Zeta Function Regularization of Photon Polarization Tensor for a Magnetized Vacuum

Abstract

In this paper, we have developed a systematic technique to regularize double summations of Landau levels and analytically evaluated the photon vacuum polarization at an external magnetic field. The final results are described by Lerch transcendent \(\Phi(z,s,v)\) or its \(z\)-derivation. We have found that the tensor of vacuum polarization is split into not only longitudinal and transverse parts but also another mixture component. We have obtained 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 counter term, all three scalar functions are limited to the usual photon polarization tensor without turning on magnetic field. In the strong \(B\)-fields, the calculations under Lowest Landau Level approximation are only valid at the region \(M^2\gg q_{\shortparallel}^2\), but not correct while \(q_{\shortparallel}^2\gg M^2\), where, an imaginary part has been missed. It reminds us, a recalculation of the gap equation under a full consideration of all Landau Levels is necessary in the next future.