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

The Schwarzschild-Tangherlini metric from scattering amplitudes in various dimensions