  There are strong interests in considering ultra-light scalar fields
(especially axion) around a rapidly rotating black hole because of the
possibility of observing gravitational waves from axion condensate (axion
cloud) around black holes. Motivated by this consideration, we propose a new
method to study the dynamics of an ultra-light scalar field with
self-interaction around a rapidly rotating black hole, which uses the dynamical
renormalization group method. We find that for relativistic clouds, saturation
of the superradiant instability by the scattering of the axion due to the
self-interaction does not occur in the weakly non-linear regime when we
consider the adiabatic growth of the cloud from a single superradiant mode.
This may suggest that for relativistic axion clouds, an explosive phenomenon
called the Bosenova may inevitably happen, at least once in its evolutionary
history.
