Covariant Color-Kinematics Duality

TL;DR

Cheung and Mangan develop a field-level, covariant form of color-kinematics duality by reformulating dynamics in terms of currents and field strengths, enabling a covariant double copy that maps BAS/GBAS, NLSM, YM, SG, BI, and GR. Central ideas include covariant propagators with $1/D^2$, a diffeomorphism-based kinematic algebra for NLSM, and a Lorentz-based kinematic algebra for the $F^3$ theory, yielding closed-form BCJ numerators and field-strength decompositions that produce all tree-level amplitudes across YM, GR, NLSM, SG, and BI. A covariant double copy then relates GR to EYM ⊗ F^3 and YM to GBAS ⊗ F^3, with a classical variant tying GR solutions to YM in curved backgrounds via tetrads. The work provides analytic, gauge-invariant amplitude formulas and reveals a structured, covariant route to the double copy beyond standard amplitude-level constructions. It opens avenues for loops, higher-dimension operators, SUSY extensions, and curved-space generalizations within a unified EOM-centric framework.

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.

Figures (2)

• Figure 1: A typical Feynman diagram that might appear in GBAS theory, where the solid and curly lines depict biadjoint scalars and gauge fields, respectively. This has a dual interpretation in YM theory, where the solid and curly lines depict field strengths and gauge fields, respectively. The root leg is depicted on the left, with all leaf legs to the right.
• Figure 2: The three- and four-point scattering amplitudes of YM theory derived from the one-point correlator of the field strength in Eq. \ref{['Greens_function_YM']} using the Feynman rules derived from the equation of motion in Eq. \ref{['EOM_YM']}. The solid and curly lines depict field strengths and gauge fields, respectively. Embedded within in each diagram is a subdiagram of solid lines describing a field strength that branches purely through $F^3$ interactions. Like minimally coupled scalars, these field strengths repeatedly emit gauge fields that in turn cascade down into other gauge fields via the nonlinear interactions of YM theory. Since the gauge fields never branch back into field strengths, they are effectively background fields and induce no back-reaction. Note that the root leg is always a field strength, while the leaf legs can be either gauge fields or field strengths, since these are generated at asymptotic infinity by the source and its derivative, respectively.