On Genera of Curves from High-loop Generalized Unitarity Cuts

TL;DR

This paper uses computational algebraic geometry to classify the topology of algebraic curves arising from high-loop unitarity cuts in Feynman diagrams, aiding in understanding their complexity and parameterization.

Contribution

It introduces a method to compute the genus of curves from two and three-loop unitarity cuts, revealing their topological properties and potential for rational parameterization.

Findings

Calculated genera of curves from two and three-loop cuts.

Classified the topology of degenerate on-shell equations.

Provided insights into rational parameterization of solutions.

Abstract

Generalized unitarity cut of a Feynman diagram generates an algebraic system of polynomial equations. At high-loop levels, these equations may define a complex curve or a (hyper-)surface with complicated topology. We study the curve cases, i.e., a 4-dimensional L-loop diagram with (4L-1) cuts. The topology of a complex curve is classified by its genus. Hence in this paper, we use computational algebraic geometry to calculate the genera of curves from two and three-loop unitarity cuts. The global structure of degenerate on-shell equations under some specific kinematic configurations is also sketched. The genus information can also be used to judge if a unitary cut solution could be rationally parameterized.