Proca Q tubes and their coupling to gravity

The Einstein-Proca system is studied in the case of a complex vector field self-interacting through an appropriate potential with a global U(1) symmetry. The corresponding equations for a static, cylindrically symmetric metric and matter fields are then constructed and solved. In the probe limit (no gravity), it is shown that the equations admit at least two classes of regular solutions distinguished by the asymptotic behavior of the matter fields. One of these classes corresponds to lumps of vector fields localized in a cylindrical region; we naturally call these solutions "Proca Q tubes". They constitute the cylindrical counterparts of spherical Proca Q balls constructed recently; they can be characterized by finite mass and charge per unit length of the tube. The domain of existence of these Proca Q tubes with respect to the coupling constants determining the potential is studied in detail. Finally, the gravitating Proca Q tubes are constructed and studied.