Medium polarization effects on $^1S_0$ neutron pairing, dineutron correlations, and the BCS-BEC crossover in nuclear and neutron matter

Neutron pairing in the $^1S_0$ channel in symmetric nuclear matter and pure neutron matter has been characterized at the mean-field level with finite-range Gogny pairing forces, giving the pairing gap $\Delta(k)$, the anomalous density $\kappa_k$, the two-neutron density correlation function $D(0)$, and a BCS-BEC crossover window $k_{Fn}\in[0.15,0.63]$ fm$^{-1}$ with no true dineutron BEC state. The open problem is to determine quantitatively how medium polarization effects (dynamical self-energy and induced vertex corrections from particle-hole excitations), together with clustering and virial physics at very low densities, modify these observables and the crossover boundaries, and to build a consistent momentum-dependent treatment with finite-range interactions that reconciles relativistic Hartree-Bogoliubov, contact-regularized gap-equation, quantum Monte Carlo, and virial-expansion predictions across the full density range. Low-density benchmarks (Gor'kov-Melik-Barkhudarov suppression, quantum Monte Carlo gaps) now semi-quantitatively constrain the answer, but the screening-versus-antiscreening discrepancy at moderate densities and the absence of any polarization-corrected finite-range calculation of the pair observables leave the problem open.

Background

Pairing of neutrons in the $^1S_0$ partial wave (spin-singlet, zero orbital angular momentum) is the longest-established superfluid instability of dilute nuclear matter. In homogeneous matter it is described by a BCS-type gap equation for the momentum-dependent gap function $\Delta(k)$, with the normal density $\rho_k$ and the anomalous density $\kappa_k$ (the off-diagonal density carrying the condensate-like Cooper-pair amplitude) as companions. The neutron-neutron scattering length is large and negative (about $-18.5$ fm), so dilute neutron matter lies near the unitary regime of universal Fermi gases (the limit where the scattering length exceeds the interparticle spacing and low-energy observables become independent of its detailed value); this motivates the analogy with the BCS-BEC crossover studied in ultracold atoms, i.e. the evolution from weakly bound Cooper pairs to compact bosonic dimers. A dineutron is a spatially compact two-neutron correlation; since it is unbound in vacuum by about 60 keV, any dineutron-like object in matter is a medium effect. Practical diagnostics are $\kappa_k$, the two-neutron density correlation function $D(0)$ (the probability of finding two neutrons at the same point), and the ratio of the Fermi-surface gap $\Delta_{F_n}$ to the Fermi energy $e_{F_n}$.

Sun and Pan (2013) computed these quantities for symmetric nuclear matter and pure neutron matter in the relativistic Hartree-Bogoliubov (RHB) framework: relativistic mean-field theory in the particle-hole channel (the PK1 parametrization) plus a BCS pairing equation in the particle-particle channel, with the finite-range Gogny force (the D1, D1S, D1N two-Gaussian parametrizations) as the pairing interaction. They found that the three Gogny forces give Fermi-surface gaps smaller at low density than those from the bare Bonn-B potential (a standard one-boson-exchange nucleon-nucleon interaction), that no true dineutron BEC state occurs at any density, that the most BEC-like state appears near $k_{Fn}\sim0.3$ fm$^{-1}$, and that the BCS-BEC crossover is confined to $k_{Fn}\in[0.15,0.63]$ fm$^{-1}$ by their relativistic criteria (the dineutron probability and the ratio $\Delta_{F_n}/e_{F_n}$). The companion RHB study of Sun, Sun, and Meng (2012) had shown that artificially scaling the bare pairing strength by more than 1.10 produces a dineutron BEC state in the low-density limit, while scaling below 0.85 keeps the system entirely in the weak-coupling BCS regime, so the conclusion is sensitive to the effective pairing strength.

