Multiple Series Representations of $N$-fold Mellin-Barnes Integrals

Mellin-Barnes (MB) integrals are well-known objects appearing in many branches of mathematics and physics, ranging from hypergeometric functions theory to quantum field theory, solid-state physics, asymptotic theory, etc. Although MB integrals have been studied for more than one century, until now there has been no systematic computational technique of the multiple series representations of $N$-fold MB integrals for $N>2$. Relying on a simple geometrical analysis based on conic hulls, we show here a solution to this important problem. Our method can be applied to resonant (i.e., logarithmic) and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones. When convergent series are obtained, the method also allows, in general, the determination of a single “master series” for each series representation, which considerably simplifies convergence studies and/or numerical checks. We provide, along with this Letter, a Mathematica implementation of our technique with examples of applications. Among them, we present the first evaluation of the hexagon and double box conformal Feynman integrals with unit propagator powers.

Figure 1

Conic hulls $C_{1,3}$ (top left), $C_{3,5}$ (top middle), and $C_{4,5}$ (top right) and their intersection (orange region, bottom left) with edges along ${\mathbf{e}}_3$ and ${\mathbf{e}}_5$ which corresponds in fact to $C_{3,5}$. Bottom right: convergence regions, given in Eq. (7), of the five series representations of the MB integral in Eq. (2). The well-known Appell $F_1$ double hypergeometric series converges in ${\mathcal{R}}_1$, its four analytic continuations in the other regions.