Reduction to master integrals via intersection numbers and polynomial expansions

TL;DR

This paper introduces a novel method for decomposing Feynman integrals into master integrals using intersection numbers and polynomial expansions, simplifying calculations without integral transformations.

Contribution

The paper presents a new rational-operation-based approach for computing intersection numbers, eliminating the need for integral transformations or basis changes.

Findings

Implemented the algorithm over finite fields

Successfully applied to one- and two-loop Feynman integrals

Removed explicit dependence on analytic regulators

Abstract

Intersection numbers are rational scalar products among functions that admit suitable integral representations, such as Feynman integrals. Using these scalar products, the decomposition of Feynman integrals into a basis of linearly independent master integrals is reduced to a projection. We present a new method for computing intersection numbers that only uses rational operations and does not require any integral transformation or change of basis. We achieve this by systematically employing the polynomial series expansion, namely the expansion of functions in powers of a polynomial. We also introduce a new prescription for choosing dual integrals, de facto removing the explicit dependence on additional analytic regulators in the computation of intersection numbers. We describe a proof-of-concept implementation of the algorithm over finite fields and its application to the decomposition of Feynman integrals at one and two loops.