Cuts and Isogenies

TL;DR

This paper investigates genus-one curves arising in Feynman integrals, demonstrating a consistent geometric structure across different computational methods and exploring their isogeny relations.

Contribution

It shows that the same genus-one geometry appears in various Feynman integral computations, suggesting an invariant geometric notion independent of the calculation method.

Findings

The genus-one curves are consistent across multiple computational approaches.

These curves are related by isogenies, indicating deeper geometric connections.

The invariants of these curves are comparable, supporting a unified geometric perspective.

Abstract

We consider the genus-one curves which arise in the cuts of the sunrise and in the elliptic double-box Feynman integrals. We compute and compare invariants of these curves in a number of ways, including Feynman parametrization, lightcone and Baikov (in full and loop-by-loop variants). We find that the same geometry for the genus-one curves arises in all cases, which lends support to the idea that there exists an invariant notion of genus-one geometry, independent on the way it is computed. We further indicate how to interpret some previous results which found that these curves are related by isogenies instead.