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2014

The Polynomial Form of the Scattering Equations

Dolan, Louise, Goddard, Peter

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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.

The Polynomial Form of the Scattering Equations · Around

Built on

I.R. Shafarevich, Basic Algebraic Geometry,
1974
Earlier work this paper cites.
D. J. Gross and P. F. Mende, String Theory Beyond the Planck Scale,
1988
Earlier work this paper cites.
I.M. Gel’fand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants,
1994
Earlier work this paper cites.

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Then

D.B. Fairlie and D.E. Roberts, Dual Models without Tachyons - a New Approach,
2009
Later among the works it cites.
L. Dolan and P. Goddard, Gluon Tree Amplitudes in Open Twistor String Theory,
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2009
Later among the works it cites.
M. Spradlin and A. Volovich, From Twistor String Theory To Recursion Relations,
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