Spherically symmetric horizonless solutions and their frozen states in Bardeen spacetime with Proca field

In this paper, we construct a static spherical symmetric Bardeen-Proca star (BPS) model, which consists of the electromagnetic field and Proca field minimally coupled with gravity. The introduction of the Proca field disrupts the formation of event horizons, ensuring that these solutions are globally regular throughout the spacetime. We obtain families of BPS solutions under several magnetic charge conditions. Based on these results, we further investigate the ADM mass, Noether charge, and energy density distribution of them. We find that when the magnetic charge is sufficiently large, solutions with a critical horizon $r_{cH}$ emerge as $\omega \rightarrow 0$, and the time component of the metric approaches zero inside $r_{cH}$. To an observer at infinity, the collapse process of the matter near the critical horizon appears frozen. Consequently, we refer to the solution with $r_{cH}$ as the frozen Bardeen-Proca star (FBPS). Additionally, we also investigate the circular geodesic orbits of BPS. For the light ring, we find that the light rings always appear in pairs, located on both sides of the critical horizon and moving further apart as the frequency $\omega$ decreases. For timelike circular orbits, we investigate their distribution in the spacetime of BPSs and highlight four representative families of BPS solutions.