The hypertriton binding-energy puzzle: a definitive $B_\Lambda(^3_\Lambda\mathrm{H})$ and a consistent $\Lambda N$/$\Lambda NN$ interaction set for the $s$-shell hypernuclei

The $\Lambda$-separation energy of the hypertriton, $B_\Lambda(^3_\Lambda\mathrm{H})$, is currently measured to mutually incompatible values by different methods: $0.523\pm0.076$ MeV by MAMI decay-pion spectroscopy, $0.406\pm0.163$ MeV by STAR, $0.102\pm0.092$ MeV by ALICE, $0.23\pm0.12$ MeV by deep-learning emulsion analysis, and roughly $0.05$–$0.15$ MeV implied by the J-PARC E73 $(K^-,\pi^0)$ production-ratio measurement, whose distorted-wave analysis directly conflicts with the MAMI value. The problem is to settle $B_\Lambda(^3_\Lambda\mathrm{H})$ with systematics controlled independently of the $^4_\Lambda\mathrm{H}$ decay-pion reference chain, and on that basis to determine quantitatively which combinations of the $\Lambda p$ spin-singlet (${}^1S_0$) and spin-triplet (${}^3S_1$) scattering lengths and which $\Lambda NN$ three-body-force content are simultaneously consistent with all strangeness $S=-1$ $s$-shell observables, including the implied conclusions on whether the neutral $nn\Lambda$ system is bound and whether a bound excited state of $^3_\Lambda\mathrm{H}$ exists. As of 2026 no analysis resolves any of these questions; the experimental discrepancy has sharpened, and even the assignment of the MAMI pion line has been publicly contested.

Background

Hypernuclei are nuclei in which one or more $\Lambda$ hyperons (the lightest baryon carrying strangeness, $uds$) replace or join nucleons, giving access to the hyperon–nucleon ($YN$) interaction, which cannot be mapped by ordinary nuclear spectroscopy. The hypertriton, $^3_\Lambda\mathrm{H}$, is the lightest hypernucleus: a proton, a neutron, and a $\Lambda$ bound chiefly by the $\Lambda$–deuteron attraction. Its $\Lambda$-separation energy $B_\Lambda(^3_\Lambda\mathrm{H})$ is the energy required to remove the $\Lambda$, leaving the deuteron; because the $\Lambda N$ attraction is weak, $B_\Lambda$ is only of order 0.1–0.5 MeV, making the hypertriton the lightest halo nucleus (a bound system with one weakly bound, spatially extended component) and exquisitely sensitive to the low-energy $\Lambda N$ interaction.

Classically $B_\Lambda(^3_\Lambda\mathrm{H})\approx0.13\pm0.05$ MeV came from nuclear-emulsion experiments of the 1960s–70s, in which the hypernucleus stops in the emulsion and its decay products leave tracks whose lengths are converted to energies through a range–energy relation; systematic biases of that conversion, and a persistent ~0.2 MeV offset between two-body ($\pi^-$-mediated) and three-body decay analyses, have never been fully quantified. Modern methods measure instead the momentum of the monochromatic pion from the two-body weak decay $^3_\Lambda\mathrm{H}\to{}^3\mathrm{He}+\pi^-$ of hypernuclei at rest (decay-pion spectroscopy), from which the hypernuclear mass and hence $B_\Lambda$ follow by two-body kinematics; heavy-ion experiments (STAR at RHIC, ALICE at the LHC) reconstruct the same quantity from decay-product momenta in flight. These routes disagree: the MAMI A1 collaboration, referencing the $^3_\Lambda\mathrm{H}$ pion momentum to the $^4_\Lambda\mathrm{H}\to{}^4\mathrm{He}+\pi^-$ line, reports $0.523\pm0.013\pm0.075$ MeV; ALICE reports $0.102\pm0.063\pm0.067$ MeV (with lifetime $\tau=253\pm13$ ps, consistent with the free $\Lambda$ and resolving the earlier lifetime side of the 'hypertriton puzzle'); the J-PARC E73 $(K^-,\pi^0)$ cross-section-ratio evaluation favors a loosely bound system.

On the theory side, the low-energy $\Lambda p$ interaction is parametrized by the spin-singlet (${}^1S_0$) and spin-triplet (${}^3S_1$) S-wave scattering lengths $a_s$ and $a_t$. Existing $\Lambda p$ scattering cross sections are unpolarized and scarce, and cannot separate $a_s$ from $a_t$; the hypertriton binding energy is therefore used as the datum that fixes this spin dependence in realistic potentials and in chiral effective field theory (EFT), the low-energy expansion of quantum chromodynamics organized in powers of momentum with couplings fitted to data. Faddeev-type three-body calculations (the standard exact treatment of three interacting particles, decomposed into coupled equations for the pairwise amplitudes) show $B_\Lambda(^3_\Lambda\mathrm{H})$ is controlled almost entirely by the $\Lambda N$ ${}^1S_0$ attraction, so a deeper hypertriton demands a more attractive $a_s$ with a compensatingly weaker ${}^3S_1$ term — a combination that Le, Haidenbauer, Meißner and Nogga showed can be accommodated within existing $\Lambda p$ scattering bounds and even improves the predicted $A=4$ separation energies. A possible $\Lambda NN$ three-body force, computable systematically in chiral EFT or pionless EFT, adds further parameters whose role is framework-dependent rather than a single degenerate knob: in the NLO chiral setting of Le et al. no $\Lambda NN$ force is fitted and its contribution to the hypertriton binding is estimated at not more than 50 keV, while in leading-order pionless EFT (Schäfer et al.) the $\Lambda NN$ force appears as channel-dependent low-energy constants in distinct isospin–spin channels ($I=0,S=1/2$; $I=1,S=1/2$; $I=0,S=3/2$), each fixed by different few-body data ($B_\Lambda(^3_\Lambda\mathrm{H})$, the $^4_\Lambda\mathrm{H}$ $0^+$ and $1^+$ energies), and in chiral EFT the YNN low-energy constants cannot be determined from the scarce hypernuclear data in the foreseeable future. Two linked near-threshold questions follow from the same spin dependence: whether the neutral isotriplet three-body system $nn\Lambda$ (claimed as bound by the HypHI collaboration at GSI, unconfirmed; searched for without observation by Jefferson Lab experiment E12-17-003, which set upper limits of 21 and 31 nb/sr at 90% confidence) is bound, virtual (an unbound near-threshold pole of the scattering amplitude), or only a resonance — few-body calculations split between 'no bound state' and 'bound if the hypertriton is deeply bound' — and whether the spin-flip $J^\pi=3/2^+$ excited state of $^3_\Lambda\mathrm{H}$, predicted to be extremely shallow or virtual, exists as a bound level. Femtoscopic $\Lambda d$ and $pp\Lambda$ correlation functions — correlations of particle pairs and triples emitted at small relative momentum, which probe the final-state $\Lambda$-nucleon interaction — measured in heavy-ion collisions provide an additional, increasingly precise observable that constrains the same two- and three-body $\Lambda N$ inputs.

Problem Statement

The problem is to resolve the hypertriton binding-energy puzzle and the $\Lambda NN$ interaction sector it fixes, as one linked objective:

1. Establish the $\Lambda$-separation energy $B_\Lambda(^3_\Lambda\mathrm{H})$ by adjudicating the published determinations, with their systematics controlled independently of the $^4_\Lambda\mathrm{H}$ decay-pion reference momentum (e.g. by an independent absolute calibration of the $^4_\Lambda\mathrm{H}$ line, an independent production-based determination, or a quantified treatment of the legacy emulsion systematics; where no independently calibrated experimental input yet exists, an explicit and uncertainty-propagated re-analysis of the published determinations is admissible, provided the value adopted is stated and defended rather than substituted silently), at a precision sufficient to decide between the currently incompatible determinations — the shallow values around $0.1$ MeV (ALICE, E73-implied) and the deeper values around $0.4$–$0.5$ MeV (STAR, MAMI) — and to settle the assignment of the observed decay-pion lines to specific hypernuclei.

2. On that basis, determine quantitatively which combinations of the $\Lambda p$ spin-singlet (${}^1S_0$) and spin-triplet (${}^3S_1$) S-wave scattering lengths and which $\Lambda NN$ three-body-force content are simultaneously consistent with all available strangeness $S=-1$ observables: the $A=3,4,5$ $\Lambda$ separation energies, the hypertriton lifetime, the $\Lambda p$ scattering data, and femtoscopic $\Lambda d$ and $pp\Lambda$ correlation functions. The interaction framework must be explicitly specified (EFT order, regulator, and treatment of truncation uncertainty), with the $\Lambda NN$ force resolved into the isospin–spin channels required by the observables used — in particular the $I=1$ channel for $nn\Lambda$ and the $I=0$, $S=3/2$ channel for the $^3_\Lambda\mathrm{H}$ $3/2^+$ state — rather than reduced to a single universal strength.

