Nonrelativistic Proca stars: Spherical stationary and multi-frequency states

In this paper we follow an effective theory approach to study the nonrelativistic limit of a selfgravitating and selfinteracting massive vector field. Our effective theory is characterized by three parameters: the field's mass $m_0$ and the selfinteraction constants $\lambda_n$ and $\lambda_s$. For definiteness, we focus on a systematic study of the equilibrium configurations, commonly referred to as Proca stars when they have finite energy. We identify two different types of Proca stars, depending on the specific sector of the effective theory that we are exploring. In the generic sector, defined by $\lambda_s\neq 0$, all equilibrium configurations are stationary states described by wave functions that evolve harmonically in time. However, in the symmetry-enhanced sector, for which $\lambda_s=0$, there exist multi-frequency states whose wave functions oscillate with two or three distinct frequencies in addition to the stationary states. We determine the conditions under which a ground state configuration with fixed particle number exists. When these conditions are met, we prove that the lowest energy is reached by a stationary spherically symmetric configuration of constant polarization that is linear or circular depending on the sign of $\lambda_s$. We numerically construct some illustrative examples of spherical stationary and multi-frequency solutions, analyze their properties, and compare them with our analytical predictions. Unlike stationary states and other soliton configurations, which form a discrete set in the solution space associated with fixed particle number, the symmetry-enhanced sector exhibits a continuum of solutions with multi-frequency states connecting stationary states of constant polarization.