The maximal ADM mass for (mini-)boson stars (BSs) -- gravitating solitons ofEinstein's gravity minimally coupled to a free, complex, mass $\mu$,Klein-Gordon field -- is $M_{\rm ADM}^{\rm max}\sim M_{Pl}^2/\mu$. Addingquartic self-interactions to the scalar field theory, described by theLagrangian $\mathcal{L}_I=\lambda |\Psi|^4$, the maximal ADM mass becomes$M_{\rm ADM}^{\rm max}\sim \sqrt{\lambda}M_{Pl}^3/\mu^2$. Thus, for mini-BSs,astrophysically interesting masses require ultra-light scalar fields, whereasself-interacting BSs can reach such values for bosonic particles with StandardModel range masses. We investigate how these same self-interactions affect Kerrblack holes with scalar hair (KBHsSH) [1], which can be regarded as (spinning)BSs in stationary equilibrium with a central horizon. Remarkably, whereas theADM mass scales in the same way as for BSs, the \textit{horizon mass} $M_H$does not increases with the coupling $\lambda$, and, for fixed $\mu$, it ismaximized at the "Hod point", corresponding to the extremal Kerr black holeobtained in the vanishing hair limit. This mass is always $M_{\rm H}^{\rm max}\sim M_{\rm Pl}^2/\mu$. Thus, introducing these self-interactions, the blackhole spacetimes may become considerably "hairier" but the trapped regionscannot become "heavier". We present evidence this observation also holds in amodel with $\mathcal{L}_I= \beta|\Psi|^6-\lambda|\Psi|^4$; if it extends to\textit{general} scalar field models, KBHsSH with astrophysically interestinghorizon masses \textit{require} ultra-light scalar fields. Their existence,therefore, would be a smoking gun for such (beyond the Standard Model)particles.
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