Proton pairing in neutron stars from chiral effective field theory

We study the ${}^{1}S_0$ proton pairing gap in $\beta$-equilibrated neutron star matter within the framework of chiral effective field theory. We focus on the role of three-body forces, which strongly modify the effective proton-proton spin-singlet interaction in dense matter. We find that three-body forces generically reduce both the size of the pairing gap and the maximum density at which proton pairing may occur. The pairing gap is computed within Bardeen-Cooper-Schrieffer theory using a single-particle dispersion relation calculated up to second order in perturbation theory. Model uncertainties are estimated by varying the nuclear potential (its order in the chiral expansion and high-momentum cutoff) and the choice of single-particle spectrum in the gap equation. We find that a second-order perturbative treatment of the single-particle spectrum suppresses the proton ${}^{1}S_0$ pairing gap relative to the use of a free spectrum. We estimate the critical temperature for the onset of proton superconductivity to be $T_c=\left(3.2\text{–}5.1\right)\times {10}^9$ K, which is consistent with previous theoretical results in the literature and marginally within the range deduced from a recent Bayesian analysis of neutron star cooling observations.

Figure 1

Diagrammatic contributions to the nucleon self-energy in nuclear matter. The wavy line represents a medium-dependent effective $NN$ interaction derived from two- and three-body chiral forces in isospin-asymmetric nuclear matter.

Figure 2

Equation of state of nuclear matter in beta equilibrium from the chiral two- and three-nuclear forces used in this work.

Figure 3

Proton fraction as a function of density for $\beta$-equilibrated nuclear matter for $n\ge 0.5n_0$. Results are shown for the five density-dependent nuclear interactions at N2LO and N3LO.

Figure 4

Single-particle energies as a function of momentum for protons and neutrons in beta-equilibrated nuclear matter at $k_F^p=0.4{\mathrm{fm}}^{-1}$. The self-consistent second-order approximation to the single-particle energy, shown in Eq. (5), is employed.

Figure 5

Density-dependent pairing gap (as a function of the proton Fermi momentum) from chiral two-body forces. The single-particle energies in the gap equation, Eq. (1), are parametrized in terms of a density-independent effective mass $M^{*}$.

Figure 6

Proton-proton pairing gap in $\beta$-equilibrated nuclear matter from the n3lo450 chiral nuclear potential, including three-body forces. The dotted vertical line represents the proton Fermi momentum at the neutron star core-crust boundary. Three approximations were employed for the single-particle energy spectrum: (i) free spectrum (dotted line), (ii) Hartree-Fock spectrum (dashed line), and (3) self-consistent second-order spectrum (solid line).

Figure 7

Density-dependent proton-proton pairing gap from the n3lo450 chiral nuclear potential for different values of the proton fraction $Y_p$ and for nuclear matter in beta equilibrium. A Hartree-Fock single-particle spectrum is employed.

Figure 8

Proton ${}^{1}S_0$ pairing gap as a function of the Fermi momentum $k_F$ for the five chiral potentials considered in the present work.

Figure 9

${}^{1}S_0$ proton pairing curves from nuclear model. Our EFT calculation (red band) gives similar pairing gaps and smaller range of Fermi momentum where the pairing is available. For comparison, it is shown the previous BCS calculations, Chao et al. [32] (‘CCY'), Takatsuka [33] (‘T'), Amundsen and Østgaard [34] (‘AO'), Baldo et al. [35] (‘BCLL'), Chen et al. [36] (‘CCDK'), Elgarøy et al. [37] (‘EEHO'), Zuo et al. [38] (‘Zuo et al.'), and Baldo and Schulze [90] (‘BS').

Figure 10

Calculated proton pairing gaps using the $e_k^{\left(1\right)}$ single-particle energy (open symbols) and the $e_k^{\left(2\right)}$ single-particle energy (filled symbols) together with the parametrization (solid lines) in Eq. (14).