Quantitative link between the in-medium scalar-vector balance and pseudospin symmetry restoration in relativistic nuclear functionals

In covariant density functional theory, pseudospin symmetry (PSS) — the quasi-degeneracy of single-particle doublets $(n,l,j=l+1/2)$ and $(n-1,l+2,j=l+3/2)$ — is governed by the balance between the attractive scalar potential $S(r)$ and the repulsive vector potential $V(r)$ of the Dirac nucleon self-energy, with exact PSS tied to $S(r)+V(r)=0$ or, more generally, $d[S(r)+V(r)]/dr=0$ . Geng, Li, Long, Niu, and Chang (Phys. Rev. C 100, 051301 (2019)) showed within relativistic Hartree-Fock theory that the Fock terms, above all the $\rho$ -meson tensor coupling, restructure the $\sigma$ - $\omega$ balance and force the isoscalar couplings $g_\sigma$ and $g_\omega$ away from the density dependence common to other functionals, and that this restructuring is what restores PSS for high-angular-momentum pseudospin doublets and eliminates spurious shell closures at $N,Z=58$ and $92$ . Their account, however, is qualitative: mean-field only, without particle-vibration coupling, and without a derived relation between the radial variation of $S+V$ (or the coupling density dependence) and the pseudo-orbital-angular-momentum ( $l'$ ) dependence of pseudospin-orbit splittings. Subsequent work (DD-LZ1 in 2020; tensor-coupling EDFs DDT/DDTC in 2024) completed Hartree-level refits guided by this mechanism and eliminated the spurious shells, but the quantitative, formal, and beyond-mean-field questions remain open. This problem asks for a quantitative theory connecting the in-medium balance to PSS restoration: a derived and benchmarked relation between $d[S(r)+V(r)]/dr$ and the coupling density dependence on one side and $l'$ -dependent pseudospin-orbit splittings on the other; its robustness under particle-vibration coupling and other beyond-mean-field corrections; and, as an optional extension beyond the open points of Geng et al. (2019), its extension to the isovector sector and the wider nuclear chart, including relativistic Hartree-Fock refits with varied $\rho$ -tensor density dependence.

Background

In nuclear structure, covariant density functional theory (CDFT) describes nucleons as Dirac particles moving in self-consistent fields generated by meson exchange; its mean-field limit, retaining only the direct (Hartree) terms, is the relativistic mean field (RMF) model, while the relativistic Hartree-Fock (RHF) approach also keeps the exchange (Fock) terms. The Dirac self-energy splits into an attractive scalar potential $S(r)$ of a few hundred MeV and a repulsive vector potential $V(r)$ of similar magnitude, dominated respectively by the isoscalar $\sigma$ -meson scalar ( $\sigma$ -S) and $\omega$ -meson vector ( $\omega$ -V) couplings; the binding of nuclei reflects a delicate cancellation between them. Pseudospin symmetry (PSS) is the empirical quasi-degeneracy of the single-particle doublets $(n,l,j=l+1/2)$ and $(n-1,l+2,j=l+3/2)$ , which become the pseudo-spin partners $(n',l'=l+1,j'=l'\pm1/2)$ of a spin- $1/2$ pseudo-nucleon with pseudo-orbital angular momentum $l'$ . Ginocchio (1997) identified PSS as a relativistic symmetry of the Dirac Hamiltonian that is exact when $S(r)+V(r)=0$ , and Meng et al. (1999) generalized the exact-symmetry condition to $d[S(r)+V(r)]/dr=0$ ; the splitting within a pseudospin doublet, the pseudospin-orbit (PSO) splitting $\Delta E_{\mathrm{PSO}}$ , therefore measures PSS breaking and is expected to track the radial variation of $S(r)+V(r)$ . Conventional RMF functionals badly violate PSS for high- $l'$ doublets, producing spurious shell closures (incorrectly predicted magic nucleon numbers) at neutron or proton numbers 58 and 92; the RHF functional PKA1, which includes the $\pi$ -meson pseudoscalar coupling and the $\rho$ -meson tensor ( $\rho$ -T) coupling (with tensor-to-vector strength ratio $\kappa_\rho=f_\rho/g_\rho$ ) via the Fock terms, removes them and restores PSS near the Fermi surface, with consequences for magic numbers, halos, and superheavy nuclei (Long et al., 2007). Geng, Li, Long, Niu, and Chang (Phys. Rev. C 100, 051301 (2019)) traced this to the in-medium balance: the strongly attractive $\rho$ -T Fock term reshapes the $\sigma$ -S/ $\omega$ -V balance, and the density dependencies of $g_\sigma$ and $g_\omega$ in PKA1 deviate from the nearly parallel behavior ( $g_\sigma/g_\omega\approx$ constant) shared by other density-dependent functionals; tentative parametrizations ( $1.0\kappa_\rho$ , $0.0\kappa_\rho$ ) confirmed the interpretation by deforming the balance in controlled steps. The authors stressed, however, that their analysis is at the mean-field level and provides only qualitative guidance for modeling: particle-vibration coupling (PVC), the coupling of a single-particle state to collective surface vibrations that shifts single-particle levels toward the Fermi surface, was only estimated qualitatively, and no quantitative relation between $d[S(r)+V(r)]/dr$ (equivalently, the density dependence of $g_\sigma$ , $g_\omega$ , and $\kappa_\rho$ ) and the observed $l'$ -dependence of PSO splittings was derived.

