Applying Groebner Bases to Solve Reduction Problems for Feynman Integrals

TL;DR

The paper addresses the problem of reducing Feynman integrals to a finite set of master integrals using integration-by-parts identities. It introduces a generalized Buchberger algorithm for polynomials of shift operators, constructing sector-specific $s$-bases that incorporate boundary conditions to enable systematic, finite reductions across multiple index sectors. The authors demonstrate the method on a range of examples, from one- and two-loop to a three-loop static-quark-potential family, and show that master integrals can be identified and expressed in terms of each other (with Mellin–Barnes representations provided for evaluation). This approach offers a constructive, algorithmic framework for automated reduction in perturbative quantum field theory and suggests avenues for further optimization, such as leveraging Janet bases for performance gains.

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.

Figures (2)

• Figure 1: Two-loop propagator diagram
• Figure 2: Feynman diagram contributing to the three-loop static quark potential. A wavy line denotes a propagator for the static source and the dotted line denotes the scalar propagator with the index shifted by $\varepsilon$.