Kinematic Hopf algebra for amplitudes and form factors

TL;DR

This paper introduces a new kinematic algebra framework for constructing gauge-invariant BCJ numerators in Yang-Mills theory with scalars, enabling closed-form expressions for complex amplitudes and form factors.

Contribution

It develops an algebraic structure that systematically generates BCJ numerators, incorporating flavor and kinematic parts, for arbitrary gluon and scalar configurations.

Findings

Closed-form BCJ numerators for any number of gluons and scalars.

Manifest gauge invariance of constructed numerators.

New relations among numerators derived from the algebra.

Abstract

We propose a kinematic algebra for the Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes and form factors in Yang-Mills theory coupled with bi-adjoint scalars. The algebraic generators of the algebra contain two parts: the first part is simply the flavour factor of the bi-adjoint scalars, and the second part that maps to non-trivial kinematic structures of the BCJ numerators obeys extended quasi-shuffle fusion products. The underlying kinematic algebra allows us to present closed forms for the BCJ numerators with any number of gluons and two or more scalars for both on-shell amplitudes and form factors that involve an off-shell operator. The BCJ numerators constructed in this way are manifestly gauge invariant and obey many novel relations that are inherited from the kinematic algebra.