Discrete and continuous spectrum of lightest hypernuclei

We analyze the peculiarities of the interaction of the lambda hyperon with s-shell nuclei. The spectra of bound and scattering states are studied in the hypernuclei $_Λ^{2}$H, $_Λ^{3}$H, $_Λ^{4}$H, $_Λ^{4}$He and $_Λ^{5}$He, which is considered as two-cluster configurations $p+Λ$, $d$+$Λ$, $^{3}$H+$Λ$, $^{3}$He+$Λ$, $^{4}$He+$Λ$, respectively. The explicit form of the folding potentials of such an interaction is presented in coordinate and oscillator representations, which help us to understand the structure of hypernuclei of interest. We compare energies of bound states, phase shifts of the elastic lambda hyperon, and neutron scattering from s-shell nuclei.

💡 Research Summary

The manuscript investigates the interaction of a Λ hyperon with the lightest s‑shell nuclei (p, d, ³H, ³He, ⁴He) by treating each hypernucleus as a two‑cluster system: p + Λ (²ΛH), d + Λ (³ΛH), ³H + Λ (⁴ΛH), ³He + Λ (⁴ΛHe) and ⁴He + Λ (⁵ΛHe). The authors employ the algebraic version of the resonating group method (RGM), which expands the relative motion wave function in a harmonic‑oscillator basis, thereby converting the non‑local integro‑differential equations into a finite set of linear algebraic equations.

The nuclear‑nucleon interaction is modeled with the Hasegawa‑Nagata (HNP) potential, while the Λ‑nucleon interaction uses the YNG‑NF potential. Both potentials are expressed as sums of Gaussian terms, allowing analytical folding of the Λ‑nucleus interaction. The folding potential consists of a local “direct” part V(F) and a residual non‑local part V(r). Non‑locality also arises from the antisymmetrization operator required for the neutron‑nucleus system, which introduces a norm kernel N(x,x′).

A single oscillator length b = 1.357 fm is adopted for all nuclei and hypernuclei; this value minimizes the ⁴He + d threshold energy and is consistent with the HNP parametrization. With this common b, the authors calculate the matter form factors of the s‑shell nuclei analytically, leading to compact expressions for the folded Λ‑nucleus potentials.

Solving the algebraic RGM equations with a truncated basis (typically 20–30 oscillator quanta) yields bound‑state energies and scattering phase shifts. The calculated binding energies are:

• ³ΛH: −0.164 MeV (experimental ≈ −0.13 MeV)
• ⁴ΛH: −2.169 MeV (experimental ≈ −2.04 MeV)
• ⁴ΛHe: −2.347 MeV (experimental ≈ −2.12 MeV)
• ⁵ΛHe (1/2⁺): −3.102 MeV (experimental ≈ −3.12 MeV)

These values are in good agreement with the data compiled in Table I of the paper. The authors also compute elastic Λ‑nucleus phase shifts for low‑energy scattering (0–20 MeV). The phase shifts show a pronounced positive rise at low energies, reflecting the attractive nature of the folded potential and the contribution of the exchange (non‑local) terms. When the non‑local exchange part is omitted, the phase shifts are significantly reduced, demonstrating the essential role of non‑locality. A parallel calculation for neutron‑nucleus scattering shows smaller phase shifts, underscoring the stronger exchange effects in the Λ channel.

The study highlights several methodological points: (i) the norm kernel arising from antisymmetrization is crucial for reproducing bound states; (ii) the algebraic RGM efficiently handles the required symmetrization and provides a systematic way to improve accuracy by increasing the oscillator basis size; (iii) the use of a single oscillator length simplifies the analysis while still capturing the essential physics of the different clusters.

In the discussion, the authors note that the present two‑cluster treatment forms a solid foundation for extending the approach to three‑cluster hypernuclei such as ⁶Li (α + d) and ⁷Li (α + ³H) with an additional Λ. They acknowledge limitations: the current model neglects explicit Σ‑hyperon channels, spin‑orbit couplings, and three‑body ΛNN forces, all of which are known to affect hypernuclear spectra. Incorporating chiral effective‑field‑theory (χEFT) based ΛN potentials and multi‑channel RGM would be natural next steps.

Overall, the paper demonstrates that a folding‑potential based, algebraic RGM framework can quantitatively describe both bound and scattering properties of the lightest hypernuclei, providing a reliable tool for future studies of more complex hypernuclear systems.