Consistent many-body calculation of the ${}^{1}S_{0}$ pairing gap and superfluid transition temperature of neutron matter: screening beyond the Landau approximation, induced three-neutron forces, and self-energy effects

The superfluid transition temperature $T_c$ of pure neutron matter in the ${}^{1}S_{0}$ channel is known to be strongly modified by medium polarization (screening of the pairing interaction by particle-hole fluctuations), and it is exponentially sensitive to details of the effective interaction. Ramanan and Urban (Phys. Rev. C 98, 024314 (2018)) identified a set of uncontrolled approximations in state-of-the-art screening calculations: the Landau (zero-momentum-transfer) approximation in the random-phase approximation (RPA) summation of the induced interaction, the neglect of the quasiparticle residue $Z<1$ and of a consistent single-particle self-energy in the Nozières-Schmitt-Rink (NSR) treatment of preformed pairs, and the absence of the three- and higher-body forces induced by the density-dependent cutoffs needed to recover the Gor'kov-Melik-Barkhudarov low-density limit. Their own 2020 follow-up performed the full RPA check and showed that the anti-screening found in 2018 is largely an artifact of the Landau approximation, and the induced three-neutron force has since been computed and validated for the equation of state. What still does not exist is a single consistent calculation of $T_c$ and the corresponding ${}^{1}S_{0}$ pairing gap as a function of density that simultaneously treats the induced interaction beyond the Landau approximation, induced and bare three-neutron forces, and self-energy effects beyond Hartree-Fock. The problem is to carry out such a calculation and establish which density dependence of $T_c$ survives, in particular the sign and magnitude of the net screening correction above $k_F\sim 0.8\ \mathrm{fm}^{-1}$.

Background

Neutron matter in the inner crust and outer core of neutron stars becomes superfluid through pairing in the spin-singlet, ${}^{1}S_{0}$ neutron-neutron channel at densities below roughly $0.4$--$0.5$ of nuclear saturation density. The Bardeen-Cooper-Schrieffer (BCS) gap equation with a bare two-nucleon interaction predicts a gap of about $2$--$2.5$ MeV peaking near $k_F\approx 0.8\ \mathrm{fm}^{-1}$ (where $k_F$ is the Fermi momentum), but the medium screens the pairing interaction: the exchange of particle-hole (ph) excitations between the paired neutrons and the surrounding Fermi sea generates an induced interaction that adds to the bare force. For a dilute gas of fermions with repulsive induced interaction, Gor'kov and Melik-Barkhudarov (1961) showed that this screening reduces the weak-coupling transition temperature by the universal factor $(4e)^{-1/3}\approx 0.45$ relative to BCS. In neutron matter, the induced interaction is usually computed in the random-phase approximation (RPA), i.e. the ph response is dressed by ring diagrams built from Landau Fermi-liquid parameters; the Landau approximation keeps only the $k\to 0$ limit of the ph interaction, although the induced interaction entering the pairing vertex samples momentum transfers $0\leq k\leq 2k_F$. The bare force is often taken as the low-momentum interaction $V_{\mathrm{low}\,k}$, obtained by renormalization-group evolution of a free-space two-nucleon potential to a cutoff $\Lambda$; Ramanan and Urban (2018) showed that the Gor'kov-Melik-Barkhudarov (GMB) limit is recovered only if $\Lambda$ is scaled with $k_F$. Correlated (preformed) pairs below $T_c$ are described in the Nozières-Schmitt-Rink (NSR) framework, in which the physical density receives a correlated contribution $\rho_{\mathrm{corr}}$ computed with a Hartree-Fock subtraction of the on-shell self-energy — a subtraction that becomes unreliable when it is as large as the leading term. Three conceptually distinct later developments frame the current frontier. First, the full RPA beyond the Landau approximation was computed by Urban and Ramanan (Phys. Rev. C 101, 035803 (2020)) using Skyrme-type ph interactions: the Landau approximation is adequate only for $k_F\lesssim 0.4\ \mathrm{fm}^{-1}$, and at higher densities the full RPA strengthens screening so much that the anti-screening (pairing enhancement by the strongly attractive density-fluctuation Landau parameter $f_0$) reported in 2018 essentially disappears. Second, the three-neutron (3N) force induced by the similarity renormalization group (SRG) flow of the two-neutron $s$-wave interaction at low, $k_F$-scaled cutoffs was computed by Palaniappan, Ramanan, and Urban (Phys. Rev. C 111, 035803 (2025)); including it strongly reduces the cutoff dependence of the dilute-neutron-matter equation of state, but its effect on the pairing gap was explicitly left as work in progress, as was the inclusion of the bare 3N force at higher densities. Third, in the adjacent problem of ultracold two-component Fermi gases, Urban and Ramanan (Phys. Rev. A 113, 053320 (2026)) computed the Hartree shift and pairing gap to third order in Bogoliubov many-body perturbation theory (BMBPT, an expansion around the Hartree-Fock-Bogoliubov ground state) with $k_F$-scaled cutoffs, reproducing the GMB and Galitskii limits (the exact weak-coupling results for the pairing gap and for the self-energy shift, respectively), and stated applications to neutron matter as the next step. Independent benchmarks exist — auxiliary-field diffusion Monte Carlo calculations of the ${}^{1}S_{0}$ gap (Gandolfi et al., 2022) find a modest suppression relative to BCS, and parquet-diagram summations (a resummation treating particle-particle and particle-hole diagrams on an equal footing) treat triplet pairing (Krotscheck et al., 2024) — and reviews (Ramanan and Urban 2021; Vidaña 2021) agree that the low-density gap is broadly under control while the region around and above the gap maximum remains uncertain because of medium effects, 3N forces, and self-energy treatments. The quasiparticle residue $Z<1$ (the reduction of the pole strength of the neutron single-particle Green's function by self-energy effects, which renormalizes the pairing vertex), whose reduction of $T_c$ was flagged in the 2018 paper following Cao, Lombardo, and Schuck (Phys. Rev. C 74, 064301 (2006)), remains unquantified for neutron matter.

