Worldline description of a bi-adjoint scalar and the zeroth copy

Bi-adjoint scalars are helpful in studying properties of color/kinematics duality and the double copy, which relates scattering amplitudes of gauge and gravity theories. Here we study bi-adjoint scalars from a worldline perspective. We show how a global $G \times \tilde G$ symmetry group may be realized by worldline degrees of freedom. The worldline action gives rise to vertex operators, which are compared to similar ones describing the coupling to gauge fields and gravity, thus exposing the color/kinematics interplay in this framework. The action is quantized by path integrals to find a worldline representation of the one-loop QFT effective action of the bi-adjoint scalar cubic theory. As simple applications, we recover the one-loop beta function of the theory in six dimensions, verifying its vanishing, and compute the self-energy correction to the propagator. The model is easily extendable to that of a particle carrying an arbitrary representation of direct products of global symmetry groups, including the multi-adjoint particle, whose one-loop beta function we reproduce as well.