On the Application of Intersection Theory to Feynman Integrals: The Univariate Case

TL;DR

This paper introduces a novel intersection theory-based method for decomposing Feynman integrals into a minimal basis, simplifying high energy physics scattering amplitude calculations, focusing on the univariate case.

Contribution

It presents an innovative intersection theory approach for direct reduction of Feynman integrals, with explicit identities for maximally cut integrals in the univariate case.

Findings

Derived identities between maximally cut Feynman integrals

Demonstrated direct decomposition onto an integral basis

Focused on univariate integrals, multivariate generalization discussed elsewhere

Abstract

This document is a contribution to the proceedings of the MathemAmplitudes 2019 conference held in December 2019 in Padova, Italy. A key step in modern high energy physics scattering amplitudes computation is to express the latter in terms of a minimal set of Feynman integrals using linear relations. In this work we present an innovative approach based on intersection theory, in order to achieve this decomposition. This allows for the direct computation of the reduction, projecting integrals appearing in the scattering amplitudes onto an integral basis in the same fashion as vectors may be projected onto a vector basis. Specifically, we will derive and discuss few identities between maximally cut Feynman integrals, showing their direct decomposition. This contribution will focus on the univariate part of the story, with the multivariate generalisation being discussed in a different contribution by Gasparotto and Mandal.