Polarized Scattering Equations for 6D Superamplitudes

We introduce a spinorial version of the scattering equations, the polarized scattering equations, that incorporates spinor polarization data. They underpin new formulas for tree-level scattering amplitudes in six dimensions that directly extend to maximal supersymmetry. We find new ingredients for integrands for super Yang-Mills theory, 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 supertwistor description of 6D superambitwistor space. 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. They give a quite distinct framework from that of Cachazo et al.; in particular, the formulas do not change character from even to odd numbers of particles.