Mass Shift of Axion in Magnetic Field
  A mass-shift of the axion propagating in an external constant homogenious magnetic field is calculated. The contributions via an electron loop and a virtual photon are examined. It is shown that the virtual photon contribution dominates substantially over the electron-loop one. Under the conditions of the early Universe the electron-loop contribution to the massless axion mass-shift is equal to zero while the virtual photon contribution is finite and can be of order of the recent restrictions on the axion mass. 
N.V. Mikheev, A.Ya. Parkhomenko, and L.A. Vassilevskaya