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

Understand

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.