Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics

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.

22 pages

References in corpus (13)

• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• Sector Decomposition
• Resolution of singularities for multi-loop integrals
• On the Resolution of Singularities of Multiple Mellin-Barnes Integrals
• AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals
• Two-Loop Iteration of Five-Point N=4 Super-Yang-Mills Amplitudes
• Two-Loop Fermionic Corrections to Heavy-Quark Pair Production: the Quark-Antiquark Channel
• The NLO multileg working group: summary report
• Virtual Hadronic and Leptonic Contributions to Bhabha Scattering
• The Six-Point NMHV amplitude in Maximally Supersymmetric Yang-Mills Theory
• Virtual Hadronic and Heavy-Fermion O(alpha^2) Corrections to Bhabha Scattering
• $\mathcal{O}(\alpha {\alpha }_s)$ corrections to the $\gamma t\bar{t}$ vertex at the top quark threshold
• A Fully Numerical Approach to One-Loop Amplitudes

Cited by in corpus (3)

• SecDec: A general program for sector decomposition
• SecDec: A tool for numerical multi-loop calculations
• New proofs for the two Barnes lemmas and an additional lemma