Quantitative mechanism of pseudospin-symmetry breaking and accidental restoration in relativistic mean-field nuclei: second-order systematics and predictions for neutron-skin and bubble nuclei

Pseudospin symmetry (PSS) — the empirical near-degeneracy of nuclear single-particle doublets such as $(2d_{5/2},1g_{7/2})$ — shapes shell evolution far from stability, yet the mechanism of its breaking and of its accidental restoration in particular nuclei is only partly understood. Heitz, Ebran and Khan (Phys. Rev. C 111, 064306 (2025)) derived the first closed-form perturbative formulae for spin and pseudospin splittings in relativistic mean-field theory, built on a spin-symmetric relativistic harmonic-oscillator reference, and showed that the pseudospin splitting decomposes into a $\Sigma$ -driven term sensitive to mass number and surface diffusivity plus an almost constant $\Delta$ -driven term. Their study leaves open the quantitative, nucleus-specific account of accidental pseudospin restorations beyond leading order, and the explicit test calculations they themselves propose: pseudospin splittings in neutron-rich nuclei with large neutron skins (extreme diffusivity) and in bubble nuclei with central density depletion. The mechanism-resolved perturbative account and its extreme-diffuseness benchmarks remain unperformed.

Background

In nuclear structure theory, single-particle states are labeled $(n,\ell,j)$ by radial quantum number $n$ , orbital angular momentum $\ell$ and total angular momentum $j=\ell\pm1/2$ . The spin-orbit interaction splits the two $j$ partners of a given $(n,\ell)$ orbital into a 'spin doublet'; these splittings are large and generate the magic numbers (shell closures) that stabilize certain nuclei. In relativistic mean-field (RMF) theory, also called covariant density functional theory, nucleons obey a Dirac equation with a self-consistent scalar potential $S$ and vector potential $V$ . Two combinations control the spectrum: the central potential $\Sigma=V+S$ and their difference $\Delta=V-S$ , which drives spin-symmetry breaking (SS), i.e. the lifting of the spin-doublet degeneracy; the effective spin-orbit potential is proportional to $(1/r)\,\mathrm{d}\Delta/\mathrm{d}r$ (Ebran et al., Phys. Rev. C 94, 024304 (2016)). (Care with signs is needed: the source paper's introduction defines $\Sigma=V+S$ and $\Delta=V-S$ , but its Fig. 1 caption and conclusion write $\Delta=S-V$ ; a consistent convention must be fixed when reproducing its formulae.) The Dirac spinor has an upper component $g$ (which becomes the Schrödinger wave function in the non-relativistic limit) and a lower component $f$ .

Pseudospin symmetry (PSS), introduced in 1969, is the empirical near-degeneracy of the doublets $(n,\ell,\ell+1/2)$ and $(n-1,\ell+2,\ell+3/2)$ , which acquire a common pseudo-orbital angular momentum $\tilde{\ell}=\ell+1$ and a pseudo-spin $1/2$ . A doublet counts as accidentally restored when its splitting is anomalously small relative to the smooth $A$ -dependent trend of the surrounding isotopic or isotonic chain — a local near-degeneracy of the kind the source reports in specific nuclei — rather than merely small in absolute terms. PSS underlies superdeformed and identical rotational bands and fingerprints shell evolution in exotic nuclei (Liang, Meng and Zhou, Phys. Rep. 570, 1 (2015)). Ginocchio (Phys. Rev. Lett. 78, 436 (1997)) showed that PSS is a relativistic symmetry: exact PSS requires the central potential $\Sigma$ to vanish (or have vanishing gradient for bound states). Exact PSS is unphysical — it would render nucleons unbound — so PSS can only be realized approximately, and a PSS-preserving state cannot serve as the reference for perturbation theory. Liang et al. (Phys. Rev. C 83, 041301 (2011)) showed instead that a spin-symmetric relativistic harmonic oscillator (SS-RHO) — harmonic $\Sigma_{\mathrm{HO}}$ , constant $\Delta_0$ — is a well-defined reference state about which pseudospin breaking can be treated perturbatively.

