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

► Vacuum birefringence is studied in the presence of externally applied magnetic field. ► A general framework is given on the basis of a vacuum polarization tensor of photon. ► A resummed vacuum polarization tensor is calculated analytically and exactly. ► Contributions of all the Landau levels are obtained in the form of an infinite sum. ► Threshold behavior of real-photon decay is obtained at the each Landau level.