Improved analytic solution of black hole superradiance

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