Cloudy black-hole configurations of the composed Einstein-Yang-Mills-Higgs field theory: Analytic treatment

Non-Abelian black holes that violate the no-hair conjecture are known to exist in the Einstein-Yang-Mills-Higgs field theory. The onset of hair formation in this composed physical system is marked by the presence of ``cloudy'' black-hole spacetimes, magnetically charged Reissner-Nordstr"om black holes that support spatially regular linearized-field configurations. In the present paper, we explore, using analytical techniques, the physical and mathematical properties of the composed black-hole-linearized-field bound-state configurations of the Einstein-Yang-Mills-Higgs theory. In particular, we derive a remarkably compact analytical formula for the discrete resonance spectrum ${{\ensuremath{\mu}}_{k}{}}_{k=1}^{k=\ensuremath{\infty}}$ of the dimensionless mass-radius parameter $\ensuremath{\mu}\ensuremath{\equiv}{m}_{\mathrm{w}}{r}_{+}$ that characterizes the cloudy black-hole-field configurations in the regime $1\ensuremath{\ll}{r}_{+}\ensuremath{\ll}{m}_{\mathrm{w}}^{\ensuremath{-}1}$ of small-mass weakly magnetized black holes (here ${r}_{+}$ is the horizon radius of the central supporting black hole and ${m}_{\mathrm{w}}$ is the mass of the Yang-Mills field due to the Higgs effect). Interestingly, our results reveal that the excited black-hole-field resonances are characterized by the asymptotic $k$-independent dimensionless ratio ${\ensuremath{\mu}}_{k+1}/{\ensuremath{\mu}}_{k}={e}^{\ensuremath{-}2\ensuremath{\pi}/\sqrt{3}}$.