The modern S-matrix program focuses directly on observables like scattering amplitudes rather than any underlying actions.  One of the greatest successes of the program is the double-copy procedure which, together with color-kinematics duality, reduces scattering in harder theories, like general relativity, to scattering in easier ones, like Yang-Mills.  While a powerful computational tool, the double copy points to much deeper structure whose physical origins remain elusive.

The goal of this talk is to clarify some of this structure by showing 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 directly at the level of fields, as well as an explicit construction of the kinematic algebra and associated kinematic current.  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/p^2 propagators are replaced by covariant 1/D^2.

As an application, I give closed-form, analytic expressions for all tree-level color-kinematic numerators in Yang-Mills and NLSM for any number of particles in any spacetime dimension.  By virtue of the double copy, these constitute explicit formulas for all tree-level scattering amplitudes in Yang-Mills, general relativity, NLSM, special Galileon, and Born-Infeld.