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.
Highlights
Almost half a century ago, just nine months after the appearance of two seminal papers on the axial U (1) anomaly in an electromagnetic field [1, 2], Kimura noticed in a lesser-known but remarkable paper [3] that a similar anomalous non-conservation of the axial vector current occurs in a gravitational field
We examine a possible use of the universal formula in the calculation of the axial U (1) anomaly (1.1); we will obtain Eq (1.1) by expansion around the flat spacetime
We have examined a possible use of the universal formula for the energy– momentum tensor in gauge theory in the flat spacetime through the Yang–Mills gradient flow [17, 18]
Summary
Almost half a century ago, just nine months after the appearance of two seminal papers on the axial U (1) anomaly in an electromagnetic field [1, 2], Kimura noticed in a lesser-known but remarkable paper [3] that a similar anomalous non-conservation of the axial vector current occurs in a gravitational field. As the general argument implies, we have Eq (1.1) This shows that the universal 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. This analysis illustrates 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, because it does not automatically reproduce the translation WT relation when operators coincide. How to remedy this point in (a generalization of) the universal formula is a forthcoming challenge.
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