Scalar clouds in charged stringy black hole-mirror system

It was reported that massive scalar fields can form bound states around Kerr black holes (Herdeiro and Radu, Phys. Rev. Lett. 112:221101, 2014). These bound states are called scalar clouds; they have a real frequency $$\omega =m\Omega _\mathrm{H}$$, where $$m$$ is the azimuthal index and $$\Omega _\mathrm{H}$$ is the horizon angular velocity of Kerr black hole. In this paper, we study scalar clouds in a spherically symmetric background, i.e. charged stringy black holes, with the mirror-like boundary condition. These bound states satisfy the superradiant critical frequency condition $$\omega =q\Phi _\mathrm{H}$$ for a charged scalar field, where $$q$$ is the charge of the scalar field, and $$\Phi _\mathrm{H}$$ is the horizon’s electrostatic potential. We show that, for the specific set of black hole and scalar field parameters, the clouds are only possible for specific mirror locations $$r_\mathrm{m}$$. It is shown that analytical results of the mirror location $$r_\mathrm{m}$$ for the clouds perfectly coincide with numerical results in the $$qQ\ll 1$$ regime. We also show that the scalar clouds are also possible when the mirror locations are close to the horizon. Finally, we provide an analytical calculation of the specific mirror locations $$r_\mathrm{m}$$ for the scalar clouds in the $$qQ\gg 1$$ regime