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

We give an explicit demonstration, using the rigorous Feynman rules developed in , that the regularized trace Trγ5e −βD/ for the gravitational chiral anomaly expressed as an appropriate quantum mechanical path integral is β-independent up to two-loop level. Identities and diagrammatic notations are developed to facilitate rapid evaluation of graphs given by these rules. It is an old observation of Alvarez-Gaumé and Witten 2 that anomalies of quantum field theories, expressed in the Fujikawa 3 approach as the regulated trace of a jacobian (J ) A(nomaly) = lim β→0 TrJ e−β h̄ R̂ (1) may be represented by quantum mechanical path integrals. However to fully understand such path integrals one must carefully address issues such as; the precise definition of the measure, which action corresponds to the particular operator ordering of R̂ and the correct Feynman rules for the perturbative expansion of such path integrals. In their original exposition, Alvarez-Gaumé andWitten consider chiral anomalies for which, due to their topological nature, the expression (1) is β-independent and calculable without such subleties via a semiclassical expansion. Recently, de Boer et.al. 1 have shown explicity how to define the measure, action and Feynman rules for quantum mechanical path integrals for both bosons and fermions in curved space. The exact rules they obtain, although novel, follow directly from a rigorous treatment of the measure constructed from insertions of complete sets of coherent states. In this note we begin with their results and return to the gravitational chiral anomaly to verify through two loop order that their Feynman rules give the correct β-independent result for (1). The gravitational chiral anomaly is given by the index of the Dirac operator A = Trγ5e− β h̄ D/D/ = n+ − n− (2) where n± are the number of positive/negative parity zero modes of D/ = ea γ(∂μ + 1 4 ωμ γaγb). A may be represented by a quantum mechanical path integral A = ∫