3. Derive from that constrained interaction set the implied conclusions on the two linked near-threshold questions: whether the neutral $nn\Lambda$ system is bound, virtual (an unbound near-threshold pole of the scattering amplitude), or resonant, and whether a bound excited state of $^3_\Lambda\mathrm{H}$ (the spin-flip $3/2^+$ level) exists — with experimental tests (dedicated searches or correlation measurements) where the theory admits observation.

Each linked goal is to be addressed with its interaction framework and uncertainty treatment explicitly declared. A negative verdict on goal 2 is admissible only as a framework-conditional, universally quantified statement — that no admissible interaction within the specified framework and its stated truncation and regulator uncertainties reproduces the declared observable set; exhibiting a single inconsistent parameter combination does not resolve the problem. The "all available ... observables" of goal 2 are the enumerated set — the $A=3,4,5$ $\Lambda$ separation energies, the hypertriton lifetime, the $\Lambda p$ scattering data, and the femtoscopic $\Lambda d$ and $pp\Lambda$ correlation functions — evaluated with the value established under goal 1 supplying the $A=3$ datum and each observable required to be reproduced within its stated uncertainty; agreement with each individual conflicting raw $B_\Lambda(^3_\Lambda\mathrm{H})$ determination is not separately required. The $nn\Lambda$ and excited-state questions are to receive definite answers rather than scenario-dependent ones, a framework-conditional verdict being admissible only in the universally quantified form stated above.

