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

We consider integral kernels for functions f ( F ^ ) of a minimal second-order differential operator F ^ ( ∇ ) 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 F ^ ( ∇ ) is still contained in the standard heat kernel coefficients 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 B α [ f | σ ] and W α [ f | σ , m 2 ] , which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form exp ( − τ F ^ ν ) / ( F ^ μ + λ ) 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.