Problem Statement

Compute, within a single consistent microscopic framework, the superfluid transition temperature $T_c$ of pure neutron matter, together with the corresponding ${}^{1}S_{0}$ pairing gap $\Delta(k_F)$ (the BCS relation $\Delta_{T=0}(k_F)=1.76\,T_c$ is not assumed to hold once preformed-pair correlations are included), as a function of density over the range in which ${}^{1}S_{0}$ pairing operates in the neutron-star inner crust and outer core, in which the three known uncontrolled corrections to Hartree-Fock-Bogoliubov-level theory are treated simultaneously and at a common level of approximation: (i) the induced (screening) interaction from particle-hole fluctuations summed beyond the Landau approximation, i.e. with the full momentum dependence of the RPA ph response over momentum transfers $0\leq k\leq 2k_F$; (ii) the three-neutron forces — both the force induced by the low, density-dependent cutoffs needed to reproduce the Gor'kov-Melik-Barkhudarov low-density limit and the bare 3N interaction required at higher densities — included in the gap equation, not only in the energy; and (iii) single-particle self-energy effects beyond Hartree-Fock, namely the quasiparticle residue $Z<1$ in the pairing vertex and a correlated-density/self-energy treatment (in the Nozières-Schmitt-Rink framework or beyond) that remains controlled at the densities where the Hartree-Fock subtraction fails. The deliverable is the robust density dependence of $T_c$ and $\Delta$: the sign and magnitude of the net screening correction at densities above $k_F\approx 0.8\ \mathrm{fm}^{-1}$ (whether the anti-screening suggested by Landau-approximation calculations survives any consistent treatment), the location and height of the gap maximum, and a quantified residual cutoff- and model-dependence, benchmarked against the GMB factor $(4e)^{-1/3}\approx 0.45$ in the dilute limit and against quantum Monte Carlo gaps at intermediate densities. Consistency requirements: the same effective interaction and cutoff scheme must be used in the particle-particle and particle-hole channels and in the self-energy, without double counting of ring and ladder diagrams. A solution resolves the task only if (i), (ii) and (iii) are all present at a common level of approximation and $T_c(\rho)$ and $\Delta(k_F)$ are reported over the stated density range, with the sign and magnitude of the net screening correction above $k_F\approx 0.8\ \mathrm{fm}^{-1}$ and the location and height of the gap maximum mandatory; the residual cutoff- and model-dependence and the GMB and quantum Monte Carlo comparisons are required as reported comparisons rather than as agreement targets, and a treatment that omits any of (i)--(iii) counts as partial progress rather than a resolution.

References

1. S. Ramanan and M. Urban, 'Screening and antiscreening of the pairing interaction in low-density neutron matter', Phys. Rev. C 98, 024314 (2018), DOI 10.1103/PhysRevC.98.024314, https://arxiv.org/abs/1804.04332
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