The Kinematic Algebras from the Scattering Equations
Ricardo Monteiro, Donal O'Connell
TL;DR
This work develops a Lie-algebraic kinematic framework for the scattering equations, showing that each equation solution defines a self-dual (and an extended anti-self-dual) vertex whose structure constants generate a kinematic algebra. By constructing BCJ numerators from these algebras and employing a reference-particle frame, the authors obtain a canonical, solution-by-solution decomposition of gauge-theory amplitudes and their gravity double-copy, with explicit links to Parke–Taylor factors, Pfaffians, and the Jacobian det'Φ. They prove vanishing properties for X-amplitudes and establish a robust method to derive BCJ numerators n_α and kinematic traces τ that respect Jacobi identities and KLT orthogonality. The framework yields natural decompositions of determinants arising in amplitudes and provides a path toward loop-level generalizations and deeper string-theoretic connections. Overall, the paper integrates scattering equations, colour–kinematics duality, and kinematic algebras to produce manifest BCJ-dual amplitudes and their gravitational counterparts in any dimension.
Abstract
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.