The calculation was deliberately at mean-field level: it omits medium polarization effects (MPEs), the dynamical self-energy corrections and the induced (vertex) interaction generated by particle-hole excitations beyond mean field. This omission is consequential because the literature was and remains split on the sign and size of these corrections. Quasiparticle-interaction calculations of Wambach, Ainsworth, and Pines (1993) found a strong suppression of the neutron-matter gap; particle-vibration-coupling calculations of Barranco et al. (1999) (coupling quasiparticles to collective surface vibrations of the nucleus) found that surface vibrations enhance pairing in finite nuclei; the Brueckner (self-consistent resummation of two-body scattering into an effective G-matrix interaction) plus random-phase-approximation (RPA) calculation of Cao, Lombardo, and Schuck (2006) found that polarization quenches the gap in neutron matter but increases pairing correlations in symmetric nuclear matter; and Margueron, Sagawa, and Hagino (2007) showed with density-dependent contact interactions that a screened pairing force would drive a dineutron BEC state in symmetric matter near $k_{Fn}\sim0.2$ fm$^{-1}$.

Two further methodological limitations frame the open question. First, the regularized gap-equation analyses used to locate the crossover replace the interaction matrix element $v_{pp}(k,p)$ by a momentum-independent contact strength fixed by the scattering length $a_{nn}$, with crossover boundaries at $1/(k_{Fn}a_{nn})=\pm1$; this approximation is reliable only for dilute systems and gives poor information at higher densities, where $v_{pp}(k,p)$ and $\Delta(k)$ are strongly momentum dependent. Second, at very low densities the mean-field description omits clustering, and the appropriate tools are quantum Monte Carlo methods (stochastic ab initio simulations of the many-neutron ground state, e.g. auxiliary-field diffusion Monte Carlo) and the virial expansion (a low-density, high-temperature expansion whose coefficients are fixed directly by measured scattering data). The open problem is to put these pieces together: a quantitatively polarization-corrected, momentum-dependent, finite-range description of $^1S_0$ neutron pairing and its crossover structure that is anchored to the low-density (Gor'kov-Melik-Barkhudarov, unitary-limit, virial) benchmarks and valid across the whole density range. The Gor'kov-Melik-Barkhudarov (GMB) correction is the leading induced-interaction suppression of the critical temperature at asymptotically low density, a factor $(4e)^{-1/3}\approx0.45$.

Problem Statement

Determine, quantitatively and consistently, how medium polarization effects — the dynamical self-energy and the induced (vertex) interaction generated by particle-hole excitations beyond the mean-field relativistic Hartree-Bogoliubov approximation — modify the $^1S_0$ neutron pairing gap $\Delta(k)$, the anomalous density $\kappa_k$, and the two-neutron density correlation function $D(0)$ in symmetric nuclear matter and pure neutron matter, and whether these corrections change the mean-field conclusions that no true dineutron BEC state occurs and that the BCS-BEC crossover is confined to $k_{Fn}\in[0.15,0.63]$ fm$^{-1}$. The single overarching objective is a polarization-corrected, momentum-dependent description of neutron pairing with finite-range interactions that is valid across the full density range; its linked components are: (i) a many-body treatment that includes self-energy and vertex corrections without double counting, using the momentum-dependent Gogny-type pairing force rather than the momentum-independent contact approximation (which is reliable only in the dilute limit); (ii) an account of how clustering and virial physics at very low densities alter pairing and whether the inferred crossover window shifts, connecting the Cooper-pair description to quantum Monte Carlo and virial-expansion benchmarks; and (iii) a consistent reconciliation of the relativistic Hartree-Bogoliubov, contact-regularized gap-equation, quantum Monte Carlo, and virial-expansion predictions across the entire density range, resolving the screening-versus-antiscreening discrepancy that persists among microscopic treatments. Resolution of the task requires a self-consistent polarization-corrected (self-energy plus induced-vertex) evaluation of $\Delta(k)$, $\kappa_k$, and $D(0)$ in symmetric nuclear matter and pure neutron matter, together with an explicit statement of whether the two mean-field conclusions above change; components (ii) and (iii) are optional extensions that connect the result to clustering, virial, and cross-framework benchmarks but are not required to resolve the task.

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