Eur. Phys. J. C (2025) 85: 511
https://doi.org/10.1140/epjc/s10052-025-14228-4

Regular Article - Theoretical Physic

Non-topological soliton Bardeen boson stars and their frozen states

¹ Department of Applied Physics, Shandong University of Science and Technology, 266590, Qingdao, China
² School of Physical Science and Technology, Lanzhou University, 730000, Lanzhou, China

^a zhaozhh78@sdust.edu.cn
^b yqwang@lzu.edu.cn

Received: 24 February 2025
Accepted: 24 April 2025
Published online: 9 May 2025

Abstract

We investigate a Bardeen model coupling Einstein gravity with nonlinear electromagnetic fields and non-topological soliton complex scalar fields, governed by the magnetic charge $\tilde{q}$, the complex scalar field frequency $\tilde{\omega }$, and the self-interaction parameter $\tilde{\eta }$. Our results reveal that the magnetic charge $\tilde{q}$ exhibits $\tilde{\eta }$-dependent critical values ${\tilde{q}}_c$, beyond which ($\tilde{q}>{\tilde{q}}_c$) Bardeen boson stars (BBSs) may transition into frozen states ($\tilde{\omega }\to 0$). These frozen states are characterized by a critical horizon whose radius ${\tilde{r}}_c^{\text{H}}$ satisfies ${\tilde{r}}_{\text{inner}}^{\text{H,RN}}<{\tilde{r}}_c<{\tilde{r}}_{\text{outer}}^{\text{H,RN}}$, where ${\tilde{r}}_{\text{inner}}^{\text{H,RN}}$ and ${\tilde{r}}_{\text{outer}}^{\text{RN}}$ denote the inner and outer horizons of magnetic Reissner–Nordström (RN) black holes with equivalent mass and magnetic charge. Notably, the ADM mass of frozen BBSs is independent of $\tilde{\eta }$. Furthermore, light ring (LR) solutions exist universally across all tested combinations of $\tilde{q}$ and $\tilde{\eta }$, with all frozen BBSs exhibiting LRs whose outer radius ${\tilde{r}}_{\text{outer}}^{\text{LR}}$ is independent of $\tilde{\eta }$. Compared to magnetic RN black holes, frozen BBSs possess a smaller outer LR radius (${\tilde{r}}_{\text{outer}}^{\text{LR}}<{\tilde{r}}_{\text{outer}}^{\text{LR, RN}}$).