A three-body calculation of 23^{23}N as an inert 21^{21}N core plus two valence neutrons (Zhang, Ren, Lyu and Ji, 2015) predicted an experimentally unobserved 1/2−1/2^- excited state with an extremely small two-neutron separation energy (0.069–0.315 MeV across n–core parameter sets, mean 0.185 MeV), a giant-halo radius of about 4.3 fm, and a geometry related to the ground state by a spatial scaling factor, and proposed it as a candidate Efimov state in a nuclear system. Later invariant-mass spectroscopy observed two neutron-unbound states of 23^{23}N at excitation energies near 3.6 and 4.5–5.0 MeV, but assigned both, on shell-model and reaction-selectivity grounds, to 3/2−3/2^-; spin-parity was not measured, and no subsequent theoretical work has tested the Efimov interpretation or the modeling approximations behind the prediction. The problem is to establish whether the predicted 1/2−1/2^- state exists in 23^{23}N and whether it is an Efimov state, through spin-parity-sensitive spectroscopy and/or controlled three-body theory addressing core excitation, three-body forces, short-range input sensitivity, and convergence — or to exclude the state and refute the Efimov reading.

Background

A two-neutron halo is a nucleus whose two last neutrons orbit a compact inert core at large distances with a binding energy of only a few hundred keV; such systems are described as three-body problems (core + neutron + neutron). Near the neutron drip line (the limit of neutron excess beyond which nuclei are unbound), 23^{23}N (Z=7, N=16) is a candidate three-body (21^{21}N+nn+nn) system whose ground state is comparatively well bound: its measured matter radius is 3.41(23) fm, its ground state has spin-parity 1/2−1/2^- (from shell-model calculations, a nuclear-structure method that computes level schemes from effective nucleon-nucleon interactions in a truncated basis of nucleon configurations) and a beta-decay half-life of 14.1 ms, and the 2021 evaluated masses give a one-neutron separation energy Sn=3.12(47)S_n=3.12(47) MeV and a two-neutron separation energy S2n=4.7(4)S_{2n}=4.7(4) MeV. The several-hundred-keV halo criterion introduced above is therefore met not by the ground state but by the predicted shallow 1/2−1/2^- excited state discussed below.

The Efimov effect (Efimov, 1970) is the appearance, when at least two of the three pair interactions are resonantly weak, of an infinite ladder of three-body bound states whose sizes and binding energies are related by a fixed discrete scaling factor (≈22.7\approx 22.7 for three identical bosons, and modified by the core-to-neutron mass ratio and by the fermionic nature of the two neutrons in halo nuclei). Efimov physics is well established in cold atoms and helium trimers, but no Efimov state has been unambiguously identified in any nuclear system: in two-neutron halos the neutron-neutron scattering length is large, but the neutron-core scattering length rarely is, and Pauli antisymmetry of the two neutrons restricts the channels in which the effect can appear, so claims of Efimov-like states in nuclei remain tentative and contested.

Zhang, Ren, Lyu and Ji (2015) solved the Faddeev equations (the exact coupled equations for the three-body bound-state problem) for 23^{23}N as 21^{21}N+nn+nn, expanding the wave function in a hyperspherical basis (a basis of total grand-angular momentum KK, truncated at Kmax⁡=20K_{\max}=20) with a realistic neutron-neutron potential and a phenomenological Woods–Saxon neutron-core potential (the standard nuclear mean-field potential with diffuse surface and spin-orbit terms) fitted to the 22^{22}N spectrum in three parameter sets spanning the empirical uncertainty of the 22^{22}N neutron separation energy. Reproducing the ground state, they additionally found a 1/2−1/2^- excited state with two-neutron separation energy S2n∗=0.185S_{2n}^*=0.185 MeV (range 0.069–0.315 MeV over the parameter sets), matter radius ≈4.3\approx 4.3 fm — a 'giant halo', the size of a stable nucleus with mass number ≈80\approx 80 — and a triangular spatial configuration whose ratio of neutron-neutron distance to core-to-neutron-pair distance, rnn/rC(nn)≈1.9r_{nn}/r_{C(nn)}\approx 1.9, matches the ground state, suggesting the two states are related by a discrete spatial scaling. On these indicators they proposed the excited state as a candidate Efimov state, while acknowledging that confirmation requires further theoretical and experimental work and that the calculation treats the 21^{21}N core as inert, uses a phenomenologically fitted n–core interaction with residual parameter uncertainty, includes no three-body forces, and relies on a Kmax⁡=20K_{\max}=20 truncation whose convergence was argued, not exhaustively studied. Note that with the evaluated masses the predicted state, at Ex≈S2n−S2n∗≈4.5E_x\approx S_{2n}-S_{2n}^*\approx 4.5 MeV (about 3.4 MeV using the ground-state S2n=3.6(5)S_{2n}=3.6(5) MeV calculated in the 2015 paper), lies above the one-neutron separation energy: it is a resonance with respect to 22^{22}N+nn emission, though still bound with respect to 21^{21}N+2n2n.