Problem Statement

Establish a quantitative account of pseudospin symmetry (PSS) restoration by the in-medium balance of nuclear attraction and repulsion in covariant density functional theory, going beyond the qualitative mean-field systematics of Geng et al. (Phys. Rev. C 100, 051301 (2019)). Concretely: (i) derive and benchmark a precise relation between the radial variation of the scalar-plus-vector nucleon self-energy, $d[S(r)+V(r)]/dr$ , together with the dependence of the isoscalar couplings $g_\sigma$ and $g_\omega$ and the $\rho$ -tensor strength ratio $\kappa_\rho$ on the baryon (nuclear-matter) density $\rho_b$ , on one side, and the pseudo-orbital-angular-momentum ( $l'$ ) dependence of pseudospin-orbit splittings of pseudospin doublets on the other, so that the restoration condition becomes a quantitative design criterion rather than qualitative guidance; (ii) determine whether this balance mechanism survives beyond-mean-field correlations, in particular how particle-vibration-coupling corrections to single-particle levels and absolute PSO splittings in RHF functionals with and without the $\rho$ -tensor coupling interplay with the density dependence of $g_\sigma$ and $g_\omega$ ; and (iii) test the mechanism's scope in the isovector sector (the role of the $\delta$ -meson isovector scalar coupling and of the full $\rho$ -vector-tensor coupling, i.e. the simultaneous $\rho$ -meson vector and tensor couplings) and across the nuclear chart, including relativistic Hartree-Fock refits in which the density dependence of the $\rho$ -tensor coupling is varied and single-particle energies, radii, and masses are surveyed. Components (i) and (ii) are both required and jointly constitute the core objective — a quantitative, benchmarked, beyond-mean-field theory of how the in-medium balance restores PSS — and correspond to the open points raised by Geng et al. (2019) themselves: the absence of a derived relation between $d[S(r)+V(r)]/dr$ (equivalently the coupling density dependence) and the $l'$ -dependent PSO splittings, and the mean-field-only, qualitative treatment of PVC. Component (i) is resolved only when the derived relation is benchmarked against self-consistent relativistic Hartree-Fock calculations, i.e. its predicted $l'$ -dependence of the doublet splittings — including the high- $l'$ doublets near the Fermi surface associated with the spurious shell closures — is confronted quantitatively with the splittings computed there; stating the relation without such a comparison does not resolve (i). Component (ii) is resolved only when the particle-vibration-coupling corrections to the single-particle levels and absolute PSO splittings are used to determine quantitatively whether the density-dependent balance of $S+V$ still governs the resulting splittings; a qualitative expectation does not resolve (ii). Component (iii) is an optional extension proposed by this problem, not required for addressing the open questions of Geng et al. (2019).

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