . . . . . . 992 6 Gauge-choice independence and master equation . . . . . . . . . . . . 994 7 From Hamiltonian-BRST to BV-BRST . . . . . . . . . . . . . . . . . 997 8 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 12 The Yang-Baxter equation and the algebraic Bethe ansatz 1007 1 The Yang-Baxter equation . . . . . . . . . . . . . . . . . . . . . . . . 1007 2 The spin 1/2 Heisenberg chain . . . . . . . . . . . . . . . . . . . . . . 1017 3 Quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 4 Transfer matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025 5 The algebraic Bethe ansatz . . . . . . . . . . . . . . . . . . . . . . . . 1030 6 Solutions of the Bethe equations . . . . . . . . . . . . . . . . . . . . . 1035 7 The boundary Yang-Baxter equation . . . . . . . . . . . . . . . . . . 1041 7 CONTENTS 13 The Gribov problem 1050 1 Gribov copies in the Coulomb gauge . . . . . . . . . . . . . . . . . . 1054 2 The relativistic gauge ∂ µ Aµ = 0 . . . . . . . . . . . . . . . . . . . . . 1059 3 Inserting unity into the path integral . . . . . . . . . . . . . . . . . . 1064 4 Gribov copies in a simple toy model . . . . . . . . . . . . . . . . . . . 1066 5 No Gribov copies in perturbation theory or axial gauges 14 Supersymmetry . . . . . . . 1068 1072 1 The Poincar´e supersymmetry algebras. . . . . . . . . . . . . . . . . . 1073 2 Multiplets of states of extended susy . . . . . . . . . . . . . . . . . . 1077 3 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 4 N = 1 susy field theories x-space . . . . . . . . . . . . . . . . . . . . 1090 5 N = 1 Susy field theories in superspace . . . . . . . . . . . . . . . . . 1098 6 The gauge action in N = 1 superspace . . . . . . . . . . . . . . . . . 1101 7 The matter action in N = 1 superspace . . . . . . . . . . . . . . . . . 1106 8 Field theories in x-space with rigid N = 2 susy . . . . . . . . . . . . . 1108 9 The N = 2 hypermultiplet . . . . . . . . . . . . . . . . . . . . . . . . 1110 10 The N = 4 rigid susy model . . . . . . . . . . . . . . . . . . . . . . . 1114 11 N = 2 superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 15 Kinks, monopoles and other solitons 1154 1 The kink solution and the BPS bound . . . . . . . . . . . . . . . . . 1156 2 The supersymmetric kink . . . . . . . . . . . . . . . . . . . . . . . . . 1162 3 Quantization of collective coordinates . . . . . . . . . . . . . . . . . . 1171 4 Solitons in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186 5 The ’t Hooft-Polyakov monopole . . . . . . . . . . . . . . . . . . . . . 1193 6 Chern-Simons terms and WZW effective actions . . . . . . . . . . . . 1206 7 The winding of the Wess-Zumino term . . . . . . . . . . . . . . . . . 1215 8 SU (3) × SU (3) symmetry in QCD and the WZW term . . . . . . . . 1217 9 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224 10 The normalization of the WZW terms . . . . . . . . . . . . . . . . . 1225 8 CONTENTS 16 Renormalization of composite operators 1231 1 Examples of composite operators . . . . . . . . . . . . . . . . . . . . 1234 2 Closure under renormalization and structure of the Z matrix . . . . . 1241 3 The general solution of QX = 0 from cohomology 17 The effective potential at the one-loop level 2 . . . . . . . . . . 1253 1271 1 The Coleman-Weinberg mechanism . . . . . . . . . . . . . . . . . . . 1272 2 One-loop contributions from fermions . . . . . . . . . . . . . . . . . . 1280 3 The mass of the Higgs boson . . . . . . . . . . . . . . . . . . . . . . . 1283 4 Gauge-choice dependence of the effective potential . . . . . . . . . . . 1285 18 Finite temperature field theory 1292 1 Elements of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 1298 2 Propagators at finite temperature . . . . . . . . . . . . . . . . . . . . 1305 3 Thermal masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313 4 Phase transitions at high temperature . . . . . . . . . . . . . . . . . . 1315 5 Gauge theories, fermions and ghosts at finite T 6 Supersymmetry violation at nonzero temperature . . . . . . . . . . . 1331 7 The real-time formulation . . . . . . . . . . . . . . . . . . . . . . . . 1336 8 The canonical approach to thermal field theory . . . . . . . . . . . . 1343 . . . . . . . . . . . . 1322 19 Quantum Chern-Simons theory in 3 dimensions 1 1362 Quantum Chern-Simons theory . . . . . . . . . . . . . . . . . . . . . 1362 20 Pauli Villars regularization of gauge theories 1377 21 The infrared R∗ operation 1389 22 Parastatistics 1418 1 One bose-like oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 1420 2 One fermi-like oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 1423 3 Parastatistics for several flavors . . . . . . . . . . . . . . . . . . . . . 1426 CONTENTS 9 4 A unique vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1430 5 The Green representation . . . . . . . . . . . . . . . . . . . . . . . . 1432 6 Parastatistics and color . . . . . . . . . . . . . . . . . . . . . . . . . . 