Constructing Canonical Feynman Integrals with Intersection Theory

TL;DR

This paper introduces a new method to construct canonical Feynman integrals using intersection theory, simplifying their differential equations and enabling analysis of multi-loop scattering amplitudes with multiple scales.

Contribution

It presents a novel approach to build canonical master integrals from hypergeometric $d ext{log}$-form integrals via intersection theory, enhancing solvability of their differential equations.

Findings

Successfully applied to one- and two-loop integrals

Effective for both maximally cut and uncut integrals

Demonstrates applicability in multi-scale problems

Abstract

Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct $d\log$-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales.