Dirac-Proca stars

We consider a Dirac-Proca star (DPS) model that consists of the Einstein gravity in four-dimensional spacetime, a Proca field, and two Dirac fields through minimal coupling. By numerically solving this model, we obtain two types of solutions: synchronized frequency solutions and nonsynchronized 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 nonsynchronized frequency cases, and the two families can shift to each other when certain parameters are changed. Moreover, we calculate the binding energy of multifield solutions and give the relationship between binding energy E and synchronized frequency ω ∼ (nonsynchronized frequency ω_∼P).