Feynman Integrals and Intersection Theory

TL;DR

This paper introduces intersection theory as a tool for decomposing Feynman integrals into a minimal basis, using the Baikov representation and maximal cuts as a tractable testing ground. By pairing twisted cycles and cocycles, it constructs a basis of maximal-cut integrals, computes their intersection numbers via logarithmic and non-logarithmic formulas, and derives a master decomposition formula that expresses any maximal-cut integral in terms of the basis. It further shows how to extract Pfaffian systems of differential equations for the basis through a projection formalism, and analyzes the two-loop non-planar triangle in arbitrary $D$ and the four-dimensional limit, where elliptic structures reduce the basis. The framework promises a geometrically transparent route to basis reduction and differential equations for Feynman integrals, with potential extensions to non-maximal cuts via relative homology.

Abstract

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.