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

Recently, by considering nontrivial couplings between scalar fields and the Gauss-Bonnet invariant, the Schwarzschild black hole may allow regular scalar hairy configurations. The numerical studies of this spontaneous scalarization phenomenon 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 black hole can support regular scalar hairy configurations. Interestingly, Hod finds the coupling parameter $\bar{\eta}^+_n$ which correspond to the black hole linearized scalar field configurations has an asymptotic universal behavior $\Delta_n\equiv\sqrt{\bar{\eta}_{n+1}}-\sqrt{\bar{\eta}_{n}} \simeq2.72$. He provides a remarkably compact analytical explanation for the numerically observed universal behavior. Motivated by this interesting phenomenon, in this paper, we study the coupling parameter behavior in RN black hole with a nontrivial coupling between scalar fields and the Gauss-Bonnet invariant. Different from Schwarzschild case where the coupling parameter can only take positive values, in this case, the coupling parameter can take positive or negativie values. Therefore, it is interesting to investigate whether the coupling parameter has a similar asymptotic behavior in this situation. By examining numerical data, we find there is a similar asymptotic behavior for both positive and negative parameters. We also compare the analytical results with the numerical data. We find analytical results agree well with the the numerical data.