Chiral Splitting and $\mathcal N = 4$ Einstein--Yang--Mills Tree Amplitudes in 4d

We present a world-sheet formula for all tree level scattering amplitudes, in all trace sectors, of four dimensional $\mathcal{N} \leq 4$ supersymmetric Einstein-Yang-Mills theory, based on the refined scattering equations. This generalizes previously known formulas for all-trace purely bosonic, or supersymmetric single-trace amplitudes. We find this formula by applying a new chiral splitting formula for all CHY Pfaffians in 4d, into two determinants, of positive and negative helicity respectively. The splitting of CHY Pfaffians is shown to be a special case of the splitting of $T\mathbb{M}$ valued fermion correlators on the sphere, which does not require the scattering equations to hold, and is a consequence of the isomorphism $T\mathbb{M} \simeq \mathbb{S}^+ \otimes \mathbb{S}^-$ between the tangent bundle of Minkowski space and the left- and right-handed spin bundles. We present and prove this general splitting formula.