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Anomalies in Quantum Field Theory

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Anomalies in Quantum Field Theory Anomalies in Quantum Field Theory Marco Serone SISSA, via Bonomea 265, I-34136 Trieste, Italy Lectures given at the “37th Heidelberg Physics Graduate Days”, 10-14 October 2016 1 Contents 0.1 Introduction.................................... 3 1 Lecture I 5 1.1 Basics of Differential Geometry . 5 1.2 VierbeinsandSpinors ............................. 8 1.3 CharacteristicClasses . 12 2 Lecture II 18 2.1 SupersymmetricQuantumMechanics. .... 18 2.2 AddingGaugeFields............................... 22 2.3 SymmetriesinQFT ............................... 24 2.3.1 WTIdentitiesinQED. 28 3 Lecture III 30 3.1 TheChiralAnomaly ............................... 30 3.2 Consistency Conditions for Gauge and Gravitational Anomalies . 36 4 Lecture IV 39 4.1 The Stora–Zumino Descent Relations . ..... 39 4.2 Path Integral for Gauge and Gravitational Anomalies . ......... 42 4.3 ExplicitformofGaugeAnomaly . 46 5 Lecture V 48 5.1 Chiral and Gauge Anomalies from One-Loop Graphs . ...... 48 5.2 A Relevant Example: Cancellation of Gauge Anomalies in theSM ..... 51 5.3 ’t Hooft Anomaly Matching and the Wess-Zumino-Witten Term ...... 52 5.4 Gravitational Anomalies for Spin 3/2 and Self-Dual Tensors......... 54 2 0.1 Introduction There are different ways of regularizing a Quantum Field Theory (QFT). The best choice of regulator is the one which keeps the maximum number of symmetries of the classical action unbroken. Cut-off regularization, for instance, breaks gauge invariance and that’s why we prefer to work in the somewhat more exotic Dimensional Regularization (DR), where instead gauge invariance is manifestly unbroken. It might happen, however, that there exists no regulator that preserves a given classical symmetry. When this happens, we say that the symmetry is anomalous, namely the quantum theory necessarily breaks it, independently of the choice of regulator. Roughly speaking, anomalies can affect global or local symmetries. The latter case is particularly important, because local symmetries are needed to decouple unphysical fields. Indeed, anomalies in linearly realized local gauge symmetries lead to inconsistent theories. Theories with anomalous global symmetries are instead consistent, yet the effect of the anomaly can have important effects on the theories. Historically, the first anomaly, discovered by Adler, Bell and Jackiw [1], was associated to the non-conservation of the axial current in QCD. Among other things, the axial anomaly resolved a puzzle related to the π0 2γ decay rate, predicted by effective Lagrangian considerations to be about → three orders of magnitude smaller than the observed one. In these lectures we will study in some detail a particularly relevant class of anomalies, those associated to chiral currents (so called chiral anomaly) and their related anoma- lies in local symmetries. Particular emphasis will be given to some mathematical aspects and to the close connection between anomalies and the so called index theorems. The computation of anomalies will be mapped to the evaluation of the partition function of a certain (supersymmetric) quantum mechanical model. Although this might sound un- usual, if compared to the more standard treatments using one-loop Feynman diagrams, and requires a bit of background material to be attached, the ending result will be very re- warding. This computation will allow us to get the anomalies associated to gauge currents (so called gauge anomalies) and stress-energy tensor (so called gravitational anomalies) in any number of space-time dimensions! Moreover, it is the best way to reveal the above mentioned connection with index theorems. Since the notion of chirality is restricted to spaces with an even number of space-time dimensions, and the connection with index theorems is best seen for euclidean spaces, we will consider in the following QFT on even dimensional euclidean spaces. The plan of the lectures is as follows. Lecture I is devoted to review minimum basic notions of differential geometry. In lecture II we will first construct step by step the su- 3 persymmetric quantum mechanical model of interest, related to anomalies. Afterwards, we will review how symmetries are implemented in QFT by means of identities between correlation functions. Equipped with these notions, we finally attack anomalies in lecture 3, where the contribution to the chiral anomaly of a spinor coupled to gravity and gauge fields is computed in any even number 2n of dimensions. In lecture 4 we turn to gauge and gravitational anomalies and explain how in 2n dimensions these can be