1433 10 CONTENTS Preface Modern quantum field theory for gauge theories should be based on path integrals and BRST symmetry, and not (in first instance at least) on Feynman graphs and operator methods. This is the point of view which forms the basis for this book. Renormalizability and unitarity of QED, QCD and electroweak gauge theory, background field methods, and anomalies will all be discussed by using Ward identities and functional methods which follow from path integrals and BRST symmetry. Quantum gauge field theory is a vast subject, and only by both working out general ideas in concrete examples, and explaining concrete problems by placing them in a general context, can one begin to understand this enormous edifice. Therefore, it is equally important to reach the level of Feynman graphs, and to work out concrete problems in gauge field theory and applications to particle physics. We shall give both detailed derivations of path integrals and their symmetry properties, but also discussions of regularization issues of chiral gauge theories and infrared divergences in QED and QCD. We prove unitarity by using cutting rules for Feynman graphs and simplify Feynman graph calculations by using background field methods. We even present some ongoing experiments which test the Standard Model. We have incorporated into our presentation of quantum gauge field theory some new concepts which were developed under the stimulus of string theory but we do not discuss string theory. We discuss supersymmetric gauge theories because they give much insight into gauge theory in general, but we do not discuss supersymmetric phenomenology. We shall discuss several subjects which usually are not covered in textbooks. Our hope is that new graduate students, those specializing in string theory as well as those engaged in higher-loop calculations or parton distribution functions or modern nuclear physics or modern statistical mechanics, will enjoy this broader outlook as much as the author who taught these subjects for two and a half decades. Chapter 1 A brief history of quantum gauge field theory All fundamental interactions between particles except gravitation are nowadays very well described by quantum gauge field theories: the electroweak theory and quantum chromodynamics. The principles of gauge invariance and Lorentz invariance, together with the choice of gauge groups and some discrete symmetries, and the requirement of renormalizability, determine all interactions up to the numerical value of the coupling constants. Masses are due to spontaneous gauge symmetry breaking and are determined by the coupling constants of the Yukawa interactions. Renormalizability excludes particles with spins larger than one, and only admits minimal gauge interactions introduced by covariant derivatives Dµ = ∂µ +gAaµ Ta , but not, ¯ µν ψFµν . Particles belong to multiplets for example, Pauli couplings of the form g ψγ which form representations of semisimple nonabelian Lie algebras with generators Ta , and for a given internal symmetry group each multiplet couples with the same nonabelian coupling constant g. Electromagnetism is at present the only abelian gauge theory known to exist in nature, and particles couple with their electric charge as coupling constant. The result is the Standard Model (perhaps one might even call it the Standard Theory) with 27 free parameters: the 6 quark masses, the 3 charged 11 12 1. A BRIEF HISTORY OF QUANTUM GAUGE FIELD THEORY lepton masses, the 3 gauge coupling constants, the 4 quark mixing parameters, the Higgs mass and its vacuum expectation value, and further the 3 neutrino masses, the 4 neutrino mixing parameters, the θ angle of QCD, and, to describe classical gravity, the cosmological constant. Path integrals are used to describe quantum effects, and the quantum action for these gauge theories (the action which appears in the path integral) is obtained by adding a gauge fixing term and a ghost action to the classical gauge action, which also contains a matter part with scalars and spinors coupled to gauge fields and to themselves. There are powerful formal arguments which explain why the quantum action has this form, in particular the observation that it has a rigid symmetry which is an extension of classical gauge symmetry to the quantum level called BRST symmetry (a kind of “quantum gauge symmetry”). Furthermore, there are very good reasons for using path integrals: they allow one to describe nonperturbative as well as perturbative physics and yield simple and general methods to deduce the consequences of BRST symmetry for the correlation functions (the Ward-Takahashi-Slavnov-Taylor identities, called Ward identities in this book). However, in this chapter we shall follow the historical path and recall the experimental indications and theoretical developments which led to these gauge theories and this structure of the quantum action. This will explain how the concept of gauge theories grew out of studies of gravity, why the nongravitational gauge groups are SU (3) and SU (2) and U (1), why the gauge fields of the weak interactions couple to chiral fermions, why also the nuclear forces are described by gauge theory, why this gauge theory couples quarks to gluons, and why one moved from canonical quantization and the Schr¨odinger equation in the Coulomb or temporal gauge, via covariant quantization with Heisenberg fields, to the manifestly relativistic path integral approach. In the next chapter we shall introduce BRST symmetry. One may divide the history of particle physics [1,2] and the development of quan- 13 tum gauge field theory into three periods:1 (i) “the birth of particle physics” from 1926–19...
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