Analytic Evaluation of Multiple Mellin-Barnes Integrals

TL;DR

This paper compares two geometrical methods for analytically evaluating complex Mellin-Barnes integrals, demonstrating that triangulation is more efficient and can produce simpler hypergeometric solutions for Feynman integrals and polylogarithms.

Contribution

It introduces and compares two geometrical approaches to evaluate Mellin-Barnes integrals, highlighting the efficiency of triangulation and deriving new hypergeometric solutions.

Findings

Triangulation approach is more computationally efficient than conic hulls.

New simpler hypergeometric solutions for Feynman integrals are derived.

New convergent series for multiple polylogarithms are obtained.

Abstract

We summarize two geometrical approaches to analytically evaluate higher-fold Mellin-Barnes (MB) integrals in terms of hypergeometric functions. The first method is based on intersections of conic hulls, while the second one, which is more recent, relies on triangulations of a set of points. We demonstrate that, once automatized, the triangulation approach is computationally more efficient than the conic hull approach. As an application of this triangulation approach, we describe how one can derive simpler hypergeometric solutions of the conformal off-shell massless two-loop double box and one-loop hexagon Feynman integrals than those previously obtained from the conic hull approach. Lastly, by applying the above techniques on the MB representation of multiple polylogarithms, we show how to obtain new convergent series representations for these functions. These new analytic expressions were numerically cross-checked with GINAC.