The Schwarzschild-Tangherlini metric from scattering amplitudes in various dimensions

Understand

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.