Empirical and microscopic determination of universal hyperonic many-body repulsion: fixing the $\Lambda N$ $P$-wave degeneracy and the $YNN$/$YYN$ three-body force from joint hypernuclear, scattering, and neutron-star constraints

Analyses of $\Lambda$ binding energies ($B_\Lambda$) across the periodic table show that the repulsive hyperonic many-body force needed to keep the neutron-star equation of state (EoS) stiff enough for two-solar-mass pulsars is exactly degenerate with the poorly known odd-parity ($P$-state) part of the two-body $\Lambda N$ interaction. Whether a strong, universal hyperonic many-body repulsion actually exists — and if so, its magnitude, density dependence, and channel structure ($YNN$, $YYN$, including isospin dependence) — therefore cannot be decided from hypernuclear data alone. The open problem is a joint determination (or refutation) combining hypernuclear spectroscopy, improved $YN$ scattering and correlation constraints, and neutron-star observations. Substantial post-2017 progress (empirical $\Lambda NN$ optical-potential extraction, combined $\Lambda p$ scattering–femtoscopy analyses, chiral-EFT and quantum-Monte-Carlo three-baryon forces) has narrowed but not closed the question: the $P$-wave $\Lambda N$ content remains essentially unconstrained, three-body-force contact terms are not fixed, and no joint constraint exists that uniquely selects among two-body baselines and many-body parametrizations.

Background

A $\Lambda$ hypernucleus is a nucleus in which one nucleon is replaced by a $\Lambda$ hyperon (the lightest baryon with strangeness $S=-1$, made of $u,d,s$ quarks, with isospin zero), and the standard spectroscopic observable is the $\Lambda$ binding energy $B_\Lambda$, the separation energy of the $\Lambda$ from the core nucleus. Systematics of $B_\Lambda$ versus mass number determine the single-particle potential $U_\Lambda$ felt by a $\Lambda$ in nuclear matter, whose depth at saturation density is known empirically to be about $-28$ to $-30$ MeV (this is the empirical $\Lambda$-nucleus optical-potential depth; in-medium single-particle potentials derived from $G$-matrix or meson-exchange models follow different conventions and are typically deeper). Unlike the nucleon-nucleon case, there are essentially no hyperon-nucleon ($YN$) scattering data (no $\Lambda n$ data at all, and only old, low-statistics $\Lambda p$ data), so two-body $YN$ interaction models — such as the Nijmegen extended-soft-core family (ESC08, ESC12, ESC14, ESC16, NSC97) — differ substantially in content that scattering cannot discriminate, in particular in the odd-parity ($P$-wave, angular momentum $L=1$) partial waves.

The astrophysical stake is the hyperon puzzle: the discovery of two-solar-mass pulsars (Demorest et al. 2010; Antoniadis et al. 2013) requires a stiff equation of state (EoS, the pressure–density relation) of neutron-star matter, but the appearance of hyperons in dense matter softens the EoS and typically lowers the maximum neutron-star mass below $2M_\odot$. A widely pursued baryonic resolution is strong repulsive many-body forces (three-body forces involving hyperons, denoted $YNN$ and $YYN$ where $Y$ stands for a hyperon) that turn on at high density. The $\Lambda NN$ three-body force (two nucleons plus one $\Lambda$) is the best-constrained member of this family, since it is sampled by ordinary single-$\Lambda$ hypernuclei at nuclear density.

Isaka, Yamamoto, and Rijken (Phys. Rev. C 95, 044308 (2017)) analyzed $B_\Lambda$ values of hypernuclei with $9\le A\le 59$ in the Antisymmetrized Molecular Dynamics framework (a nuclear-structure method representing the nucleus as an antisymmetrized set of Gaussian wave packets) extended to include a $\Lambda$ (HyperAMD), using $\Lambda N$ $G$-matrix effective interactions — the $G$-matrix being the in-medium two-body effective interaction obtained by summing particle-hole ladder diagrams in the Brueckner theory — folded from the ESC14 and ESC12 two-body potentials plus a many-body effect (MBE) consisting of multi-pomeron exchange repulsion (MPP, a repulsive many-baryon force model tied to a phenomenological Pomeron exchange) and a phenomenological three-body attraction (TBA). The local density entering the $G$-matrix is supplied by the averaged-density approximation (ADA), which maps the nucleon density sampled by the $\Lambda$ wave function to an effective Fermi momentum $k_F$; for weakly bound $\Lambda$ $p$-states an empirical rescaling $k_F'=(1+\alpha)k_F$ with mass-dependent $\alpha$ was introduced. Their central result is a degeneracy: with ESC14-type models whose $P$-state $\Lambda N$ contributions almost vanish, a strong MPP repulsion (consistent with a stiff, $2M_\odot$ hyperon-mixed EoS) reproduces the $B_\Lambda$ mass dependence; with ESC12/NSC97-type models whose $P$-states are repulsive, the $P$-state contribution itself supplies the needed mass dependence and there is no room for strong three-body repulsion; with an attractive-$P$-state model (ESC04a-type) it is difficult to reproduce the $B_\Lambda$ mass dependence by adding the MPP+TBA form of many-body effect used there, and no reasonable parameter set was found. Since no scattering data fix the $\Lambda N$ $P$-waves, $B_\Lambda$ data alone cannot prove or disprove strong universal hyperonic many-body repulsion.

