The Gamow shell model (GSM) — a shell-model framework built on the Berggren ensemble of bound, resonant, and continuum single-particle states — has so far been applied to hypernuclei only in the pp-shell single-Λ\Lambda sector: after a GSM description of the neutron-rich He hyper-isotope chain from Λ6He{}^{6}_{\Lambda}\mathrm{He} to Λ9He{}^{9}_{\Lambda}\mathrm{He} (Li, Michel, Li, and Zhou, Phys. Lett. B 868, 139708 (2025)), Dassie, Hiyama, Michel, and Płoszajczak (Phys. Rev. C 114, 024331 (2026)) described pp-shell hypernuclei with a single Λ\Lambda hyperon, from Λ5He{}^{5}_{\Lambda}\mathrm{He} to Λ16O{}^{16}_{\Lambda}\mathrm{O}, using one optimized effective ΛN\Lambda N interaction (central, spin–orbit, and tensor terms on Woods–Saxon cores). The open problem, stated by the authors themselves, is to carry this framework beyond the pp-shell single-Λ\Lambda sector: to sdsd-shell and heavier hypernuclei — including quantitative predictions of Λ40,48K{}^{40,48}_{\Lambda}\mathrm{K} spectra for the JLab (e,e′K+)(e,e'K^{+}) program — and to multi-strangeness systems such as double-Λ\Lambda hypernuclei, deciding whether the fitted ΛN\Lambda N parameters (whose tensor component, comparatively strong in the fit to the Λ16O{}^{16}_{\Lambda}\mathrm{O} ground-state doublet, the source indicates will remain essential for heavier hypernuclei) transfer or must be refined, with enlarged model spaces and continuum coupling sufficient to reproduce widths of unbound states. The single scientific objective unifying these components is establishing the transferability of one continuum-coupled effective ΛN\Lambda N interaction across nuclear mass and strangeness. No GSM calculation for hypernuclei beyond the pp-shell single-Λ\Lambda case exists: sdsd-shell GSM machinery exists only for normal nuclei, and double-Λ\Lambda hypernuclei have been treated only with real-energy shell models, cluster models, and mean-field-type methods.

Background

A hypernucleus is a nucleus in which one or more nucleons are replaced by a hyperon — a baryon containing a strange quark. The lightest and most studied is the Λ\Lambda hyperon (quark content udsuds, mass about 1116 MeV, zero electric charge and zero isospin, so it is not subject to the Pauli exclusion principle with respect to nucleons and sinks deep into the nuclear interior). Hypernuclear spectroscopy constrains the baryon–baryon YNYN interaction (YY = hyperon, NN = nucleon), which cannot be mapped by ordinary nucleus–nucleus scattering and which matters for the equation of state of neutron-star matter; high-resolution γ\gamma-ray spectroscopy at KEK/BNL and kaon electroproduction experiments at JLab have supplied precise binding energies and level schemes, mainly for pp-shell Λ\Lambda hypernuclei. The (e,e′K+)(e,e'K^{+}) reaction converts a proton into a Λ\Lambda in flight and is the cleanest spectroscopic tool for heavier hypernuclei; the JLab program aims at the medium-mass region, including potassium hypernuclei such as Λ40,48K{}^{40,48}_{\Lambda}\mathrm{K}. The Gamow shell model (GSM) replaces the harmonic-oscillator basis of the standard nuclear shell model by the Berggren ensemble — a one-body basis that collects bound states, resonant states with complex energies (whose imaginary parts encode decay widths), and real scattering continuum states, all treated on an equal footing — so that weakly bound and unbound nuclei are described with explicit continuum coupling. In an ordinary shell model the continuum is absent and resonances can only be mimicked; in the GSM the Hamiltonian matrix is complex symmetric and its complex eigenvalues give positions and widths of resonant states directly. In 2026, Dassie, Hiyama, Michel, and Płoszajczak established the GSM for pp-shell single-Λ\Lambda hypernuclei with an optimized effective interaction: a Woods–Saxon mean field for the nucleons and the Λ\Lambda, plus a two-body effective ΛN\Lambda N interaction with central, spin–orbit, and tensor components (the tensor force couples spin and spatial motion and splits states of different total spin) whose parameters were optimized once against pp-shell data. With a single parameter set the calculation reproduced and predicted binding energies, excitation spectra, and densities from Λ5He{}^{5}_{\Lambda}\mathrm{He} to Λ16O{}^{16}_{\Lambda}\mathrm{O}; the tensor force was found to shift excited-state splittings appreciably in bound hypernuclei while its effect is suppressed in unbound ones by coupling to the decay continuum. The calculation was restricted to the pp shell and a single Λ\Lambda, in a model space with excitations up to 8ℏω8\hbar\omega and at most two valence particles in the continuum. Adjacent literature frames what is missing: a quantified GSM interaction exists for psdpsd-shell ordinary nuclei (Jaganathen et al., 2017) and GSM studies of sdsd-shell normal nuclei near the drip lines are routine (e.g., Li et al., 2026), but no GSM treatment of a Λ\Lambda hyperon beyond the pp shell exists; double-Λ\Lambda hypernuclei have been predicted with the standard real-energy shell model (Gal and Millener, 2011) and with cluster and mean-field-type approaches (Zhang, Hiyama, and Sagawa, 2025, for multi-Λ\Lambda systems in O and Ca isotopes), none of which treats resonances and continuum coupling through a Berggren basis.

