The kinematic algebras from the scattering equations

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.

Sudakov representation of the Cachazo-He-Yuan scattering equations formalism

“…This exhibits the double-copy structure of graviton amplitudes, already found in many other physical setups [30][31][32][33][34][35]. Similarly, biadjoint scalar amplitudes can be obtained from eq.…”

BCJ numerators from reduced Pfaffian

“…1 However, it is hard to find a well-controlled way to write down the final result of the expansion for arbitrary number of particles. The construction of BCJ numerators has been studied from the kinematic algebra [11] and the reduction of CHY integrals [12,13].…”

New ambitwistor string theories

“…Regarding the relation to previous work, we should mention that a double copy construction for Einstein-Yang-Mills amplitudes was first presented in [27] for the single trace contribution, and in [28] for the complete amplitude, with results also at loop level. These double copy constructions are based on the colour-kinematics duality [26,29], whose relation to the scattering equations has been explored in [6,30,31].…”