In relativistic (covariant) nuclear energy density functionals, treating exchange (Fock) terms explicitly induces an isovector-scalar, delta-meson-like contribution to the effective spin-orbit potential that Hartree relativistic mean-field models lack. As shown by Ebran, Mutschler, Khan, and Vretenar (Phys. Rev. C 94, 024304 (2016)), this changes the isovector structure of the spin-orbit potential — raising the spin-orbit strength ratio W1/W2W_1/W_2 from about 1 (Hartree) to about 1.8, close to the value 2 of non-relativistic Skyrme Hartree-Fock — but the strength and density dependence of this channel are not constrained by standard bulk data (masses, radii). The open problem is to determine this exchange-induced isovector-scalar channel empirically from observables sensitive to isospin-dependent single-particle structure, chiefly the systematics of spin-orbit splittings in neutron-rich nuclei, and thereby to decide from data whether this exchange-induced isovector-scalar structure — which relativistic Hartree-Fock models and, via Fierz equivalence, local covariant functionals augmented by an explicit isovector-scalar channel encode in the same way — is required in place of the purely isovector-vector structure of Hartree relativistic mean-field models. The core remains open: later work produced a properly refitted local functional with localized exchange terms (PCF-PK1, 2022) whose isovector sector is nevertheless fixed from ab initio input, an active experimental program on spin-orbit splittings in neutron-rich silicon whose interpretation stays degenerate, and 2024-2026 parity-violating electron scattering evidence for a strongly enhanced isovector spin-orbit interaction whose relativistic origin (tensor versus scalar channel) is still unsettled.

Background

Self-consistent mean-field models of nuclear structure are built on energy density functionals (EDFs), which specify the interaction energy of nuclear matter and finite nuclei as a functional of nucleon densities and currents. In the relativistic (covariant) realization, nucleons couple to meson fields or, equivalently, to point-coupling (contact) interactions; in the relativistic mean-field (RMF) approximation only direct (Hartree) terms are kept, whereas the relativistic Hartree-Fock (RHF) scheme also treats exchange (Fock) terms, which are non-local. A key advantage of the covariant framework is that the nuclear spin-orbit potential — the term in the single-particle potential proportional to l⃗⋅s⃗\vec{l}\cdot\vec{s} that splits the energies of states with the same orbital angular momentum ll but different total angular momentum j=l±1/2j=l\pm 1/2 — emerges automatically with roughly the empirical strength from the large scalar and vector fields, whereas non-relativistic functionals such as Skyrme's must add a phenomenological spin-orbit term fitted to observed splittings. Writing the effective non-relativistic spin-orbit potential as Vso(q)=[W1 dρB(q)/dr+W2 dρB(q′)/dr] l⃗⋅s⃗V_{so}^{(q)}=[W_1\, d\rho_B^{(q)}/dr + W_2\, d\rho_B^{(q')}/dr]\,\vec{l}\cdot\vec{s}, where q=n,pq=n,p, q′q' denotes the other species, and ρB\rho_B are the baryon densities, Skyrme Hartree-Fock fixes W1/W2=2W_1/W_2=2, while standard Hartree RMF models give W1/W2W_1/W_2 close to 1 because they contain only a weak isovector-vector (ρ\rho-meson) contribution. Ebran, Mutschler, Khan, and Vretenar (Phys. Rev. C 94, 024304 (2016)) analyzed the effect of exchange terms by reducing the single-nucleon Dirac equation to Schrödinger-like form for the point-coupling RMF Lagrangian, and, for RHF, by first approximating the meson-exchange RHF interaction PKO2 by a point-coupling Lagrangian and then applying a Fierz transformation — a rearrangement of the interaction vertices that trades non-local exchange terms for an equivalent local Lagrangian with additional contact channels. The result is that Fock terms raise W1/W2W_1/W_2 to about 1.8 and generate an isovector-scalar channel (in meson language, delta-meson exchange) whose strength is not an independent parameter of the local equivalent. This matters because the isovector (neutron-versus-proton) structure of the spin-orbit potential governs how spin-orbit splittings evolve along isotopic chains toward the neutron drip line. The isovector-scalar channel cannot be pinned down by standard ground-state data: in the interaction DD-MEδ (Roca-Maza et al., Phys. Rev. C 84, 054309 (2011)) its coupling was fixed from the proton-neutron Dirac effective-mass splitting computed in relativistic Brueckner theory (a microscopic many-body calculation of nuclear matter based on meson-exchange interactions), and its explicit inclusion did not improve calculated masses and radii (Kubis and Kutschera had earlier studied this channel in nuclear matter). A related empirical obstacle is that single-particle energies and spectroscopic factors (the reduction of a transfer-reaction cross section to single-particle occupation weights) are not direct observables, since nuclear correlations must be unfolded through reaction and shell-model theory (Duguet et al., Phys. Rev. C 92, 034313 (2015)). The first extraction of a two-body spin-orbit force amplitude in a nucleus was achieved by Burgunder et al. (Phys. Rev. Lett. 112, 042502 (2014)) from spectroscopic factors in 35^{35}Si and 37^{37}S, which suggested a ~25% reduction of the pp-shell spin-orbit splitting; proton density bubbles inferred spectroscopically in 34^{34}Si (Mutschler et al., Nature Physics 2016) complicate the interpretation, since a depleted central density also weakens the spin-orbit splitting. On the theory side, the Fierz-transformed equivalent of PKO2 was used without refitting its parameters, which distorts single-particle spectra — most visibly by inverting the ordering of the neutron 1i11/21i_{11/2} and 2g9/22g_{9/2} orbitals above N=126N=126 relative to the spectra of 209^{209}Pb and 211^{211}Pb. This ordering is tied to pseudospin symmetry, the approximate quasi-degeneracy of single-particle states (n,l,j=l+1/2)(n,l,j=l+1/2) and (n−1,l+2,j=l+3/2)(n-1,l+2,j=l+3/2), which Ebran et al. argued underlies the empirical kink (slope change) in the charge isotope shifts of Pb nuclei across N=126N=126.

