Dirac-Proca stars

We consider a model consists of the Einstein gravity in four-dimensional spacetime, a Proca field and two Dirac fields through minimum coupling. By numerically solving this model, we obtain two types of solutions: synchronized frequency solutions and non-synchronized frequency solutions. We find that in the case of two kinds of matter fields, different families of solutions can be obtained in both synchronized and non-synchronized frequency cases, and the two families can shift to each other when certain parameters are changed. Moreover, we calculate the binding energy of multi-field solutions and give the relationship between binding energy $E$ and synchronized frequency $\tilde{\omega}$ (non-synchronized frequency $\tilde{\omega}_P$), so as to analyze the stability of the corresponding family.
Comment: 22 pages, 9 figures