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