Feynman Integrals and Intersection Theory

TL;DR

This paper applies intersection theory to Feynman integrals, providing a new basis projection method, an algorithm for basis decomposition, and differential equations derivation, demonstrated on a two-loop non-planar triangle example.

Contribution

It introduces intersection theory tools to Feynman integrals, enabling basis projection, decomposition, and differential equations derivation in a novel way.

Findings

Developed a basis of differential forms with logarithmic singularities.

Provided an algorithm for basis decomposition using intersection numbers.

Illustrated the method on a two-loop non-planar triangle diagram.

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.