Hybrid Proca-boson stars

In this paper, we construct a hybrid boson star model that contains a complex scalar field and a Proca field. The scalar field is in the ground state, while the Proca field is in the first excited state. We numerically solve the model and obtain solution families of different coexisting states by considering both synchronized and nonsynchronized cases. By examining the relation between ADM mass and synchronized frequency $\tilde{\omega}$ or nonsynchronized frequency $\tilde{\omega}_P$, we identify several types of solution families for the hybrid boson stars. In addition to solutions that intersect the scalar field and the Proca field at each end, there are also several types of multi-branch coexisting 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 synchronized frequency $\tilde{\omega}$ and nonsynchronized frequency $\tilde{\omega}_P$. Furthermore, we obtain the stability of the corresponding hybrid star solution families from these analyses above.