Effective mass in mixed-dimensionality pseudo-Proca systems, g-factor corrections, and the radiative effects to the interaction potential

In this work, we study a model that describes the $(2+1)\mathrm{D}$ Yukawa interaction for static particles confined to the plane. In order to do so, we perform a dimensional reduction of a $(3+1)\mathrm{D}$ massive vector field whose mass term introduces a natural scale for a correlation length within the system. We derive the projected model similarly to the so-called pseudo (or reduced) quantum electrodynamics that has been used in several applications. One-loop radiative corrections to the electron and to the massive vector-field mass are computed, as well as the vertex corrections; this brings forth the quantum correction to the electron $g$ factor, demonstrating that it smoothly goes to zero when the mass of the vector field is significantly larger than the electron mass and generalizing the results obtained for graphene within pseudo-QED to a system with a correlation length. Such a limit is expected to emerge due to many-body interactions in two-dimensional materials. Furthermore, the model is likely to be relevant in the relativistic limit, when electrons move with a high velocity.