Banana diagrams as functions of geodesic distance

TL;DR

The paper analyzes n-banana Feynman diagrams in curved, harmonic spaces by recasting configuration-space integrals into a one-variable framework where the Green function depends on geodesic distance $\sigma$. It introduces the $\Lambda$ formalism to convert the Klein–Gordon equation into a simple operator equation $\left(\Lambda^2 - \lambda^2\right) G(\sigma)$, and shows that the banana diagram differential equation reduces to a determinant of a tridiagonal operator matrix, $\det(\cdots) G^n = 0$, independent of space type. For simple harmonic spaces, the Feynman parameter representation and the resulting PF equations retain their Euclidean form, with mass terms appropriately shifted by curvature, while the two-variable Green function case remains more intricate, lacking a universal determinant structure. The work connects heat-kernel methods to the algebraic structure of Feynman diagrams in curved backgrounds, offering a unified position-space approach and outlining paths to generalizations. This provides a framework for analyzing loop corrections in curved spacetimes within a configuration-space, geodesic-distance-centric formalism that aligns with Euclidean intuition in simple cases and highlights where new techniques are needed.

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.