标题： 磁化真空的光子极化张量的Zeta函数正则化 显示英文标题

标题： Zeta Function Regularization of Photon Polarization Tensor for a Magnetized Vacuum

摘要： 本文发展了一种系统的技巧以正则化朗道能级的双重求和，并解析地评估了在外加磁场下的光子真空极化。最终结果由Lerch超越函数$\Phi(z,s,v)$或其$z$- 导数描述。我们发现真空极化的张量被分裂为纵向部分、横向部分以及另一种混合成分。我们首次得到了任意运动学区域和任意磁场强度下磁化的光子真空极化的完整表达式。在弱$B$- 场中，经过消除一个对数反项后，所有三个标量函数都退化为通常的无磁场开启时的光子极化张量。在强$B$- 场中，最低朗道能级近似的计算仅在区域$M^2\gg q_{\shortparallel}^2$内有效，而在$q_{\shortparallel}^2\gg M^2$区域不正确，因为在该区域遗漏了一个虚部。这提醒我们，在未来的工作中，需要在全面考虑所有朗道能级的情况下重新计算间隙方程。 显示英文摘要

摘要： 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.