The polarized scattering equations for 6d superamplitudes

We introduce a spinorial version of the scattering equations, the \emph{polarized scattering equations}, that incorporates spinor polarization data. They lead to new formulae for tree-level scattering amplitudes in six dimensions that directly extend to maximal supersymmetry. They give a quite distinct framework from that of Cachazo et al.; in particular, the formulae do not change character from even to odd numbers of particles. We find new ingredients for integrands for maximally supersymmetric Yang-Mills, gravity, M5 and D5 branes. We explain how the polarized scattering equations and supersymmetry representations arise from an ambitwistor-string with target given by a super-twistor description of the geometry of super-ambitwistor space for six dimensions. On reduction to four dimensions the polarized scattering equations give rise to massive analogues of the 4d refined scattering equations for amplitudes on the Coulomb branch. At zero mass this framework naturally generalizes the twistorial version of the ambitwistor string in four dimensions.