Hybrid Proca-boson stars

In this paper, we construct a hybrid boson star model composed of a complex scalar field and a Proca field. The scalar field and the Proca field are both in their ground state. The model is numerically solved to obtain families of solutions for different mixed states, considering both synchronized and nonsynchronized cases. By examining the relation between the ADM mass and the synchronized frequency ω or the nonsynchronized frequency ωP, we identify several types of solution families for the hybrid Proca-boson stars. In addition to solutions that intersect with the scalar field and the Proca field at each end, there are also several types of multibranch mixed-state solutions. The characteristics of various solutions are analyzed and discussed in detail. We calculate the binding energy E of the hybrid Proca-boson stars and provide the relationship between E and both the synchronized frequency ω and the nonsynchronized frequency ωP. Furthermore, we obtain the stability of the corresponding hybrid star solution families from the above analyses.