Energy-momentum tensor and effective Lagrangian of scalar QED with a nonminimal coupling in 2D de Sitter spacetime

We have studied the induced one-loop energy-momentum tensor of a massive complex scalar field within the framework of nonperturbative quantum electrodynamics (QED) with a uniform electric field background on the Poincar\'e patch of the two-dimensional de Sitter spacetime ($\mathrm{dS_{2}}$). We also consider a direct coupling the scalar field to the Ricci scalar curvature which is parameterized by an arbitrary dimensionless nonminimal coupling constant. We evaluate the trace anomaly of the induced energy-momentum tensor. We show that our results for the induced energy-momentum tensor in the zero electric field case, and the trace anomaly are in agreement with the existing literature. Furthermore, we construct the one-loop effective Lagrangian from the induced energy-momentum tensor.