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 [Detweiler, Phys. Rev. D 22, 2323 (1980)] is widely used in the analysis of the stability of black holes as well as the search of axionlike particles. In this work, we confirm a a missing factor 1/2 in this solution, which was first identified in [Pani et al., Phys. Rev. D 86, 104017 (2012)]. 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 rg mu. Especially the percentage error is & LSIM;10% in the region of rg mu similar to 0.35. The NLO solution also has a compact form and could be used straightforwardly.