Breakdown of the classical double copy for the effective action of dilaton-gravity at NNLO

We demonstrate that a recently proposed classical double-copy procedure to construct the effective action of two massive particles in dilaton-gravity from the analogous problem of two color-charged particles in Yang-Mills gauge theory fails at next-to-next-to-leading order perturbative expansions, i.e., in the third order of the post-Minkowskian and the second order in the post-Newtonian expansions.