Heitz, Ebran and Khan (Phys. Rev. C 111, 064306 (2025)) turned this into closed formulae: acting with self-consistent $\Sigma$ and $\Delta$ potentials from relativistic Hartree-Bogoliubov (RHB: RMF plus pairing) calculations with the DD-MEV parametrization (a covariant density functional whose fit includes single-particle data) on the SS-RHO reference, they obtained first- and second-order expressions for the doublet splittings. For spin doublets only $\Delta$ and the lower component $f$ contribute; for pseudospin doublets both potentials and both components contribute, and after integration by parts the first-order pseudospin splitting is $\Delta E_{\mathrm{PSS}}=\int \mathrm{d}r\,(G_a-G_b)\,\mathrm{d}\Sigma/\mathrm{d}r+\int \mathrm{d}r\,(F_a-F_b)\,\mathrm{d}\Delta/\mathrm{d}r$ , where $F$ ( $G$ ) are radially cumulative integrals of $|f|^2$ ( $|g|^2$ ) over the doublet partners $a,b$ . The $\Sigma$ -driven term $\Delta E_g$ grows roughly linearly with mass number $A$ while the $\Delta$ -driven term $\Delta E_f$ stays almost constant and negative. Surface diffuseness of $\Sigma$ (how gradually the potential falls off at the nuclear surface) is the controlling knob: a more diffuse surface widens $\mathrm{d}\Sigma/\mathrm{d}r$ , making it probe the negative core lobe of $G_a-G_b$ and reducing the splitting. In the benchmark neutron $(2d_{5/2},1g_{7/2})$ doublet along the $N=50$ isotonic chain (nuclei with fixed neutron number $N=50$ ), second order acts as a near-uniform shift toward the fully self-consistent RHB values, so first order already captures the generic trends. Two density-profile extremes make the mechanism testable: neutron-rich nuclei develop a neutron skin — a dilute neutron excess at the surface, i.e. extreme diffusivity (Dobaczewski et al., Phys. Rev. Lett. 72, 981 (1994)) — and 'bubble' (or semibubble) nuclei have a depleted central density, giving $\mathrm{d}\Sigma/\mathrm{d}r<0$ at the center where $G_a-G_b<0$ , which the mechanism predicts should enhance the pseudospin splitting.

Problem Statement

Within the covariant perturbative framework of Heitz, Ebran and Khan (SS-RHO reference, self-consistent $\Sigma$ and $\Delta$ potentials), or an explicitly comparable mechanism-resolved covariant treatment, establish a quantitative theory of pseudospin-symmetry breaking that (1) explains, nucleus by nucleus across the nuclear chart, the observed splittings and the accidental (near-)restorations of PSS — identifying which features (surface diffusivity and shape of $\Sigma$ , nodal structure of the doublet wave functions, shell occupancy) produce near-degeneracy, and delimiting where first order suffices and where the second-order refinement is required; and (2) delivers explicit, falsifiable predictions for the pseudospin splittings of neutron-rich nuclei with developed neutron skins (the large-diffusivity limit) and of bubble nuclei with central density depletion (where the mechanism predicts an enhanced splitting), confronted with self-consistent RHB calculations and with available spectroscopic data. These two components serve one objective — a predictive, mechanism-level account of pseudospin breaking and restoration — since (1) certifies the framework that (2) stress-tests at the density-profile extremes. A solution to component (1) states the finite set of nuclei and doublets it treats, which must include at least the source's neutron $(2d_{5/2},1g_{7/2})$ pair along the $N=50$ isotonic chain together with one neutron-rich (large-skin) nucleus and one bubble nucleus, and, for each, confronts the first- and second-order splittings with the corresponding fully self-consistent RHB value, stating which order already reproduces it.

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