Coupling between superfluid neutrons and superfluid protons in the elementary excitations of neutron star matter

Several phenomena occurring in neutron stars are affected by the elementary excitations that characterize the stellar matter. In particular, low-energy excitations can play a major role in the emission and propagation of neutrinos, neutron star cooling, and transport processes. In this paper, we consider the elementary modes in the star region where both proton and neutron components are superfluid. We study the overall spectral functions of protons, neutrons, and electrons on the basis of the Coulomb and nuclear interactions. This paper is performed in the framework of the random-phase approximation, generalized to superfluid systems. The formalism we use ensures that the generalized Ward's identities are satisfied. We focus on the coupling between neutrons and protons. On one hand, this coupling results in collective modes that involve simultaneously neutrons and protons; on the other hand, it produces a damping of the excitations. Both effects are especially visible in the spectral functions of the different components of the matter. At high densities while the neutrons and protons tend to develop independent excitations as indicated by the spectral functions, the neutron-proton coupling still produces a strong damping of the modes.

Figure 1

Spectral functions of neutrons (black thick lines), proton (red thin lines), and electrons (green dashed lines) calculated at the saturation density of the neutron star matter. The neutron and proton pairing gaps are 1.5 and 1 MeV, respectively. In panel (a), the neutron-proton interaction has been suppressed, whereas in panel (b), it is included. For convenience, the neutron strength function has been divided by 15.

Figure 2

Spectral functions of neutrons (black thick lines), proton (red thin lines), and electrons (green dashed lines) calculated at the saturation density of the neutron star matter. The neutron and proton pairing gaps are 0.5 and 1 MeV, respectively. In panel (a), the neutron-proton interaction has been suppressed, whereas in panel (b), it is included. For convenience, the neutron strength function has been divided by 15.

Figure 3

Diagram (a) illustrates the coupling between the neutron and the proton pairing gap fluctuations. Diagram (b) illustrates the coupling between the neutron pair fluctuations and the electron excitations. See the discussion in the text. The labels n, p, and e stand for neutrons, protons, and electrons, respectively.

Figure 4

Spectral functions of neutrons (black thick lines), protons (red thin lines), and electrons (green dashed lines) calculated at saturation density of the neutron star matter with the same pairing gaps as in Fig. 2 but at higher momentum. In panel (a), the neutron-proton interaction has been suppressed, whereas in panel (b), it is included. For convenience, the neutron strength function has been divided by 15.

Figure 5

Spectral functions of neutrons (black thick lines), protons (red thin lines), and electrons (green dashed lines) calculated at twice saturation density of the neutron star matter with the same pairing gaps as in Fig. 2 for two different momenta. In both cases, the neutron-proton interaction has been included. For convenience, the neutron strength function has been divided by 15.

Figure 6

Spectral functions of neutrons (black thick lines), protons (red thin lines), and electrons (green dashed lines) calculated at twice saturation density of the neutron star matter with proton gap as in Fig. 2 but taking the neutron pairing gap as vanishing. In both cases, the neutron-proton interaction has been included. For convenience, the neutron strength function has been divided by 15.