A 2025 distorted-wave impulse-approximation calculation reproduced the ⁶Li(K⁻,π⁻) spectrum of the hypernucleus ⁶_ΛLi, including the narrow 1⁺ peak at E_Λ = 13.8 MeV near the ³He+d+Λ threshold, but had to omit the coherent ΛN–ΣN-coupling-induced ΛNN–ΣNN three-body force, treat the two dominant cluster channels as decoupled, and build the Λ–nucleus potential phenomenologically from cluster densities. The open problem is to compute the fine structure of these high-lying (³He–d)+Λ states with explicit ΛNN–ΣNN coupling, a self-consistent microscopic description of the ⁵Li core (including α-cluster breakup and Λ-induced shrinkage), a Λ–nucleus potential derived from modern hyperon–nucleon interactions, and a unified coupled-channel reaction treatment, thereby quantifying the predicted YNN-induced energy shifts and widths and establishing what YN and YNN interaction information ⁶_ΛLi spectroscopy can actually extract beyond the ~4 MeV FWHM resolution of the existing data.

Background

A hypernucleus is a nucleus in which one or more nucleons are replaced by a hyperon — a baryon containing a strange quark, here the Λ (lambda) particle; the subscript Λ in ⁶_ΛLi denotes a ⁵Li nuclear core bound to a Λ. Hypernuclei are the main source of information on the hyperon–nucleon (YN) interaction, for which direct scattering data are scarce. The reaction ⁶Li(K⁻,π⁻) replaces a neutron in ⁶Li by a Λ; at beam momentum 790 MeV/c and 0° it is near-recoilless and populates substitutional neutron-hole–Λ-particle configurations. The only existing spectrum, measured by Bertini et al. (1981), has an energy resolution of roughly 4 MeV FWHM (full width at half maximum). Harada and Hirabayashi (2025) calculated this spectrum in the distorted-wave impulse approximation (DWIA — a reaction framework treating the projectile–target and ejectile–residual-system distortions to first order in an elementary K⁻n→π⁻Λ amplitude averaged over nuclear Fermi motion), combining it with the Green's function method, which sums Λ bound, resonance, and continuum states of the ⁵Li+Λ system in one spectral function. Their Λ–nucleus interaction was a folding-model potential — obtained by folding an effective ΛN force into the ⁵Li density built from α+p and ³He+d cluster wave functions — and the coupled-channel (CC) equations decoupled the (α−p)+Λ and (³He−d)+Λ channels. This framework reproduces the data, including the narrow peak at E_Λ = 13.8 MeV with spin-parity J^P = 1⁺ near the ³He+d+Λ threshold, arising from interference between ⁵Li(3/2⁺)⊗(0s_{1/2})Λ and ⁵Li(1/2⁺)⊗(0s{1/2})_Λ components. However, the authors stress five linked deficiencies. First, the ⁵Li core is described at the cluster-model level without self-consistency, although near the core-breakup thresholds — where the α cluster breaks into 3N+N — and under the Λ-induced shrinkage of the core (the contraction of nuclear clusters when a Λ is added, studied in three-body models of A = 6–7 hypernuclei by Hiyama et al. (1996)), the folding densities become inaccurate for highly excited states. Second, the three-body YNN force generated by coherent ΛN–ΣN coupling — a coupling of ΛNN to ΣNN components in the wave function driven by the coherent Λ–Σ coupling mechanism studied by Akaishi et al. (2000), whereby, for example, ³He+Λ couples to ³H+Σ⁺ and ³He+Σ⁰, and d+Λ to nn+Σ-type configurations — is entirely absent, although it is expected to shift the fine structure of exactly these high-lying (³He−d)+Λ states. Third, explicit Σ-production channels and Σ admixtures are neglected. Fourth, the (³He−d)+Λ channel has rearrangement partners, chiefly (³He−Λ)+d, that are not treated on the same footing in the reaction dynamics. Fifth, the folding potential is phenomenological rather than derived from modern microscopic ΛN (and YNN) interactions. In parallel, chiral effective field theory (an organization of nuclear forces in powers of momentum over a breakdown scale) now supplies consistent two-body YN and three-body ΛNN/ΣNN forces: Le et al. (2025, Eur. Phys. J. A) benchmarked the chiral ΛNN matrix elements and quantified their contributions in A = 3–5 hypernuclei, and Le et al. (2025, Phys. Rev. Lett.) included chiral ΛNN and ΣNN forces in no-core shell model (NCSM — a diagonalization of the many-body Hamiltonian in a harmonic-oscillator basis treating all nucleons equivalently) calculations of bound-state separation energies of ³_ΛH, ⁴_ΛH/⁴_ΛHe, ⁵_ΛHe, and ⁷_ΛLi; the review by Haidenbauer, Meißner, and Nogga (2026) summarizes ab initio hypernucleus calculations up to A = 7 (NCSM) and A = 16 (nuclear lattice effective field theory). All of these are bound-state results: no existing calculation combines explicit ΛNN–ΣNN dynamics with a continuum reaction treatment of ⁶_ΛLi.

Problem Statement

Determine, quantitatively and within a single microscopic framework, how the coherent ΛN–ΣN-coupling-induced ΛNN–ΣNN three-body force modifies the fine structure — excitation energies, widths, and production strengths — of the high-lying (³He−d)+Λ excited states of ⁶_ΛLi near the ³He+d+Λ threshold (including the J^P = 1⁺ states around E_Λ = 13.8 MeV), and establish what hyperon–nucleon (YN) and three-baryon (YNN) interaction information can be extracted from ⁶_ΛLi spectroscopy at the ~4 MeV FWHM resolution of the existing data and at the higher resolution that would be required to resolve the predicted fine structure. The open aspects of this objective, corresponding to the deficiencies of the existing baseline calculation, are: (1) a self-consistent microscopic description of the ⁵Li core appropriate near its α-cluster (3N+N) breakup thresholds, including Λ-induced core shrinkage, feeding a consistent Λ–nucleus potential; (2) a Λ–nucleus potential derived from modern microscopic ΛN/YN interactions (with short-range correlations and explicit YNN terms) rather than a purely phenomenological folding parametrization; (3) explicit ΛNN–ΣNN coupling in the structure and in the reaction mechanism, with an assessment of Σ-admixture and direct Σ-production channel effects; and (4) a unified coupled-channel reaction treatment in which (α−p)+Λ, (³He−d)+Λ, and the (³He−Λ)+d rearrangement channel enter on the same footing. These four aspects set the required framework rather than four separate deliverables: a sufficient answer quantifies the ΛNN–ΣNN-induced energy shifts, width changes, and production-strength changes of these states with their theoretical uncertainties, including explicit ΛNN–ΣNN coupling (aspect 3), a Λ–nucleus potential derived from modern microscopic YN interactions (aspect 2), and a ⁵Li-core description valid near the ³He+d+Λ threshold (aspect 1); the rearrangement channel of aspect (4) and the direct Σ-production channels may be treated at a controlled level of approximation, provided their residual impact on the reported shifts, widths, and strengths is estimated and included in the stated uncertainties. The answer must be benchmarked against the existing 790 MeV/c (0°) spectrum with its reported ~4 MeV FWHM resolution, state the predicted YNN-induced energy shifts and width changes with their theoretical uncertainties, and identify which of them are resolvable at that ~4 MeV FWHM resolution and, for those that are not, the energy resolution (in MeV FWHM) that would be required to resolve them.

References

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