Symmetry restoration beyond the intrinsic mean field and direct observable tests of the proton halo in $^{12}$N within the deformed relativistic Hartree-Bogoliubov theory in continuum

The first microscopic description of a proton halo in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) — applied to the proton drip-line nucleus $^{12}$N — reproduces the available proton separation energies and empirical matter radii of the $N=5$ isotones without adjustable parameters and predicts an extended proton density forming an oblate halo around a prolate core. The description, however, stops at the intrinsic (symmetry-breaking) mean-field level and rests on reaction-model-dependent empirical radii, leaving its central predictions untested against laboratory-frame and reaction observables. This problem asks to carry out the extension the source itself poses as a future direction: extend angular momentum projection (or equivalent symmetry-restoration and configuration-mixing techniques) in the continuum covariant framework to odd-mass and odd-odd nuclei and compute the electromagnetic moments and low-lying spectra of $^{12}$N in the laboratory frame. The source also notes limitations whose resolution defines complementary open directions — the residual overestimation of the $^{12}$N proton separation energy, the reaction-model dependence of the empirical radii used for validation, and the untested implications of the predicted halo and core-halo shape decoupling for reaction observables computed from the microscopic densities. No resolution of the core problem, or of the complementary directions, is known in the literature to date.

Background

A nuclear halo forms when one or two valence nucleons are weakly bound (separation energy of order a few hundred keV to ~1 MeV) and tunnel far into the classically forbidden region, producing a spatially extended, dilute density tail. Halos were discovered through the anomalously large interaction cross section of $^{11}$Li in 1985, and roughly twenty halo nuclei or candidates are now known. Proton halos are intrinsically harder to form and to describe than neutron halos because the repulsive Coulomb barrier confines the proton wave function and suppresses long-range tunneling; only a handful of light, weakly bound proton-rich nuclei qualify. The nucleus $^{12}$N, the most proton-rich bound isotope of nitrogen, sits at the proton drip line — the boundary of the chart of nuclides beyond which nuclei are unbound to proton emission. It was classified as a proton halo nucleus on the basis of an enhanced reaction cross section on a Si target, a large one-proton removal cross section, and remeasured interaction cross sections, as summarized in the 2013 review by Tanihata et al. Its electric quadrupole and magnetic dipole moments have been measured and are compiled in the evaluated tables of Stone.

The theoretical tool of the source work is covariant density functional theory (CDFT), in which the nucleus is described by a relativistic (meson-mediated or point-coupling) energy density functional solved in the Hartree-Bogoliubov quasiparticle formalism, so that mean field and pairing correlations are treated self-consistently. Its deformed continuum extension, the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), treats axial deformation and continuum coupling on equal footing by expanding quasiparticle wave functions in a spherical Dirac Woods-Saxon basis with correct asymptotic behavior; over fifteen years it has given self-consistent descriptions of deformed neutron halos ($^{17,19}$B, $^{15,19,22}$C, $^{31}$Ne, $^{37}$Mg) and underpins an international mass-table collaboration. Before 2025, however, no fully microscopic, self-consistent calculation for $^{12}$N simultaneously included deformation, pairing, and the continuum: earlier treatments used phenomenological cluster-core or Woods-Saxon single-particle models, spherically constrained Skyrme Hartree-Fock, or relativistic mean-field calculations that neglected pairing.

Zhang and Lu (Phys. Lett. B 871 (2025) 139989) provided the first DRHBc description of the proton halo phenomenon, taking $^{12}$N as the example. Without any parameter fitting, the calculation reproduces the proton separation energies and empirical matter root-mean-square (rms) radii of the $N=5$ isotones from $^{9}$Be to $^{12}$N reasonably well, captures the abrupt increase of both proton and matter rms radii at $^{12}$N, and attributes the halo to the occupation of a weakly bound valence orbital, $(1/2)^-_2$ (the subscript 2 labels it as the second orbital with the same angular-momentum projection $\Omega=1/2$ and negative parity, in order of increasing energy), with more than 90% $1p_{1/2}$ content; this halo proton contributes about 90% of the total proton density at large distance from the center. The calculation predicts a well-defined shape decoupling — a prolate core (quadrupole deformation $\beta_2 = 0.198$) surrounded by an oblate halo ($\beta_2 = -0.325$), a phenomenon discussed for deformed halos since Misu, Nazarewicz and Aberg (1997), where the halo deformation is governed by the valence particle's wave function.

