Automatized analytic continuation of Mellin-Barnes integrals

By Michal Czakon. Computer Physics Communications, 2005

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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 naive 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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/489b1c23facb3a6ba7a1abff0d07d0b2689dfb32

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Citations in the corpus

Soft triple-real radiation for Higgs production at N3LO (2013, methodology): “We proceed in the standard way: we apply the packages MB [90], MBresolve [91] and barnesroutines [92] to resolve singularities in ǫ and to expand the resulting integrals”
A two-loop five-gluon helicity amplitude in QCD (2013, methodology): “2I 8 2 4;5123[1] + 2(2 )tr5 I 10 2 5;12345[1] : (5.2) We use two techniques for the numerical integration. Firstly we use the Mellin-Barnes approach with the help of AMBRE [66]1, M. Czakon’s MB.m[67], A. V. Smirnov’s MBresolve.m [68] and D. A. Kosower’s barnesroutines.m. The second approach uses se”
When does the gluon reggeize? (2013, methodology): “The author thanks Tristan Dennen for convincing him to use the Mellin-Barnes approach for this problem, and for providing initial results obtained with the help of the MB package [65].”
Scattering amplitudes, Wilson loops and the string/gauge theory correspondence (2008, background): “For the state of the art in the evaluation of Feynman integrals we refer the reader to [94, 95, 96, 97].”
From correlation functions to scattering amplitudes (2010, methodology): “The precision of the numerical evaluation used by the package [43] was therefore good: For all kinematic points in our (rather general) sample set we obtained values like 0.”
Cuts of Feynman Integrals in Baikov representation (2017, methodology): “An alternative is based on MellinBarnes representation [44, 45], implemented in [46]2.”
Calculation of the quark and gluon form factors to three loops in QCD (2010, methodology): “Compared to [56] we increased the numerical precision of the remaining coefficients, both for A9,2 and A9,4, by means of conventional packages like MB.m [ 81 ].”
Three loop analysis of the critical O (N) models in 6-ε dimensions (2014, methodology): “5) The -expansion of the above result can also be verified indirectly with the mathematica packages MBTools implementing the Mellin-Barnes representation [43].”
Collider physics at the precision frontier (2020, background): “[701], massive propagators lead to an oscillatory behaviour of the integrand which complicates the numerical integration.”
Two-loop fermionic corrections to heavy-quark pair production: the quark-antiquark channel (2008, methodology): “The final result for these two MIs was then checked by calculating their t → 0 limit with the Mellin Barnes technique, using the Mathematica packages Ambre [43] and MB [44].”