Applying Color-Kinematics Duality in Pure Yang-Mills at Three Loops

TL;DR

This paper demonstrates the first successful application of color-kinematics duality at three loops in pure Yang-Mills theory, providing a new method for constructing complex amplitudes in non-supersymmetric gauge theories.

Contribution

It extends the minimal deformation approach to three-loop calculations, achieving a compact, Lorentz-invariant, and local representation of numerators in non-supersymmetric Yang-Mills theory.

Findings

Achieved a compact, Lorentz-invariant numerator representation at three loops.

Extended the minimal deformation approach to non-supersymmetric Yang-Mills theory.

Provided a promising method for constructing three-loop amplitudes and gravity via double copy.

Abstract

We present the first application of color-kinematics (CK) duality at the three-loop level in non-supersymmetric pure Yang-Mills (YM) theory. Building on the minimal deformation approach introduced in \cite{Li:2023akg}, we extend its use to the three-loop Sudakov form factor. Although three classes of unitarity cuts fail under the globally off-shell CK-dual ansatz, a compact and elegant solution is achieved by deforming a single master numerator. The final numerators exhibit Lorentz invariance in $d$ dimensions and take a local form. This method harnesses CK duality's full potential by enforcing a subset of off-shell dual Jacobi identities for the deformation, offering a promising path toward constructing three-loop amplitudes in non-supersymmetric YM theory and gravity through CK duality and double copy.