Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics

TL;DR

This paper presents two numerical algorithms for evaluating tensor Feynman integrals in Euclidean kinematics, utilizing Mellin-Barnes representations and sector decomposition, with automated tools and publicly available software.

Contribution

It introduces two automated numerical methods for tensor Feynman integrals in Euclidean space, extending existing packages and comparing their efficiency.

Findings

Algorithms successfully evaluate planar tensor Feynman integrals.

Automated tools streamline the preparation of integrals for numerical evaluation.

Publicly available packages facilitate broader use in the community.

Abstract

For the investigation of higher order Feynman integrals, potentially with tensor structure, it is highly desirable to have numerical methods and automated tools for dedicated, but sufficiently 'simple' numerical approaches. We elaborate two algorithms for this purpose which may be applied in the Euclidean kinematical region and in d=4-2*epsilon dimensions. One method uses Mellin-Barnes representations for the Feynman parameter representation of multi-loop Feynman integrals with arbitrary tensor rank. Our Mathematica package AMBRE has been extended for that purpose, and together with the packages MB (M. Czakon) or MBresolve (A. V. Smirnov and V. A. Smirnov) one may perform automatically a numerical evaluation of planar tensor Feynman integrals. Alternatively, one may apply sector decomposition to planar and non-planar multi-loop epsilon-expanded Feynman integrals with arbitrary tensor rank. We automatized the preparations of Feynman integrals for an immediate application of the package sector_decomposition (C. Bogner and S. Weinzierl) so that one has to give only a proper definition of propagators and numerators. The efficiency of the two implementations, based on Mellin-Barnes representations and sector decompositions, is compared. The computational packages are publicly available.