It has recently been revealed [Gervalle and Volkov, Phys. Rev. Lett. 133, 171402 (2024)] that spatially regular bound-state matter configurations which are made of electroweak fields can be supported by magnetically charged black holes of the Einstein-Weinberg-Salam field theory. Intriguingly, it has been demonstrated numerically that the dimensionless mass parameter $\mu \equiv m_{\mathrm{w}}{r}_{+}$ of cloudy black-hole-field configurations, magnetically charged Reissner-Nordström black holes that support linearized electroweak fields, is characterized by a discrete resonant spectrum $\{{\mu }_k(n){\}}_{k=0}^{k=\infty }$ (here $\{r_{+},n\}$ are respectively the horizon radius and the discrete magnetic parameter of the central black hole and $m_{\mathrm{w}}$ is the mass of the supported W-boson field). In the present paper we explore the excited resonant spectrum of the Einstein-Weinberg-Salam field theory. In particular, using analytical techniques we derive a remarkably compact formula for the low-mass (${\mu }_k\ll 1$) resonant spectrum of composed weakly magnetized-black-hole-linearized-electroweak-field bound-state configurations. Interestingly, it is shown that the excited black-hole-field resonances are characterized by the simple universal ($\{m_{\mathrm{w}},r_{+}\}$-independent) relation ${\mu }_{k+1}/{\mu }_k=e^{-2\pi /\sqrt{2n-1}}$. Finally, it is explicitly shown that our analytically derived formula for the discrete resonance spectrum of the composed black-hole-field system agrees remarkably well with the corresponding numerical data that recently appeared in the physics literature.

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• Gravitation