Intersection Numbers, Polynomial Division and Relative Cohomology

TL;DR

This paper introduces a simplified recursive algorithm for computing intersection numbers of differential forms, leveraging delta-forms and polynomial division, which streamlines the analysis of Feynman integrals and twisted period relations.

Contribution

The paper combines delta-forms and polynomial division to simplify the recursive computation of intersection numbers, bypassing evanescent regulator issues.

Findings

Efficient decomposition of two-loop Feynman integrals into master integrals.

New relations among twisted period integrals derived.

Simplified algorithm reduces computational complexity.

Abstract

We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential $n$-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology groups and the polynomial division technique, recently proposed in the literature. We show that delta-forms capture the leading behaviour of the intersection numbers in presence of evanescent analytic regulators, whose use is, therefore, bypassed. This simplified algorithm is applied to derive the complete decomposition of two-loop planar and non-planar Feynman integrals in terms of a master integral basis. More generally, it can be applied to derive relations among twisted period integrals, relevant for physics and mathematical studies.