Algebras for Amplitudes
N. E. J. Bjerrum-Bohr, Poul H. Damgaard, Ricardo Monteiro, Donal O'Connell
TL;DR
The paper presents a systematic framework to recast tree-level gauge-theory amplitudes in a color-kinematics dual form using cubes-vertex constructions, guided by KK and BCJ relations. It builds a basis of auxiliary amplitudes from cubic-vertex theories with dual numerators, enabling gauge-theory amplitudes to be expressed as CK-dual linear combinations with symmetric coefficients. By exploiting the dual structure, it derives a dual-amplitude decomposition and shows how gravity amplitudes arise as a double copy of gauge-theory data, with kinematic traces playing the role of color traces. The work also discusses how finite-dimensional auxiliary algebras (e.g., SU(N)) can realize the kinematic algebra, offering practical paths to compute and interpret gravity amplitudes within a CK-dual paradigm.
Abstract
Tree-level amplitudes of gauge theories are expressed in a basis of auxiliary amplitudes with only cubic vertices. The vertices in this formalism are explicitly factorized in color and kinematics, clarifying the color-kinematics duality in gauge theory amplitudes. The basis is constructed making use of the KK and BCJ relations, thereby showing precisely how these relations underlie the color-kinematics duality. We express gravity amplitudes in terms of a related basis of color-dressed gauge theory amplitudes, with basis coefficients which are permutation symmetric.