BCJ relations from a new symmetry of gauge-theory amplitudes

TL;DR

This work reveals a new color-factor symmetry for gauge-theory amplitudes, where the color factors are shifted by momentum-dependent amounts and the full amplitude remains invariant. The authors connect this symmetry to BCJ relations and color-kinematic duality through the radiation vertex expansion, derive generalized constraints on kinematic numerators, and demonstrate the symmetry across gluonic and matter-containing amplitudes, including loop-level generalizations. The results explain BCJ relations from a purely Lagrangian, symmetry-based perspective and extend to bi-adjoint scalar theory, yielding null eigenvectors of the propagator matrix and rank reductions. The loop extension shows that if a theory admits color-kinematic dual numerators, one-loop amplitudes exhibit similar color-factor invariance and integrand relations, underscoring the broad applicability of the color-factor symmetry in gauge theories. Overall, the paper strengthens the foundational link between gauge invariance, color algebra, and kinematic structures, offering a versatile framework for exploring BCJ-type relations in diverse theories and at higher orders.

Abstract

We introduce a new set of symmetries obeyed by tree-level gauge-theory amplitudes involving at least one gluon. The symmetry acts as a momentum-dependent shift on the color factors of the amplitude. Using the radiation vertex expansion, we prove the invariance under this color-factor shift of the $n$-gluon amplitude, as well as amplitudes involving massless or massive particles in an arbitrary representation of the gauge group with spin zero, one-half, or one. The Bern-Carrasco-Johansson relations are a direct consequence of this symmetry. We also introduce the cubic vertex expansion of an amplitude, and use it to derive a generalized-gauge-invariant constraint on the kinematic numerators of the amplitude. We show that the amplitudes of the bi-adjoint scalar theory are invariant under the color-factor symmetry, and use this to derive the null eigenvectors of the propagator matrix. We generalize the color-factor shift to loop level, and prove the invariance under this shift of one-loop $n$-gluon amplitudes in any theory that admits a color-kinematic-dual representation of numerators. We show that the one-loop color-factor symmetry implies known relations among the integrands of one-loop color-ordered amplitudes.

Figures (7)

• Figure 1: Attaching a gluon to the legs of a cubic vertex. These form parts of the color factors $c_{(1)}$, $c_{(2)}$, and $c_{(3)}$, respectively.
• Figure 2: Diagram for the half-ladder color factor ${\bf c} _{1 \gamma(2) \cdots \gamma(n-1) n }$.
• Figure 3: Diagrams with color factors $c_{1 \cdots \sigma(b-1) [a\sigma(b)] \sigma(b+1) \cdots n}$ and ${\bf c} _{1\cdots \sigma(b-1)a\sigma(b) \cdots n}$
• Figure 4: Some of the diagrams to which a gluon is attached to obtain the one-loop four-point cubic decomposition.
• Figure 5: Diagrams ${[14]23}$, ${1[24]3}$, ${12[34]}$, ${1234}$, ${1423}$, and ${1243}$.