New model of spontaneous scalarization of black holes induced by curvature and matter

We propose a new model of black hole spontaneous scalarization that combines a scalar--Gauss--Bonnet interaction with a non--minimal coupling to a U(1) gauge field (a dark photon or an electromagnetic field). This construction generalizes earlier single-coupling setups and allows both curvature--induced and matter--induced scalarization within one framework, which allows us to overcome the limitations of each mechanism alone. We focus on charged, spherically symmetric black holes and demonstrate that our model substantially expands the range of black hole masses and charges that permit scalar hair. Negative Gauss--Bonnet couplings, previously associated only with near--extremal charges or rapidly spinning black holes, now trigger scalarization for much broader charge intervals. We develop a numerical procedure to solve the field equations, and investigate the various properties of these black holes. This results in new branches emerging at distinct mass thresholds, a behavior not seen in the pure Einstein--scalar--Gauss--Bonnet or Einstein--scalar--Gauss--Bonnet--Ricci models. The scalar charge depends sensitively on the coupling parameters and on the $U(1)$ charge. Our analysis also shows that these black holes have larger entropy than their Reissner--Nordstr\"om counterparts and can become overcharged, surpassing the usual extremal limit of charge-to-mass ratio. Analyzing the scalar charge behavior suggests that adding matter-coupling appears to stabilize solutions that were previously prone to higher-order instabilities in pure Gauss--Bonnet models with quadratic coupling and broadens the range of possible configurations, making this model a promising candidate for further studies in strong gravity.