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Recently, by considering nontrivial couplings between scalar fields and theGauss-Bonnet invariant, the Schwarzschild black hole may allow regular scalarhairy configurations. The numerical studies of this spontaneous scalarizationphenomenon show if the coupling parameter $\bar{\eta}$ belongs to discrete sets$\bar{\eta}\in\{\bar{\eta}^-_n,\bar{\eta}^+_n\}^{n=\infty}_{n=0}$, the blackhole can support regular scalar hairy configurations. Interestingly, Hod findsthe coupling parameter $\bar{\eta}^+_n$ which correspond to the black holelinearized scalar field configurations has an asymptotic universal behavior$\Delta_n\equiv\sqrt{\bar{\eta}_{n+1}}-\sqrt{\bar{\eta}_{n}} \simeq2.72$. Heprovides a remarkably compact analytical explanation for the numericallyobserved universal behavior. Motivated by this interesting phenomenon, in thispaper, we study the coupling parameter behavior in RN black hole with anontrivial coupling between scalar fields and the Gauss-Bonnet invariant.Different from Schwarzschild case where the coupling parameter can only takepositive values, in this case, the coupling parameter can take positive ornegativie values. Therefore, it is interesting to investigate whether thecoupling parameter has a similar asymptotic behavior in this situation. Byexamining numerical data, we find there is a similar asymptotic behavior forboth positive and negative parameters. We also compare the analytical resultswith the numerical data. We find analytical results agree well with the thenumerical data.

Spontaneous Scalarization of Charged Gauss-Bonnet Black Holes: Analytic Treatment