Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams

TL;DR

This paper investigates the algebraic curves arising from three-loop Feynman diagrams using algebraic geometry, specifically computing their genus to aid in amplitude reduction.

Contribution

It introduces a method to compute the geometric genus of algebraic curves from three-loop diagrams using Riemann-Hurwitz and convex hull techniques, advancing amplitude analysis.

Findings

Computed genus for various three-loop diagrams

Identified properties of genus across loop orders

Provided a geometric approach for amplitude reduction

Abstract

This paper studies the global structure of algebraic curves defined by generalized unitarity cut of four-dimensional three-loop diagrams with eleven propagators. The global structure is a topological invariant that is characterized by the geometric genus of the algebraic curve. We use the Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with the help of techniques involving convex hull polytopes and numerical algebraic geometry. Some interesting properties of genus for arbitrary loop orders are also explored where computing the genus serves as an initial step for integral or integrand reduction of three-loop amplitudes via an algebraic geometric approach.