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Anisotropic Propagator for the Goldstone Modes in Color-flavor Locked Phase in the Presence of a Magnetic Field

We consider the phase diagram of QCD at very high baryon density and at zero temperature in the presence of a strong magnetic field. The state of matter at such high densities and low temperatures is believed to be a phase known as the color-flavor locked phase which breaks color and electromagnetic gauge invariance leaving a linear combination of them unbroken. Of the 9 quarks (three flavors and three colors), five are neutral under this unbroken generator and four are oppositely charged. In the presence of a magnetic field corresponding to the unbroken generator however, the properties of the condensate changes and a new phase known as the magnetic color flavor locked (MCFL)phase is realized. This phase breaks some of the color-flavor symmetry of the Lagrangian spontaneously, giving rise to 6 Goldstone modes, 5 of which are pseudo Goldstone modes. These Goldstone modes are composed of excitations that correspond to both neutral quarks and charged quarks. Hence it is natural to expect that the propagators of these Goldstone modes get affected in the presence of a magnetic field and their speed becomes considerably anisotropic. Although this anisotropy is self-evident from symmetry arguments, it has not been quantified yet. We calculate this anisotropy in the speed of the Goldstone modes using an NJL model type of interaction between the quarks and comment on the impact of such anisotropic modes on the transport properties of the MCFL phase.