Two scalarizations of magnetically charged black holes

We study two scalarizations of magnetically charged black holes in the Einstein-Gauss-Bonnet-scalar theory with the nonlinear electrodynamics (NED) term. For this purpose, two scalar coupling functions $f_1(\phi)$ and $f_2(\phi)$ are introduced to couple to Gauss-Bonnet (GB) and NED terms. The bald black hole is described by mass ($M$) and action parameter ($\alpha$) with magnetic charge ($P$), which becomes the quantum Oppenheimer-Snyder black hole for $P=M$ whose action is still unknown. We derive Smarr formula which describes a correct thermodynamics for the charged black hole. Two Davies points of heat capacity are not identified with two critical onset mass and action parameter for GB$^-$ scalarization with $f_1(\phi)$. Furthermore, the shadow radius analysis of charged black hole is performed to include naked singularities and it is compared to the EHT observation. There is no constraints on the action parameter, whereas new mass constraints are found. Finally, the NED$^+$ spontaneous scalarization with $f_2(\phi)$ leads to infinite branches of scalarized charged black holes.