Robustness of medium-polarization screening in neutron-matter pairing: finite-range particle-hole input, momentum-dependent effective mass, and the triplet channel

Urban and Ramanan (2020) computed the superfluid transition temperature of pure neutron matter in the $^1S_0$ channel including medium polarization (screening) evaluated in the full random-phase approximation (RPA) beyond the Landau approximation, using zero-range Skyrme forces for the particle-hole channel. They found that the Landau approximation fails above about $0.002\ \mathrm{fm}^{-3}$, that screening is stronger than previously thought, and that the antiscreening obtained at Landau level essentially disappears — but they could not decide whether these results, in particular an anomalous large-momentum-transfer behavior of the static density and spin responses, are realistic properties of neutron matter or artefacts of the Skyrme parametrization. The open problem is to settle this question with particle-hole input not limited by zero-range, quadratic-in-momentum forces: benchmark the screening against genuine finite-range interactions (e.g., Gogny) or ab initio/quantum Monte Carlo response data at momenta up to $\sim 2k_F$, quantify the effect of a momentum-dependent effective mass, treat the high-density regime where common functionals develop spin-channel instabilities, and extend the medium-polarization treatment to the coupled $^3P_2$–$^3F_2$ channel of the neutron-star core. Later literature has built parts of the required machinery (parquet-diagrammatic triplet-pairing calculations claiming radical suppression of the $^3P_2$–$^3F_2$ gap, ab initio spin responses in the long-wavelength limit, microscopically constrained Skyrme functionals) but the central benchmark and the momentum-dependent effective-mass screening calculation remain undone.

Background

Pure neutron matter — a degenerate Fermi gas of neutrons, as realized in the inner crust and outer core of neutron stars — becomes superfluid below a critical temperature $T_c$. At crustal densities, pairing occurs in the spin-singlet, isotropic $^1S_0$ partial wave; at the higher densities of the core, where the $^1S_0$ neutron-neutron interaction becomes repulsive, pairing is expected in the spin-triplet channels, chiefly the coupled $^3P_2$–$^3F_2$ waves (total angular momentum $J=2$, coupled orbital waves $L=1$ and $L=3$). The pairing gaps and transition temperatures are key microphysical inputs for neutron-star cooling, neutrino emission, and glitch modeling.

In a medium, the pairing interaction between two neutrons is modified by medium polarization, usually called screening: each member of the pair excites density or spin-density fluctuations of the surrounding Fermi sea — particle-hole (ph) excitations — and the exchange of these fluctuations adds an induced interaction to the bare pairing force. The standard resummation of these ph excitations is the random-phase approximation (RPA), a geometric series of ph bubbles. The widely used Landau approximation replaces the ph interaction inside the bubbles by its zero-momentum-transfer limit, expressed through Landau Fermi-liquid parameters (the dimensionless coefficients $F_0$, $G_0$, etc. that characterize the quasiparticle interaction on the Fermi surface); this is justified only for momentum transfers $q \ll k_F$, where $k_F$ is the Fermi momentum, whereas the induced interaction involves $0 \le q \lesssim 2k_F$.

Urban and Ramanan (2020) reexamined the screened $^1S_0$ transition temperature, evaluating the RPA series exactly with the bare pairing interaction taken as $V_{\rm low\,k}$ (a low-momentum two-body interaction obtained by renormalization-group softening of the Argonne $V_{18}$ potential) and with the ph interaction and effective mass taken from Skyrme energy-density functionals (phenomenological zero-range forces truncated at second order in relative momentum, whose effective mass $m^*$ is momentum-independent). They found that the Landau approximation is adequate only for $k_F \lesssim 0.4\ \mathrm{fm}^{-1}$, where the exactly known low-density limit of induced-interaction screening is reproduced; at higher densities the full RPA screens more strongly, and the antiscreening (enhancement of pairing) found in earlier Landau-level studies is lost. Their full-RPA static responses also behaved anomalously at large $q/k_F$: the density response falls below the Lindhard function (the response of a noninteracting Fermi gas computed with the effective mass), while the spin response is enhanced above it. The authors state explicitly that they cannot decide whether this behavior is realistic or an artefact of the Skyrme form, that a comparison with a genuine finite-range ph interaction such as the Gogny force (a phenomenological Gaussian-mixture nucleon-nucleon interaction) would be desirable but was beyond the scope of their study, that quantum Monte Carlo (QMC) data suggest the density response should approach the free response at large $q$, that a momentum-dependent effective mass — beyond the reach of standard Skyrme functionals — could modify the loop integrals and the screening, that several Skyrme parametrizations develop unphysical ferromagnetic (spin-channel) instabilities above saturation density (the equilibrium density of symmetric nuclear matter, $\rho_0 \approx 0.16\ \mathrm{fm}^{-3}$), and that a realistic treatment of the $^3P_2$–$^3F_2$ channel in the neutron-star core will require methods that build medium correlations systematically, for which they suggest the in-medium similarity renormalization group (IM-SRG), a renormalization-group method that decouples low-energy particle and hole degrees of freedom directly in the medium.

Understanding which of these limitations matter is not a technicality: the quantitative value of the screened gaps controls the heat capacity and pair-breaking neutrino emissivity of neutron-star interiors, so the difference between screening and antiscreening at $k_F \sim 0.8$–$1.2\ \mathrm{fm}^{-1}$ propagates directly into neutron-star cooling and glitch phenomenology.

