Multiple Mellin-Barnes integrals and triangulations of point configurations

TL;DR

This paper introduces a new triangulation-based method for analytically evaluating complex Mellin-Barnes integrals, significantly improving computational efficiency and enabling calculations of high-fold integrals in quantum field theory.

Contribution

The authors develop a novel triangulation technique for Mellin-Barnes integrals, enhancing speed and capability over previous conic hulls methods, and implement it in a Mathematica package.

Findings

Faster computation of high-fold Mellin-Barnes integrals

Successful evaluation of a 15-point Feynman integral with 104 folds

New results for conformal hexagon and double box integrals

Abstract

We present a novel technique for the analytic evaluation of multifold Mellin-Barnes (MB) integrals, which commonly appear in physics, as for instance in the calculations of multi-loop multi-scale Feynman integrals. Our approach is based on triangulating a set of points which can be assigned to a given MB integral, and yields the final analytic results in terms of linear combinations of multiple series, each triangulation allowing the derivation of one of these combinations. When this technique is applied to the computation of Feynman integrals, the involved series are of the (multivariable) hypergeometric type. We implement our method in the Mathematica package MBConicHulls.wl, an already existing software dedicated to the analytic evaluation of multiple MB integrals, based on a recently developed computational approach using intersections of conic hulls. The triangulation method is remarkably faster than the conic hulls approach and can thus be used for the calculation of higher-fold MB integrals as we show here by computing triangulations for highly complicated objects such as the off-shell massless scalar one-loop 15-point Feynman integral whose MB representation has 104 folds. As other applications we show how this technique can provide new results for the off-shell massless conformal hexagon and double box Feynman integrals, as well as for the hard diagram of the two loop hexagon Wilson loop.