General framework for the spontaneous scalarization of regular black holes

We investigate the spontaneous scalarization of generic, static, and spherically symmetric regular black holes supported by nonlinear electrodynamics. Starting from an arbitrary seed metric, we employ the $P$-dual formalism to reconstruct the electromagnetic sector and subsequently couple a real scalar field nonminimally. As a worked example, we apply the framework to the regular Balart-Vagenas black hole, showing that scalarized and scalar-free branches can coexist in a region where the scalarized configurations are entropically preferred. We further assess possible observational imprints, finding percent-level deviations in both the shadow size and the fundamental scalar quasinormal modes ($<10%$ for small charge-to-mass ratios), indicating that current electromagnetic and gravitational-wave observations do not rule out these solutions. Our construction thus provides a general route to explore scalarization on top of nonlinear-electrodynamics-supported spacetimes, extending beyond specific Reissner-Nordstr"om-like cases.