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

TL;DR

The paper demonstrates a classical double-copy construction by computing the YM-based effective action for two color-charged masses, applying a BCJ-like replacement to obtain the dilaton-gravity effective action, and validating the result with an independent DG calculation. It shows agreement at next-to-leading order in the weak-field expansion and reproduces the scalar-tensor theory Hamiltonian in the post-Newtonian limit, thereby establishing a proof of concept for using BCJ duality beyond scattering amplitudes in classical gravity. The approach leverages a trivalent worldline graph representation and field redefinitions to manage complexity, hinting at substantial simplifications for higher-order computations in gravitational physics. This work broadens the applicability of the double-copy framework to gauge-covariant, classical gravitational dynamics and motivates future extensions to radiation, spin, and higher perturbative orders.

Abstract

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.