Finding linear dependencies in integration-by-parts equations: A Monte Carlo approach

Abstract

The reduction of a large number of scalar integrals to a small set of master integrals via Laporta’s algorithm is common practice in multi-loop calculations. It is also a major bottleneck in terms of running time and memory consumption. It involves solving a large set of linear equations where many of the equations are linearly dependent. We propose a simple algorithm that eliminates all linearly dependent equations from a given system, reducing the time and space requirements of a subsequent run of Laporta’s algorithm. Program summaryProgram title: ICE—the IBP Chooser of EquationsCatalogue identifier: AESF_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESF_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 3137No. of bytes in distributed program, including test data, etc.: 366461Distribution format: tar.gzProgramming language: Haskell.Computer: any system that hosts the Haskell Platform.Operating system: GNU/Linux, Windows, OS/X.Classification: 4.4, 4.8, 5, 11.1.Nature of problem: find linear dependencies in a system of linear equations with multivariate polynomial coefficients. To be used on Integration-By-Parts identities before running Laporta’s Algorithm.Solution method: map the system to a finite field and solve there, keeping track of the required equations.Restrictions: typically less than the restrictions imposed by the requirement of being able to process the output with Laporta’s Algorithm.Unusual features: complexity increases only very mildly with the number of kinematic invariants.Running time: depends on the individual problem. Fractions of a second to a few minutes have been observed in tests.