Winter 2007: Anomalies

As far as textbooks go, many of this quarter's topics are discussed in Weinberg Vol. 2, Ch. 22., and to a lesser extent in Zee, Ch. IV.7 and Peskin & Schroeder Ch. 19.

Abstracts

Paul Chesler
A symmetry transformation in a quantum field theory is said to be anomalous if it is a symmetry of the Lagrangian but not a symmetry of the quantum theory. These anomalies play an important role in defining a mathematically consistent field theory and have many observable consequences. After motivating that anomalies are a natural occurrence in a QFT (and not necessarily a pathology as the name might suggest), I will discuss several examples of well known anomalies and then discuss some of their consequences. In particular I will focus on one aspect of the mathematical consistency of gauge theories (i.e. no anomalous currents coupled to gauge fields) as well as some experimental consequences such as the meson spectrum of QCD and Pi0 decay. Finally I will give a brief outline of the next few talks for the quarter.

Amy Nicholson

Kristan Jensen
When a classical symmetry is anomalous in its quantum cousin, the measure of the path integral must transform nontrivially under the anomalous symmetry. We will make this fact explicit by finding the action of the axial transformation on the measure of SU(n) Yang-Mills coupled to massive fermions in arbitrary dimension. From here, we'll discuss the conditional convergence of the divergence of the axial current. Its integral will naturally lead us into an analysis of the zero modes of D-slash and thus the Atiyah-Singer Index Theorem. We will conclude by connecting these results to the general analysis of Dirac spectral flow.

Rob Schabinger
I will first discuss why the leading order contribution to the chiral anomaly in massless QED (the triangle diagram) is, in fact, the full answer and show how the dimensional regularization scheme leads to this answer. Using the result of this calculation, I will then derive an approximate formula for the partial width of pi_0 to two photons in terms of physical observables.
References: Peskin and Schroeder pgs. 661-664,668-670,673-676; t'Hooft and Veltman, "Regularization and Renormalization of Gauge Fields," Nucl. Phys. B44 (1972) 189-213; Adler and Bardeen, "Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation," Phys. Rev. 182 (1969) 1517-1536

Andrew Lytle
First I will discuss chiral gauge theories, and how gauge anomalies place constraints on which theories are sensible. Then I will talk about a way to simulate such theories, using a four dimensional slice of a vector-like theory in five dimensions. We will see that in these models the gauge anomalies are given by charge flow from the extra dimension.
My main reference was Kaplan's paper "A Method For Simulating Chiral Fermions on the Lattice" hep-lat/9206013

Ethan Thompson
In the large-N limit, the solution to the U(1) problem takes on a more clear physical picture. The axial anomaly is suppressed by 1/N and the eta' becomes massless. This will be made evident via consideration of the N dependence of the theta angle. The necessary machinery of the 1/N expansion will be developed from the ground up.
References: E. Witten, Current Algebra Theorems for the U(1) "Goldstone Boson", Nucl. Phys. B156, (1979), 269-283

Kristan Jensen
Geometry, Bundles, and Global Anomalies
This journal club will be a departure from the normal routine: the majority of the talk will be a survey of basic differential geometry. We will start with the tangent and cotangent structures on smooth manifolds and use them to discuss bundles in some generality. From here, we naturally extend to tensors and forms, from which we can define differentiation (exterior, covariant). This will allow us to discuss curvature (Riemannian, form) and the connection to gauge theory. With this machinery, we can discuss spinors and the construction of spin bundles. The nonexistence of spin bundles is intimately related to the existence of global anomalies, which arise as a consequence of global topological properties of spacetime and gauge groups. Notes will be provided.
Refs: For the differential geometry, pick your favorite text. I'll be relying on Lee's 'Intro. to Smooth Manifolds' up until the Riemannian section. There I'll switch to a mixture of Frenkel's 'Geometry of Physics' and Nakahara. For the final section on global anomalies, consider: Witten, Phys. Lett. 117B (1982), 324; Friedan and Windey, Nucl.Phys.B 235 (1984), 395.

Steve Paik
In 2n dimensional Yang-Mills theories with chiral fermions it is possible that classical gauge invariance no longer holds under quantization. I will describe how to characterize such an anomalous gauge invariance and explain a consistency condition for the anomaly. Simple differential equations in 2n+2 dimensions are used to obtain an expression for the anomaly, up to a normalization factor. This suggests that the non-abelian anomaly should somehow emerge from the index theorem in 2n+2 dimensions. The goal will be to make this connection precise and give a topological understanding for its existence.
My main ref. was Alvarez-Gaume and Ginsparg, Nucl.Phys.B243:449,1984.

Andy O'Bannon
My talk is about anomalous diffeomorphisms (coordinate transformations) in theories of matter fields coupled to gravity. I will start by revealing that the familiar axial current has two anomalies: not only the one we've seen, from background gauge fields, but also one from background gravitational fields! From there we will learn how to couple fermions to gravity and how to write general relativity as a gauge theory in analogy with Yang-Mills theory. We can then steal Steve's results from last week to write the anomaly down immediately. Along the way we will learn some general facts about gravitational anomalies: that they can only occur in D = 4k + 2 dimensions and that spin 1/2, spin 3/2 and self-dual form fields can contribute. We will see that, indeed, in string theory, anomalies from fields of different spin must cancel one another for the anomaly to vanish. This "miraculous" cancellation was part of the so-called "First Superstring Revolution" in 1984.
References: Steve's notes from last week; Green, Schwarz, Witten Volume 2, sections 10.2, 12.1 and 13.3 - 13.5; Polchinski Volume 2, section 12.2; Alvarez-Gaume and Witten, Nucl.Phys.B234:269,1984; Alvarez-Gaume and Ginsparg, Annals Phys.161:423,1985

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Organizers: Andy O'Bannon, Paul Chesler, Andrew Lytle, Steve Paik, Ethan Thompson
Last modified: 17 Mar 2007