AMBRE—a Mathematica package for the construction of Mellin–Barnes representations for Feynman integrals

Abstract

The Mathematica toolkit AMBRE derives Mellin–Barnes (MB) representations for Feynman integrals in d = 4 − 2 ε dimensions. It may be applied for tadpoles as well as for multi-leg multi-loop scalar and tensor integrals. The package uses a loop-by-loop approach and aims at lowest dimensions of the final MB representations. The present version works fine for planar Feynman diagrams. The output may be further processed by the package MB for the determination of its singularity structure in ε. The AMBRE package contains various sample applications for Feynman integrals with up to six external particles and up to four loops. Program summary Program title:AMBRE Catalogue identifier:ADZR_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZR_v1_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:21 387 No. of bytes in distributed program, including test data, etc.:100 004 Distribution format:tar.gz Programming language:MATHEMATICA v.5.0 and later versions Computer:all Operating system:all RAM:sufficient for a typical installation of MATHEMATICA Classification:5; 11.1 External routines:The examples in the package use: – MB.m [M. Czakon, Comput. Phys. Commun. 175 (2006) 559 (CPC Cat. Id. ADYG)], for expansions in ε; – CUBA [T. Hahn, Comput. Phys. Commun. 168 (2005) 78 (CPC Cat. Id. ADVH)], for numerical evaluation of multidimensional integrals; – CERNlib [CERN Program Library, http://cernlib.web.cern.ch/cernlib/], for the implementation of Γ and Ψ functions in Fortran. Nature of problem:Derivation of a representation for a Feynman diagram with L loops and N internal lines in d dimensions by Mellin–Barnes integrals; the subsequent evaluation, after an analytical continuation in ε = ( 4 − d ) / 2 , has to be done with other packages. Solution method:Introduction of N Feynman parameters x i , integration over the loop momenta, and subsequent integration over x, introducing thereby representations of sums of monomials in x by Mellin–Barnes integrals. Restrictions:Limited by the size of the available storage space. Running time:Depending on the problem; usually seconds.