It is worth noting that the term “linearized clouds” is commonly used in the physics literature (see [15, 16, 17, 18, 19] and references therein) to describe spatially regular linearized field configurations supported by central black holes.

We shall use natural units in which $G=c=1$.

See [31] for a recent study of the composed magnetic-black-hole-linearized-electroweak-field cloudy configurations in the regime $m_{\mathrm{w}}{r}_{+}\gg 1$ of large black-hole-field masses.

One may assume, without loss of generality, the relation $n\ge 0$ for the discrete magnetic parameter [which corresponds to $P\ge 0$, see Eq. (5)].

Note that Eq. (12) maps the radial coordinate $r\in [r_{+},\infty ]$ to $y\in [-\infty ,\infty ]$.

In Sec. 4 we shall determine the exact regime of validity of the inequalities (18).

Note that the normalization prefactor of the linearized electroweak eigenfunction (24) is chosen, without loss of generality, to be 1.

Note that the characteristic asymptotic behavior ${}_2{F}_1(a,b;c;z)\to 1$ for $(ab/c)·z\to 0$ of the hypergeometric function [see Eq. (15.1.1) of [35] ] implies that the radial solution (24) for the electroweak eigenfunction is well behaved at the horizon of the central supporting black hole. In particular, one finds from Eqs. (24) and (25) the near-horizon relation $\psi (x\to 0)\to 1$.

Note that $\ell (\ell +1)=-n/2$.

Here we have used the leading-order relation $x=r/r_{+}+O(1)$ in the regime $r\gg r_{+}$ [see Eq. (25)].

Note that the characteristic asymptotic behavior $U(a,b,z)\to z^{-a}[1+O(z^{-1})]$ for $z\to \infty$ of the confluent hypergeometric function [see Eq. (13.1.7) of [35] ] implies that the radial eigenfunction (33) of the bound-state electroweak field decays asymptotically as $\psi (r\to \infty )\sim e^{-m_{\mathrm{w}}r}$.

Here we have used the relation $1=e^{-i2\pi k}$, where the integer $k=1,2,3,\dots$ is the discrete resonance parameter of the composed magnetic-black-hole-electroweak-field system.

Note that for $k=2$ one finds from (46) the relation ${\mu }_2(n=2)\simeq 7.3\times {10}^{-4}$ which respects the requirement (45) [in general, one finds that ${\mu }_k(n=2)$ satisfies the requirement (45) for $k\ge 2$], whereas for $k=1$ one finds from (46) the relation ${\mu }_1(n=2)\simeq 2.8\times {10}^{-2}$ which marginally violates the requirement (45). Interestingly, however, as we shall explicitly demonstrate below (see Table 1 in Sec. 5), the analytically derived relation (46) for $n=2$ agrees quite well with the exact numerically computed values of [26, 27] already for $k\ge 1$.

Here we assume the relation $Q/r_{+}\lesssim {10}^{-1}$ [see Eq. (18)].

Note that the two requirements $k\gtrsim \sqrt{2n}$ and $k\lesssim \sqrt{n}·[37-\ln (n)]/(\sqrt{2}\pi )$ for $n\gg 1$ can be satisfied simultaneously in the regime $n\lesssim {10}^{13}$.

Here we have used Eq. (6.1.23) of [35].

See Sec. 4 for the regime of validity of the analytically derived resonance formula (48).

Here we have used the relation $g^{\prime }\equiv \sin {\theta }_{\mathrm{w}}\simeq \sqrt{0.223}$ which implies $g\equiv \cos {\theta }_{\mathrm{w}}\simeq 0.881$ [27]. Taking cognizance of Eq. (7) one finds the relation $m_{\mathrm{w}}\simeq 0.623$ which is used to compute from Table I of [27] the $k$-dependent numerical values of the dimensionless mass-radius parameter $\mu \equiv m_{\mathrm{w}}{r}_{+}$.

As emphasized in Sec. 4, for the minimally allowed value $n=2$ of the magnetic parameter, the analytically derived resonance formula (48) is valid in the regime $k\lesssim 9$.