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.

Automated summary

Mellin-Barnes (MB) integrals are widely studied objects in mathematics and physics with various applications in different areas. A systematic computational technique for N-fold MB integrals had been lacking until a solution was proposed based on a simple geometrical analysis. This method can handle resonant and nonresonant cases, providing both convergent and diverging series representations depending on the integrand.