# Consequences of anomalies

I'm trying to understand the phenomenological consequences of an anomalous global symmetry. In 't Hooft's "Symmetry breaking through Bell-Jackiw anomalies", he states in the abstract:

In models of fermions coupled to gauge fields certain current-conservation laws are violated by Bell-Jackiw anomalies. In perturbation theory the total charge corresponding to such currents seems to be still conserved,

and he goes to show that non-perturbative effects can lead to symmetry violating processes suppressed by the factor $e^{-8\pi^2/g^2}$. Let's assume that at interesting energies $g$ is small and these processes are suppressed beyond relevance.

Can anyone give more details about 't Hooft's claim about the non-violation of charges in perturbation theory? Does this mean that there are almost no consequences of a given symmetry being anomalous (neglecting the highly-suppressed non-perturbative processes)? Or are there other phenomena that would be forbidden which are now possible?

How does this relate to the ubiquitous calculation of the decay rate for the neutral pion, where a classically forbidden process is computed via the anomaly through a one-loop process? Are there other one loop processes like this which are now allowed?

Thanks for any help or literature recommendations!

• Have a look at this question and its answer. Although calculated by a one-loop diagram, the anomalous non-conservation is not a perturbative effect, since it does not depend on the perturbative parameter. – ACuriousMind Aug 2 '16 at 9:10
• Thanks for that. I can see the connection to instantons and non-trivial gauge configurations, and I'm happy to call it a non-perturbative effect. However, I'm mainly interested in the phenomenological consequences of the anomaly, and it does seem to lead to effects in conventional perturbation theory which happen at relatively fast rates, for example a previously forbidden decay for the neutral pion. Is this due to some quirk of the treatment of the pion, or is this a generic effect? In general should we expect the presence of an anomaly to lead to unsuppressed symmetry violating processes? – GEB24 Aug 2 '16 at 9:52