Mixed scalarization of charged black holes: from spontaneous to non-linear 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 non-linear 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. The objective of this work is twofold: first, study mixed scalarization on a family of Einstein-Maxwell-scalar models; and second, study the effect of the counter scalarization that occurs when one of the coupling parameters has a sign opposite to the one that generates scalarization. Both objectives are addressed by constructing and examining the mixed scalarization's domain of existence. An overall dominance of the spontaneous scalarization over the non-linear scalarization is observed. Thermodynamically, an entropical preference for mixed over the standard scalarization (spontaneous or non-linear) 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.Comment: 14 pages, 5 figure