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

A. O. Barvinsky1,2,*, A. E. Kalugin1,†, and W. Wachowski1,‡

  • *Contact author: barvin@td.lpi.ru
  • †Contact author: kalugin.ae@phystech.edu
  • ‡Contact author: vladvakh@gmail.com
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Phys. Rev. D 113, 045005 – Published 4 February, 2026

DOI: https://doi.org/10.1103/l112-5cz3

Abstract

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 Bα[f|σ] and Wα[f|σ,m2], which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form exp(−τF^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.

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Phys. Rev. D 113, 045005– Published 4 February, 2026
  • Received 4 December 2025
  • Accepted 20 January 2026
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