Scattering equations: from projective spaces to tropical grassmannians

Abstract

Abstract

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1. The standard, k = 2 Mandelstam invariants, sab, are generalized to completely symmetric tensors

s

a 1

a 2

…

a k

$$ {\mathrm{s}}_{a_1{a}_2\dots {a}_k} $$ subject to a ‘massless’ condition

s

a 1

a 2

…

a

k − 2

b b

= 0

$$ {\mathrm{s}}_{a_1{a}_2\dots {a}_{k-2}bb}=0 $$ and to ‘momentum conservation’. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all ‘biadjoint amplitudes’ for (k, n) = (3, 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.