Three limitations of this description motivate the problem below: the first is the open question explicitly posed by the source, while the second and third, together with the source's remark that the halo structure may have significant implications for nuclear reaction observables, define complementary directions. First, DRHBc works with intrinsic wave functions that break rotational symmetry of the nuclear Hamiltonian, whereas the measured electromagnetic moments and low-lying spectra are defined in the laboratory frame; the standard remedy is angular momentum projection (AMP) — restoring eigenstates of total angular momentum by integrating the intrinsic wave function over rotations — optionally combined with configuration mixing in the generator coordinate method, which builds correlated many-body states by mixing mean-field wave functions computed at different deformation values; AMP has so far been implemented within DRHBc only for even-even nuclei (Sun and Zhou 2021). Second, the empirical matter radii used for comparison were extracted from reaction cross sections through Glauber-model analyses (the Glauber model is a high-energy eikonal reaction theory relating cross sections to matter densities), and are therefore not fully model independent. Third, the calculated proton separation energy $S_p$ of $^{12}$N is slightly overestimated, indicating a residual quantitative deficiency in the covariant functionals or pairing treatment for this proton drip-line system.

Problem Statement

Complete the first microscopic (DRHBc) description of the $^{12}$N proton halo by carrying its intrinsic mean-field picture into laboratory-frame observables and testing it there. Concretely, the objective is to establish a quantitative correspondence between the predicted microscopic proton-halo structure of $^{12}$N (extended proton density from the weakly bound $(1/2)^-_2$ orbital with dominant $1p_{1/2}$ content; prolate core with $\beta_2 \approx 0.198$ and oblate halo with $\beta_2 \approx -0.325$) and its measured laboratory-frame observables:

(1) Core objective — symmetry restoration for open (odd-mass and odd-odd) systems: extend angular momentum projection, or equivalent symmetry-restoration and configuration-mixing techniques, within the DRHBc (continuum covariant density-functional) framework to the general class of odd-mass and odd-odd nuclei, and apply that extension to $^{12}$N by computing its laboratory-frame electromagnetic moments (electric quadrupole and magnetic dipole) and low-lying spectra for direct comparison with the measured values. The objective is resolved by producing these symmetry-restored laboratory-frame observables and confronting them with the measured values; agreement with the measurements is the physical test, not a precondition for resolution. This is the extension the source explicitly poses as highly desirable for the future; AMP within DRHBc exists so far only for even-even nuclei.

The source further notes limitations whose resolution defines the following complementary open directions:

(2) Separation-energy accuracy: the source notes that the calculated proton separation energy of $^{12}$N is slightly overestimated; identifying the origin of this overestimation and reducing it would make the binding energy that governs proton-halo formation quantitatively reliable at the proton drip line.

(3) Model-independent experimental validation: the source notes that the current empirical matter radii were extracted from reaction cross sections through Glauber-model analyses and are therefore not fully model independent; a more direct, less model-dependent experimental test of the predicted extended proton density of $^{12}$N would strengthen the validation beyond these radii.

(4) Reaction-observable tests: the source remarks that the halo structure may have significant implications for nuclear reaction observables, citing a microscopic fusion study, but no such calculation has been performed for $^{12}$N; computing reaction observables for $^{12}$N from the microscopic DRHBc densities — such as reaction/interaction cross sections, one-proton removal cross sections and longitudinal momentum distributions, or elastic/inelastic proton scattering — and confronting them with the existing cross-section data would directly test the predicted halo structure and the prolate-core/oblate-halo shape decoupling rather than only inferring them from radii trends.

The directions are jointly linked, not independent: (1) supplies the only route from the predicted intrinsic densities to the measured moments and spectra; (4) converts the same densities into reaction observables; (2) quantifies the separation energy that controls the halo's spatial extension; and (3) anchors the density tail experimentally without reaction-model intermediaries. The open problem posed by the source is the core objective (1); contributions on (2)-(4) complement it, and progress there, however substantial, does not by itself resolve it.

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