# When is an anomaly one-loop exact?

There are many examples of quantum anomalies that are one-loop exact, and many examples of anomalies that have contributions to all orders in perturbation theory. I haven't been able to identify a pattern though:

• Is there any way to tell, from first principles, whether a given (potential) anomaly is $$n$$-loop exact (for a certain finite $$n\in\mathbb N$$)? Or does this require a case-by-case analysis?

• Is there any example of an anomaly that is $$n$$-loop exact, for some $$n>1$$?

• Have you seen the possible duplicate physics.stackexchange.com/q/27002/50583, where the answer is basically given by equating 1-loop exactness to being non-perturbative? – ACuriousMind Oct 23 '18 at 16:26
• @ACuriousMind Yes, it is a great answer, but it seems to be rather specific to that particular system. For more general problems, I presume dimensional analysis does not fix the anomaly up to an unknown function $f(g)$, so the argument breaks down. Also, I'm pretty sure there are examples of one-loop exact anomalies that do not correspond to any topological invariant (although I cannot come up with an example right now...) – AccidentalFourierTransform Oct 23 '18 at 16:32

Before answering the question, let me remark that number of loops required to obtain the anomaly exactly isn't an invariant quantity; the anomaly is physical, but the number of loops is not.

In the case of chiral anomaly, we need to evaluate a loop diagram because we are using Grassmann fields as coordinates the fermions infinite dimensional configuration space. It is very difficult to do quantum field theory otherwise, but we must remember that the Grassmann fields are just coordinates, thus of no physical significance. The chiral anomaly can be obtained at tree level from the classical Lagrangian of a Weyl particle, i.e., classically, please see the following work by Stone and Dwivedi $$^1$$. This work is based on the seminal work by: Stephanov and Yin (Chiral kinetic theory).

The essence of their construction can be formulated in quite ingenious simple phase space arguments as given by Kharzeev based on a deep observation by Gribov.

A deep examination of Stephanov and Yin's (or Stone and Dwivedi's) argument shows that the only additional information needed beyond the classical Lagrangian of a Weyl particle is that it obeys the Fermi-Dirac statistics (Here we don't have the Grassmann variables to take care of this part) and that we are working in the infinite volume limit.

Given the above, the exceptional phenomena about the anomaly is that it can be computed exactly (regardless of the number of loops). Certainly, there are topological explanation of this exactness in the case of the chiral anomaly, but the topological arguments do not cover all the cases where quantities can be computed exactly in perturbation theory.

The deep reason is supersymmetry.

In mathematics, this phenomenon is known by equivariant localization, based on the seminal work by Duistermaat and Heckman, please see the following physically oriented review by Szabo. (The rigorous mathematical results exist mainly for the finite dimensional cases; the path integral applications were introduced in the physical literature; they are less rigorous but led to marvelous results especially by Witten).

Basically, localization means that instead of performing the integral over an entire phase space, the result can be obtained by summing the contributions from a smaller subset which can be discrete, or in the case of path integrals can be a finite dimensional manifold (instead of the infinite dimensional path space). Please see the following presentation by Hosomichi. The existence of supersymmetry is responsible for solvability of the $$N=2$$ supersymmetric gauge theories in $$4$$ dimensions.

Now, how the chiral anomaly is related to the above: According to Schwinger's proper time method, processes described by excitations of quantum fields can be expressed as quantum mechanical (i.e., in $$0+1$$ dimensions) path integrals, please see the following review by Bastianelli and van Nieuwehuizen. This is true for example to the process described by the triangle diagram. The quantum mechanical action describes a spinning particle in $$0+1$$ dimensions which is supersymmetric, therefore can be solved by the localization techniques. This is just what Friedan and Windey did in their seminal work.

$$^1$$ Although Stone and Dwivedi remark that the alignment of the spin along the angular momentum of a massless particle is a quantum phenomenon; it can be obtained entirely in classical mechanics, please see Duval and Horvathy.

• A tour de force answer, bravo. What is your take on the "new SU(2) anomaly" by Juven Wang, Xiao-Gang Wen, Edward Witten , here: arxiv.org/abs/1810.00844 – MadMax Oct 29 '18 at 18:07
• @MadMax This a good point, since I did not mention global anomalies. Global gauge anomalies can be obtained through the embedding of the gauge group in a larger gauge group. The larger gauge group theory has an ordinary (infinitesimal) anomaly, which can be computed using the above methods. I think that no one has implemented the chiral kinetic theory techniques yet for the Witten's $SU(2)$ anomaly and it would be a nice project to do. – David Bar Moshe Nov 1 '18 at 14:44
• cont. As for the new article, from a superficial inspection, the new anomaly looks like a type of global gauge-gravity anomaly, but I don't have enough knowledge to comment beyond this. – David Bar Moshe Nov 1 '18 at 14:45