Constructing Canonical Feynman Integrals with Intersection Theory

TL;DR

This work tackles constructing canonical Feynman integrals for multi-loop scattering amplitudes using an intersection-theory framework applied to Baikov based hypergeometric representations. The authors develop univariate and multivariate $dlog$-form constructions that, via twisted cohomology and intersection numbers, project onto canonical master integrals whose differential equations are in the $epsilon$-form. They demonstrate the method on nontrivial two-loop multi-scale topologies, including maximally cut and uncut cases such as massless and massive double boxes, obtaining complete canonical bases. The approach is constructive and amenable to automation, with promising extensions to elliptic sectors and deeper geometric interpretations in planar theories.

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.

Figures (1)

• Figure 1: The integral topology considered as an example in the text.