In the relativistic Hartree-Fock approximation (RHFA) for finite nuclei, the exchange (Fock) term is the only channel through which the pion — coupled pseudovectorially to nucleons — enters the covariant energy functional, and with it the pion tensor force (PTF), the non-central component of the nucleon-nucleon interaction that drives spin-orbit splittings and shell-gap evolution. Marcos, López-Quelle, Niembro and Savushkin showed that phenomenologically quenching the PTF to about one third of its RHFA value simultaneously improves the spectrum of 56^{56}Ni, the binding energies of the Pb chain, and the proton 1d1d spin-orbit attenuation along Z=14Z=14, but the quenching is imposed by hand: RHFA contains no short-range correlations, 56^{56}Ni and heavy-nucleus systematics prefer different PTF strengths, and the neutron 2p3/2−1f7/22p_{3/2}-1f_{7/2} gap quenching along N=28N=28 toward 42^{42}Si is overpredicted regardless of the PTF. The open problem is to derive microscopically — within a relativistic many-body framework containing short-range correlations — the magnitude of the in-medium screened PTF (a density dependence may emerge from the derivation rather than being a required deliverable), incorporate it into covariant functionals, and test whether one microscopically determined PTF strength consistently describes both the 56^{56}Ni spin-orbit and N=28N=28-gap observables and the Pb binding-energy systematics. Tensor strengths in covariant functionals are still fixed by fitting the nuclear data rather than derived microscopically, so the core remains open.

Background

Covariant (relativistic) density functional theory describes nuclei as Dirac nucleons interacting through the exchange of mesons (σ\sigma, ω\omega, ρ\rho, π\pi) and the photon. In the relativistic mean-field (RMF) approach only the direct (Hartree) terms are kept; adding the exchange (Fock) terms gives the relativistic Hartree-Fock approximation (RHFA, also called RHF). The Fock sector is the only place where the pion, taken with a pseudovector πN\pi N coupling, contributes; it brings in the tensor force, the non-central part of the nucleon-nucleon interaction proportional to the operator S12=3(σ1⋅q^)(σ2⋅q^)−σ1⋅σ2S_{12}=3(\sigma_1\cdot\hat q)(\sigma_2\cdot\hat q)-\sigma_1\cdot\sigma_2 built from the nucleon spins σ1,2\sigma_{1,2} and the momentum transfer direction q^\hat q. Tensor forces drive spin-orbit (SO) splittings of single-particle levels and the evolution of magic shell gaps along isotopic chains, for example the erosion of the neutron N=28N=28 gap (2p3/2−1f7/22p_{3/2}-1f_{7/2} separation) from 48^{48}Ca to 42^{42}Si. Because the bare pion tensor force (PTF) is very strong at short inter-nucleon distances, short-range nucleon-nucleon correlations — correlation holes in the wave functions, absent at the mean-field level — are expected to screen (quench) it in nuclear matter. Marcos, López-Quelle, Niembro and Savushkin [Phys. At. Nucl. 76, 562 (2013); Phys. At. Nucl. 77, 299 (2014)] developed two approximate schemes to isolate the PTF contribution to the RHFA energy: taking the chiral limit mπ→0m_\pi\to 0, in which the finite-range pion contribution to the energy flows entirely through the PTF, and reading the PTF off the diagonal pion-exchange matrix element of the Hamiltonian. In their 2014 analysis [EPJ Web Conf. 66, 02067 (2014)] they found that reducing the PTF by hand — models with none (PVee, PVe′e') or about one third (PVc′c') of the full PTF — improves the SO splittings and N=28N=28 gap of the spin-unsaturated midweight nucleus 56^{56}Ni and the binding energy per particle of the Pb isotopic chain, and reproduces the measured 1.94 MeV attenuation of the proton 1d1d SO splitting along Z=14Z=14. They stated explicitly that quenching by hand is justified only heuristically, since RHFA lacks the short-range correlations responsible for the screening. Three further limitations were acknowledged: 56^{56}Ni appears to need somewhat more PTF than heavy-nucleus binding-energy systematics (so a single global reduction is inconsistent); the calculated quenching of the neutron 2p3/2−1f7/22p_{3/2}-1f_{7/2} gap along N=28N=28 overshoots experiment for every PTF strength, so missing physics — deformation, higher-order correlations, other interaction components — must be at work; and the spherical symmetry assumed for possibly deformed nuclei such as 42^{42}Si blocks the study of the PTF-deformation interplay. A parallel school treats the pion in density-dependent RHF (DDRHF) functionals (PKO1, PKA1), where the πN\pi N coupling fπf_\pi carries a fitted density dependence: there, the π\pi-pseudovector tensor dominates the tensor channel but is largely canceled by ρ\rho-tensor contributions [W. H. Long et al., EPL 82, 12001 (2008); Lalazissis et al., Phys. Rev. C 80, 041301(R) (2009); Z. Wang et al., Phys. Rev. C 98, 034313 (2018)], making tensor strengths hard to determine from nuclear masses. Relativistic ab initio Brueckner-Hartree-Fock (RBHF) calculations with realistic one-boson-exchange interactions, which resum short-range correlations microscopically, now provide in-medium benchmarks for tensor effects [Shen et al., Prog. Part. Nucl. Phys. 109, 103713 (2019); S. Wang et al., Phys. Rev. C 100, 064319 (2019); S. Wang et al., Sci. Bull. 69, 2166 (2024)], but no density-dependent screened PTF functional for finite nuclei has been extracted from them.

Problem Statement

Derive, within a relativistic many-body framework that contains short-range nucleon-nucleon correlations (for example a relativistic Brueckner/G-matrix treatment of the π\pi-exchange channel or an equivalent resummation), the magnitude of the in-medium screened pion tensor force — replacing the ad hoc reduction to about one third of the full PTF imposed by hand in the relativistic Hartree-Fock approximation (RHFA) on the stated ground that RHFA does not contain the short-range correlations expected to weaken the strong short-distance PTF — and incorporate the derived screened PTF self-consistently into the covariant Hartree-Fock description of finite nuclei. The derived PTF must then be validated against the two classes of observables that motivated the quenching and exposed its inconsistency, namely (i) the spin-orbit splittings and magic gaps of spin-unsaturated midweight nuclei, in particular the ν1f\nu 1f spin-orbit splitting and N=28N=28 gap of 56^{56}Ni, which in the one-third-quenching model PVc′c' require somewhat more PTF contribution than the heavy-nucleus fit, and (ii) the binding-energy-per-particle systematics of heavy nuclei such as the Pb isotopic chain, for which the one-third quenching gives the best agreement; no single global PTF reduction currently fits both (i) and (ii). The question is whether one single microscopically determined PTF strength — either a single density-independent value or, if the derivation yields one, a single density-dependent screened strength — can describe both consistently, without re-tuning that strength separately for each of the two observable classes. A density or in-medium dependence of the screened strength may be part of what the microscopic derivation produces, but it is not a required deliverable. The source's further result for the neutron 2p3/2−1f7/22p_{3/2}-1f_{7/2} gap along N=28N=28 from 48^{48}Ca to 42^{42}Si — its calculated quenching is considerably larger than the experimental one in all models, and removing the PTF relieves the overshoot by only about 0.5 MeV, so the PTF has a meager influence on the evolution of that gap — is retained as background on the limits of spherical PTF tuning, not as a validation requirement.

References

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