FeynGKZ: A Mathematica package for solving Feynman integrals using GKZ hypergeometric systems

Abstract

In the Lee-Pomeransky representation, Feynman integrals can be identified as a subset of Euler-Mellin integrals, which are known to satisfy Gel'fand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. Here we present an automated package to derive the associated GKZ system for a given Feynman diagram and solve it in terms of hypergeometric functions using two equivalent algorithms, namely the triangulation method and the Gröbner deformation method. We present our code in the form of a Mathematica package FeynGKZ.wl which requires the softwares polymake, Macaulay2 and TOPCOM, and the packages AMBRE and Olsson.wl as dependencies. As applications of the package, we find series solutions to the GKZ systems of several one-loop and two-loop Feynman integrals. These are included in the file Examples.nb that can be downloaded along with the package from GitHub. Program summaryProgram Title:FeynGKZ.wl, version 1.0CPC Library link to program files:https://doi.org/10.17632/m8zds756vb.1Developer's repository link:https://github.com/anant-group/FeynGKZLicensing provisions: GNU General Public License 3Programming language:WolframMathematica version 13.0 or higherExternal routines/libraries:Macaulay2 version 1.20, TOPCOM version 0.17.8, polymake version 4.6, AMBRE version 2.1.1 and Olsson.wlNature of problem: Deriving the GKZ system associated with a given Feynman integral, and solving it in terms of multivariate hypergeometric functions.Solution method: Automating the triangulation and Gröbner deformation methods for obtaining Γ-series solutions to the GKZ hypergeometric system associated with a given Feynman integral.