Abstract
We present a world-sheet formula for all tree level scattering amplitudes, in all trace sectors, of four dimensional 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 Tmathbb{M} valued fermion correlators on the sphere, which does not require the scattering equations to hold, and is a consequence of the isomorphism Tmathbb{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.
Highlights
Moduli space which determine the states and interactions look very different in the CHY and twistor representations
We present a world-sheet formula for all tree level scattering amplitudes, in all trace sectors, of four dimensional N ≤ 4 supersymmetric Einstein-Yang-Mills theory, based on the refined scattering equations
This generalizes previously known formulas for alltrace 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
Summary
It is well known that in four dimensions the scattering equations split into R-charge sectors, known as N k−2MHV sectors. We outline the idea of the proof here and point the interested reader to section A for details: at first glance it seems as though the right hand side depends on the splitting of the 2n points into the two halves b, bc and b, bc respectively, which would be at odds with the manifest S2n antisymmetry of the Pfaffian on the left. This tension is resolved by the surprising fact that the combination det ij zi−zj i∈b j∈bc (2.3). We demonstrate the chiral splitting formula eq (2.1) by translating various CHY formulae into 4d spinor helicity variables
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