Study of boson stars with wormhole

In this paper, we study the mixed system of boson stars (BSs) with wormholes at their center. The boson star is obtained by employing a complex scalar field without self-interaction or a complex scalar field with quartic self-interaction and the wormhole is obtained by employing a phantom field. Utilizing the numerical method, we successfully obtain both symmetric and asymmetric solutions within the two asymptotically flat regions. The key focus of our study involves the systematic exploration of variations in results by varying the throat parameter eta 0, and the parameter c, representing the quartic term in potential. In the ground state, we find the mass M and Noether charge Q versus the scalar field frequencies omega are multi-valued curves when the eta 0 is small or the self-interaction is strong, the multi-valued curves will turn into single-valued curves as eta 0 or c increases. Furthermore, we observe that asymmetric solutions can transition into symmetric solutions at specific scalar field frequencies omega for certain settings of eta 0 and c. In addition, when it comes to the excited state, the properties of symmetric solutions remain akin to those in the ground state, while asymmetrical results display different characteristics from the ground state. We also present the wormhole spacetime geometry to investigate the properties of this model.