Kerr-Newman scalar clouds

Massive complex scalar fields can form bound states around Kerr black holes. These bound states---dubbed scalar clouds---are generically nonzero and finite on and outside the horizon; they decay exponentially at spatial infinity, have a real frequency and are specified by a set of integer ``quantum'' numbers $(n,l,m)$. For a specific set of these numbers, the clouds are only possible along a one-dimensional subset of the two-dimensional parameter space of Kerr black holes, called an existence line. In this paper we make a thorough investigation of the scalar clouds due to neutral (charged) scalar fields around Kerr(-Newman) black holes. We present the location of the existence lines for a variety of quantum numbers, their spatial representation and compare analytic approximation formulas in the literature with our exact numerical results, exhibiting a sometimes remarkable agreement.