Quantitative microscopic consistency of the $^1S_0$ pairing gap in finite nuclei from chiral pion-nucleon dynamics: medium-polarization (induced-interaction) corrections, particle-hole/particle-particle effective-mass consistency, and missing collective correlations

The 2012 work of Finelli, Nikšić and Vretenar mapped a chiral nucleon-nucleon potential (N3LO two-body plus N2LO three-body sector) onto a separable pairing force adjusted to the $^1S_0$ gap of symmetric nuclear matter, and used it in the particle-particle channel of a relativistic Hartree-Bogoliubov (RHB) calculation with a chiral pion-nucleon-based covariant functional (FKVW) in the particle-hole channel. The mapping left a precise, unresolved core: the quantitative role of medium-polarization (screening/induced-interaction) corrections to the pairing interaction, explicitly deferred to a forthcoming paper, and of missing particle-vibration-coupling contributions, named in the source as potentially important. The framework additionally combines an effective mass in the BCS gap equation with the different single-particle spectrum generated by the covariant particle-hole functional, an inconsistency present in the source calculation though not flagged there as an open item. Independent ab initio work through 2026 shows that chiral interactions still underpredict empirical pairing gaps in finite nuclei, so determining these missing-correlation effects is an open problem whose resolution would anchor the pairing sector of nuclear energy density functionals in chiral dynamics.

Background

In the $^1S_0$ (spin-singlet, orbital angular momentum zero) partial wave, the nuclear interaction is strongly attractive, and nucleons in the vicinity of the Fermi surface form Cooper pairs, making open-shell nuclei superfluid. In infinite isospin-symmetric nuclear matter this is described by the Bardeen–Cooper–Schrieffer (BCS) gap equation, whose solution $\Delta(k_F,k_F)$ — the pairing gap at the Fermi momentum $k_F$ — depends on the two-body nucleon-nucleon (NN) interaction, on three-nucleon (3N) forces, and on the single-particle spectrum, usually approximated by a quadratic form with an effective nucleon mass $M^*(k_F)$. Chiral effective field theory (an organization of nuclear forces in powers of momentum over the chiral-symmetry-breaking scale, with pion exchange plus contact terms; here the N3LO, i.e. next-to-next-to-next-to-leading-order, NN potential together with the N2LO, i.e. next-to-next-to-leading-order, 3N sector) provides these forces with a connection to low-energy QCD.

To compute finite nuclei one commonly uses the relativistic Hartree-Bogoliubov (RHB) framework — a covariant mean-field (density functional) treatment of the particle-hole (ph) channel combined with a pairing field in the particle-particle (pp) channel. Finelli, Nikšić and Vretenar (Phys. Rev. C 86, 034327) used, in the ph channel, the FKVW covariant functional (the Finelli–Kaiser–Vretenar–Weise energy density functional, whose density-dependent couplings are constrained by in-medium QCD sum rules and one- and two-pion exchange), and, in the pp channel, a finite-range pairing force separable in momentum space, of the Tian–Ma–Ring Gaussian form $\langle k|V|k'\rangle = -G\,e^{-a^2k^2}e^{-a^2k'^2}$, whose two parameters $(G,a)$ were adjusted to reproduce the density dependence of the chiral $^1S_0$ gap of symmetric matter computed with the bare N3LO NN force plus a density-dependent effective two-body form of the N2LO 3N interaction. The calculated average pairing gaps in Ni, Sn and Pb isotopic chains and N = 28, 50, 82 isotonic chains reproduce empirical trends, with 3N forces sizably reducing the gaps, but the analysis was explicitly incomplete.

Empirical pairing gaps are deduced from odd-even mass staggering, e.g. the five-point formula $\Delta^{(5)}(N_0) = -\frac{1}{8}[E(N_0+2)-4E(N_0+1)+6E(N_0)-4E(N_0-1)+E(N_0-2)]$ built from experimental binding energies (the source work itself used the analogous three-point mass differences centered on odd-mass nuclei). Particle-vibration coupling (PVC) — the coupling of single quasiparticles to collective surface vibrations of the nucleus, which exchange phonons and can generate an additional, state-dependent component of the pairing field — was cited as a missing contribution, and medium-polarization (screening) corrections, i.e. induced interactions generated by the exchange of density and spin-density fluctuations of the medium, were deferred.

Problem Statement

For the microscopic description of the $^1S_0$ pairing gap of finite nuclei built from chiral pion-nucleon dynamics — a separable pairing interaction mapped from the N3LO two-body plus N2LO three-body chiral forces and used in the pp channel of a relativistic Hartree-Bogoliubov calculation whose ph channel is a chiral covariant (FKVW-type) functional — determine, quantitatively: (a) how medium-polarization (screening/induced-interaction) corrections to the pairing interaction modify the $^1S_0$ gap in symmetric nuclear matter and, through the mapping, the average pairing gaps of finite nuclei — the calculation the source explicitly deferred to a forthcoming paper; and (b) the size of the missing collective contributions, in particular particle-vibration coupling, that the source named as potentially important for bringing calculated gaps into quantitative agreement with empirical odd-even mass-staggering gaps. A further feature inherited from the source framework is the mismatch between the effective mass $M^*(k_F)$ entering the nuclear-matter gap equation and the single-particle spectrum generated by the covariant ph functional in the RHB solver; this inconsistency is present in the source calculation but is not flagged there as an open item, and an answer must treat it explicitly when interpreting the corrections of (a) and (b), either by quantifying its effect on those corrections or by stating explicitly why it is left untreated. These questions jointly serve one objective — a quantitatively predictive, microscopically consistent chiral-based account of nuclear pairing — and the mandatory deliverables are: (a) and (b) above; an explicit treatment of the ph/pp effective-mass mismatch; and an explicit statement of the induced-interaction treatment and its double-counting relative to the 3N forces. An answer must state whether the corrections considered suffice to bring the calculated gaps into quantitative agreement with the empirical odd-even mass-staggering gaps, this sufficiency verdict being judged over the chains studied in the source (the Ni, Sn and Pb isotopic chains and the N = 28, 50, 82 isotonic chains, or an explicitly justified substituted set) and taking 'quantitative agreement' to mean a residual deviation within about 0.2 MeV or about 10%, whichever the solver justifies; a larger residual must be reported as unresolved rather than as agreement, and the verdict must not presuppose that any particular mechanism restores agreement.

References

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