Problem Statement

Extend the Gamow shell model framework with its optimized effective ΛN\Lambda N interaction beyond the pp-shell single-Λ\Lambda sector in which it was established (Dassie et al., Phys. Rev. C 114, 024331 (2026)): (i) construct GSM calculations for sdsd-shell and heavier single-Λ\Lambda hypernuclei, requiring at minimum quantitative predictions of binding energies, excitation spectra, and widths for Λ40,48K{}^{40,48}_{\Lambda}\mathrm{K} accessible via the (e,e′K+)(e,e'K^{+}) reaction at JLab; (ii) extend the framework to multi-strangeness systems, in particular double-Λ\Lambda hypernuclei; and (iii) determine along the way whether the fitted ΛN\Lambda N interaction parameters — whose tensor component, comparatively strong in the fit, the Λ16O{}^{16}_{\Lambda}\mathrm{O} description indicates will remain essential for heavier systems — transfer unchanged to heavier and multi-strange systems or require refinement, using model spaces enlarged beyond the 8ℏω8\hbar\omega / two-continuum-particle limit and continuum coupling improved enough to reproduce widths of unbound states (such as the 1/2−1/2^{-} state of the normal nucleus 9Be{}^{9}\mathrm{Be}) that the original model space describes poorly. These components serve one overarching objective: establishing whether a single continuum-coupled effective ΛN\Lambda N interaction, validated in the pp shell, describes hypernuclear structure across nuclear mass and strangeness. The problem is complete when quantitative GSM spectra and an explicit transferability verdict (parameters transfer, or specified re-optimized values with the quality of fit to heavier and double-Λ\Lambda data stated) exist for sdsd-shell and heavier single-Λ\Lambda hypernuclei — at minimum, quantitative predictions for Λ40,48K{}^{40,48}_{\Lambda}\mathrm{K} for the JLab (e,e′K+)(e,e'K^{+}) program — and for a double-Λ\Lambda system.

References

  1. Alan Cruz Dassie, Emiko Hiyama, Nicolas Michel, Marek Płoszajczak, 'Gamow shell model description of the hypernuclei', Phys. Rev. C 114, 024331 (2026), DOI 10.1103/qksp-p1jy, arXiv:2607.26961, https://arxiv.org/abs/2607.26961
  2. Y. Jaganathen, R. M. Id Betan, N. Michel, W. Nazarewicz, M. Płoszajczak, 'Quantified Gamow Shell Model interaction for psd-shell nuclei', Phys. Rev. C 96, 054316 (2017), DOI 10.1103/PhysRevC.96.054316, arXiv:1710.04727, https://arxiv.org/abs/1710.04727
  3. Kun-Hao Li, Pei-Yan Wang, Jian-Guo Li, Nicolas Michel, Meng-Ran Xie, Chun-Wang Ma, Wei Zuo, 'Mechanisms of mirror energy difference for states exhibiting Thomas–Ehrman shift: Gamow shell model case studies of 18Ne/18O and 19Na/19O', Nuclear Science and Techniques 37, 109 (2026), DOI 10.1007/s41365-026-01944-z, arXiv:2407.00884, https://arxiv.org/abs/2407.00884
  4. A. Gal, D. J. Millener, 'Shell-model predictions for ΛΛ\Lambda\Lambda hypernuclei', Phys. Lett. B 701, 342–345 (2011), DOI 10.1016/j.physletb.2011.05.069, arXiv:1104.3429, https://arxiv.org/abs/1104.3429
  5. Ying Zhang, Emiko Hiyama, Hiroyuki Sagawa, 'Lambda-Lambda interaction in multi-Lambda hypernuclei of O and Ca isotopes', Eur. Phys. J. A 61, 278 (2025), DOI 10.1140/epja/s10050-025-01757-5, https://doi.org/10.1140/epja/s10050-025-01757-5
  6. Satoshi N. Nakamura, 'Spectroscopy of electro-produced hypernuclei at JLab', AIP Conf. Proc. 2130, 020012 (2019), DOI 10.1063/1.5118380, https://doi.org/10.1063/1.5118380
  7. Xin Li, N. Michel, J. G. Li, Xian-Rong Zhou, 'Gamow shell model description of neutron-rich He hyper-isotopes', Phys. Lett. B 868, 139708 (2025), arXiv:2408.16223, https://arxiv.org/abs/2408.16223