Color-Kinematics Duality for QCD Amplitudes

TL;DR

This paper extends color-kinematics (CK) duality to tree-level QCD amplitudes with massive, distinctly flavored quarks by introducing a Melia-basis color decomposition that is valid for arbitrary gauge groups and representations. It demonstrates that kinematic numerators can be chosen to satisfy Jacobi-like identities mirroring color factors, yielding BCJ relations, and identifies a reduced BCJ basis of size $\beta(n,k)=(n-3)!(2k-2)/k!$ for $k\ge2$ (and $(n-3)!$ for $k=0,1$). The work provides explicit all-multiplicity formulas and a mixed decomposition that combine color- and kinematics-based bases, strengthening evidence that QCD obeys CK duality at tree level and suggesting extensions to SUSY and D-dimensional theories as well as potential loop implications via unitarity. Overall, the results offer a compact, representation-agnostic framework for high-multiplicity amplitudes and could improve analytic and numerical approaches to precision QCD phenomenology.

Abstract

We show that color-kinematics duality is present in tree-level amplitudes of quantum chromodynamics with massive flavored quarks. Starting with the color structure of QCD, we work out a new color decomposition for n-point tree amplitudes in a reduced basis of primitive amplitudes. These primitives, with k quark-antiquark pairs and (n-2k) gluons, are taken in the (n-2)!/k! Melia basis, and are independent under the color-algebra Kleiss-Kuijf relations. This generalizes the color decomposition of Del Duca, Dixon, and Maltoni to an arbitrary number of quarks. The color coefficients in the new decomposition are given by compact expressions valid for arbitrary gauge group and representation. Considering the kinematic structure, we show through explicit calculations that color-kinematics duality holds for amplitudes with general configurations of gluons and massive quarks. The new (massive) amplitude relations that follow from the duality can be mapped to a well-defined subset of the familiar BCJ relations for gluons. They restrict the amplitude basis further down to (n-3)!(2k-2)/k! primitives, for two or more quark lines. We give a decomposition of the full amplitude in that basis. The presented results provide strong evidence that QCD obeys the color-kinematics duality, at least at tree level. The results are also applicable to supersymmetric and D-dimensional extensions of QCD.

Figures (7)

• Figure 1: Color vertices with planar ordering consistent with the color-stripped Feynman rules.
• Figure 2: Color-algebra relations in the adjoint (a) and fundamental representation (b). The color-kinematics duality requires that the kinematic numerators satisfy the corresponding kinematic-algebra relations, which can be represented by the same graphs.
• Figure 3: Multi-peripheral cubic diagram for the color factors in formulas \ref{['QuarkLineDecomposition']} and \ref{['DDM']}. All permuted legs are gluons, while the horizontal line can be either a quark or a gluon line.
• Figure 4: Feynman diagrams for the six-quark amplitude ${\cal A}^{\text{tree}}_{6,3}(\underline{1},\overline{2},\underline{3},\overline{4},\underline{5},\overline{6})$.
• Figure 5: Feynman diagrams for the four-quark one-gluon amplitude ${\cal A}^{\text{tree}}_{5,2}(\underline{1},\overline{2},\underline{3},\overline{4},5)$.