Gauss-Bonnet scalarization of charged qOS-black holes

The Gauss-Bonnet (GB) scalarization for charged quantum Oppenheimer-Snyder (cqOS)-black holes is investigated in the Einstein-Gauss-Bonnet-scalar theory with the nonlinear electrodynamics (NED) term. Here, the scalar coupling function to GB term is given by $f(\phi)=2\lambda \phi^2$ with a coupling constant $\lambda$. Three parameters of mass ($M$), action parameter ($\alpha$), and magnetic charge ($P$) are necessary to describe the cqOS-black hole, and it may become the qOS-black hole when $P=M$. The GB scalarization of cqOS-black holes comes into two cases GB$^\pm$, depending on the sign of GB term which triggers the different phenomena. For $\alpha=0$ and $\lambda>0$, GB$^+$ scalarization is allowed, while for $\alpha\not=0$ and $\lambda<0$, GB$^-$ scalarization appears for a narrow band of $3.5653\le \alpha\le 4.6875$. After discussing the onset GB$^-$ scalarization, we construct scalarized cqOS-black holes which belong to the single branch. The scalar field decays much more rapidly compared to the GB$^+$ case. Stability analysis shows these scalarized black holes are linearly stable under scalar perturbations.