Automatized analytic continuation of Mellin–Barnes integrals

Abstract

This paper describe a package written in MATHEMATICA that automatizes typical operations performed during evaluation of Feynman graphs with Mellin–Barnes (MB) techniques. The main procedure allows to analytically continue a MB integral in a given parameter without any intervention from the user and thus to resolve the singularity structure in this parameter. The package can also perform numerical integrations at specified kinematic points, as long as the integrands have satisfactory convergence properties. It is demonstrated that, at least in the case of massive graphs in the physical region, the convergence may turn out to be poor, making naïve numerical integration of MB integrals unusable. Possible solutions to this problem are presented, but full automatization in such cases may not be achievable. Program summary Title of program: MB Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYG_v1_0 Catalogue identifier: ADYG_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computers: All Operating systems: All Programming language used: MATHEMATICA, Fortran 77 for numerical evaluation Memory required to execute with typical data: Sufficient for a typical installation of MATHEMATICA. No. of lines in distributed program, including test data, etc.: 12 013 No. of bytes in distributed program, including test data, etc.: 231 899 Distribution format: tar.gz Libraries used: CUBA [T. Hahn, Comput. Phys. Commun. 168 (2005) 78] for numerical evaluation of multidimensional integrals and CERNlib [CERN Program Library, obtainable from: http://cernlib.web.cern.ch/cernlib/] for the implementation of Γ and ψ functions in Fortran. Nature of physical problem: Analytic continuation of Mellin–Barnes integrals in a parameter and subsequent numerical evaluation. This is necessary for evaluation of Feynman integrals from Mellin–Barnes representations. Method of solution: Recursive accumulation of residue terms occurring when singularities cross integration contours. Numerical integration of multidimensional integrals with the help of the CUBA library. Restrictions on the complexity of the problem: Limited by the size of the available storage space. Typical running time: Depending on the problem. Usually seconds for moderate dimensionality integrals.