Collective excitations in a superfluid of color-flavor locked quark matter

By using the Nambu-Jona-Lasinio (NJL) model in the mean-field approximation, we derive the spectra both for the normal and color-flavor locked (CFL) superfluid phases at zero and finite temperature. In the normal phase, we obtain zero sound as a low-lying collective mode in the particle-hole (vector) channel. In the CFL phase, when the excitation energy, $\omega$, is smaller than the threshold given by twice the pairing gap $\Delta$ ($\omega<2\Delta$), a phonon corresponding to fluctuations in the U(1) phase of $\Delta$ appears as a sharp peak in the particle-particle ($H$) channel. The excitation energy goes up and asymptotically approaches $\omega=2\Delta$ with increasing momentum of the phonon. Above the threshold for pair excitations ($\omega>2\Delta$), zero sound manifests itself in the vector channel. We also find that in the long wavelength limit the phonon mode has its spectral weight in the $H$ channel alone, while the spectral function vanishes in the vector channel.