# Vector Space of Feynman Integrals and Multivariate Intersection Numbers

## TL;DR

This paper introduces a general algorithm for constructing multivariate intersection numbers of Feynman integrals, enabling new methods for integral reduction, deriving functional equations, and simplifying multi-loop integrals in quantum field theory.

## Contribution

It presents the first application of intersection numbers to solve integral reduction and derive functional equations for Feynman integrals, advancing computational techniques in quantum field theory.

## Key findings

- Developed a general algorithm for multivariate intersection numbers
- Applied to derive contiguity relations for special functions
- Decomposed multi-loop Feynman integrals as initial steps for broader applications

## Abstract

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the derivation of contiguity relations for special functions admitting multi-fold integral representations, and to the decomposition of a few Feynman integrals at one- and two-loops, as first steps towards potential applications to generic multi-loop integrals.