Scalar hairy black holes in Einstein-Maxwell-conformally coupled scalar theory

We obtain scalar hairy black holes from Einstein-Maxwell-conformally coupled scalar (EMCS) theory with the scalar coupling parameter $\alpha$ to the Maxwell term. In case of $\alpha=0$, the $\alpha=0$ EMCS theory provides constant (charged) scalar hairy black hole and charged BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole where the former is stable against full perturbations, while the latter remains unstable because it belongs to an extremal black hole. It is noted that for $\alpha\not=0$, the unstable Reissner-Nordstr\"{o}m black holes without scalar hair imply infinite branches of $n=0(\alpha \ge 8.019),1(\alpha \ge 40.84),2(\alpha \ge 99.89),\cdots$ scalarized charged black holes. In addition, for $\alpha>0$, we develop a single branch of scalarized charged black hole solutions inspired by the constant scalar hairy black hole. Finally, we obtain the numerical charged BBMB black hole solution from the $\alpha=0$ EMCS theory.