Tree-Level Color-Kinematics Duality Implies Loop-Level Color-Kinematics Duality up to Counterterms

TL;DR

This paper demonstrates that the Yang-Mills action can be reformulated to manifest color-kinematics duality at all loop levels, establishing a direct proof of the double copy relation between gauge and gravity theories.

Contribution

It shows how to recast the complete Yang-Mills action to manifest CK duality, including loop integrands and gauge sectors, and proves the double copy at all loop orders.

Findings

Loop integrands are automatically CK-dual up to counterterms.

CK duality can be embedded at the action level, making it a symmetry.

Provides a method to fuse CK duality manifest actions into consistent theories.

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.