Low-density neutron matter (NM) is a strongly interacting Fermi system because the neutron-neutron scattering length in the spin-singlet 1S0^1S_0 channel, ann≃−18.9a_{nn}\simeq-18.9 fm, is much larger than the interaction range, so dilute NM lies close to the unitary limit. Building on the program laid out by Sammarruca (Phys. Rev. C 108, 044310 (2023)), the open problem is to replace the currently model-dependent predictions of neutron-matter pairing with quantitatively reliable, uncertainty-quantified, model-independent results. Concretely: (a) determine the 1S0^1S_0 neutron-neutron pairing gap beyond the Bardeen-Cooper-Schrieffer (BCS) approximation, systematically including medium-polarization (screening) effects, vertex corrections, and self-consistent single-particle energies with consistent two- and three-nucleon forces; and (b) settle the existence and size of the spin-triplet 3P2^3P_2–3F2^3F_2 gap at neutron-star-core densities, where published calculations still disagree on whether a gap exists at all. The same source also revisits the unitarity-derived lower bound on the symmetry energy and its propagation into (L,Esym,0)(L, E_{sym,0}) constraints, but presents that propagation as a qualitative demonstration; it is retained here as background context rather than as a separately posed open deliverable. Literature through 2026 shows both pairing components remain open: the beyond-BCS 1S0^1S_0 gap retains a method-dependent spread, and the triplet-channel results range from near-complete suppression to gaps of 0.010.01–1010 MeV.

Background

Neutron matter — an idealized uniform system of neutrons — is the reference system for the inner crust and core of neutron stars, with densities from roughly 10−310^{-3} fm−3^{-3} in the crust up to and beyond nuclear saturation density ρ0≈0.16\rho_0\approx0.16 fm−3^{-3}. In the spin-singlet 1S0^1S_0 channel (spin S=0S=0, orbital angular momentum L=0L=0, total angular momentum J=0J=0) the neutron-neutron interaction is attractive but too weak to bind the dineutron, producing instead a virtual state and a huge negative scattering length, ann≃−18.9a_{nn}\simeq-18.9 fm. When ∣ann∣|a_{nn}| far exceeds the effective range (re≈2.75r_e\approx2.75 fm), dilute NM behaves like a unitary Fermi gas — the a→−∞a\to-\infty limit in which all length scales drop out and the energy per particle is ξEF\xi E_F with the universal Bertsch parameter ξ≈0.37\xi\approx0.37 and EFE_F the Fermi energy. This universality links NM to ultracold-atomic experiments and underlies the use of unitarity as a lower bound on the NM energy per particle, ENM(ρ)≥ξEFE_{NM}(\rho)\ge\xi E_F (equality only at a=∞a=\infty) — a bound proposed by Tews et al., consistent with existing calculations but not proved in general — and hence on the nuclear symmetry energy Esym(ρ)=ENM(ρ)−ESNM(ρ)E_{sym}(\rho)=E_{NM}(\rho)-E_{SNM}(\rho), where SNM is symmetric nuclear matter (equal neutrons and protons). Modern nuclear forces are derived from chiral effective field theory (EFT), an expansion in powers of Q/ΛbQ/\Lambda_b (typical momentum over breakdown scale) organized by power counting, which yields two-nucleon forces (2NF) plus three-nucleon forces (3NF) with low-energy constants (LECs) fixed in few-nucleon systems; the truncation order, cutoff/regulator choice, and 3NF LEC uncertainties propagate into every NM prediction.

The 1S0^1S_0 attraction makes the dripped neutrons of a neutron star's inner crust superfluid through Cooper pairing, described at the mean-field level by the BCS gap equation, which yields a peak gap of roughly 2.52.5–33 MeV near kF∼0.8k_F\sim0.8 fm−1^{-1} (kFk_F the Fermi momentum). This superfluid governs the thermal evolution and rotational dynamics of the crust, including pulsar glitches — the sudden spin-ups observed in rotating neutron stars. Beyond BCS, the pairing interaction is modified by the surrounding medium: particle-hole polarization diagrams screen or anti-screen the attraction, vertex corrections and the self-energy renormalize the quasiparticle spectrum (effective mass m∗m^*, quasiparticle residue ZZ), and 3NFs add density-dependent contributions. Published beyond-BCS treatments — from early screening calculations that suppressed the gap almost completely, to random-phase-approximation and Brueckner-based studies finding moderate reduction, to quantum Monte Carlo (QMC) methods (stochastic sampling of the many-body Schrödinger equation) finding moderate suppression with large error bars — still do not agree on a model-independent 1S0^1S_0 gap with quantified uncertainties.

