Improved ring potential of QED at finite temperature and in the presence   of weak and strong magnetic field
  Using the general structure of the vacuum polarization tensor $\Pi_{\mu\nu}(k_{0},\mathbf{k})$ in the infrared (IR) limit, $k_{0}\to 0$, the ring contribution to QED effective potential at finite temperature and non-zero magnetic field is determined beyond the static limit, $(k_{0}\to 0,\mathbf{k}\to \mathbf{0})$. The resulting ring potential is then studied in weak and strong magnetic field limit. In the limit of weak magnetic field, at high temperature and for $\alpha\to 0$, the improved ring potential consists of a term proportional to $T^{4}\alpha^{5/2}$, in addition to the expected $T^{4}\alpha^{3/2}$ term arising from the static limit. Here, $\alpha$ is the fine structure constant. In the limit of strong magnetic field, where QED dynamics is dominated by the lowest Landau level (LLL), the ring potential includes a novel term consisting of dilogarithmic function $(eB){Li}_{2}(-\frac{2\alpha}{\pi}\frac{eB}{m^{2}})$. Using the ring improved (one-loop) effective potential including the one-loop effective potential and ring potential in the IR limit, the dynamical chiral symmetry breaking of QED is studied at finite temperature and in the presence of strong magnetic field. The gap equation, the dynamical mass and the critical temperature of QED in the regime of LLL dominance are determined in the improved IR as well as in the static limit. For a given value of magnetic field, the improved ring potential is shown to be more efficient in decreasing the critical temperature arising from one-loop effective potential. 
N. Sadooghi and K. Sohrabi Anaraki