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

TL;DR

This work computes the photon polarization tensor $oldsymbol{ ilde{ u}}$ in a constant magnetic field at finite temperature and density, decomposing it into X- and O-mode components and evaluating all Landau levels. Analytically, it performs transverse momentum integrals and then numerically sums Landau levels up to $k_{ ext{max}}=100$, obtaining the imaginary part from Cutkosky-type constraints and reconstructing the real part via the Kramers–Kronig relation with careful treatment of UV behavior at $k^2=0$. The results show the imaginary part reproduces known results, reveal how decay rates depend on field strength (X-mode dominant at weak $B$, O-mode at strong $B$), and demonstrate density-induced polarization-state changes through the Stokes parameters. The methodology provides a robust framework for understanding electromagnetic probes in hot, dense, magnetized QCD matter and has potential applications to magnetar X-ray polarization observations.

Abstract

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.

Figures (10)

• Figure 1: Tadpole diagram for calculating the expectation value of the density.
• Figure 2: One-loop diagram for the photon polarization tensor.
• Figure 3: Imaginary part of the polarization tensor components as functions of the photon momentum parallel to the magnetic field for $T=5m_e,\mu=5m_e$, and two fixed magnetic fields: $|eB| = 4m_e^2$ (left panel) and $|eB| = 25m_e^2$ (right panel).
• Figure 4: Imaginary part of the $33$-component of the polarization tensor component as a function of the photon momentum parallel to the magnetic field at $|eB|=4m_e^2$. (Left panel) Dependence on four different values of $T$ at $\mu=0$. (Right panel) Dependence on four different values of $\mu$ at $T=m_e$.
• Figure 5: Photon splitting process into an electron and a positron