Cancellation of quantum mechanical higher loop contributions to the gravitational chiral anomaly

TL;DR

This paper demonstrates that higher-loop quantum corrections do not contribute to the gravitational chiral anomaly up to two loops, using rigorous Feynman rules and path integral methods.

Contribution

It provides an explicit, diagrammatic proof that higher-loop contributions cancel out in the gravitational chiral anomaly calculation.

Findings

Higher-loop contributions to the anomaly are absent up to two loops.

Develops diagrammatic notation for efficient evaluation of quantum graphs.

Establishes $eta$-independence of the trace up to two loops.

Abstract

We give an explicit demonstration, using the rigorous Feynman rules developed in~$\0^{1}$, that the regularized trace $\tr \gamma_5 e^{-\beta \Dslash^2}$ for the gravitational chiral anomaly expressed as an appropriate quantum mechanical path integral is $\beta$-independent up to two-loop level. Identities and diagrammatic notations are developed to facilitate rapid evaluation of graphs given by these rules.