Multiple Mellin-Barnes integrals with polygamma functions

TL;DR

This paper extends geometric methods for evaluating Mellin-Barnes integrals to include those with polygamma functions, enabling systematic series solutions for more complex integrals in physics.

Contribution

It introduces new techniques and algorithms for computing Mellin-Barnes integrals with polygamma functions using conic hulls and triangulations, implemented in a Mathematica package.

Findings

Extended conic hull and triangulation methods to polygamma functions.

Developed two algorithms for straight and non-straight contours.

Implemented algorithms in MBConicHulls.wl package.

Abstract

Mellin-Barnes (MB) integrals appear in various branches of physics and mathematics and are, in particular, used as a standard tool for evaluating multi-loop, multi-scale Feynman integrals both analytically and numerically. Recent geometric approaches based on conic hulls and triangulations provide a systematic framework for computing multiple MB integrals in terms of multivariate series. These approaches have so far been limited to MB integrals whose integrands are ratios of products of Euler's gamma functions only. However, in Feynman integral calculus, MB integrals with polygamma functions naturally arise, for instance, after resolving singularities in the dimensional-regularisation parameter $\epsilon$ and expanding the MB integrand in powers of $\epsilon$, as done by the public codes MB.m and MBresolve.m. In this paper, we extend the conic hull and triangulation methods to the computation of MB integrals having polygamma functions in their integrand. We show that the arguments of polygamma functions can be treated in a similar way to the arguments of gamma functions when applying the conic hull and triangulation techniques to identify poles that would contribute to different series solutions. However, since the singularity structure of the polygamma function is different from that of the gamma function, we propose two different ways to compute MB integrals involving polygamma functions, depending on whether the MB integral has straight or non-straight contours. We have implemented these algorithms in an updated version of the Mathematica package MBConicHulls.wl, which can be found at https://github.com/SumitBanikGit/MBConicHulls/, and we illustrate their use with a set of examples from Feynman integral calculus.