Color-factor symmetry and BCJ relations for QCD amplitudes

TL;DR

The paper demonstrates that BCJ relations for tree-level amplitudes with gluons and fundamental matter arise directly from a color-factor symmetry: invariance under momentum-dependent shifts of color factors in the cubic decomposition. By formalizing an $(n-3)!/k!$-parameter shift space per external gluon and establishing the Johansson-Ochirov color-factor basis (Melia basis), the authors show that these shifts imply BCJ relations among Melia primitives and produce gauge-invariant constraints on kinematic numerators via a cubic vertex expansion. The results extend color-kinematic duality ideas to amplitudes with multiple fundamental pairs, clarifying why BCJ relations exist only when gluons are present and connecting the symmetry to underlying gauge and Poincaré invariances. This symmetry-based perspective provides a unifying, Lagrangian-grounded explanation for BCJ relations in QCD-like theories and suggests avenues for loop-level generalizations.

Abstract

Tree-level $n$-point gauge-theory amplitudes with $n-2k$ gluons and $k$ pairs of (massless or massive) particles in the fundamental (or other) representation of the gauge group are invariant under a set of symmetries that act as momentum-dependent shifts on the color factors in the cubic decomposition of the amplitude. These symmetries lead to gauge-invariant constraints on the kinematic numerators. They also directly imply the BCJ relations among the Melia-basis primitive amplitudes previously obtained by Johansson and Ochirov.

Figures (6)

• Figure 1: Diagrammatic form of the operator $\Xi^\textsf{a}_l$.
• Figure 2: Color factors $c_1$ through $c_5$ for ${\cal A} _{5,2}$.
• Figure 3: Color factor $C_{13542}$ for the amplitude ${\cal A} _{5,2}$.
• Figure 4: Commutator $C_{\cdots [cn] \cdots}$ for gluon $g_c$.
• Figure 5: Commutator $C_{\cdots [cn] \cdots}$ for ${\bar{\psi}}_c$.