Electroweak clouds supported by magnetically charged black holes: Analytic treatment along the existence-line

It has recently been revealed [R. Gervalle and M. S. Volkov, Phys. Rev. Lett. (2024)] that magnetically charged black holes of the composed Einstein-Weinberg-Salam field theory can support bound-state hairy configurations of electroweak fields. In the present paper we study, using {\it analytical} techniques, the physical and mathematical properties of the supported linearized electroweak fields (spatially regular electroweak ‘clouds’) in the dimensionless large-mass $\mu \equiv m_{\text{w}}{r}_{+}\gg 1$ regime of the composed black-hole-field system (here $m_{\text{w}}$ is the mass of the supported W-boson field and $r_{+}$ is the outer horizon radius of the central supporting black hole). In particular, we derive a remarkably compact formula for the discrete resonance spectrum $\{{\mu }_k(n){\}}_{k=0}^{k=\mathrm{\infty }}$ that characterizes the composed black-hole-linearized-field configurations, where the integer $n\equiv 2Pe\in \mathbb{Z}$ characterizes the discrete charge parameter $P$ of the central magnetic Reissner-Nordström black hole and $e$ is the electron charge. The physical significance of the analytically derived resonant spectrum stems from the fact that, in the dimensionless large-charge $n\gg 1$ regime, the fundamental (largest) eigenvalue ${\mu }_0(n)$ determines the critical existence-line of the composed Einstein-Weinberg-Salam theory, a boundary line that separates hairy magnetic-black-hole-electroweak-field bound-state configurations from bald magnetic Reissner-Nordström black holes.