Effective action of dilaton gravity as the classical double copy of Yang-Mills theory

We compute the classical effective action of color charges moving along worldlines by integrating out the Yang-Mills gauge field to next-to leading order in the coupling. An adapted version of the Bern-Carrasco-Johansson (BCJ) double-copy construction known from quantum scattering amplitudes is then applied to the Feynman integrands, yielding the prediction for the classical effective action of point-masses in dilaton gravity. We check the validity of the result by independently constructing the effective action in dilaton gravity employing field redefinitions and gauge choices that greatly simplify the perturbative construction. Complete agreement is found at next-to-leading order. Finally, upon performing the post-Newtonian expansion of our result, we find agreement with the corresponding action of scalar-tensor theories known from the literature. Our results represent a proof of concept for the classical double-copy construction of the gravitational effective action and provides another application of a BCJ-like double copy beyond scattering amplitudes