Investigation of Pontryagin trace anomaly using Pauli-Villars regularization

In this paper, we investigate the Pontryagin trace anomaly for chiral fermions in a general curved background using Pauli-Villars regularization. We use both Feynman diagram method and Fujikawa's method to calculate the parity-odd contribution (Pontryagin term). Our result indicates that the trace anomaly of energy-momentum tensor for chiral fermions has Pontryagin term $P=\frac{i}{1536\pi^2}\epsilon_{\nu\sigma\kappa\lambda}R^{\nu\sigma}_{\ \ \rho\mu}R^{\rho\mu\kappa\lambda}$ which agrees with the work of Bonora et al \cite{Bonora:2014qla}.