Induced energy-momentum tensor of the scalar field in 3D de Sitter QED
In this work, we derive the renormalized expectation value of the energy--momentum tensor of a quantized charged scalar field in three-dimensional de Sitter spacetime $\mathrm{dS}_{3}$ in the presence of a uniform electric field. Using the adiabatic regularization method, ultraviolet divergences are systematically removed, yielding finite expressions for all components of the induced tensor. We analyze the behavior of the renormalized energy--momentum tensor in both the strong-field and infrared regimes. In the strong-field limit, the induced energy density exhibits a quadratic dependence on the electric field strength; however, this leading contribution is dominated by vacuum polarization effects rather than directly by the Bogoliubov particle-production component. In the infrared regime, the tensor shows a pronounced inverse-mass dependence, indicating strong infrared sensitivity characteristic of light scalar fields in de Sitter spacetime. In the conformally coupled massless limit, the renormalized trace vanishes, as expected from the absence of a genuine Weyl anomaly in odd-dimensional spacetimes. These results provide a precise characterization of vacuum polarization and infrared effects in three-dimensional de Sitter scalar QED.