Intersection Numbers from Companion Tensor Algebra

TL;DR

This paper introduces a new tensor algebra-based method using companion matrices to efficiently compute intersection numbers, which are crucial in evaluating complex integrals in physics and mathematics.

Contribution

It develops a novel tensor structure framework combined with the fibration method, enabling direct projection of Feynman integrals to master integrals using intersection numbers.

Findings

Successfully applied to two-loop integrals from planar five-point massless functions

Enables efficient numerical decomposition of complex Feynman integrals

Advances the computational techniques for intersection numbers in theoretical physics

Abstract

Twisted period integrals are ubiquitous in theoretical physics and mathematics, where they inhabit a finite-dimensional vector space governed by an inner product known as the intersection number. In this work, we uncover the associated tensor structures of intersection numbers and integrate them with the fibration method to develop a novel evaluation scheme. Companion matrices allow us to cast the computation of the intersection numbers in terms of a matrix operator calculus within the ambient tensor space. For illustrative purposes, our algorithm has been successfully applied to the numerical decomposition of a sample of two-loop integrals, coming from planar five-point massless functions, representing a significant advancement for the direct projection of Feynman integrals to master integrals via intersection numbers.