The only spectroscopy of 23^{23}N excited states to date (Jones et al., 2017) populated neutron-unbound states by proton knockout from an 83.4 MeV/u 24^{24}O beam on a liquid deuterium target and reconstructed the 22^{22}N+nn decay by invariant-mass spectroscopy (determining a resonance's energy from the measured momenta of its decay products) with the MoNA-LISA neutron arrays at NSCL. Two states were observed at excitation energies near 3.6 and 4.5–5.0 MeV; on the basis of shell-model calculations (WBP and WBT, two widely used phenomenological shell-model interactions) and the selectivity of proton removal from the proton p3/2p_{3/2} orbit (written πp3/2\pi p_{3/2}), both were assigned as 3/2−3/2^-, with no gamma-ray detection or direct spin-parity measurement, and the evaluators of the A=23 mass chain stress that the level ordering could not be disentangled. The predicted 1/2−1/2^- candidate was thus neither confirmed nor excluded: its expected energy window overlaps the observed resonances, but its defining quantum numbers remain unmeasured, and the Efimov interpretation has never been subjected to the standard Efimov tests by any later calculation.

Problem Statement

Determine whether 23^{23}N possesses the predicted shallow 1/2−1/2^- excited state of the 21^{21}N+nn+nn three-body system — a state with two-neutron separation energy in the predicted 0.069-0.315 MeV range (equivalently Ex≈3.4E_x\approx 3.4-4.54.5 MeV depending on the ground-state separation energy adopted), bound against 21^{21}N+2n2n breakup but neutron-unbound — and, if it exists, whether it is an Efimov state. The single objective is to establish the existence and nature of this state; the following complementary approaches serve that objective. Approach (1) settles only the existence question — identifying the state or excluding it — while approach (2), or the combination of (1) and (2), is required to settle whether the state is Efimovian: (1) Experimental identification: a spin-parity-sensitive measurement of the 23^{23}N spectrum that identifies a 1/2−1/2^- state — distinguishing it from the nearby shell-model 3/2−3/2^- resonances observed near 3.6 and 4.5-5.0 MeV — and determines its decay energy and width, or a documented non-observation with sensitivity limits covering the predicted energy window. (2) Theoretical assessment: a controlled calculation of the 23^{23}N spectrum that tests the Efimov interpretation against the standard Efimov criteria (discrete scale invariance between states of the same channel, universality of geometry and scaling ratio, robustness of the state to the short-range n-n and n-core physics) while addressing the approximations under which the prediction was made: coupling to excited states of the 21^{21}N core, the phenomenological n-core input and its residual parameter uncertainty, sensitivity to three-body forces and to the choice of n-n interaction, and convergence and uncertainty quantification beyond the Kmax⁡=20K_{\max}=20 truncation. Admissible conclusions include confirmation (the 1/2−1/2^- state exists and is Efimovian), partial resolution (the state exists but is an ordinary halo resonance, refuting the Efimov reading), or refutation (the 1/2−1/2^- state is shown not to exist in a controlled calculation, or is experimentally excluded). The problem is not narrowed to any particular reaction, parameter set, or answer form.

References

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