Multiple Mellin-Barnes integrals and triangulations of point configurations

Abstract

Mellin-Barnes (MB) integrals are a well-known type of integrals appearing in diverse areas of mathematics and physics, such as in the theory of hypergeometric functions, asymptotics, quantum field theory, solid-state physics, etc. Although MB integrals have been studied for more than a century, it is only recently that, due to a remarkable connection found with conic hulls, N-fold MB integrals can be computed analytically for N>2 in a systematic way. In this article, we present an alternative novel technique by unveiling a new connection between triangulations of point configurations and MB integrals, to compute the latter. To make it ready to use, we have implemented our new method in the package oniculls.wl, an already existing software dedicated to the analytic evaluation of MB integrals using conic hulls. The triangulation method is remarkably faster than the conic hull approach and can thus be used for the calculation of higher-fold MB integrals, as we show here by testing our code on the case of the off-shell massless scalar one-loop N-point Feynman integral up to N=15, for which the MB representation has 104 folds. Among other examples of applications, we present new simpler solutions for the off-shell one-loop massless conformal hexagon and two-loop double-box Feynman integrals, as well as for some complicated 8-fold MB integrals contributing to the hard diagram of the two-loop hexagon Wilson loop in general kinematics. Published by the American Physical Society 2024