Lack of Debye and Meissner screening in strongly magnetized quark matter at intermediate densities

We study the static responses of cold quark matter in the intermediate baryonic density region (characterized by a chemical potential $\mu$) in the presence of a strong magnetic field. It is shown that in that region the phase materialized by an inhomogeneous condensate formed by a particle-hole pair, the so-called Magnetic Dual Chiral Density Wave (MDCDW) phase, is more stable in the weak-coupling regime than the one considered in the magnetic catalysis of chiral symmetry braking phenomenon (MC$\chi$SB), which is formed by a constant condensate produced by a particle-antiparticle pairing, and even than the chiral symmetric phase without any condensate that was expected to be realized at sufficiently high baryonic chemical potential. The different components of the photon polarization operator of the MDCDW phase in the one-loop approximation are calculated. We found that in the MDCDW phase there is no Debye screening neither Meissner effect. The obtained Debye length depends on the amplitude $m$ and modulation $b$ of the inhomogeneous condensate and it is only different from zero if the relation $| \mu -b| > m$ holds. But, we found that in the region of interest this inequality is not satisfied. Thus, no Debye screening takes place in this dense medium. On the other hand, since the particle-hole condensate is electrically neutral, the U(1) electromagnetic group is not broken by the ground state and consequently there is no Meissner effect. These results can be of interest for the astrophysics of neutron stars.