| Summary: | Abstract We present a world-sheet formula for all tree level scattering amplitudes, in all trace sectors, of four dimensional N ≤ 4 $$ \mathcal{N}\le 4 $$ 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 M $$ T\mathbb{M} $$ valued fermion correlators on the sphere, which does not require the scattering equations to hold, and is a consequence of the isomorphism T M ≃ S + ⊗ S − $$ T\mathbb{M}\simeq {\mathbb{S}}^{+}\otimes {\mathbb{S}}^{-} $$ between the tangent bundle of Minkowski space and the left- and right-handed spin bundles. We present and prove this general splitting formula.
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