Ring formation from black hole superradiance through repeated particle production on bound orbits

Ultralight bosonic fields around a rotating black hole can extract energy and angular momentum through the superradiant instability and form a dense cloud. We investigate the scenario involving two scalar fields, $ϕ$ and $χ$, with a coupling term 1/2λϕχ², which is motivated by the multiple-axion framework. The ultralight scalar $ϕ$ forms a cloud that efficiently produces $χ$ particles non-perturbatively via parametric resonance, with a large mass hierarchy, μ_χ μ_ϕ. Rather than escaping the system as investigated by previous studies, these $χ$ particles remain bound, orbiting the black hole. Moreover, the particle production occurs primarily at the peak of the cloud's profile, allowing $χ$ particles in quasi-circular orbits to pass repeatedly through resonant regions, leading to sustained amplification. This selective process naturally forms a dense ring of $χ$ particles, with a mass ratio to the cloud fixed by (μ_ϕ/μ_χ)². Our findings reveal a novel mechanism for generating bound-state particles via parametric resonance, which also impacts the evolution of the cloud. This process can be probed through its imprint on binary dynamics and its gravitational-wave signatures.