Integration-by-parts reductions from the viewpoint of computational algebraic geometry

TL;DR

This paper presents a novel, efficient method for generating integration-by-parts reductions in quantum field theory calculations, utilizing algebraic geometry techniques to simplify the process.

Contribution

It introduces a new approach that employs generalized-unitarity cuts and syzygy equations to streamline IBP reductions, enhancing computational efficiency.

Findings

Reduces computational complexity of IBP reductions

Uses algebraic geometry tools for polynomial equations

Facilitates faster basis integral computations

Abstract

Integration-by-parts reductions play a central role in perturbative QFT calculations. They allow the set of Feynman integrals contributing to a given observable to be reduced to a small set of basis integrals, and they moreover facilitate the computation of those basis integrals. We introduce an efficient new method for generating integration-by-parts reductions. This method simplifies the task by making use of generalized-unitarity cuts and turns the problem of finding the needed total derivatives into one of solving certain polynomial (so-called syzygy) equations.