Feynman integral reduction using Gr\"obner bases

TL;DR

This paper presents a novel method for reducing Feynman integrals to master integrals using Gr"obner bases within a rational algebra, enabling systematic and potentially more efficient reductions.

Contribution

It introduces a Gr"obner basis approach to Feynman integral reduction and develops an algebraic ansatz for complex cases where computations are expensive.

Findings

Successfully applied to several examples.

Introduced first-order normal-form IBP relations.

Developed a linear algebra-based ansatz for complex cases.

Abstract

We investigate the reduction of Feynman integrals to master integrals using Gr\"obner bases in a rational double-shift algebra Y in which the integration-by-parts (IBP) relations form a left ideal. The problem of reducing a given family of integrals to master integrals can then be solved once and for all by computing the Gr\"obner basis of the left ideal formed by the IBP relations. We demonstrate this explicitly for several examples. We introduce so-called first-order normal-form IBP relations which we obtain by reducing the shift operators in Y modulo the Gr\"obner basis of the left ideal of IBP relations. For more complicated cases, where the Gr\"obner basis is computationally expensive, we develop an ansatz based on linear algebra over a function field to obtain the normal-form IBP relations.