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-Nordström 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 $\{{\mu }_k{\}}_{k=1}^{k=\infty }$ of the dimensionless mass-radius parameter $\mu \equiv m_{\mathrm{w}}{r}_{+}$ that characterizes the cloudy black-hole-field configurations in the regime $1\ll r_{+}\ll m_{\mathrm{w}}^{-1}$ 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 ${\mu }_{k+1}/{\mu }_k=e^{-2\pi /\sqrt{3}}$.

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• Gravitation