Tree-level color–kinematics duality implies loop-level color–kinematics duality up to counterterms

Abstract

Color–kinematics (CK) duality is a remarkable symmetry of gluon amplitudes that is the key to the double copy which links gauge theory and gravity amplitudes. Here we show that the complete Yang–Mills action itself, including its gauge-fixing and ghost sectors required for quantization, can be recast to manifest CK duality using a series of field redefinitions and gauge choices. Crucially, the resulting loop-level integrands are automatically CK-dual, up to potential Jacobian counterterms required for unitarity. While these counterterms may break CK duality, they exist, are unique and, since the tree-level is unaffected, may be deduced from the action or the integrands. Consequently, CK duality is a symmetry of the action like any other symmetry, and it is anomalous in a controlled and mostly harmless sense. Our results apply to any theory with CK-dual tree-level amplitudes. We also show that two CK duality-manifesting parent actions may be factorized and fused into a consistent quantizable offspring, with the double copy as the prime example. This provides a direct proof of the double copy to all loop orders.