Adjacent microscopic developments frame the current frontier. Leading three-baryon forces have been derived in $SU(3)$ chiral effective field theory (EFT, the low-energy expansion of QCD in terms of baryons and Goldstone mesons) by Petschauer et al., and used both in Brueckner–Hartree–Fock neutron-star calculations (Logoteta et al.) and, most recently, in no-core shell model studies of light hypernuclei with consistent chiral $\Lambda NN$ and $\Sigma NN$ forces (Le et al. 2025) — but the contact (short-range) three-body couplings of the chiral $\Lambda NN$ force are not fixed by any data at the orders currently accessible. On the empirical side, Friedman and Gal fitted a $\Lambda$-nucleus optical potential with linear-density ($\Lambda N$) and quadratic-density ($\Lambda NN$) terms to $\Lambda$ single-particle binding energies across $A=12$–$208$, extracting a repulsive $\Lambda NN$ depth of about $+10$ MeV at nuclear saturation density, and a first combined analysis of low-energy $\Lambda p$ scattering and ALICE $p\Lambda$ correlation functions (Mihaylov et al. 2024) has tightened the $S$-wave $\Lambda N$ constraint, indicating weaker attraction than previously accepted. None of these inputs has yet been combined into a single joint determination, and the hyperon puzzle remains under active debate in neutron-star theory.

Problem Statement

Establish or refute the existence of a strong universal hyperonic many-body repulsion and, if established, determine its strength and density dependence, through a joint constraint that combines (i) hypernuclear $\Lambda$ binding energies and spectroscopy across the periodic table, (ii) improved $YN$ scattering and correlation-function constraints including the hitherto undetermined $P$-wave $\Lambda N$ content, and (iii) the neutron-star EoS requirement that hyperon-mixed matter be stiff enough to support two-solar-mass stars. Here "universal" is used in the sense of the source work: a many-baryon repulsion that acts across baryon channels ($NNN$, $YNN$, $YYN$), as in the multi-pomeron exchange potential (MPP) of the Nijmegen models, rather than a channel-specific force; if the repulsion is established, whether it requires channel-resolved or isospin-dependent refinements is part of determining its structure. A complete answer must resolve the demonstrated degeneracy between repulsive $\Lambda N$ $P$-state contributions and three-body repulsion in the $B_\Lambda$ mass dependence — i.e. decide whether two-body-only baselines (e.g. ESC12/NSC97-type repulsive $P$-states) or strong many-body repulsion on vanishing-$P$-state baselines (e.g. ESC14+MPP) is selected when all three data classes are imposed simultaneously — or demonstrate that this degeneracy cannot be lifted. The latter outcome completes the refutation branch rather than failing the task: a demonstration that the three data classes, even jointly, do not select among the alternatives answers the source's stated limitation that the existence of strong many-body repulsion is difficult to prove while the two-body interaction model is not finely determined; what is not sufficient is a selection claim resting on hypernuclear data alone, which the source shows admit both alternatives.

