Adler-Bardeen theorem and cancellation of gauge anomalies to all orders in nonrenormalizable theories

TL;DR

This paper proves that gauge anomalies in a broad class of theories, including nonrenormalizable ones, cancel to all orders if they vanish at one loop, using advanced regularization techniques.

Contribution

It extends the Adler-Bardeen theorem to nonrenormalizable gauge theories with a novel proof combining chiral dimensional and higher-derivative regularizations.

Findings

Anomaly cancellation at one loop implies all-order cancellation within truncations.

Super-renormalizability when higher-derivative terms are beyond the truncation.

Standard Model with quantum gravity is anomaly-free to all orders.

Abstract

We prove the Adler-Bardeen theorem in a large class of general gauge theories, including nonrenormalizable ones. We assume that the gauge symmetries are general covariance, local Lorentz symmetry and Abelian and non-Abelian Yang-Mills symmetries, and that the local functionals of vanishing ghost number satisfy a variant of the Kluberg-Stern--Zuber conjecture. We show that if the gauge anomalies are trivial at one loop, for every truncation of the theory there exists a subtraction scheme where they manifestly vanish to all orders, within the truncation. Outside the truncation the cancellation of gauge anomalies can be enforced by fine-tuning local counterterms. The framework of the proof is worked out by combining a recently formulated chiral dimensional regularization with a gauge invariant higher-derivative regularization. If the higher-derivative regularizing terms are placed well beyond the truncation, and the energy scale $\Lambda$ associated with them is kept fixed, the theory is super-renormalizable and has the property that, once the gauge anomalies are cancelled at one loop, they manifestly vanish from two loops onwards by simple power counting. When the $\Lambda$ divergences are subtracted away and $\Lambda$ is sent to infinity, the anomaly cancellation survives in a manifest form within the truncation and in a nonmanifest form outside. The standard model coupled to quantum gravity satisfies all the assumptions, so it is free of gauge anomalies to all orders.