  We derive the static Schwarzschild-Tangherlini metric by extracting the
classical contributions from the multi-loop vertex functions of a graviton
emitted from a massive scalar field. At each loop orders the classical
contribution is proportional to a unique master integral given by the massless
sunset integral. By computing the scattering amplitudes up to three-loop order
in general dimension, we explicitly derive the expansion of the metric up to
the fourth post-Minkowskian order $O(G_N^4)$ in four, five and six dimensions.
There are ultraviolet divergences that are cancelled with the introduction of
higher-derivative non-minimal couplings. The standard Schwarzschild-Tangherlini
is recovered by absorbing their effects by an appropriate coordinate
transformation induced from the de Donder gauge condition.
