Genus drop involving non-hyperelliptic curves in Feynman integrals

TL;DR

This paper investigates the phenomenon of genus drop in Feynman integrals, revealing a geometric mechanism involving algebraic curve coverings that explains genus changes and spacetime symmetries.

Contribution

It reformulates the genus drop mechanism as an unramified double covering and shows its application to non-hyperelliptic to hyperelliptic curve transitions in three-loop diagrams.

Findings

Genus drop corresponds to a specific algebraic curve covering mechanism.

The origin of discrete spacetime symmetry is linked to the genus drop process.

Some non-hyperelliptic Feynman integrals do not exhibit genus drops.

Abstract

For both theoretical and phenomenological studies, it is important to analyze the function types of Feynman integrals. The phenomenon of genus drop between different representations of hyperelliptic Feynman integrals was discussed in \cite{Marzucca2024Genusdrop}. In this paper, we reformulate the extra-involution mechanism of \cite{Marzucca2024Genusdrop} as a special case of an unramified double covering between algebraic curves, and show that this covering mechanism also explains genus drops accompanied by a curve-type change from non-hyperelliptic to hyperelliptic for a class of three-loop Feynman diagrams. We also demonstrate that within a specific framework, the origin of the discrete spacetime symmetry that leads to the genus drop in hyperelliptic cases is manifest. This work also points out that there exist non-hyperelliptic Feynman integrals that exhibit no apparent genus drop.