Nuclear pairing from microscopic forces: Singlet channels and higher-partial waves

Background: An accurate description of nuclear pairing gaps is extremely important for understanding static and dynamic properties of the inner crusts of neutron stars and to explain their cooling process.

Purpose: We plan to study the behavior of the pairing gaps ${\Delta }_F$ as a function of the Fermi momentum $k_F$ for neutron and nuclear matter in all relevant angular momentum channels where superfluidity is believed to naturally emerge. The calculations will employ realistic chiral nucleon-nucleon potentials with the inclusion of three-body forces and self-energy effects.

Methods: The superfluid states of neutron and nuclear matter are studied by solving the BCS gap equation for chiral nuclear potentials using the method suggested by Khodel et al., where the original gap equation is replaced by a coupled set of equations for the dimensionless gap function $\chi \left(k\right)$ defined by $\Delta \left(k\right)={\Delta }_F\chi \left(k\right)$ and a nonlinear algebraic equation for the gap magnitude ${\Delta }_F=\Delta \left(k_F\right)$ at the Fermi surface. This method is numerically stable even for small pairing gaps, such as that encountered in the coupled ${}^{3}P{F}_2$ partial wave.

Results: We have successfully applied Khodel's method to singlet ($S$) and coupled channel ($SD$ and $PF$) cases in neutron and nuclear matter. Our calculations agree with other ab initio approaches, where available, and provide crucial inputs for future applications in superfluid systems.

Figure 1

The effective mass in the case of nuclear (red line) and neutron (blue dashed line) matter as a function, respectively, of the total nucleon density $\rho$ or the neutron density ${\rho }_n$ (see Ref. [36]).

Figure 2

The ${}^1{S}_0$ gap for nuclear matter computed with the realistic chiral potential of Refs. [7, 34] at N3LO (red line) and the corresponding $V_{\mathrm{lowk}}$ potential (blue dashed line). With the green dashed-dotted line we include, as a benchmark, a similar calculation was performed by Hebeler et al. [38].

Figure 3

The ${}^1{S}_0$ gap for neutron matter computed with the realistic chiral potential of [7, 34] at N3LO plus the three-body contribution of Eq. (11) and the inclusion of the effective mass in Eq. (12). As a comparison, we include a similar calculation by Hebeler [38] with a green dashed line and a set of ab initio simulations with different many-body techniques: AFDMC (blue circles) [41], QMC Green's functions (green squares) [44], and CBF (yellow diamonds) [46]. See the text for additional details.

Figure 4

The ${}^1{S}_0$ gap for neutron matter. In this figure we show all the contributions to the pairing gap ${\Delta }_F$ starting from the inclusion of the bare two-body potential (red dotted line) or $V_{\mathrm{lowk}}$ (blue dotted line) and then including effective three-body forces (dashed lines) and a density-dependent effective mass (solid lines).

Figure 5

The gap in the ${}^{3}S{D}_1$ channel. We plot our calculations with the N3LO interaction (red line) in comparison to results obtained employing BONN-A potential [47] (blue curve) and OPEG [26] (yellow line). All results suggest a very large pairing gap (around 10 MeV), but complete calculations including three-body forces and effective masses [see Eqs. (11) and (12)], shown in the dashed red curve indicate a substantial reduction and a sizable modification of the gap's shape.

Figure 6

The evolution of the pairing gap in the ${}^{3}S{D}_1$ channel with SRG-evolved interactions. We employed two different evolution operators $G_s$ : $T_{\text{rel}}$ (red band) and $\text{PHP}+\text{QHQ}$ (blue band). The arrows denote the flow variable $\lambda$ (related to $s$ through $\lambda \equiv s^{-1/4}$), which is varied from 4.7 down to 1.1 ${\mathrm{fm}}^{-1}$.

Figure 7

The gap in the ${}^{3}P{F}_2$ channel obtained from the N3LO [34] (red line) interaction in comparison to several realistic NN potentials taken from Ref. [5]. Chiral potentials, by definition, can be trusted only up to momenta close to the cutoff (beyond the cutoff, the pairing gap is symbolized with a dashed line).

Figure 8

The gap in the ${}^{3}P{F}_2$ channel with only N3LO potential (dotted red line), with three-body forces (dashed red line) and including also self-energy effects (solid red line). In comparison we plot the results of recent BHF calculations [49] (blue curve).

Figure 9

The gap in the ${}^{3}P{F}_2$ channel as a function of the resolution scale in the Juelich nucleon-nucleon interactions [8]. The scales refer, respectively, to the cutoffs (units of MeV) in the Lippmann-Schwinger equation (${\Lambda }_{\text{LS}}$) and the spectral function regulator in multipion exchange loop diagrams (${\Lambda }_{2\pi }$). It appears that the magnitude of the gap's maximum is very sensitive to ${\Lambda }_{2\pi }$ and, to a lesser content, to ${\Lambda }_{\text{LS}}$.