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.
Highlights
Only N − 3 of the conditions (1.1) are needed to restrict to its set of solutions
The scattering equations for N particles are shown to be equivalent to a Mobius invariant system of N − 3 equations, hm = 0, 2 ≤ m ≤ N − 2, in N variables, where hm is a homogeneous polynomial of degree m, with the exceptional property of being linear in each variable taken separately
Expressions are given for the tree amplitudes in terms of the hm and hm
Summary
The set of N −3 homogeneous polynomial equations (2.4) is equivalent to the scattering equations (1.1) but this form is not convenient, e.g. for taking advantage of the Mobius invariance to fix z1 at ∞. Hm is a homogeneous polynomial of degree m with the special property that, it is of degree m in the za, it is linear in each one of them taken individually, i.e. the monomials that it comprises are square-free This considerably simplifies various calculations and constructions. The scattering equations (1.1) are equivalent to p(z), as given by (1.15), vanishing everywhere as a function of z for za, a ∈ A, satisfying these equations, and this statement is equivalent to the vanishing of the polynomial of degree N − 2,. HSm0 is the appropriate form of hm for the polynomial scattering equations for n+1 particles with variables za, a ∈ Sand 0, associated with momenta ka, a ∈ S, and kS, while hSm∞ is the appropriate form of hm for the polynomial scattering equations for N − n + 1 particles with variables zb, b ∈ S, and ∞, associated with momenta kb, b ∈ S, and kS, demonstrating that the polynomial equations factorize as they should
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