Off-shell Currents and Color-Kinematics Duality

TL;DR

The paper develops a diagrammatic framework for off-shell color-kinematics (C/K) duality in gauge theories with matter, showing that Jacobi violations arise from subdiagrams and vanish when external legs are on-shell. An off-shell decomposition in axial gauge isolates anomaly terms into a finite set of effective vertices, enabling a constructive procedure to obtain dual numerators for higher-point amplitudes in four dimensions (via a Four Dimensional Formulation) and in $d$ dimensions (through the FDH/FDF scheme). The authors validate the approach with explicit tree-level examples, notably $gg\to q\bar q g$, and extend the analysis to dimensionally regulated amplitudes, providing a path toward loop-level unitarity calculations. The work clarifies the origin of C/K violations, derives consistency constraints among kinematic factors and anomalies, and establishes a practical algorithm for generating dual representations with controlled gauge freedom, including monodromy-like relations.

Abstract

We elaborate on the color-kinematics duality for off-shell diagrams in gauge theories coupled to matter, by investigating the scattering process $gg\to ss, q\bar q, gg$, and show that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a color-kinematics violating term due to the contributions of sub-graphs only. Such anomaly vanishes when the four particles connected by the Jacobi relation are on their mass shell with vanishing squared momenta, being either external or cut particles, where the validity of the color-kinematics duality is recovered. We discuss the role of the off-shell decomposition in the direct construction of higher-multiplicity numerators satisfying color-kinematics identity in four as well as in $d$ dimensions, for the latter employing the Four Dimensional Formalism variant of the Four Dimensional Helicity scheme. We provide explicit examples for the QCD process $gg\to q\bar{q}g$.

Figures (11)

• Figure 1: The Jacobi combination: it can be applied both to Lie group structure constants and to the kinematic part of numerators.
• Figure 2: Embedding of the Jacobi combination into either higher-point or multi-loop diagrams.
• Figure 3: Jacobi combination for $gg \to ss$.
• Figure 4: Off-shell color-kinematics duality for gluons and scalars. The Jacobi combination of tree-level numerators (l.h.s) is expressed in terms of subdiagrams only (r.h.s.).
• Figure 5: Jacobi combination for $gg\to q\bar{q}$.