Scattering Equations and KLT Orthogonality

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.

21 pages

References in corpus (13)

• New Relations for Gauge-Theory Amplitudes
• Minimal Basis for Gauge Theory Amplitudes
• The Momentum Kernel of Gauge and Gravity Theories
• Scattering in Three Dimensions from Rational Maps
• Gravity and Yang-Mills Amplitude Relations
• Gravity from Rational Curves
• New Identities among Gauge Theory Amplitudes
• A simple formula for gravitational MHV amplitudes
• A "Twistor String" Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA
• Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
• A Direct Proof of BCFW Recursion for Twistor-Strings
• Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation
• On the Factorisation of the Connected Prescription for Yang-Mills Amplitudes

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