# Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers

## TL;DR

This paper introduces a novel method for decomposing Feynman integrals on maximal cuts using intersection numbers, applicable to complex multi-loop and multi-leg integrals, and verifies its consistency with traditional techniques.

## Contribution

It develops a new intersection number-based approach for Feynman integral decomposition, extending to 2-form and n-form representations, and connects to differential equations and recurrence relations.

## Key findings

- Successfully decomposes multi-loop Feynman integrals using intersection numbers.
- Derives contiguity relations for special functions like hypergeometric functions.
- Matches results with traditional IBP-based methods for validation.

## Abstract

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss ${}_2F_1$ hypergeometric function, and the Appell $F_1$ function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to $n$-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.