| Summary: | Abstract Recent numerical studies have revealed the physically intriguing fact that charged black holes whose charge-to-mass ratios are larger than the critical value $$(Q/M)_{\text {crit}}=\sqrt{2(9+\sqrt{6})}/5$$ ( Q / M ) crit = 2 ( 9 + 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 $$Q/M\gtrsim (Q/M)_{\text {crit}}$$ Q / M ≳ ( Q / M ) crit regime. In particular, we derive an analytical resonance formula that describes the charge-dependence of the dimensionless coupling parameter $$\bar{\eta }_{\text {crit}}=\bar{\eta }_{\text {crit}}(Q/M)$$ η ¯ crit = η ¯ 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 $$-\bar{\eta }_{\text {crit}}(Q/M)\gg 1$$ - η ¯ crit ( Q / M ) ≫ 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.
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