Towards a self-interacting complex scalar field boson-star model

We propose a simple model intended to address boson stars (BSs) in a theory of self-interacting massive charged scalar fields coupled to the electromagnetic (EM) gauge field and gravity. We first consider standard scalar electrodynamics (SED) in 3 + 1 flat space-time dimensions in which only a complex scalar field and the EM field are present in the Lagrangian of the system. In order to better understand scalar field dynamics in (3 + 1)-dimensions we reduce the space dimensions to 2 and solve a problem consisting of embedded interacting charged scalar particles moving on the polar plane in the presence of certain linear static EM fields. If the (3 + 1)-dimensional system is consistently coupled to gravity, we comment on the possible relevance of this approach to study BS dynamics.