Kerr black holes with self-interacting scalar hair: Hairier but not heavier

The maximal Arnowitt-Deser-Misner (ADM) mass for (mini)boson stars (BSs)---gravitating solitons of Einstein's gravity minimally coupled to a free, complex, mass μ, Klein-Gordon field---is MADMᵐᵃˣ~MPl²/μ. Adding quartic self-interactions to the scalar field theory, described by the Lagrangian LI=λ|Ψ|⁴, the maximal ADM mass becomes MADMᵐᵃˣ~√λMPl³/μ². Thus, for mini-BSs, astrophysically interesting masses require ultralight scalar fields, whereas self-interacting BSs can reach such values for bosonic particles with Standard Model range masses. We investigate how these same self-interactions affect Kerr black holes with scalar hair (KBHsSH) [C. A. R. Herdeiro and E. Radu, Kerr Black Holes with Scalar Hair, Phys. Rev. Lett. 112, 221101 (2014).], which can be regarded as (spinning) BSs in stationary equilibrium with a central horizon. Remarkably, whereas the ADM mass scales in the same way as for BSs, the horizon mass MH does not increases with the coupling λ, and, for fixed μ, it is maximized at the ``Hod point,'' corresponding to the extremal Kerr black hole obtained in the vanishing hair limit. This mass is always MHᵐᵃˣ~MPl²/μ. Thus, introducing these self-interactions, the black hole spacetimes may become considerably ``hairier'' but the trapped regions cannot become ``heavier.'' We present evidence that this observation also holds in a model with LI=β|Ψ|⁶-λ|Ψ|⁴; if it extends to general scalar field models, KBHsSH with astrophysically interesting horizon masses require ultralight scalar fields. Their existence, therefore, would be a smoking gun for such (beyond the Standard Model) particles.