Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities

TL;DR

This paper computes three-loop sunrise diagrams in four dimensions using configuration space, finds numerical agreement with momentum space results, and derives new integral identities involving Bessel functions and elliptic functions.

Contribution

It introduces a configuration space method for three-loop diagrams, derives new Bessel integral identities, and connects these to elliptic functions, advancing computational techniques in quantum field theory.

Findings

Numerical agreement between configuration and momentum space calculations.

New integral identities involving Bessel functions and elliptic functions.

Expressed moments of Bessel products in terms of polylogarithms.

Abstract

We integrate three-loop sunrise-type vacuum diagrams in $D_0=4$ dimensions with four different masses using configuration space techniques. The finite parts of our results are in numerical agreement with corresponding three-loop calculations in momentum space. Using some of the closed form results of the momentum space calculation we arrive at new integral identities involving truncated integrals of products of Bessel functions. For the non-degenerate finite two-loop sunrise-type vacuum diagram in $D_0=2$ dimensions we make use of the known closed form $p$-space result to express the moment of a product of three Bessel functions in terms of a sum of Claussen polylogarithms. Using results for the nondegenerate two-loop sunrise diagram from the literature in $D_0=2$ dimensions we obtain a Bessel function integral identity in terms of elliptic functions.