consistently extracted from the form of the chiral anomaly in n+2 dimensions. Lecture 5 is devoted to some more physical applications of anomalies: we will discuss the ’t Hooft anomaly match- ing condition, anomalies in QCD and their matching to the low energy pion Lagrangian, cancellation of anomalies in the Standard Model. We will also briefly mention to gravita- tional anomalies induced by other fields and how these cancel in a certain 10-dimensional QFT. These lecture notes do assume that the reader has a basic knowledge of QFT. Quan- tizations of spin 0, spin 1/2 and spin 1 fields are assumed, as well as basic notions of the path integral formulation of QFT (including Berezin integration for fermions), Feynman rules, basic knowledge of renormalization and the notion of functional generators of Green functions. Although some basics of differential geometry will be given in Lecture I, some familiarity with fundamental notions such as metric, Levi-Civita connection, curvature tensors and general relativity will be assumed. Some exercises are given during the lectures (among rows and in bold face in the text). They are simple, yet useful, and the reader should be able to solve them without any problem. I have essentially followed the review paper [2], which in turn is mostly based on the seminal paper [3] by Alvarez-Gaum`eand Witten. Some other parts are taken from the QFT course given by the author at SISSA [4]. Basic, far from being exhaustive, references to some of the main original papers are given in the text. 4 Chapter 1 Lecture I 1.1 Basics of Differential Geometry We remind here some very basic knowledge of differential geometry that will be useful in understanding anomalies in QFT. We warn the reader that in no way this should be considered a comprehensive review. We will mention the basic notions needed for our purposes, without giving precise mathematical definitions, presented in a “physical” way with no details. There are several excellent introductory books in differential geometry. See e.g. ref.[5] for a perspective close to physics. Given a manifold of dimension d, a tensor of type (p,q) is given by M µ1...µp ∂ ∂ ν1 νq T = T ν ...νq . dx ...dx , (1.1.1) 1 ∂xµ1 ∂xµp where µi,νi = 1,...,d. A manifold is called Riemannian if it admits a positive definite µ ν symmetric (0,2) tensor. We call such tensor the metric g = gµν dx dx . On a general non-trivial manifold, a vector acting on a (p,q) tensor does not give rise to a well-defined tensor, e.g. if Vν are the components of a (0, 1) tensor, ∂µVν are not the components of a (0, 2) tensor. This problem is solved by introducing the notion of connection and covariant derivative. If has no torsion (as will be assumed from now on), then ∂ V V , M µ ν → ∇µ ν where 1 V = ∂ V Γρ V , Γρ = gρσ(∂ g + ∂ g ∂ g ) . (1.1.2) ∇µ ν µ ν − µν ρ µν 2 µ νσ ν µσ − σ µν ρ The (not tensor) field Γµν is called the Levi-Civita connection. In terms of it, we can construct the fundamental curvature (or Riemann) tensor Rµ = ∂ Γµ ∂ Γµ +Γα Γµ Γα Γµ . (1.1.3) νρσ ρ νσ − σ νρ σν αρ − ρν ασ 5 Other relevant tensors constructed from the Riemann tensor are the Ricci tensor Rνσ = µ νσ R νµσ and the scalar curvature R = g Rνσ. Particular interesting for our purposes will be the completely antisymmetric tensors of type (0,p), also denoted differential forms. A differential form ωp of type (0,p) is commonly denoted a p-form and is represented as 1 ω = ω dxµ1 . dxµp . (1.1.4) p p! µ1...µp ∧ ∧ The wedge product in eq.(1.1.4) represents the completely antisymmetric product of the ∧ basic 1-forms dxµi : dxµ1 . dxµp = sign (P )dxµP (1) ...dxµP (p) , (1.1.5) ∧ ∧ PX∈Sp where P denotes a permutation of the indices and Sp is the group of permutations of p objects. For each point in a manifold of dimension m, the dimension of the vector space spanned by ωp equals d d! = . (1.1.6) p (d p)!p! ! − The total vector space spanned by all p-forms, p = 0,...,d, equals d d = 2d . (1.1.7) p=0 p ! X Given a p-form ω and a q-form χ , their wedge product ω χ is a (p + q) form. From p q p ∧ q eq.(1.1.5) one has pq ωp χq = ( ) χq ωp ∧ − ∧ (1.1.8) (ω χ ) η = ω (χ η ) . p ∧ q ∧ r p ∧ q ∧ r One has clearly ω χ =0 if p + q > d and ω ω =0 if p is odd. Given a p-form ω , p ∧ q p ∧ p p we define the exterior derivative dp as the (p + 1)-form given by 1 d ω = ∂ ω dxµp+1 dxµ1 . dxµp . (1.1.9) p p p! µp+1 µ1...µp ∧ ∧ ∧ Notice that in eq.(1.1.9) we have the ordinary, rather than the covariant derivative, because upon antisymmetrization their action is the same: ω = ∂ ω Γα ω + . = ∂ ω , (1.1.10) ∇[µp+1 µ1...µp] [µp+1 µ1...µp] − [µp+1µ1 α|µ2...µp] [µp+1 µ1...µp] the Levi-Civita connection being symmetric in its two lower indices.
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