Improved Analytic Solution of Black Hole Superradiance

The approximate solution of the Klein-Gordon equation for a real scalar field of mass $\mu$ in the geometry of a Kerr black hole obtained by Detweiler \cite{Detweiler:1980uk} is widely used in the analysis of the stability of black holes as well as the search of axion-like particles. In this work, we confirm a missing factor $1/2$ in this solution, which was first identified in Ref.~\cite{Pani:2012bp}. The corrected result has strange features that put questions on the power-counting strategy. We solve this problem by adding the next-to-leading order (NLO) contribution. Compared to the numerical results, the NLO solution reduces the percentage error of the LO solution by a factor of 2 for all important values of $r_g \mu$. Especially the percentage error is $\lesssim 10\%$ in the region of $r_g\mu \lesssim 0.35$. The NLO solution also has a compact form and could be used straightforwardly.