References

1. M. Isaka, Y. Yamamoto, Th. A. Rijken, 'Effects of a hyperonic many-body force on $B_\Lambda$ values of hypernuclei', Phys. Rev. C 95, 044308 (2017). DOI: 10.1103/PhysRevC.95.044308; arXiv:1703.03117, https://arxiv.org/abs/1703.03117
2. E. Friedman, A. Gal, 'Constraints from $\Lambda$ hypernuclei on the $\Lambda NN$ content of the $\Lambda$-nucleus potential', Phys. Lett. B 837, 137669 (2023). DOI: 10.1016/j.physletb.2023.137669; arXiv:2204.02264, https://arxiv.org/abs/2204.02264
3. E. Friedman, A. Gal, '$\Lambda$ hypernuclear potentials beyond linear density dependence', Nucl. Phys. A 1039, 122725 (2023). DOI: 10.1016/j.nuclphysa.2023.122725; arXiv:2306.06973, https://arxiv.org/abs/2306.06973
4. E. Friedman, A. Gal, '$\Lambda NN$ input to neutron stars from hypernuclear data', PoS EXA-LEAP2024, 060 (2025). DOI: 10.22323/1.480.0060; arXiv:2411.11751, https://arxiv.org/abs/2411.11751
5. D. L. Mihaylov, J. Haidenbauer, V. Mantovani Sarti, 'Constraining the $p\Lambda$ interaction from a combined analysis of scattering data and correlation functions', Phys. Lett. B 850, 138550 (2024). DOI: 10.1016/j.physletb.2024.138550; arXiv:2312.16970, https://arxiv.org/abs/2312.16970
6. H. Le, J. Haidenbauer, U.-G. Meißner, A. Nogga, 'Light $\Lambda$ Hypernuclei Studied with Chiral Hyperon-Nucleon and Hyperon-Nucleon-Nucleon Forces', Phys. Rev. Lett. 134, 072502 (2025). DOI: 10.1103/PhysRevLett.134.072502; arXiv:2409.18577, https://arxiv.org/abs/2409.18577
7. D. Logoteta, I. Vidaña, I. Bombaci, 'Impact of chiral hyperonic three-body forces on neutron stars', Eur. Phys. J. A 55, 207 (2019). DOI: 10.1140/epja/i2019-12909-9; arXiv:1906.11722, https://arxiv.org/abs/1906.11722
8. S. Petschauer, N. Kaiser, J. Haidenbauer, U.-G. Meißner, W. Weise, 'Leading three-baryon forces from SU(3) chiral effective field theory', Phys. Rev. C 93, 014001 (2016). DOI: 10.1103/PhysRevC.93.014001; arXiv:1511.02095, https://arxiv.org/abs/1511.02095
9. S. Petschauer, J. Haidenbauer, N. Kaiser, U.-G. Meißner, W. Weise, 'Density-dependent effective baryon–baryon interaction from chiral three-baryon forces', Nucl. Phys. A 957, 347 (2017). DOI: 10.1016/j.nuclphysa.2016.09.010; arXiv:1607.04307, https://arxiv.org/abs/1607.04307
10. D. Lonardoni, A. Lovato, S. Gandolfi, F. Pederiva, 'Hyperon Puzzle: Hints from Quantum Monte Carlo Calculations', Phys. Rev. Lett. 114, 092301 (2015). DOI: 10.1103/PhysRevLett.114.092301; arXiv:1407.4448, https://arxiv.org/abs/1407.4448
11. J. Haidenbauer, U.-G. Meißner, A. Nogga, H. Le, 'Hyperon–nucleon interaction in chiral effective field theory at next-to-next-to-leading order', Eur. Phys. J. A 59, 63 (2023). DOI: 10.1140/epja/s10050-023-00960-6; arXiv:2301.00722, https://arxiv.org/abs/2301.00722
12. H. Kamada, M. Kohno, K. Miyagawa, 'Faddeev Calculation of Hypertriton Including with YNN Three-Body Force', PoS CD2024, 094 (2025). DOI: 10.22323/1.479.0094, https://doi.org/10.22323/1.479.0094
13. H.-F. Zhu, G.-Z. Liu, X. Wu, Y.-F. Yuan, 'New solution to the hyperon puzzle of neutron stars: Quantum many-body effects', Phys. Rev. D 113, 043050 (2026). DOI: 10.1103/gxyr-v5h7; arXiv:2602.07939, https://arxiv.org/abs/2602.07939
14. M. Isaka, Y. Yamamoto, Th. A. Rijken, 'Competing effects of nuclear deformation and density dependence of the $\Lambda N$ interaction in $B_\Lambda$ values of hypernuclei', Phys. Rev. C 94, 044310 (2016). DOI: 10.1103/PhysRevC.94.044310; arXiv:1609.06009, https://arxiv.org/abs/1609.06009