Non-equatorial scalar rings supported by magnetized Schwarzschild-Melvin black holes

It has recently been demonstrated that magnetized black holes in composed Einstein-Maxwell-scalar-Gauss-Bonnet field theories with a non-minimal negative coupling of the scalar field to the Gauss-Bonnet curvature invariant may support spatially regular scalar hairy configurations. In particular, it has been revealed that, for Schwarzschild-Melvin black-hole spacetimes, the onset of the near-horizon spontaneous scalarization phenomenon is marked by the numerically computed dimensionless critical relation $(BM)_{\text{crit}}\simeq0.971$, where $\{M,B\}$ are respectively the mass and the magnetic field of the spacetime. In the present paper we prove, using analytical techniques, that the boundary between bald Schwarzschild-Melvin black-hole spacetimes and hairy (scalarized) black-hole solutions of the composed Einstein-Maxwell-scalar-Gauss-Bonnet theory is characterized by the exact dimensionless relation $(BM)_{\text{crit}}=\sqrt{{{\sqrt{6}-2}\over{2\sqrt{6}}}+\sqrt{{{\sqrt{6}-1}\over{2}}}}$ for the critical magnetic strength. Intriguingly, we prove that the critical dimensionless magnetic parameter $(BM)_{\text{crit}}$ corresponds to magnetized black holes that support a pair of linearized non-minimally coupled thin scalar rings that are characterized by the non-equatorial polar angular relation $(\sin^2\theta)_{\text{scalar-ring}}={{690-72\sqrt{6}+4\sqrt{3258\sqrt{6}-7158}}\over{789}}<1$. It is also proved that the classically allowed angular region for the negative-coupling near-horizon spontaneous scalarization phenomenon of magnetized Schwarzschild-Melvin spacetimes is restricted to the black-hole poles, $\sin^2\theta_{\text{scalar}}\to0$, in the asymptotic large-strength magnetic regime $BM\gg1$.