Vacuum polarization energy losses of high energy cosmic rays
  The process of the vacuum polarization energy losses of high energy cosmic rays propagating in the extragalactic space is considered. The process is due to the polarization of Cosmic Background Radiation by a moving charged particle. With the goal of the description of the process, the photon mass, refractive indices and permittivity function for low and high energy photons are found. Calculations show the rather noticeable level of the energy losses for propagating protons with the enegies more than 10^6 - 10^7 GeV. The influence of the polarization energy losses on propagation of cosmic rays is discussed. 
V.A. Maisheev