Quantum Field Theory II: Gauge Theories Lecture in WS 2019/20 at the KFU Graz Axel Maas 2 Contents 1 Introduction 1 2 Path integral 3 2.1 Heuristicintroduction ............................. 3 2.2 Functionalanalysis ............................... 7 2.3 Fermions..................................... 11 2.3.1 Grassmannvariables .......................... 11 2.3.2 Fermionicmatter ............................ 14 2.4 Free particles and generating functionals . .. 15 3 Abelian gauge theories 18 3.1 Globalsymmetries ............................... 18 3.2 ClassicalMaxwelltheory ............................ 21 3.3 QuantizationofMaxwelltheory . 23 3.4 Gauge-dependentcorrelationfunctions . ... 29 3.5 Matterfields................................... 31 4 Classical non-Abelian gauge theories 33 4.1 Somebasicsofgrouptheory .......................... 33 4.2 Orbitstructure ................................. 38 4.2.1 Orbits .................................. 38 4.2.2 (Non-)linear representations . 39 4.2.3 The little group . 40 4.3 Yang-Millstheory................................ 41 4.4 Yang-Millstheoryasgeometry. 45 4.5 Matter, representations, and coupling universality . ..... 47 5 Perturbative quantization of Yang-Mills theory 52 5.1 Quantization .................................. 52 i ii Contents 5.2 BRSTandasymptoticstates. .. .. 55 5.2.1 BRSTsymmetry ............................ 56 5.2.2 Constructingthephysicalstatespace . 59 5.2.2.1 Negative-normstates. 59 5.2.2.2 BRSTalgebra......................... 59 5.2.2.3 Statespace .......................... 61 5.2.2.4 Asymptoticspace....................... 62 5.3 Generalanti-fieldapproach. 65 5.3.1 Structureofgaugesymmetries. 65 5.3.1.1 Yang-Mills type . 65 5.3.1.2 Closure only on-shell . 66 5.3.1.3 Reducible gauge theories . 66 5.3.1.4 Reducibility on-shell . 67 5.3.2 BRSTconstructionrevisited . 67 5.3.3 Observables ............................... 70 5.3.4 Application as gauge-fixing to the path integral . 73 5.3.5 BeyondYang-Millstheory . 75 5.3.6 QuantumMasterEquation. 77 5.4 Ward-Takahashi and Slavnov-Taylor identities . ... 80 5.4.1 Ward-Takahashiidentities . 81 5.4.2 Slavnov-Taylor identities . 83 6 Perturbative treatment 86 6.1 Perturbationtheory............................... 86 6.1.1 Cross-sectionsanddecays . 86 6.1.2 Generalconstruction .......................... 88 6.2 Renormalization................................. 99 6.2.1 Cutoffregularization . .. .. 100 6.2.2 Renormalization prescription . 101 6.2.3 Counter-termstructure. 102 6.2.4 Renormalizationschemes. 105 6.2.5 Renormalizationgroup . 107 6.2.6 Gauge-symmetry-breakingregulators . 110 6.2.7 Dimensional regularization . 112 6.2.8 Lattice and infrared regularization . 114 6.2.9 Compositeoperators . .. .. 116 6.3 Examples of perturbative calculations . 117 Contents iii 6.3.1 Ghostself-energy ............................ 117 6.4 Running couplings, Landau poles, and asymptotic freedom . 122 7 Quantization beyond perturbation theory 125 7.1 The Gribov-Singer ambiguity . 126 7.2 Non-perturbativeBRSTsymmetry . 134 7.3 Gauge-invariance beyond perturbation theory . 138 7.3.1 Abelian gauge theories and the Dirac phase . 139 7.3.2 Composite states as physical degree of freedoms . 141 7.3.3 Gaugeinvarianceasredundancy. 143 7.3.4 Asymptoticstatespace. 143 8 Topology 145 8.1 Topological excitations . 145 8.1.1 Instantonsinquantummechanics . 145 8.1.1.1 Leadingorder......................... 145 8.1.1.2 Higher-ordercorrections . 148 8.1.1.3 Correctionsattwolooporder . 152 8.1.1.4 Instanton-Anti-Instanton interactions . 153 8.1.1.5 Fermions ........................... 153 8.1.2 Yang-Millstheory............................ 154 8.1.2.1 Topology ........................... 154 8.1.2.2 Tunneling . 155 8.1.2.3 θ vacua ............................ 161 8.1.2.4 Tunneling amplitude . 162 8.1.3 Lightfermions.............................. 162 8.2 Other topological excitations . 163 9 Anomalies 165 9.1 Introduction................................... 165 9.2 Globalanomalies ................................ 165 9.2.1 Classicallevel .............................. 166 9.2.2 One-loop violation . 168 9.3 Localanomalies................................. 171 9.3.1 Anomaliesasaquantumeffect. 172 9.3.2 Fullexpressionfortheanomaly . 173 9.3.3 Anomaly cancellation . 176 iv Contents 9.4 Relationtotopology .............................. 177 9.5 Wittenanomalies ................................ 179 9.6 AnomaliesintheBRSTframework . 