Multiple Mellin-Barnes integrals with polygamma functions

TL;DR

This paper extends geometric MB techniques (conic-hull and triangulation) to MB integrals that include polygamma functions, which arise after $\epsilon$-resolution in Feynman integral calculations. By treating polygamma arguments analogously to gamma-function arguments and using a reflection-based residue approach, the authors adapt residue extraction to the polygamma singularity structure, with separate handling for straight and non-straight contours. The methods are implemented in an updated MBConicHulls.wl package, and demonstrated on two-fold MB integrals from Higgs production calculations, yielding explicit series and closed-form expressions in terms of zeta-values (and Euler’s constant) that agree with known results and numerical MB approaches. This work enables a largely automated analytic evaluation of MB integrals containing polygamma functions, integrating epsilon-resolution, pole analysis, and series summation into a unified workflow for multi-loop, multi-scale Feynman integrals.

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 $ε$ and expanding the MB integrand in powers of $ε$, 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.