Homotopy double copy and the Kawai-Lewellen-Tye relations for the non-abelian and tensor Navier-Stokes equations

TL;DR

This paper explores the double copy structure of a non-abelian Navier-Stokes equation with colour-kinematics duality, demonstrating how to construct its solutions and scattering amplitudes using homotopy algebra and Berends-Giele currents.

Contribution

It provides a new homotopy algebraic perspective on the double copy formulation of non-abelian Navier-Stokes equations, including explicit construction of currents and KLT relations.

Findings

Double copy can be realized at the level of perturbiner expansions.

Colour-dressed currents can be used to generate double copied currents.

Tree-level amplitudes satisfy Kawai-Lewellen-Tye relations.

Abstract

Recently, a non-abelian generalisation of the Navier-Stokes equation that exhibits a manifest duality between colour and kinematics has been proposed by Cheung and Mangan. In this paper, we offer a new perspective on the double copy formulation of this equation, based on the homotopy algebraic picture suggested by Borsten, Kim, Jur\v{c}o, Macrelli, Saemann, and Wolf. In the process, we describe precisely how the double copy can be realised at the level of perturbiner expansions. Specifically, we will show that the colour-dressed Berends-Giele currents for the non-abelian version of the Navier-Stokes equation can be used to construct the Berends-Giele currents for the double copied equation by replacing the colour factors with a second copy of kinematic numerators. We will also show a Kawai-Lewellen-Tye relation stating that the full tree-level scattering amplitudes in the latter can be written as a product of tree-level colour ordered partial amplitudes in the former.