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

Groote, S.  (Füüsika Instituut, Tartu Ülikool, W. Ostwaldi 1, Tartu, EE-50411, Estonia) ; Körner, J.G. (Institut für Physik der Johannes-Gutenberg-Universität, Staudinger Weg 7, Mainz, D-55099, Germany)

29 November 2018

Abstract: We integrate three-loop sunrise-type vacuum diagrams in D0=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 D0=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 Clausen polylogarithms. Using results for the nondegenerate two-loop sunrise diagram from the literature in D0=2 dimensions we obtain a Bessel function integral identity in terms of elliptic functions.

Published in: Nuclear Physics B 938 (2019) 416-425