Abstract
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 × overset{sim }{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.
Highlights
In a purely classical setting, the bi-adjoint scalar has served as the basis to find maps between classical solutions in gauge and gravity at both non-perturbative [6,7,8,9] and perturbative [10,11,12,13] levels
We show how a global G×Gsymmetry group may be realized by worldline degrees of freedom
Seeds of color/kinematic relations, and corresponding double copy structure that reproduces gravity amplitudes from the gauge ones, can be traced back to the origin of string theory, when it was noticed that the Veneziano amplitude could be related in a simple way to the Virasoro-Shapiro amplitude
Summary
A recurrent topic in the worldline approach to quantum field theory is the derivation of parameter integrals which sum up whole classes of Feynman diagrams. With a color factor associated to a different symmetry group G, carrying its own color variables dα (τ ) and dα (τ ) taken in the fundamental and antifundamental representation, respectively This replacement produces the vertex operator V (0)[k, a, α; x, c, c, d, d] = dτ ca (τ )(T a)a b cb (τ ) dα (τ )(T α)α β dβ (τ ) eik·x(τ) , (2.7). We see that this replacement produces a vertex operator for the coupling of the particle to the plane wave of a bi-adjoint scalar, reminding the classical double copy of [6] This vertex operator can be used to obtain an effective action of the form (2.1) for a particle coupled to a background bi-adjoint scalar field Ψ(0) ∼ Φaα
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