At higher densities (kF≳1.3k_F\gtrsim1.3 fm−1^{-1}), the most attractive channel becomes the spin-triplet 3P2^3P_2–3F2^3F_2 coupled channel (S=1S=1, J=2J=2, with the tensor force coupling L=1L=1 and L=3L=3 partial waves), which would make the neutrons of a neutron-star core superfluid. Whether this gap exists, and how big it is, directly controls neutron-star cooling: the onset of neutron 3P2^3P_2 pairing has been invoked to explain the observed rapid cooling of the Cassiopeia A neutron star. Here the situation is far more unsettled than in the singlet channel: the results are extremely sensitive to the 3NF (with one Urbana-IX-family parametrization supporting a gap and another removing it), to in-medium renormalization of the spin-orbit interaction, and to short-range correlations; recent diagrammatic calculations find an almost complete suppression of the gap, while calculations including induced (Kohn-Luttinger-type) interactions can generate gaps anywhere from 1010 keV to 1010 MeV depending on the interaction.

Finally, the unitarity bound on the symmetry energy is converted into constraints on the slope parameter L=dEsym/dρ∣ρ0L=dE_{sym}/d\rho|_{\rho_0} and value Esym,0=Esym(ρ0)E_{sym,0}=E_{sym}(\rho_0) by expanding the SNM energy about saturation, ESNM(u)=E0+K0(u−1)2/18+Q0(u−1)3/162+…E_{SNM}(u)=E_0+K_0(u-1)^2/18+Q_0(u-1)^3/162+\dots with u=ρ/ρ0u=\rho/\rho_0, incompressibility K0K_0, and skewness Q0Q_0; LL and Esym,0E_{sym,0} set neutron-star radii and tidal deformabilities. Present treatments truncate this expansion at a poorly known Q0Q_0, and the resulting bounds have only been demonstrated qualitatively, without quantified higher-order or chiral-EFT uncertainties. The open problem announced by the source (Section V) is the systematic prediction of uncertainty-quantified, model-independent pairing gaps in both the 1S0^1S_0 and the 3P2^3P_2–3F2^3F_2 channel; the unitarity-bound exercise above is motivating background from the same paper rather than part of the announced open goal.

Problem Statement

The problem, as posed by the source program (Sammarruca, Phys. Rev. C 108, 044310 (2023), Section V), is to replace the present model-dependent predictions for neutron-matter superfluidity with quantitatively reliable, model-independent results carrying controlled uncertainty budgets — one objective with two jointly supporting deliverables:

  1. (1S0^1S_0 gap beyond BCS) Compute the 1S0^1S_0 neutron-neutron pairing gap Δ(kF)\Delta(k_F) in pure neutron matter beyond the BCS approximation, systematically including medium-polarization (screening/anti-screening) effects, vertex corrections, self-consistently determined single-particle energies, and consistent two- and three-nucleon chiral-EFT interactions, such that the spread among reasonable modern treatments is explained and the residual theoretical uncertainty is quantified (from chiral truncation, regulator, 3NF low-energy constants, and many-body truncation) rather than merely exhibited.

  2. (3P2^3P_2–3F2^3F_2 triplet gap) Determine whether a 3P2^3P_2–3F2^3F_2 pairing gap exists in neutron matter at neutron-star-core densities (kF≳1.3k_F\gtrsim1.3 fm−1^{-1}) and, if so, its magnitude and density profile — resolving the present conflict between calculations that find gaps of order 0.010.01–1010 MeV and those finding near-complete suppression, including the role of the 3NF parametrization, in-medium renormalization of the spin-orbit interaction, and induced interactions.

The completion boundary is a set of predictions for these observables whose theoretical error bands are quantified and defensible: gap functions for both channels over the relevant density ranges — the 1S0^1S_0 gap over the inner-crust range 0.05≲kF≲1.30.05\lesssim k_F\lesssim1.3 fm−1^{-1} and the 3P2^3P_2–3F2^3F_2 statement over kF≳1.3k_F\gtrsim1.3 fm−1^{-1} — with uncertainties attributed per source, and a definitive statement on triplet-channel neutron superfluidity (if the 3P2^3P_2–3F2^3F_2 gap is absent, that definitive negative statement takes the place of the triplet gap function). Relative language such as 'model-independent' is meant in the established sense of the field: predictions whose variation across interactions, regulators, and many-body schemes at fixed chiral-EFT order is accounted for and small compared with the claimed accuracy, not the absence of any model input.

References

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