Critical scalarization and descalarization of black holes in a generalized scalar-tensor theory

We study the critical dynamics in scalarization and descalarization in the fully nonlinear dynamical evolution in the class of theories with a scalar field coupling with both Gauss-Bonnet (GB) invariant and Ricci scalar. We explore the manner in which the GB term triggers black hole (BH) scalarization. A typical type I critical phenomenon is observed, in which an unstable critical solution emerges at the threshold and acts as an attractor in the dynamical scalarization. For the descalarization, we reveal that a marginally stable attractor exists at the threshold of the first-order phase transition in shedding off BH hair. This is a new type I critical phenomenon in the BH phase transition. Implications of these findings are discussed from the perspective of thermodynamic properties and perturbations for static solutions. We examine the effect of scalar-Ricci coupling on the hyperbolicity in the fully nonlinear evolution and observe that such coupling can suppress the elliptic region and enlarge parameter space in computations.