Bardeen spacetime with charged scalar field

Recently, Ref. \cite{Wang:2023tdz} investigated the model of Einstein-Bardeen theory coupled to a free complex scalar field. The introduction of the scalar field prevents the formation of the event horizon, and when the magnetic charge exceeds a certain critical value, the frozen Bardeen-boson star can be obtained with the frequency $\omega \rightarrow 0$. In this paper, we extend the investigation of the Einstein-Bardeen model with a charged scalar field and obtain two types of solutions: the small $q$ solution and the large $q$ solution. Specifically, for the small $q$ solution, we find that there exists a maximum value for the charge $q$, the introduction of the charge makes it possible to obtain solutions for frozen stars without the frequency to be approached to zero. For the large $q$ solution, the charge can tend toward infinity, and as $q \rightarrow \infty$, the large $q$ solution gradually becomes the pure Bardeen solution. Similar to Ref. \cite{Wang:2023tdz}, the event horizon is not found in our results.