Zeta function regularization of the photon polarization tensor for a magnetized vacuum

In this paper, we develop a systematic technique to regularize double summations of Landau levels and analytically evaluate the photon vacuum polarization at an external magnetic field. The final results are described by Lerch transcendent $\mathrm{\Phi }(z,s,v)$ or its $z$ derivation. We find that the tensor of vacuum polarization is split into not only longitudinal and transverse parts but also another mixture component. The results remain gauge covariant as required by the Ward identity. We obtain a complete expression of the magnetized photon vacuum polarization at any kinematic regime and any strength of magnetic field for the first time. In the weak $B$ fields, after canceling out a logarithmic counterterm, all three scalar functions are limited to the usual photon polarization tensor without turning on a magnetic field. In the strong $B$ fields, the calculations under the lowest Landau level approximation are only valid at the region $M^2\gg q_{\parallel }^2$, but not correct when $q_{\parallel }^2\gg M^2$, where an imaginary part has been missed. It reminds us that a recalculation of the gap equation under a full consideration of all Landau levels will be necessary in the near future.