Photon Self-Energy and Electric Susceptibility in a Magnetized   Three-flavor Color Superconductor
  We study the photon self-energy for the in-medium photon in a three-flavor color superconductor in the presence of a magnetic field. At strong magnetic field, the quark dynamics becomes $(1+1)$-dimensional and the self-energy tensor only has longitudinal components. In this approximation there is no Debye or Meissner screenings at zero temperature, but the electric susceptibility is nonzero and highly anisotropic. In the direction transverse to the applied field, the electric susceptibility is the same as in vacuum, while in the longitudinal direction it depends on the magnitude of the magnetic field. Such a behavior is a realization in cold-dense QCD of the magnetoelectric effect, which was first discovered in condensed matter physics. The magnetic permeability remains equal to that in vacuum for both transverse and longitudinal components. We discuss the importance of the Pauli-Villars regularization to get meaningful physical results in the infrared limit of the polarization operator. We also find the covariant form of the polarization operator in the reduced (1+1)-D space of the lowest Landau level and proves its transversality. 
Bo Feng, Efrain J. Ferrer and Vivian de la Incera
hep-th hep-ph