Mixed scalarization of charged black holes: From spontaneous to nonlinear scalarization

Scalarized black holes (BH) have been shown to form dynamically in extended-scalar-tensor theories, either through spontaneous scalarization---when the BH is unstable against linear perturbations---or through a nonlinear scalarization. In the latter, linearly stable BHs can ignite scalarization when sufficiently perturbed. These phenomena are, however, not incompatible and mixed scalarization is also possible. We explore two aspects of the Einstein-Maxwell-scalar model: solutions containing, simultaneously, linear (also known as standard) and nonlinear scalarization; and the effects of having one of the coupling constants with an ``opposite sign'' to the one leading to scalarization. Both points are addressed by constructing and examining the mixed scalarization's domain of existence. An overall dominance of the spontaneous scalarization over the nonlinear scalarization is observed. Thermodynamically, an entropical preference for mixed over the standard scalarization (spontaneous or nonlinear) exists. In the presence of counter scalarization, a quench of the scalarization occurs, mimicking the effect of a scalar particle's mass/positive self-interaction term.