Integral reduction with Kira 2.0 and finite field methods

Abstract

We present the new version 2.0 of the Feynman integral reduction program Kira and describe the new features. The primary new feature is the reconstruction of the final coefficients in integration-by-parts reductions by means of finite field methods with the help of FireFly. This procedure can be parallelized on computer clusters with MPI. Furthermore, the support for user-provided systems of equations has been significantly improved. This mode provides the flexibility to integrate Kira into projects that employ specialized reduction formulas, direct reduction of amplitudes, or to problems involving linear system of equations not limited to relations among standard Feynman integrals. We show examples from state-of-the-art Feynman integral reduction problems and provide benchmarks of the new features, demonstrating significantly reduced main memory usage and improved performance w.r.t. previous versions of Kira. New version program summaryProgram title:KiraCPC Library link to program files:https://doi.org/10.17632/v3cmsnfrnn.2Developer's repository link:https://gitlab.com/kira-pyred/kiraLicensing provisions: GNU General Public License 3 (GPL)Programming language:C++Journal Reference of previous version: P. Maierhöfer, J. Usovitsch and P. Uwer, Kira—A Feynman integral reduction program, Comput. Phys. Commun.230 (2018) 99 [1705.05610].Does the new version supersede the previous version?: Yes.Reasons for the new version: Implementation of new features, some of which improve the performance significantly for many problems.Summary of revisions: The primary new feature is the reconstruction of the final coefficients in integration-by-parts reductions by means of finite field methods with the help of FireFly [1,2]. This procedure can be parallelized on computer clusters with MPI. Further improvements include the expanded support for user-provided systems of equations as well as a new feature to reduce the main memory usage when generating the system of equations.Nature of problem: The reduction of Feynman integrals to a smaller set of master integrals is a central strategy for high precision calculations in theoretical particle physics, e.g. for cross sections. Furthermore, the reduction is a key ingredient in many methods to calculate the master integrals themselves.Solution method:Kira generates a system of equations employing integration-by-parts [3,4] and Lorentz-invariance identities [5] as well as symmetry relations. Linearly dependent equations are removed and master integrals are identified by solving the system over a finite field. The resulting system can be solved with two methods: Since version 1.0, Kira can solve the system algebraically with the help of Fermat [6]. New in this version is the strategy of solving the system many times over finite fields and reconstructing the master integral coefficients with the help of FireFly [1,2]. Both strategies can also be applied to arbitrary homogeneous linear systems of equations.