Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities
S. Groote, J. G. Körner
TL;DR
This work develops a configuration-space framework to compute two- and three-loop sunrise-type vacuum diagrams with multiple masses, and uses a Gaussian-regulator splitting to isolate divergent parts in dimensional regularization ($D=D_0-2epsilon$, $D_0=4$). It defines truncated Bessel integrals that render the finite parts amenable to numerical evaluation and cross-checks against known momentum-space results, yielding new integral identities. For $D_0=4$ the authors derive seven truncated Bessel identities and verify them numerically, while for $D_0=2$ they obtain a Bessel-elliptic identity linking moments of Bessel functions to Clausen and elliptic polylogarithms; these results are validated against existing p-space calculations. The method provides robust cross-checks of high-loop vacuum diagrams and uncovers a wealth of potentially reusable Bessel-function identities with potential extension to higher loops and more intricate mass configurations.
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.