ar X iv : h ep - p h / 06 11 14 0 v 1 10 N ov 2 00 6 Light plasmon mode in the CFL phase

The self-energies and the spectral densities of longitudinal and transverse gluons at zero temperature in the color-flavor-locked (CFL) phase are calculated. There appears a collective excitation, a light plasmon, at energies smaller than two times the gap parameter and momenta smaller than about eight times the gap. The minimum in the dispersion relation of this mode at some nonzero value of momentum corresponds to the van Hove singularity. In cold and dense quark matter, due to asymptotic freedom, at quark chemical potentials μ ≪ λQCD single-gluon exchange is the dominant interaction between quarks. Since this interaction is attractive in the color-antitriplet channel, therefore, quark matter is a color superconductor [1]. While there are, in principle, many different colorsuperconducting phases, corresponding to the different possibilities to form quark Cooper pairs, the ground state of color-superconducting quark matter is the so-called color-flavor-locked (CFL) phase [2]. At asymptotically large μ , the QCD coupling constant g ≪ 1, thus, the gluon selfenergy is dominated by the contributions from one quark and one gluon loop. The quark loop is ∼ g2μ2, while the gluon loops are ∼ g2T 2. Since the color-superconducting gap parameter is φ ∼ μ exp(−1/g) ≪ μ [3], and since the transition temperature to the normal conducting phase is Tc ∼ φ , for temperatures where quark matter is in the colorsuperconducting phase, T less than Tc ≪ μ , the gluon loop contribution can be neglected. The full description for the 2SC phase is given in Sec. II of Ref. [8]. The full energymomentum dependence of the one-loop gluon self-energy has also been computed, but so far only for the 2SC phase [7, 8]. Here we want to do the same calculations for the CFL phase. The detailed computation of the individual components and projections can be found in the appendix of Ref.[9]. Fig. 1 shows the imaginary part of several components of the gluon self-energy for a gluon momentum p = 4φ as a function of the gluon energy p0. The corresponding results for the gluon self-energy in the “hard-dense loop” (HDL) limit, Π 0 , are also shown with the dotted lines. The imaginary parts are quite similar to those of the 2SC case, cf. Fig. 1 of Ref. [8]. Nevertheless, there are subtle differences due to appearance of two kinds of gapped quark excitations, one so-called singlet excitation with a gap φ1, and eight so-called octet excitations with a gap φ8 ≡ φ [2]. In weak coupling, the singlet gap is approximately twice as large as the octet gap, φ1 ≃ 2φ8 ≡ 2φ [11, 10]. Therefore, the one-loop gluon self-energy in the CFL phase has two types of contributions, depending on whether the quarks in the loop correspond to singlet or octet excitations, cf. Eq. (23b) of Ref. [6]. For the first type, both quarks in the loop are octet excitations, and for the