Elementary excitations in homogeneous superfluid neutron star matter: Role of the neutron-proton coupling

The thermal evolution of neutron stars is affected by the elementary excitations that characterize the stellar matter. In particular, the low-energy excitations, with a spectrum linear in momentum, can play a major role in the emission and propagation of neutrinos. In this paper, we focus on the elementary modes in the region of proton superfluidity, where the neutron component is expected to have a very small or zero pairing gap. We study the overall spectral functions of protons, neutrons, and electrons on the basis of the Coulomb and nuclear interactions. This study is performed in the framework of the random phase approximation, generalized in order to describe the response of a superfluid system. The formalism we use ensures that the generalized Ward's identities are satisfied. Despite their relative small fraction, the protons turn out to modify the neutron spectral function as a consequence of the nuclear neutron-proton interaction. This effect is particularly evident at the lower density, just below the crust for a density close to the saturation value, while at increasing density the neutrons and the protons are mainly decoupled. The proton spectral function is characterized by a pseudo-Goldstone mode below $2\mathrm{\Delta }$, twice the pairing gap, and a pair-breaking mode above $2\mathrm{\Delta }$. The latter merges in the sound mode of the normal phase at higher momenta. The neutron spectral function develops a collective sound mode only at the higher density. The electrons have a strong screening effect on the proton-proton interaction at the lower momenta and decouple from the protons at higher momenta.

Figure 1

Spectral functions of neutrons (black thick line), protons (red thin line), and electrons (green dashed line). In panel (a), the spectral functions when the Coulomb and nuclear interactions are introduced are reported. For comparison, in panel (b) the spectral functions when the neutron-proton interaction is neglected are reported. For a closer comparison, the neutron spectral function of this case is also reported in panel (a) (blue dotted line).

Figure 2

Spectral functions as in Fig. 1, for other two different momenta.

Figure 3

Spectral functions for the small pairing gap $\mathrm{\Delta }=0.002\mathrm{MeV}$. Symbols like in Figs. 1 and 2.

Figure 4

Energy of the zero of the real part of the inverse of the response function at different momenta $q$. The pairing gap is $\mathrm{\Delta }=$ 1 MeV. The red full line and full squares correspond to the case when the neutrons and the proton are uncoupled. The green dashed line and the full circles correspond to the case when the interaction between neutrons and protons is switched on.

Figure 5

Spectral functions for the Skyrme force Sly230 (a), and for the functional NRAPR (b). The pairing gap is $\mathrm{\Delta }=1\mathrm{MeV}$. Symbols as in Figs. 1, 2, 3.

Figure 6

Spectral function within the microscopic approach to the effective nuclear interaction without neutron-proton coupling at twice saturation density (a). The same, but with the neutron-proton interaction included (b). The pairing gap is $\mathrm{\Delta }=1\mathrm{MeV}$. Symbols as in Figs. 1, 2, 3.