Abnormal number of Nambu-Goldstone bosons in the color-asymmetric dense color superconducting phase of a Nambu-Jona-Lasinio-type model

We consider an extended Nambu-Jona-Lasinio model including both (q (q) over bar) and (qq) interactions with two light-quark flavors in the presence of a single (quark density) chemical potential. In the color superconducting phase of the quark matter the color SUc(3) symmetry is spontaneously broken down to SUc(2). If the usual counting of Goldstone bosons would apply, five Nambu-Goldstone (NG) bosons corresponding to the five broken color generators should appear in the mass spectrum. Unlike that expectation, we find only three gapless diquark excitations of quark matter. One of them is an SUc(2) singlet; the remaining two form an SUc(2) (anti)doublet and have a quadratic dispersion law in the small momentum limit. These results are in agreement with the Nielsen-Chadha theorem, according to which NG bosons in Lorentz-noninvariant systems, having a quadratic dispersion law, must be counted differently. The origin of the abnormal number of NG bosons is shown to be related to a nonvanishing expectation value of the color charge operator Q(8) reflecting the lack of color neutrality of the ground state. Finally, by requiring color neutrality, two massive diquarks are argued to become massless, resulting in a normal number of five NG bosons with the usual linear dispersion laws.