The 2025 calculation of Harada and Hirabayashi reproduced the measured continuum Λ spectrum of Λ6Li^6_\Lambda\mathrm{Li} from the 6Li(K−,π−)^6\mathrm{Li}(K^-,\pi^-) reaction at 790 MeV/c using a distorted-wave impulse-approximation treatment with a phenomenological folding-model Λ–nucleus potential, and identified a narrow JP=1+J^P=1^+ state at EΛ=13.8E_\Lambda=13.8 MeV associated with a high-lying (³He–d)-Λ cluster configuration. The authors flag as unresolved the quantitative effect of the three-body YNN force generated by coherent ΛN–ΣN coupling (the ΛNN–ΣNN force) on the fine structure of these high-lying states — expected shifts are too small to resolve at the roughly 4 MeV FWHM (full width at half maximum) resolution of the existing 1981 data — together with the enabling theory ingredients: a Λ–nucleus potential derived from modern microscopic ΛN interactions, self-consistent core structure including Λ-induced shrinkage and α-cluster breakup, and a unified fully coupled reaction treatment of the (α–p)-Λ, (³He–d)-Λ, and rearrangement (³He–Λ)-d channels. No published work has quantified the ΛNN–ΣNN fine-structure shifts in the Λ6Li^6_\Lambda\mathrm{Li} continuum or provided the required coupled-channel framework.

Background

A hypernucleus is a nucleus in which one nucleon is replaced by a hyperon YY — a baryon carrying strangeness, here the Λ\Lambda (mass 1116 MeV, isospin 0). The two-body ΛN\Lambda N interaction is reasonably mapped, but its coherent coupling to ΣN\Sigma N channels and the resulting three-baryon (YNNYNN) dynamics remain poorly quantified, and light Λ\Lambda hypernuclei are the cleanest laboratory for them. In a (K−,π−)(K^-,\pi^-) strangeness-exchange reaction a kaon converts a neutron into a Λ\Lambda (K−n→π−ΛK^- n\to\pi^-\Lambda); at near-recoilless kinematics (small momentum transfer) the reaction preferentially substitutes the Λ\Lambda into an existing orbital, and the outgoing-pion spectrum maps the Λ\Lambda bound, resonant, and continuum states.

Harada and Hirabayashi (2025) calculated the continuum Λ\Lambda spectrum of Λ6Li^6_\Lambda\mathrm{Li} produced in 6Li(K−,π−)^6\mathrm{Li}(K^-,\pi^-) at pK−=790p_{K^-}=790 MeV/c. They used the distorted-wave impulse approximation (DWIA — the reaction is treated as a single quasi-free K−n→π−ΛK^- n\to\pi^-\Lambda conversion with distorted incoming and outgoing waves) with a Fermi-averaged (in-medium energy-averaged) production amplitude, and obtained the spectral distribution from the continuum Green's function method (the spectrum follows from the single-Λ\Lambda propagator in a Λ–nucleus potential, giving bound, resonant, and continuum states on one footing). The Λ–nucleus potential was a folding model — an effective ΛN\Lambda N interaction folded over the 5Li^5\mathrm{Li} core density — with the 5Li^5\mathrm{Li} core built from (α–p) and (³He–d) two-cluster wave functions, where α is the 4He^4\mathrm{He} nucleus and d the deuteron. The calculation reproduces the measured spectrum (Bertini et al. 1981), including a narrow peak from a high-lying JP=1+J^P=1^+ (spin-parity) state at EΛ=13.8E_\Lambda=13.8 MeV (energies measured from the 5Li^5\mathrm{Li} ground-state +Λ+\Lambda threshold) near the ³He+d+Λ breakup threshold — a (³He–d)-Λ cluster state.

Three limitations define the open problem. First, because the Λ\Lambda and the Σ\Sigma hyperon (mass 1193 MeV) are close in mass and the ΛN\Lambda N and ΣN\Sigma N two-baryon systems share total isospin I=1/2I=1/2 channels, the two-body ΛN\Lambda N interaction coherently mixes ΛN\Lambda N and ΣN\Sigma N; in a nucleus this generates an effective three-baryon force, the ΛNN\Lambda NN–ΣNN\Sigma NN force (denoted YNNYNN), which is distinct from the Fujita–Miyazawa-type two-pion-exchange ΛNN\Lambda NN force and is expected to shift the fine structure of the high-lying (³He–d)-Λ states, whose spin-isospin structure is inherited from the 5Li^5\mathrm{Li} core. The expected shifts lie far below the roughly 4 MeV FWHM (full width at half maximum) resolution of the existing data. Second, the potential used is phenomenological, adjusted to the Λ6Li^6_\Lambda\mathrm{Li} ground state; modern microscopic ΛN\Lambda N interactions now exist, and the Λ\Lambda also compresses the nuclear core (Λ-induced shrinkage), changing the very density that generates the folding potential, especially near breakup thresholds where the α cluster breaks up. Third, the reaction was treated in a single-step framework that does not couple the production channels (α–p)-Λ and (³He–d)-Λ with their rearrangement channel (³He–Λ)-d on the same footing, nor include explicit Σ-production channels.

