Applying Groebner Bases to Solve Reduction Problems for Feynman Integrals

TL;DR

This paper introduces a Groebner bases-based algorithm for reducing Feynman integrals to master integrals, demonstrated on various loop integral families including three-loop static quark potential.

Contribution

It presents a novel application of a generalized Buchberger algorithm to Feynman integral reduction problems, expanding computational tools in quantum field theory.

Findings

Successfully reduced one- and two-loop Feynman integrals.

Solved the reduction problem for three-loop static quark potential integrals.

Demonstrated the effectiveness of Groebner bases in complex integral reductions.

Abstract

We describe how Groebner bases can be used to solve the reduction problem for Feynman integrals, i.e. to construct an algorithm that provides the possibility to express a Feynman integral of a given family as a linear combination of some master integrals. Our approach is based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. We illustrate it through various examples of reduction problems for families of one- and two-loop Feynman integrals. We also solve the reduction problem for a family of integrals contributing to the three-loop static quark potential.