Proca Q balls and their coupling to gravity

Extending the Proca Lagrangian of a massive complex-valued vector field by self-interaction potential, we construct a large class of spherically symmetric solutions in flat Minkowski background as well as in the self-gravitating case. Our solutions encompass Proca Q balls and Proca stars known in the literature, but present new features and go beyond the limited region of parameter space charted so far. A special emphasis is set on the domain of existence of the solutions in relation with the coupling constants of the potential and to the critical phenomena limiting this domain.