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

TL;DR

This paper analytically investigates electroweak field configurations supported by magnetically charged black holes, deriving a compact resonance spectrum that delineates the boundary between hairy black holes and bald Reissner-Nordström black holes.

Contribution

It provides an analytical derivation of the resonance spectrum for electroweak clouds around magnetically charged black holes in the large-mass and large-charge regimes.

Findings

Derived a compact formula for the discrete resonance spectrum.

Identified the critical eigenvalue determining the existence boundary of hairy black holes.

Connected the spectrum to the physical boundary separating different black hole configurations.

Abstract

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 $\mu\equiv m_{\text{w}} r_+\gg1$ 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 $\{\mu_k(n)\}_{k=0}^{k=\infty}$ 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-Nordstr\"om 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 $n\gg1$ 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-Nordstr\"om black holes.