Graphical Scattering Equations
Barbara Betti, Viktoriia Borovik, Bella Finkel, Bernd Sturmfels, Bailee Zacovic
TL;DR
This work develops a graph-based generalization of CHY scattering by encoding momentum conservation on a fixed graph $G$ via the kinematic space $\mathcal{K}_G$ and the scattering correspondence $\mathcal{V}_G$. It introduces the scattering matroid $\mathcal{S}(G)$ and four equivalent notions of copiousness (geometric, matroidal, algebraic, topological), linking them to the ML degree $\mu(G)$ through multidegree data and partial compactifications $\mathcal{M}_G$ of $\mathcal{M}_{0,n}$; for many graphs, $\mu(G)$ is computable via topological invariants such as Euler characteristics and chromatic polynomials. The paper develops the scattering ideal $I_G$, establishes cases where it is prime, and connects the ML degree to both algebraic and combinatorial structures, including a conjectured closed-form formula $\mu(G)=\left|\sum_{s=3}^n (-1)^{s-3}(s-3)\!\,\!\!\! (s-3)!\,\pi_G(s)\right|$ counting independent-set partitions. Through extensive computations up to $n\le 9$ and analysis of discriminants and hypertrees (notably bipyramids), the authors illuminate the rich geometry governing graphical scattering, with implications for moduli spaces, very affine varieties, and the combinatorics of ML degrees.
Abstract
The CHY scattering equations on the moduli space $M_{0,n}$ play a prominent role at the interface of particle physics and algebraic statistics. We study the scattering correspondence when the Mandelstam invariants are restricted to a fixed graph on $n$ vertices.