A numerical algorithm for efficiently obtaining a Feynman parameter representation of one-gluon loop QCD Feynman diagrams for a large number of external gluons

Abstract

A numerical program is presented which facilitates a computation pertaining to the full set of one-gluon loop diagrams (including ghost loop contributions), with M attached external gluon lines in all possible ways. The feasibility of such a task rests on a suitably defined master formula, which is expressed in terms of a set of Grassmann and a set of Feynman parameters. The program carries out the Grassmann integration and performs the Lorentz trace on the involved functions, expressing the result as a compact sum of parametric integrals. The computation is based on tracing the structure of the final result, thus avoiding all intermediate unnecessary calculations and directly writing the output. Similar terms entering the final result are grouped together. The running time of the program demonstrates its effectiveness, especially for large M. Program summary Program title:DILOG2 Program identifier:ADXN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXN_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Programming language:FORTRAN 90 Computer(s) for which the program has been designed:Personal Computer Operating system(s) for which the program has been designed: Windows 98, XP, LINUX Number of processors used:one No. of lines in distributed program, including test data, etc.:2000 No. of bytes in distributed program, including test data, etc.:16 249 Distribution format:tar.gz External routines/libraries used:none CPC Program Library subprograms used:none Nature of problem:The computation of one gluon/ghost loop diagrams in QCD with many external gluon lines is a time consuming task, practically beyond reasonable reach of analytic procedures. We apply recently proposed master formulas towards the computation of such diagrams with an arbitrary number ( M) of external gluon lines, achieving a final result which reduces the problem to one involving integrals over the standard set, for given M, of Feynman parameters. Solution method:The structure of the master expressions is analyzed from a numerical computation point of view. Using the properties of Grassmann variables we identify all the different forms of terms that appear in the final result. Each form is called “structure”. We calculate theoretically the number of terms belonging to every “structure”. We carry out the calculation organizing the whole procedure into separate calculations of the terms belonging to every “structure”. Terms which do not contribute to the final result are thereby avoided. The final result, extending to large values of M, is also presented with terms belonging to the same “structure” grouped together. Restrictions:M is coded as a 2-digit integer. Overflow in the dimension of used array is expected to appear for M ⩾ 20 in a processor that uses 4-bytes integers or for M ⩾ 34 in a processor with 8-bytes integers. Running time:Depends on M, see enclosed figures.