Trace Anomaly of Weyl Fermions via the Path Integral

TL;DR

This paper calculates the anomalies of a Weyl fermion in a gravitational background using path integral methods, clarifying the trace anomaly and confirming the absence of certain parity-odd curvature terms, thus resolving recent controversies.

Contribution

It provides a detailed path integral computation of Weyl fermion anomalies, establishing the vanishing of the Pontryagin density and clarifying symmetry properties in various dimensions.

Findings

Trace anomaly is half that of a Dirac fermion.

The parity-odd curvature term $R ilde R$ vanishes.

Regularisation does not induce spurious anomalies.

Abstract

We compute the trace, diffeomorphism and Lorentz anomalies of a free Weyl fermion in a gravitational background field by path integral methods. This is achieved by regularising the variation of the determinant of the Weyl operator building on earlier work by Leutwyler. The trace anomaly is found to be one half of the one of a Dirac fermion. Most importantly we establish that the potential parity-odd curvature term $R \tilde R$, corresponding to the Pontryagin density, vanishes. This is to the contrary of some recent findings in the literature which gave rise to a controversy. We verify, that the regularisation does not lead to (spurious) anomalies in the Lorentz and diffeomorphism symmetries. We argue that in $d = 2\;(\textrm{mod } 4)$ $P$- and $CP$-odd terms cannot appear and that for $d = 4\;(\textrm{mod } 4)$ they are absent at least at leading order.