Abstract

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.

Highlights

  • Universite Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France and Universite Lyon, Universite Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622 Villeurbanne, France

  • 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

  • Our method can be applied to resonant and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones

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Summary

Shayan Ghosh

Helmholtz-Institut für Strahlen- und Kernphysik, D-53115 Bonn, Germany and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany (Received 1 January 2021; revised 18 June 2021; accepted 27 July 2021; published 5 October 2021). 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. 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. 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 allows, in general, the determination of a single “master series” for each series representation, which considerably simplifies convergence studies and/or numerical checks. We present the first evaluation of the hexagon and double box conformal Feynman integrals with unit propagator powers

Γai Γbj
Published by the American Physical Society
One therefore obtains
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