Geometrical methods for the analytic evaluation of multiple Mellin-Barnes integrals

TL;DR

This paper introduces two geometrical methods for analytically evaluating complex Mellin-Barnes integrals, enabling faster and more efficient calculations of challenging Feynman integrals in theoretical physics.

Contribution

It presents two novel geometrical approaches for Mellin-Barnes integral evaluation, implemented in a Mathematica package, improving efficiency in calculating complex Feynman integrals.

Findings

First analytic calculation of massless off-shell conformal hexagon

Conic hulls method effectively computes complex integrals

Triangulation method offers faster computation

Abstract

Two recently developed techniques of analytic evaluation of multifold Mellin-Barnes (MB) integrals are presented. Both approaches rest on the definition of geometrical objets conveniently associated with the MB integrands, which can then be used along with multivariate residues analysis to derive series representations of the MB integrals. The first method is based on introducing conic hulls and considering specific intersections of the latter, while the second one rests on point configurations and their regular triangulations. After a brief description of both methods, which have been automatized in the MBConicHulls.wl Mathematica package, we review some of their applications. In particular, we show how the conic hulls method was used to obtain the first analytic calculation of complicated Feynman integrals, such as the massless off-shell conformal hexagon and double-box. We then show that the triangulation method is even more efficient, as it allows one to compute these nontrivial objects and harder ones in a much faster way.