The Reissner-Nordstr\"om-Tangherlini solution from scattering amplitudes of charged scalars

The metric and the electromagnetic potential generated by a static, spherically symmetric charged massive object in any dimension are given by the Reissner-Nordstr\"om-Tangherlini solution. We derive the expansion of this solution up to third post-Minkowskian order by computing the classical contribution of scattering amplitudes describing the emission of either a graviton or a photon from a massive charged scalar field up to two loops. In four and five dimensions these amplitudes develop ultraviolet divergences that are cancelled by higher-derivative counterterms in a way that generalises what was recently shown to happen in the chargeless case. This renormalisation procedure produces logarithmic terms that match exactly those produced in the post-Minkowskian expansion of the classical solution in de Donder gauge.