References

1. R. Kino et al., Precise measurement of the $\Lambda$-binding energy difference between $^3_\Lambda\mathrm{H}$ and $^4_\Lambda\mathrm{H}$ via decay-pion spectroscopy at MAMI, Phys. Rev. Lett. 136, 152301 (2026), arXiv:2601.08694, DOI 10.1103/19gd-jqw2, https://arxiv.org/abs/2601.08694
2. J. Adam et al. (STAR Collaboration), Measurement of the mass difference and the binding energy of the hypertriton and antihypertriton, Nature Physics 16, 409–412 (2020), arXiv:1904.10520, DOI 10.1038/s41567-020-0799-7, https://arxiv.org/abs/1904.10520
3. S. Acharya et al. (ALICE Collaboration), Measurement of the Lifetime and $\Lambda$ Separation Energy of $^3_\Lambda\mathrm{H}$, Phys. Rev. Lett. 131, 102302 (2023), arXiv:2209.07360, DOI 10.1103/PhysRevLett.131.102302, https://arxiv.org/abs/2209.07360
4. S. Acharya et al. (ALICE Collaboration), First Measurement of $A=4$ Hypernuclei and Antihypernuclei at the LHC, Phys. Rev. Lett. 134, 162301 (2025), arXiv:2410.17769, DOI 10.1103/PhysRevLett.134.162301, https://arxiv.org/abs/2410.17769
5. M. Abdallah et al. (STAR Collaboration), Measurement of $^4_\Lambda\mathrm{H}$ and $^4_\Lambda\mathrm{He}$ binding energy in Au+Au collisions at $\sqrt{s_{NN}}=3$ GeV, Phys. Lett. B 834, 137449 (2022), arXiv:2207.00778, DOI 10.1016/j.physletb.2022.137449, https://arxiv.org/abs/2207.00778
6. A. Kasagi et al., Binding Energy of $^3_\Lambda\mathrm{H}$ and $^4_\Lambda\mathrm{H}$ via Image Analyses of Nuclear Emulsions Using Deep-Learning, Prog. Theor. Exp. Phys. 2025, 083D01 (2025), arXiv:2504.01601, DOI 10.1093/ptep/ptaf097, https://arxiv.org/abs/2504.01601
7. T. Akaishi et al., Measurement of $^{3,4}\mathrm{He}(K^-,\pi^0)_{\Lambda}^{3,4}\mathrm{H}$ reaction cross section and evaluation of hypertriton $\Lambda$ binding energy, Phys. Lett. B 873, 140163 (2026), arXiv:2509.16967, DOI 10.1016/j.physletb.2026.140163, https://arxiv.org/abs/2509.16967
8. T. Harada and Y. Hirabayashi, Sensitivity of the $^{3,4}\mathrm{He}(K^-,\pi^0)$ production ratio to the $\Lambda$ binding energy of $^3_\Lambda\mathrm{H}$, arXiv:2604.16181 (2026), https://arxiv.org/abs/2604.16181
9. A. Gal, Remarks on MAMI's recent determination of $B_\Lambda({}_{\Lambda}^{3}\mathrm{H})$, arXiv:2604.18259 (2026), https://arxiv.org/abs/2604.18259
10. R. Kino, P. Achenbach, P. Klag, S. Nagao, S. N. Nakamura, K. Nishi et al., Response to the $^7_\Lambda\mathrm{He}$ interpretation of MAMI's recent determination of $B_\Lambda({}_{\Lambda}^{3}\mathrm{H})$, arXiv:2604.27748 (2026), https://arxiv.org/abs/2604.27748
11. H. Le, J. Haidenbauer, U.-G. Meißner, A. Nogga, Implications of an increased $\Lambda$-separation energy of the hypertriton, Phys. Lett. B 801, 135189 (2020), arXiv:1909.02882, DOI 10.1016/j.physletb.2019.135189, https://arxiv.org/abs/1909.02882
12. M. Schäfer, B. Bazak, N. Barnea, A. Gal, J. Mareš, Consequences of increased hypertriton binding for $s$-shell $\Lambda$-hypernuclear systems, Phys. Rev. C 105, 015202 (2022), arXiv:2108.13900, DOI 10.1103/PhysRevC.105.015202, https://arxiv.org/abs/2108.13900
13. T. Y. Htun and Y. Yan, $\Lambda nn$ bound state and resonance, Phys. Rev. C 105, 064001 (2022), arXiv:2211.01693, DOI 10.1103/PhysRevC.105.064001, https://arxiv.org/abs/2211.01693
14. E. Hiyama, S. Ohnishi, B. F. Gibson, Th. A. Rijken, Three-body structure of the $nn\Lambda$ system with $\Lambda N$–$\Sigma N$ coupling, Phys. Rev. C 89, 061302(R) (2014), arXiv:1405.2365, DOI 10.1103/PhysRevC.89.061302, https://arxiv.org/abs/1405.2365
15. J. Haidenbauer, U.-G. Meißner, A. Nogga, Ab initio description of hypernuclei, Prog. Part. Nucl. Phys. 149, 104242 (2026), arXiv:2508.05243, DOI 10.1016/j.ppnp.2026.104242, https://arxiv.org/abs/2508.05243
16. K. N. Suzuki et al., The cross-section measurement for the $^3\mathrm{H}(e,e'K^+)nn\Lambda$ reaction, Prog. Theor. Exp. Phys. 2022, 013D01 (2022), arXiv:2110.09104, DOI 10.1093/ptep/ptab158, https://arxiv.org/abs/2110.09104
17. J. Haidenbauer, Exploring the $\Lambda$-deuteron interaction via correlations in heavy-ion collisions, Phys. Rev. C 102, 034001 (2020), arXiv:2005.05012, DOI 10.1103/PhysRevC.102.034001, https://arxiv.org/abs/2005.05012
18. E. Meoto and M. Lekala, Hypertriton states from inverse scattering theory, arXiv:2010.05070, https://arxiv.org/abs/2010.05070
19. S. N. Nakamura, High-precision spectroscopy of $\Lambda$ hypernuclei with electron and meson beams, Few Body Syst. 67, 25 (2026), DOI 10.1007/s00601-026-02044-9, https://doi.org/10.1007/s00601-026-02044-9
20. C. Rappold et al., Study of light hypernuclei in Europe: The hypertriton and $nn\Lambda$ puzzles, EPJ Web Conf. 290, 09007 (2023), DOI 10.1051/epjconf/202329009007, https://doi.org/10.1051/epjconf/202329009007
21. E. Garrido, A. Kievsky, M. Gattobigio, M. Viviani, L. E. Marcucci, R. Del Grande, L. Fabbietti, D. Melnichenko, $p\Lambda$ and $pp\Lambda$ correlation functions, Phys. Rev. C 110, 054004 (2024), arXiv:2408.01750, DOI 10.1103/PhysRevC.110.054004, https://arxiv.org/abs/2408.01750
22. T. Gogami, F. Garibaldi, P. Markowitz, J. Reinhold, S. Nagao, S. N. Nakamura, Lambda Hypernuclear Spectroscopy by Electron Scattering at JLab, PoS(QCHSC24) 230 (2025), DOI 10.22323/1.483.0230, https://doi.org/10.22323/1.483.0230
23. X. Zeng, R.-Y. Zheng, Z.-W. Liu, L.-S. Geng, X.-R. Zhou, Hypernuclear structure with a new leading-order covariant chiral hyperon-nucleon force, Phys. Rev. C 113, 014310 (2026), arXiv:2509.16878, DOI 10.1103/9s7s-ypl3, https://arxiv.org/abs/2509.16878