Problem Statement

Can the isovector-scalar, delta-meson-like contribution to the effective spin-orbit potential of relativistic nuclear structure models — the channel generated by exchange (Fock) terms, which raises the spin-orbit strength ratio from W1/W2≈1W_1/W_2\approx 1 in Hartree relativistic mean-field models to W1/W2≈1.8W_1/W_2\approx 1.8, close to the Skyrme value 2 — be determined empirically, in strength and density dependence with quantified uncertainties (the strength being the explicit isovector-scalar coupling of a local covariant functional that includes the channel; the Fierz-equivalent coefficient of the exchange-induced channel is a fixed prediction to be compared against, not a fitted parameter), from observables that probe isospin-dependent single-particle structure, chiefly the systematics of spin-orbit splittings (and associated spectroscopic factors) in neutron-rich nuclei, given that single-particle energies and occupation numbers are not direct observables? Since standard bulk ground-state data (masses, radii) cannot discern between the isovector-vector and isovector-scalar channels, the determination must rest on isospin-dependent single-particle observables; and since relativistic Hartree-Fock models and local covariant functionals augmented by an explicit isovector-scalar channel are Fierz-equivalent routes to the same exchange physics, the decisive question is whether the data require this exchange-induced isovector-scalar structure in place of the purely isovector-vector structure of Hartree relativistic mean-field models. The interpretation must take into account that Fierz-transformed equivalent parameters used without refitting distort single-particle spectra (such as the inversion of the neutron 1i11/21i_{11/2}–2g9/22g_{9/2} ordering near N=126N=126). The determination therefore concerns the data-supported strength and density dependence of the isovector-scalar channel, with its quantified uncertainty and the observables responsible, and whether the data require, rule out, or leave undetermined the exchange-induced isovector-scalar structure relative to the Hartree relativistic mean-field isovector structure.

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