Schwarzschild-Tangherlini Metric from Scattering Amplitudes

We present a general framework with which the Schwarzschild-Tangherlini metric of a point particle in arbitrary dimensions can be derived from a scattering amplitude to all orders in the gravitational constant, $G_N$, in covariant gauge (i.e. $R_{\xi }$-gauge) with a generalized de Donder-type gauge function, $G_{\sigma }$. The metric is independent of the covariant gauge parameter $\xi$ and obeys the classical gauge condition $G_{\sigma }=0$. We compute the metric with the generalized gauge choice explicitly to second order in $G_N$ where gravitational self-interactions become important and these results verify the general framework to one-loop order. Interestingly, after generalizing to arbitrary dimension, a logarithmic dependence on the radial coordinate appears in space-time dimension $D=5$.