Functorial properties of Schwinger-DeWitt expansion and Mellin-Barnes representation

Abstract

We consider integral kernels for functions $f(\hat F)$ of a minimal second-order differential operator $\hat F(\nabla)$ on a curved spacetime. We show that they can be expanded in a functional series, analogous to the DeWitt expansion for the heat kernel, by integrating the latter term-by-term. This procedure leads to a separation of two types of data: all information about the bundle geometry and the operator $\hat F(\nabla)$ is still contained in the standard HaMiDeW coefficients $\hat a_k[F | x,x']$ (we call this property ``off-diagonal functoriality''), while information about the function $f$ is encoded in some new scalar functions $\mathbb{B}_\alpha[f | \sigma]$ and $\mathbb{W}_\alpha[f | \sigma, m^2]$, which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form $\exp(-\tau\hat F^\nu)/(\hat F^\mu + \lambda)$ as multiple Mellin--Barnes integrals. The article also discusses subtle issues such as the validity of the term-by-term integration, the regularization of IR divergent integrals, and the physical interpretation of the resulting expansions.