Photon polarization tensor and gauge dependence in three-dimensional quantum electrodynamics

We calculate the three-dimensional QED (${\mathrm{QED}}_3$) photon polarization tensor using dressed fermion propagators and a fermion-photon vertex that satisfies the Ward-Takahashi identity. Irrespective of the structural details of the transverse part of the fermion-photon vertex the photon remains massless; i.e., there is no photon mass generation in the manner of the Schwinger mechanism. Our calculation suggests that ${\mathrm{QED}}_3$ is confining when the polarization tensor is calculated using a dressed fermion propagator and fermion-photon vertex. The gauge parameter dependence of the fermion propagator, fermionphoton vertex, and photon polarization tensor is discussed in connection with the Landau-Khalatnikov gauge transformation laws.