Phonons and gluons in the crystalline color superconducting phase of QCD

The high density effective theory formalism is used to calculate the low energy properties of the phonons and gluons in the Larkin-Ovchinnikov-Fulde-Ferrell phase of two-flavor QCD. The effective phonon Lagrangian for the cubic crystal structure, which is favored according to a recent study, depends, at the second order in the derivatives, on three parameters which we calculate in this paper. We also compute for soft momenta the effective Lagrangian for the gluons of the unbroken $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_c$ group, both for a single plane wave structure and for the cube. In both cases the Meissner gluon mass vanishes as in the case of complete isotropy; on the other hand, there is a partial Debye screening due to the existence of blocking regions on the Fermi spheres. For the single plane wave structure the gluon dielectric tensor is nonisotropic, while it is isotropic for the cubic crystal, in spite of the intrinsic anisotropy of the structure.