Scattering equations: from projective spaces to tropical grassmannians

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, s(ab), are generalized to completely symmetric tensors sa1a2...ak subject to a massless' condition sa1a2...ak-2bb=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.