Color-Kinematics Duality with Minimal Deformation: Two-Loop Four-Gluon Amplitudes in Pure Yang-Mills Revisited

The conjectured duality between color and kinematics has significantly advanced our understanding of both gauge and gravitational theories. However, constructing numerators that manifest the color-kinematics (CK) duality, even for the two-loop four-gluon amplitude in pure Yang-Mills, has been challenging. In this paper, we revisit this amplitude and show that the difficulty of applying CK duality can be overcome by introducing a simple deformation. Our approach distinguishes itself from previous studies by maximizing the use of off-shell CK duality while maintaining a compact ansatz. In particular, the deformation we introduce satisfies a subset of off-shell dual Jacobi relations. The resulting numerators are presented in $d$-dimensionally Lorentz invariant local form and are applicable to all helicities of external gluons. The solution we provide can be directly employed to construct the corresponding gravitational amplitude through double copy. Our findings suggest a novel and efficient strategy for constructing high-loop gauge and gravitational amplitudes using CK duality.