IBP reduction via Gr\"obner bases in a rational double-shift algebra

TL;DR

This paper introduces a novel method for IBP reduction using Gr"obner bases within a noncommutative algebra framework, successfully reducing complex integrals to master integrals and analyzing performance and challenges.

Contribution

It develops a new algebraic approach for IBP reduction employing Gr"obner bases in a noncommutative rational double-shift algebra, demonstrating its effectiveness on a one-loop massless box example.

Findings

Achieved full reduction to master integrals for the example

Analyzed implementation performance and bottlenecks

Outlined future directions for complex cases

Abstract

We report on an approach to integration-by-parts reduction based on Gr\"obner bases. We establish the underlying noncommutative rational double-shift algebra wherein the integration-by-parts relations form a left ideal. We describe in detail the one-loop massless box as an example where we achieved the full reduction to master integrals by means of the Gr\"obner basis approach, and report on the performance of the implementation. We also identify potential bottlenecks in more complicated examples and elaborate on interesting further directions.