Abstract
We elaborate on the color–kinematics duality for off-shell diagrams in gauge theories coupled to matter, by investigating the scattering process gg→ss,qq¯,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→qq¯g.
Highlights
We study the role of color-kinematics duality within off-shell currents, which enter the construction of both higher-multiplicity tree-level and multi-loop ampli
We focus on four-point tree-level amplitudes, showing that C/K-duality can be established for all interactions involving particles propagating within the four dimensional formulation (FDF) framework
In this letter we investigated, from a diagrammatic point of view, the off-shell color-kinematics duality for amplitudes in gauge theories coupled with matter in four as well as d dimensions, within the Four Dimensional Formulation variant of the Four Dimensional Helicity scheme
Summary
The Jacobi relation for the numerators in axial gauge, which, in the case of on-shell tree-amplitudes, is identically zero and resolves the C/K-duality, in the off-shell case, can be expressed in terms of contact interactions, which we explicitly identify for the first time for the processes at hand and represent one of the main result of this communication This decomposition, developed in the canonical formalism of Feynman diagrams, shows that the C/K-duality for high-multiplicity diagrams naturally holds when the four particles entering the Jacobi combination are cut, since, in this case, the contribution of the subdiagrams trivially vanishes. We discuss how our result, which provides a precise identification of the anomalies which should be absorbed into the redefinition of the trivalent numerators, can be used, together with generalized gauge transformations [1, 50], in order to re-shuffle contact terms between diagrams and build on-shell C/K-dual representations for higherpoint tree-level amplitudes. Algebraic manipulations and numerical evaluations have been carried out by using the mathematica packages FeynCalc [55] and S@M [56]
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