Charged Gauss-Bonnet black holes supporting non-minimally coupled scalar clouds: analytic treatment in the near-critical regime

Recent numerical studies have revealed the physically intriguing fact that charged black holes whose charge-to-mass ratios are larger than the critical value (Q/M)${}_{crit}$ = $\sqrt{2(9+\sqrt{6})}$/5 can support hairy matter configurations which are made of scalar fields with a non-minimal negative coupling to the Gauss-Bonnet invariant of the curved spacetime. Using analytical techniques, we explore the physical and mathematical properties of the composed charged-black-hole-nonminimally-coupled-linearized-massless-scalar-field configurations in the near-critical Q/M $\gtrsim$ (Q/M)${}_{crit}$ regime. In particular, we derive an analytical resonance formula that describes the charge-dependence of the dimensionless coupling parameter ${\overline{\eta }}_{crit}$ = ${\overline{\eta }}_{crit}$(Q/M) of the composed Einstein-Maxwell-nonminimally-coupled-scalar-field system along the existence-line of the theory, a critical border that separates bald Reissner-Nordström black holes from hairy charged-black-hole-scalar-field configurations. In addition, it is explicitly shown that the large-coupling -${\overline{\eta }}_{crit}$(Q/M) $\gg$ 1 analytical results derived in the present paper for the composed Einstein-Maxwell-scalar theory agree remarkably well with direct numerical computations of the corresponding black-hole-field resonance spectrum.