On Genera of Curves from High-loop Generalized Unitarity Cuts

TL;DR

<3-5 sentence high-level summary>We address the topology of solution spaces arising from generalized unitarity cuts by treating the maximal-cut equations as complex algebraic curves in four dimensions. The authors develop and apply a computational algebraic-geometry framework to compute the geometric genus for one-, two-, and three-loop diagrams, using the relation between arithmetic genus and geometric genus and the Riemann-Hurwitz formula to determine genus, as well as degeneracy analyses under special kinematics. They report genera ranging from 0 up to 13 across varied topologies (planar and non-planar) and show how degenerate kinematics decompose curves into multiple lower-genus branches, informing rational parametrization and branch-by-branch coefficient extractions in unitarity methods. This genus information provides a practical guide for parametrization feasibility and the organization of integrand bases in multi-loop amplitude calculations.

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.

Figures (10)

• Figure 1: Two-loop diagrams with $7$ propagators: (a) planar pentagon-triangle diagram, (b) planar double-box diagram, (c) non-planar crossed-box diagram. All external momenta are out-going and massive. The loop momenta are denoted by $\ell_1,\ell_2$.
• Figure 2: Topological pictures of on-shell equations from the two-loop double-box diagram under specific kinematic configurations. The pictures should be understood as complex curves, or two-dimensional real surfaces. (a) the curve is irreducible and the solution set is a torus (b) the curve has 2 irreducible branches, (c) the curve has 4 irreducible branches, (d) the curve has 6 irreducible branches. For general kinematics the curve is genus 1. In degenerate limit, tubes shrink to points along dashed lines. The resulting Riemann surfaces for each branch can only be a sphere. This explains why we get Riemann spheres connected by points and linked in a chain in degenerate limits.
• Figure 3: Topological pictures of degenerate on-shell equations from non-planar two-loop crossing-box diagram under specific kinematic configurations. (a) the curve is irreducible, the geometric genus is $3$, (b) the curve has 2 irreducible branches, each branch is genus 1, and they are connected by two points, (c) the curve has 2 irreducible branches, each branch is genus 0, and they are connected by four points, (d) no kinematic configuration corresponds to this topological picture.
• Figure 4: Topological pictures of degenerate on-shell equations from non-planar two-loop crossing-box diagram under specific kinematic configurations, where the solution set has more than $2$ branches. These $5$ pictures includes the degeneracies under all possible kinematic configurations. Each irreducible branch is genus 0 sphere, and they are connected by points along dashed lines where tubes have been contracted.
• Figure 5: Planar three-loop diagrams with $11$ propagators: (a) pentagon-box-box diagram, (b) box-pentagon-box diagram. All external momenta are out-going and massive. The loop momenta are denoted by $\ell_1,\ell_2,\ell_3$.