180 Index 182 Chapter 1 Introduction Gauge theories play a fundamental role in all of modern fundamental physics, but are at the same time enigmatic and challenging. The aim of this lecture is to provide an introduction to their quantum version, in the form of quantum gauge theories. It turns out that a suitable language for these kind of theories is the path integral, which will therefore be introduced in chapter 2. This will be done in a first step for non-gauge theories, as this substantially simplifies it. Gauge theories will be then introduced only in chapter 3, starting with the extension of classical electrodynamics to quantum electro dynamics (QED). QED is a special case of a gauge theory, as it is related to an Abelian structure. The generalization to so-called non- Abelian gauge theories, or Yang-Mills theories, is necessary in particle physics, as they turn out to be a building block of the standard model. In fact, even classical general relativity can be considered to be a gauge theory of related type, and thus will any quantum version of it. Classical Yang-Mills theory will be introduced in chapter 4. Only in chapter 5 will Yang-Mills theory be quantized, as this requires much more techniques due to the non-Abelian nature. In fact, it will only be possible at this stage to quantize the theory at the level of perturbation theory. However, as both in practice and for the sake of concept formation perturbation theory is invaluable, this is anyhow a necessary step. This will be continued in chapter 6, where various perturbative processes will be discussed in detail. As at loop level also in gauge theories divergences arise, this will not be possible without an extensive discussion of this topic. After this, it is possible to delve into gauge theories beyond perturbation theory in chapter 7. This issue is indeed challenging, and not entirely solved until today. This will lead also to questions what are the physical degrees of freedom of gauge theories beyond perturbation theory, and will show that gauge symmetry leads to deeper ontological problems as well. Here QED will be again a role model, as it is possible to spell out both 1 2 the problem and solution in this case explicitly, although at the price of exponentially more complicated calculations. It will then be seen how this problem translates to non-Abelian gauge theories, and where the resolution becomes more involved. This lecture will not be able to cover actual calculations beyond perturbation theory, as the necessary methods are quiet involved, and thus deserve lectures of their own. These are available for lattice gauge theory and for functional methods, although this is not an exhaustive list of possible methods. Some other methods will be encountered in more specialized lectures. Finally, two generically important aspects, topology in chapter 8 and anomalies in chapter 9, will round out the discussion of gauge theories. However, gauge theories are a vast subject, and thus many issues can only be treated superficially. Especially, concrete realizations of gauge theories as theories of, e. g., the strong and weak forces require their own dedicated lecture. There is also a vast literature on gauge theories. In the preparation of this course, the following books and review articles have been used: Peskin et al. “An introduction to quantum field theory”, Perseus B¨ohm et al. “Gauge theories”, Teubner O’Raifertaigh, “Group structure of gauge theories”, Cambridge Weinberg, “The quantum theory of fields I+II”, Cambridge DeWitt, “The global approach to quantum field theory”, Oxford Haag, “Local quantum physics”, Springer Lavelle et al., “Constituent quarks from QCD”, Phys. Rept. 279 Maas, “Gauges bosons at zero and finite temperatures”, Phys. Rept. 524 Maas, “Brout-Englert-Higgs physics”, PPNP 106 Of course, these are my personal choices, and not necessarily the best ones to learn the topic from. Thus, I strongly recommend that any student should test various books and choose the one best suited for their own purposes. Chapter 2 Path integral While gauge theories can be formulated in canonical quantization this is at least cumber- some. As a consequence, it is used little in practice, and not every issue has even been treated in this approach. The path integral is a much more natural framework for gauge theories, which will become especially visible in chapter 5. Thus, it will be used exclusively in this lecture. In this chapter, the basic formulation of the path integral will be briefly reviewed, together with the necessary tools to treat both bosonic and fermionic particles. The application of perturbation theory and the derivation of Feynman rules will be discussed in chapter 6. 2.1 Heuristic introduction The path integral formulation is as axiomatic as is canonical quantization, it cannot be derived. However, it is possible to motivate it. This is done most easily by defining it as a limiting procedure, using a so-called lattice regularization. As the name implies, it works by replacing space-time with a finite lattice of discrete points. This technique will reappear again later