Photon polarization tensor at finite temperature and density in a magnetic field

We present analytical and numerical calculations for the photon polarization tensor at finite temperature and density in a constant magnetic field. We first discuss the tensor decomposition in the presence of the magnetic field, which breaks rotational symmetry. Then, we analytically perform all the momentum integrations and numerically take the Landau level sum. We confirm that the imaginary part of the photon polarization tensor correctly reproduces the known result from the independent calculation. We utilize the Kramers-Kronig relation to estimate the real part numerically as a function of the momenta, the chemical potential, and the finite temperature. As an application, we consider the real photon limit and estimate the photon decay rate and the Stokes parameter in the hot and dense medium. We specifically quantify the difference between the X-mode and the O-mode with the polarization orthogonal and parallel to the magnetic field. As long as the magnetic field is weak, the decay rate of the X-mode photon is larger than that of the O-mode photon, while the O-mode becomes dominant due to the Landau level suppression of the X-mode at a strong magnetic field. We also find that the eigenmodes of the propagating photon change their polarization state with increasing density.