Decomposition of Feynman Integrals by Multivariate Intersection Numbers

TL;DR

This paper introduces a novel method for decomposing Feynman integrals using multivariate intersection numbers, enabling direct derivation of differential equations and offering multiple algorithms for practical computation, advancing multi-loop integral analysis.

Contribution

It presents a new approach combining intersection theory with unitarity methods for Feynman integral decomposition and provides algorithms for direct computation of master integrals.

Findings

Developed recursive algorithm for multivariate intersection numbers.

Introduced three decomposition approaches: straight, bottom-up, top-down.

Applied methods to explicit Feynman integrals demonstrating effectiveness.

Abstract

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.