Critical phenomena of charged Einstein-Gauss-Bonnet black holes with charged scalar hair

Understand

Einstein-Gauss-Bonnet-gravity (EGB) coupled minimally to a $U(1)$ gauged, massive scalar field possesses -- for appropriate choices of the $U(1)$ charge -- black hole solutions that carry charged scalar hair if the frequency of the harmonic time-dependence of the scalar field is equal to the upper bound on the superradiant frequency.

• The existence of these solutions has first been discussed in \cite{Grandi:2017zgz}.
• In this paper, we demonstrate that the critical value of the scalar charge results from the requirement of non-extremality of the charged black hole solutions and the fact that the scalar field should not escape to infinity.
• Moreover, we investigate the hairy black holes in more detail and demonstrate that the branch of these solutions joins the branch of the corresponding charged EGB black hole for vanishing scalar field, but is {\it not} connected to the branch of boson stars in the limit of vanishing horizon radius.