Problem Statement

Determine quantitatively the effect of the three-body YNNYNN force generated by coherent ΛN\Lambda N–ΣN\Sigma N coupling (the ΛNN\Lambda NN–ΣNN\Sigma NN force) on the fine structure of the high-lying (³He–d)-Λ states of Λ6Li^6_\Lambda\mathrm{Li} near the ³He+d+Λ and α+p+Λ breakup thresholds, and the signature of that effect in the 6Li(K−,π−)^6\mathrm{Li}(K^-,\pi^-) production spectrum. The components of this objective are jointly required and not separately answerable: (a) the Λ–nucleus potential must be constructed from a modern microscopic ΛN\Lambda N interaction (including short-range correlations and explicit YNYN dynamics) rather than fitted phenomenologically, since the YNNYNN energy shifts are only identifiable relative to a microscopic baseline; (b) the 5Li^5\mathrm{Li} core structure must be treated self-consistently, including Λ-induced shrinkage and α-cluster breakup channels, since these change the folding potential precisely for states near threshold; and (c) the production reaction must be described in a unified, fully coupled treatment that places the (α–p)-Λ, (³He–d)-Λ, and rearrangement (³He–Λ)-d channels on the same footing — including explicit Λ–Σ coupling and, if the required sensitivity demands it, direct Σ-production channels — since the fine structure is observable only through the reaction. A complete answer states the predicted energy shifts and changed level pattern of the high-lying states (with JPJ^P spin-parity assignments), the resulting spectrum at realistic experimental resolution, and whether the YNNYNN effect is resolvable in a next-generation high-resolution measurement.

References

  1. T. Harada and Y. Hirabayashi, 'Continuum Λ spectra for a ⁶ΛLi hypernucleus in the ⁶Li(K⁻,π⁻) reaction', Physical Review C 111, 064617 (2025), DOI 10.1103/9c6g-dsq2, arXiv:2505.06844, https://arxiv.org/abs/2505.06844
  2. T. Harada and Y. Hirabayashi, 'Evidence for ⁴ΛHe hyperfragment production in the ⁶Li(π⁺,K⁺) reaction', Physics Letters B 868, 139657 (2025), DOI 10.1016/j.physletb.2025.139657, https://doi.org/10.1016/j.physletb.2025.139657
  3. M. Knöll and R. Roth, 'Hyperon-nucleon interaction constrained by light hypernuclei', Physics Letters B 846, 138258 (2023), DOI 10.1016/j.physletb.2023.138258, arXiv:2307.11577, https://arxiv.org/abs/2307.11577
  4. J. Haidenbauer, U.-G. Meißner and A. Nogga, 'Ab initio description of hypernuclei', Progress in Particle and Nuclear Physics 149, 104242 (2026), DOI 10.1016/j.ppnp.2026.104242, arXiv:2508.05243, https://arxiv.org/abs/2508.05243
  5. E. Friedman and A. Gal, 'Constraints from Λ hypernuclei on the ΛNN content of the Λ-nucleus potential', Physics Letters B 837, 137669 (2023), DOI 10.1016/j.physletb.2023.137669, arXiv:2204.02264, https://arxiv.org/abs/2204.02264
  6. E. Friedman and A. Gal, 'Hypernuclear constraints on ΛN and ΛNN interactions', Acta Physica Polonica B 57, 2-A16 (2026), DOI 10.5506/aphyspolb.57.2-a16, https://doi.org/10.5506/aphyspolb.57.2-a16
  7. H. Kamada, M. Kohno and K. Miyagawa, 'Faddeev calculation of Λ³H incorporating the 2π-exchange ΛNN interaction', Physical Review C 108, 024004 (2023), DOI 10.1103/PhysRevC.108.024004, https://doi.org/10.1103/PhysRevC.108.024004
  8. M. Kohno and H. Kamada, 'Faddeev calculations of low-energy Λ-deuteron scattering and momentum correlation function', Physical Review C 110, 044005 (2024), DOI 10.1103/PhysRevC.110.044005, https://doi.org/10.1103/PhysRevC.110.044005
  9. R. Bertini, O. Bing, P. Birien, K. Braune, W. Brückner, A. Chaumeaux, M. A. Faessler, R. W. Frey, D. Garreta, T. J. Ketel, K. Kilian, B. Mayer, J. Niewisch, B. Pietrzyk, B. Povh, H. G. Ritter and M. Uhrmacher, 'Neutron hole states in Λ6Li, Λ7Li, Λ9Be and Λ12C hypernuclei', Nuclear Physics A 368, 365 (1981), DOI 10.1016/0375-9474(81)90761-2, https://doi.org/10.1016/0375-9474(81)90761-2