Feynman Integral Reductions by Intersection Theory with Orthogonal Bases and Closed Formulae

TL;DR

This paper introduces a method for constructing orthogonal bases of differential forms for quadratic twisted period integrals, providing a new closed formula for intersection numbers, which simplifies Feynman integral reductions at one-loop.

Contribution

It presents a novel approach to choosing orthogonal bases and a closed formula for intersection numbers, enhancing Feynman integral reduction techniques.

Findings

Systematic construction of orthonormal bases for twisted period integrals.

New closed formula for intersection numbers beyond d log forms.

Application to all one-loop Feynman diagrams.

Abstract

We present a prescription for choosing orthogonal bases of differential $n$-forms belonging to quadratic twisted period integrals, with respect to the intersection number inner product. To evaluate these inner products, we additionally propose a new closed formula for intersection numbers beyond $\mathrm{d} \log$ forms. These findings allow us to systematically construct orthonormal bases between twisted period integrals of this type. In the context of Feynman integrals, this represents all diagrams at one-loop.