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 $(Q/M)_{\text{crit}}=\sqrt{2(9+\sqrt{6})}/5$ 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 {\it 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 $Q/M\gtrsim (Q/M)_{\text{crit}}$ regime. In particular, we derive an analytical resonance formula that describes the charge-dependence of the dimensionless coupling parameter $\bar\eta_{\text{crit}}=\bar\eta_{\text{crit}}(Q/M)$ of the composed Einstein-Maxwell-nonminimally-coupled-scalar-field system along the {\it existence-line} of the theory, a critical border that separates bald Reissner-Nordstr\"om black holes from hairy charged-black-hole-scalar-field configurations. In addition, it is explicitly shown that the large-coupling $-\bar\eta_{\text{crit}}(Q/M)\gg1$ 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.