Scattering Equations and KLT Orthogonality

TL;DR

This paper reveals a universal set of scattering equations linking kinematic invariants to puncture data on the n-punctured sphere, valid in any spacetime dimension. It provides an inductive algorithm to obtain all $(n-3)!$ solutions and proves KLT Orthogonality, showing that Parke-Taylor–type vectors from different solutions are orthogonal under the KLT bilinear form. The results underpin a dimension-agnostic framework for constructing gauge and gravity amplitudes, and they illuminate deep connections to string theory in the high-energy limit. Together, these findings propose a powerful, dimension-independent backbone for tree-level massless S-matrix computations and their relations via KLT/BCJ structures.

Abstract

Several recent developments point to the fact that rational maps from n-punctured spheres to the null cone of D dimensional momentum space provide a natural language for describing the scattering of massless particles in D dimensions. In this note we identify and study equations relating the kinematic invariants and the puncture locations, which we call the scattering equations. We provide an inductive algorithm in the number of particles for their solutions and prove a remarkable property which we call KLT Orthogonality. In a nutshell, KLT orthogonality means that "Parke-Taylor" vectors constructed from the solutions to the scattering equations are mutually orthogonal with respect to the Kawai-Lewellen-Tye (KLT) bilinear form. We end with comments on possible connections to gauge theory and gravity amplitudes in any dimension and to the high-energy limit of string theory amplitudes.