Problem Statement

Determine whether medium-polarization (screening) effects on neutron-matter superfluidity — as computed beyond the Landau approximation in the full RPA — are robust properties of neutron matter rather than artefacts of zero-range, quadratic-in-momentum Skyrme particle-hole forces, and extend the treatment to the channels and regimes the source study could not reach. Concretely, for pure neutron matter: (1) compute or benchmark the static density and spin responses that enter the induced pairing interaction using genuine finite-range ph interactions (e.g., Gogny) or ab initio/QMC response data, in particular at momentum transfers $q$ up to $\sim 2k_F$, and establish whether the full-RPA behavior found with Skyrme forces — density response suppressed below the free Lindhard function and spin response enhanced at large $q/k_F$ — is realistic or an artefact; (2) quantify how a momentum-dependent effective mass $m^*(p)$ modifies the generalized Lindhard functions, the induced interaction, and the screened transition temperature $T_c(k_F)$; (3) establish the treatment of screening in the high-density regime, above saturation density, where commonly used functionals develop spin-channel (ferromagnetic) instabilities; and (4) extend the medium-polarization treatment to the coupled $^3P_2$–$^3F_2$ triplet channel of the neutron-star core with medium correlations built in systematically (e.g., by in-medium similarity renormalization group or self-consistent finite-range ph interactions). These four targets serve one objective: a quantitatively reliable, model-independent account of screening in neutron-star superfluidity, delivered as the screened pairing interaction, gaps $\Delta(k_F)$ or critical temperatures $T_c(k_F)$, together with an assessment of the residual model dependence of those quantities. Each of the four targets is an independently reportable deliverable with its own resolution standard: a definite result on any one of them — confirming, refuting, or bounding the effect in question, including a partial or negative outcome — resolves that target and stands as progress toward the objective by itself, whereas the objective as a whole is resolved only once all four are settled. A solution that resolves a proper subset of the targets must state which targets it resolves and which it leaves open; a bare survey of the four targets, without a definite result on any of them, resolves none.

References

1. M. Urban and S. Ramanan, 'Neutron pairing with medium polarization beyond the Landau approximation', Phys. Rev. C 101, 035803 (2020). DOI: 10.1103/PhysRevC.101.035803, arXiv:1911.10772, https://arxiv.org/abs/1911.10772
2. 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, arXiv:1804.04332, https://arxiv.org/abs/1804.04332
3. E. Krotscheck and J. Wang, 'Variational and parquet-diagram calculations for neutron matter. III. S-wave pairing', Phys. Rev. C 103, 035808 (2021). DOI: 10.1103/PhysRevC.103.035808, arXiv:2010.13194, https://arxiv.org/abs/2010.13194
4. E. Krotscheck and J. Wang, 'Variational and parquet-diagram calculations for neutron matter. IV. Spin-orbit interactions and linear response', Phys. Rev. C 105, 034345 (2022). DOI: 10.1103/PhysRevC.105.034345, arXiv:2111.12051, https://arxiv.org/abs/2111.12051
5. E. Krotscheck, P. Papakonstantinou, and J. Wang, 'Variational and parquet-diagram calculations for neutron matter. V. Triplet pairing', Phys. Rev. C 109, 015803 (2024). DOI: 10.1103/PhysRevC.109.015803, arXiv:2308.00873, https://arxiv.org/abs/2308.00873
6. P. Papakonstantinou, E. Krotscheck, and J. Wang, 'Triplet pairing in neutron matter in a comprehensive diagrammatic approach', EPJ Web of Conferences 378, 03002 (2026). DOI: 10.1051/epjconf/202637803002, arXiv:2512.00099, https://arxiv.org/abs/2512.00099
7. J. E. Sobczyk, W. Jiang, and A. Roggero, 'Spin Response of Neutron Matter in Ab Initio Approach', Phys. Rev. Lett. 134, 192701 (2025). DOI: 10.1103/PhysRevLett.134.192701, arXiv:2407.20986, https://arxiv.org/abs/2407.20986
8. M. Buraczynski and A. Gezerlis, 'Static Response of Neutron Matter', Phys. Rev. Lett. 116, 152501 (2016). DOI: 10.1103/PhysRevLett.116.152501, arXiv:1510.06417, https://arxiv.org/abs/1510.06417
9. M. Buraczynski, S. Martinello, and A. Gezerlis, 'Skyrme-based extrapolation for the static response of neutron matter', Phys. Rev. C 105, 025807 (2022). DOI: 10.1103/PhysRevC.105.025807, arXiv:2111.08056, https://arxiv.org/abs/2111.08056
10. M. Buraczynski, N. Ismail, and A. Gezerlis, 'Nonperturbative Extraction of the Effective Mass in Neutron Matter', Phys. Rev. Lett. 122, 152701 (2019). DOI: 10.1103/PhysRevLett.122.152701, arXiv:1901.00870, https://arxiv.org/abs/1901.00870
11. M. Duan and M. Urban, 'New Skyrme parametrizations to describe finite nuclei and neutron star matter with realistic effective masses', Phys. Rev. C 110, 065806 (2024). DOI: 10.1103/PhysRevC.110.065806, arXiv:2401.15916, https://arxiv.org/abs/2401.15916
12. M. Duan and M. Urban, 'New Skyrme parametrizations to describe finite nuclei and neutron star matter with realistic effective masses. II. Adjusting the spin-dependent terms', Phys. Rev. C 112, 045808 (2025). DOI: 10.1103/zrtg-2gs5, arXiv:2501.03411, https://arxiv.org/abs/2501.03411
13. S. Gandolfi, G. Palkanoglou, J. Carlson, A. Gezerlis, and K. E. Schmidt, 'The 1S0 Pairing Gap in Neutron Matter', Condens. Matter 7, 19 (2022). DOI: 10.3390/condmat7010019, arXiv:2201.01308, https://arxiv.org/abs/2201.01308