标题： ζ函数正则化磁化真空中的光子极化张量 显示英文标题

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

摘要： 在本文中，我们开发了一种系统的技术来正则化Landau能级的双重求和，并在外部磁场下解析地评估了光子真空极化。 最终结果由Lerch-Zeta函数$\Phi(z,s,v)$或其$z$-导数描述。 我们发现真空极化张量不仅被分为纵向和横向部分，还有另一个混合成分。 我们首次得到了任何运动学区域和任何强度磁场下的磁化光子真空极化的完整表达式。 在弱$B$-场中，抵消了一个对数反常项后，所有三个标量函数都限制为通常的光子极化张量，而没有开启磁场。 在强$B$-场中，在最低Landau能级近似下的计算仅在区域$M^2\gg q^2$有效，而在$q^2\gg M^2$时不正确，其中遗漏了一个虚部。 这提醒我们，有必要在未来全面考虑所有Landau能级重新计算间隙方程。 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-Zeta function $\Phi(z,s,v)$ 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 $M^2\gg q^2$, but not correct while $q^2\gg M^2$, 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.