FIRE4, LiteRed and accompanying tools to solve integration by parts relations

Abstract

We present the new version of the Mathematica code FIRE and ideas behind it. It can be applied together with the recently developed code LiteRed by Lee in order to provide an integration by parts reduction to master integrals for quite complicated families of Feynman integrals. As an example, we consider four-loop massless propagator integrals for which LiteRed provides reduction rules and FIRE assists to apply these rules. So, as a by-product one obtains a four-loop variant of the well-known algorithm MINCER. The existence of these explicit reduction rules shows that any four-loop massless propagator integral can be reduced to a linear combination of master integrals in the sense of a mathematical theorem. We also describe various algebraic ways to find additional relations between master integrals associated with several families of Feynman integrals. Program summaryProgram title: FIRE4Catalogue identifier: AEPW_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPW_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 2No. of lines in distributed program, including test data, etc.: 102107No. of bytes in distributed program, including test data, etc.: 3539091Distribution format: tar.gzProgramming language: Wolfram Mathematica 6.0 or higher.Computer: Workstation or PC.Operating system: Unix, Linux, Windows, Mac OS X.RAM: Depends on the complexity of the problemClassification: 4.4, 4.8, 5, 20.External routines: QLink [1], FLink [2], LiteRed [3]Nature of problem:Reducing Feynman integrals to master integrals can be treated as a task to solve a huge system of sparse linear equations with polynomial coefficients.Solution method:Since the matrix of equations is very specific, none of the standard methods of solving linear equations can be applied efficiently. The program approaches solving those equations with a special version of Gauss elimination. It stores data on disk with the use of the QLink [1] routine and can speed up calculations (evaluation of GCD) with FLink [2]. The external package LiteRed [3] can be used to produce additional rules for reduction.Restrictions:The complexity of the problem is mostly restricted by the CPU time required to perform the reduction of integrals.Running time:Depends on the complexity of the problem