Pairing in high-density neutron matter including short- and long-range correlations

Pairing gaps in neutron matter need to be computed in a wide range of densities to address open questions in neutron-star phenomenology. Traditionally, the Bardeen-Cooper-Schrieffer approach has been used to compute gaps from bare nucleon-nucleon interactions. Here we incorporate the influence of short- and long-range correlations in the pairing gaps. Short-range correlations are treated, including the appropriate fragmentation of single-particle states, and substantially suppress the gaps. Long-range correlations dress the pairing interaction via density and spin modes and provide a relatively small correction. We use different interactions, some with three-body forces, as a starting point to control for any systematic effects. Results are relevant for neutron-star cooling scenarios, in particular in view of the recent observational data on Cassiopeia A.

Figure 1

Imaginary part of the self-energy around the Fermi energy at $k_F=1.33{\mathrm{fm}}^{-1}$ for the CDBonn interaction for several temperatures. Panels (a)–(c) correspond to momenta $k=0, k_F$, and $2k_F$, respectively. The $T=0$ extrapolation is shown in a solid line. Note the different vertical scales of each panel.

Figure 2

(a) Momentum distribution at $k_F=1.33{\mathrm{fm}}^{-1}$ for the CDBonn interaction at different temperatures, including the $T=0$ extrapolation (solid line). (b) Effective sp denominator, $\chi \left(k\right)$, under the same conditions.

Figure 3

(a)–(c) Energy denominator at $T=0$ as a function of momentum for different Fermi momenta, corresponding to the (a) CDBonn, (b) Av18, and (c) N3LO interactions. (d)–(f) The same function, around the Fermi surface, plotted in a logarithmic scale. See text for details.

Figure 4

(a)–(c) Energy denominator at $T=0$ in the quasiparticle limit as a function of momentum for different Fermi momenta, corresponding to the (a) CDBonn, (b) Av18, and (c) N3LO interactions.

Figure 5

(a)–(c) Ratio of denominators, Eq. (12), at $T=0$ as a function of momentum for different Fermi momenta. The three panels correspond to the (a) CDBonn, (b) Av18, and (c) N3LO interactions. (d)–(f) The actual $Z$ factor as a function of momentum in the same conditions. See text for details.

Figure 6

Landau parameters $F_0$ and $G_0$ extrapolated to $k_F=3.0{\mathrm{fm}}^{-1}$ employing the results of Ref. [52]. The corresponding value of the forward-scattering sum rule is indicated by the open circles.

Figure 7

(a), (b) Diagonal matrix elements of the Av18 interaction in the (a) ${}^{1}S_0$ and (b) ${}^{3}P_2$ channels. (c), (d) Diagonal matrix elements of the additional pairing interaction representing the low-energy medium polarization at a density corresponding to $k_F=1.6{\mathrm{fm}}^{-1}$ in the same channels. The scales are different for each panel.

Figure 8

Pairing gaps at the Fermi surface as a function of Fermi momentum in the ${}^{1}S_0$ channel. The three panels correspond to the (a) CDBonn, (b) Av18, and (c) N3LO interactions. Results for different approximations are presented: BCS (solid lines), beyond BCS with short-range correlations (open circles), beyond BCS with long-range correlations (open squares), and beyond BCS with both short- and long-range correlations included (solid circles). The dashed lines represent the fits provided in Table 1.

Figure 9

Pairing gaps in the ${}^{1}S_0$ channel as a function of momentum for several Fermi momenta. The three panels correspond to the (a) CDBonn, (b) Av18, and (c) N3LO interactions. The BCS (dashed lines) and SRC (solid lines) results are presented. The symbols represent the corresponding gaps at the Fermi surface, as shown in Fig. 8.

Figure 10

The same as Fig. 8 for the coupled ${}^{3}PF_2$ channel. The gray band on panel (c) indicates the region in which cutoff effects are relevant for the N3LO force.

Figure 11

(Left panels) Matrix elements of $\mathit{NN}$ forces in fm for the ${}^{3}P_2$ partial wave. (Right panels) The same for the ${}^{3}F_2$ wave.

Figure 12

The same as Fig. 9 for the pairing gaps in the ${}^{3}PF_2$ channel. Note the logarithmic scale on the $z$ axis.

Figure 13

Pairing gaps at the Fermi surface as a function of Fermi momentum in the ${}^{1}S_0$ (a) and ${}^{3}PF_2$ (b) channels. Results for different approximations are presented for the chiral N3LO Idaho $\mathit{NN}$ force in the BCS (dotted lined) and $\mathrm{BCS}+\mathrm{SRC}$ approximation (light circles). Results including 3NF ($2\mathrm{N}+3\mathrm{NF}$) are given in a solid (bold circles) line for the BCS ($\mathrm{BCS}+\mathrm{SRC}$) approximation. See text for details.

Figure 14

Each point in this plot corresponds to a density and temperature where ladder self-energies have been computed. Finite-temperature points are used to extrapolate to zero temperature.