The Polynomial Form of the Scattering Equations
Louise Dolan, Peter Goddard
TL;DR
The CHY scattering equations for massless particles are reformulated as a polynomial system, revealing an algebraic structure that is linear in each variable. The authors derive homogeneous polynomials $\tilde{h}_m$ of degree $m$ (with $m$ running from 2 to $N-2$) that are equivalent to the original equations and further fix Möbius invariance to obtain $h_m$ in $N-3$ variables, enabling efficient variable elimination and amplitude representations. They provide a representation-theoretic view of Möbius invariance, analyze special symmetric kinematics yielding explicit solutions, and extend the framework to massive particles and four dimensions using twistors, linking to MHV helicity sectors. The work offers a computationally tractable, dimension-agnostic approach to tree-level amplitudes and suggests rich algebro-geometric structures worth exploring further.
Abstract
The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reformulated in polynomial form. The scattering equations for $N$ particles are shown to be equivalent to a Moebius invariant system of $N-3$ equations, $\tilde h_m=0$, $2 \leq m \leq N-2$, in $N$ variables, where $\tilde h_m$ is a homogeneous polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Moebius invariance appropriately, yields polynomial equations $h_m=0$, $1 \leq m \leq N-3$, in $N-3$ variables, where $h_m$ has degree $m$. The linearity of the equations in the individual variables facilitates computation, e.g the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the $\tilde h_m$ and $h_m$. The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.