Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels

TL;DR

This paper calculates the photon polarization tensor in strong magnetic fields considering all Landau levels, providing a comprehensive, exact description of vacuum birefringence and photon decay phenomena without approximations.

Contribution

It presents an exact, all-encompassing computation of the photon polarization tensor in strong magnetic fields, including contributions from all Landau levels, extending previous approximate methods.

Findings

Derived a series representation of the polarization tensor as an infinite sum over Landau levels.

Identified conditions under which photons decay into fermion-antifermion pairs.

Applicable to any photon momentum and magnetic field strength without approximations.

Abstract

Photons propagating in strong magnetic fields are subject to a phenomenon called the "vacuum birefringence" where refractive indices of two physical modes both deviate from unity and are different from each other. We compute the vacuum polarization tensor of a photon in a static and homogeneous magnetic field by utilizing Schwinger's proper-time method, and obtain a series representation as a result of double integrals analytically performed with respect to proper-time variables. The outcome is expressed in terms of an infinite sum of known functions which is plausibly interpreted as summation over all the Landau levels of fermions. Each contribution from infinitely many Landau levels yields a kinematical condition above which the contribution has an imaginary part. This indicates decay of a sufficiently energetic photon into a fermion-antifermion pair with corresponding Landau level indices. Since we do not resort to any approximation, our result is applicable to the calculation of refractive indices in the whole kinematical region of a photon momentum and in any magnitude of the external magnetic field.