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

We consider integral kernels for functions $f(\widehat{F})$ of a minimal second-order differential operator $\widehat{F}(\mathrm{\nabla })$ 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 $\widehat{F}(\mathrm{\nabla })$ is still contained in the standard HaMiDeW coefficients ${\widehat{a}}_k[F\mathrm{\mid }x,x^{\mathrm{\prime }}]$ (we call this property ``off-diagonal functoriality’'), while information about the function $f$ is encoded in some new scalar functions ${\mathbb{B}}_{\alpha }[f\mathrm{\mid }\sigma ]$ and ${\mathbb{W}}_{\alpha }[f\mathrm{\mid }\sigma ,m^2]$, which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form $\exp (-\tau {\widehat{F}}^{\nu })\mathrm{/}({\widehat{F}}^{\mu }+\lambda )$ 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.