BCJ relations from a new symmetry of gauge-theory amplitudes

TL;DR

This paper uncovers a new symmetry in gauge-theory amplitudes that leads to BCJ relations, extending to loop levels and involving various particles, with implications for amplitude structures and constraints.

Contribution

It introduces a novel color-factor symmetry in gauge-theory amplitudes, deriving BCJ relations and generalizing the symmetry to loop levels and different particle types.

Findings

Proves invariance of tree-level amplitudes under a new color-factor shift.

Derives BCJ relations as a consequence of this symmetry.

Extends the symmetry to one-loop amplitudes, revealing new relations.

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.