Banana diagrams as functions of geodesic distance

TL;DR

This paper generalizes the analysis of banana diagrams in curved harmonic space-times, showing that their Green functions depend on geodesic distance and can be expressed through determinants, with implications for their mathematical treatment.

Contribution

It extends the study of banana diagrams to curved space-times, demonstrating the preservation of key mathematical structures and the generalization of Feynman parameter representations.

Findings

Green functions depend on geodesic distance in harmonic spaces

Coordinate differential equations can be expressed as determinants

Feynman parameter representation can be generalized for simple harmonic spaces

Abstract

We extend the study of banana diagrams in coordinate representation to the case of curved space-times. If the space is harmonic, the Green functions continue to depend on a single variable -- the geodesic distance. But now this dependence can be somewhat non-trivial. We demonstrate that, like in the flat case, the coordinate differential equations for powers of Green functions can still be expressed as determinants of certain operators. Therefore, not-surprisingly, the coordinate equations remain straightforward -- while their reformulation in terms of momentum integrals and Picard-Fuchs equations can seem problematic. However we show that the Feynman parameter representation can also be generalized, at least for banana diagrams in simple harmonic spaces, so that the Picard-Fuchs equations retain their Euclidean form with just a minor modification. A separate story is the transfer to the case when the Green function essentially depends on several rather than a single argument. In this case, we provide just one example, that the equations are still there, but conceptual issues in the more general case will be discussed elsewhere.