Covariant Color-Kinematics Duality

TL;DR

This paper demonstrates that color-kinematics duality is a fundamental property of equations of motion, leading to new formulations and explicit constructions of scattering amplitudes across various theories including gravity, gauge, and scalar models.

Contribution

It introduces covariant color-kinematics duality, enabling a field-level double copy and explicit amplitude formulas for multiple theories, extending the classical double copy framework.

Findings

Reveals covariant color-kinematics duality in YM theory.

Provides explicit formulas for all tree-level amplitudes in several theories.

Derives general relativity from YM and $F^3$ theory via double copy.

Abstract

We show that color-kinematics duality is a manifest property of the equations of motion governing currents and field strengths. For the nonlinear sigma model (NLSM), this insight enables an implementation of the double copy at the level of fields, as well as an explicit construction of the kinematic algebra and associated kinematic current. As a byproduct, we also derive new formulations of the special Galileon (SG) and Born-Infeld (BI) theory. For Yang-Mills (YM) theory, this same approach reveals a novel structure -- covariant color-kinematics duality -- whose only difference from the conventional duality is that $1/\Box$ is replaced with covariant $1/D^2$. Remarkably, this structure implies that YM theory is itself the covariant double copy of gauged biadjoint scalar (GBAS) theory and an $F^3$ theory of field strengths encoding a corresponding kinematic algebra and current. Directly applying the double copy to equations of motion, we derive general relativity (GR) from the product of Einstein-YM and $F^3$ theory. This exercise reveals a trivial variant of the classical double copy that recasts any solution of GR as a solution of YM theory in a curved background. Covariant color-kinematics duality also implies a new decomposition of tree-level amplitudes in YM theory into those of GBAS theory. Using this representation we derive a closed-form, analytic expression for all BCJ numerators in YM theory and the NLSM for any number of particles in any spacetime dimension. By virtue of the double copy, this constitutes an explicit formula for all tree-level scattering amplitudes in YM, GR, NLSM, SG, and BI.