Axial $U(1)$ anomaly in a gravitational field via the gradient flow

TL;DR

This paper explores using a universal gradient flow-based formula to compute the axial U(1) anomaly in a gravitational field, highlighting its strengths in reproducing the anomaly structure and limitations regarding Ward--Takahashi relations.

Contribution

It demonstrates the applicability of a regularization-independent formula for the energy--momentum tensor to axial U(1) anomaly calculations in gravitational backgrounds, emphasizing on-shell correlation functions.

Findings

Reproduces the correct non-local structure of the anomaly

Requires local counterterms for Ward--Takahashi relations

Applicable mainly to on-shell correlation functions

Abstract

A regularization-independent universal formula for the energy--momentum tensor in gauge theory in the flat spacetime can be written down by employing the so-called Yang--Mills gradient flow. We examine a possible use of the formula in the calculation of the axial $U(1)$ anomaly in a gravitational field, the anomaly first obtained by Toshiei Kimura [Prog.\ Theor.\ Phys.\ {\bf 42}, 1191 (1969)]. As a general argument indicates, the formula reproduces the correct non-local structure of the (axial $U(1)$ current)--(energy--momentum tensor)--(energy--momentum tensor) triangle diagram in a way that is consistent with the axial $U(1)$ anomaly. On the other hand, the formula does not automatically reproduce the general coordinate (or translation) Ward--Takahashi relation, requiring corrections by local counterterms. This analysis thus illustrates the fact that the universal formula as it stands can be used only in on-shell correlation functions, in which the energy--momentum tensor does not coincide with other composite operators in coordinate space.