Decomposition of Feynman Integrals by Multivariate Intersection Numbers

TL;DR

The paper develops a comprehensive framework to decompose Feynman integrals into a minimal set of master integrals by projecting onto twisted-cohomology bases using multivariate intersection numbers. It introduces a recursive algorithm to compute these intersection numbers and presents three decomposition strategies—straight, bottom-up, and top-down—that integrate intersection-theory with unitarity-based and integrand-decomposition approaches. The method yields linear and quadratic relations, derives differential equations for master integrals, and is demonstrated on representative one- and two-loop examples with results consistent with established IBP reductions. This work lays groundwork for scalable, regulator-aware computations and hints at extensions via relative twisted cohomology and canonical integral bases. Overall, the approach offers a principled, potentially minimal-basis pathway for handling generic multi-loop Feynman integrals.

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.

Figures (3)

• Figure 1: Massless Box
• Figure 2: QED triangle.
• Figure 3: QED Sunrise.