Completing the SU(3) chiral hyperon-nucleon interaction at N²LO: YNN three-baryon forces, charge-symmetry breaking, convergence, and empirical constraints

The 2020 chiral-EFT analysis of the strangeness S = −1 hyperon-nucleon (YN) interaction by Haidenbauer, Meißner and Nogga identified five unresolved issues: undetermined SU(3) chiral YNN three-baryon forces at N²LO; truncation and regulator dependence of the NLO nuclear-matter results; insufficient empirical constraints on the spin dependence and higher partial waves of the YN interaction; missing charge-symmetry breaking (CSB) in the ΛN potential; and the open challenge of three-body correlations in many-body calculations. Major progress has been made on the construction side — an N²LO YN potential with quantified truncation uncertainty (2023), the first consistent inclusion of chiral ΛNN and ΣNN forces in ab initio hypernuclear calculations (2024–2025), derivation and application of the leading CSB ΛN interaction (2021–2023), and new ΣN differential data from J-PARC E40 plus first direct coupled-channel ΛN–ΣN evidence from ALICE femtoscopy — but the core remains open: the YNN low-energy constants are still fixed by the same hypernuclear separation energies used for validation, chiral YNN forces have not entered nuclear-matter calculations beyond density-dependent approximations (Bethe–Faddeev with chiral three-body forces is explicitly deferred to the future), CSB is absent from the N²LO potential while the hypertriton separation energy is contested between STAR and ALICE, and the P-wave contact terms await Λp angular-distribution data from the planned J-PARC E86 program. The problem — a single, regulator-independent, empirically determined chiral YN+YNN interaction predicting scattering, hypernuclear, and nuclear-matter observables within quantified uncertainties — is therefore open.

Background

Hyperons are baryons containing one strange quark; the lightest are the $\Lambda$ (isospin 0) and the $\Sigma$ (isospin 1) particles, both carrying strangeness $S=-1$. How they interact with nucleons ($N$) — the hyperon-nucleon ($YN$) interaction — governs the structure of hypernuclei (bound systems of nucleons plus a hyperon) and whether hyperons can appear in the interior of neutron stars (the "hyperon puzzle": most interactions including hyperons soften the equation of state below the observed two-solar-mass maximum). The modern framework for nuclear forces is chiral effective field theory (chiral EFT), an expansion in powers of typical momenta over the chiral-symmetry-breaking scale of order 1 GeV, organized by the power counting suggested by Weinberg: the potential is computed order by order (LO, NLO, N²LO, ...) and then iterated in a scattering or few-body equation. In the $S=-1$ sector one can additionally exploit approximate SU(3) flavor symmetry, which relates the $\Lambda N$, $\Sigma N$, and $NN$ contact interactions; the strengths of these contact terms are low-energy constants (LECs), free parameters to be fixed by fits to data. A specific complication is the coupled channel $\Lambda N\leftrightarrow\Sigma N$: the two thresholds differ by only about 80 MeV, so the strength of the transition potential influences few- and many-body observables even though it is barely constrained by the sparse $YN$ database (roughly three dozen low-energy $\Lambda p$ and $\Sigma^{\pm}p$ cross-section points and essentially no spin-dependent observables).

In the source paper, Haidenbauer, Meißner and Nogga built two NLO SU(3) chiral $YN$ potentials (NLO13 and NLO19) that describe the scattering data equally well but differ in the strength of the $\Lambda N\leftrightarrow\Sigma N$ transition. The difference has only moderate effects in the hypertriton ${}^{3}_{\Lambda}\mathrm{H}$ (the weakly bound $pn\Lambda$ system) and the $A=4$ hypernuclei, but sizable effects on the $\Lambda$ and $\Sigma$ single-particle potentials in nuclear matter computed from the $G$-matrix (Bethe–Goldstone) equation, the standard quasiparticle interaction in infinite matter. Because two-body data cannot fix this transition strength — "arguably as a matter of principle" — compensation must come from three-baryon forces $YNN$ ($\Lambda NN$, $\Sigma NN$), which in the applied Weinberg counting first arise at N²LO. In nuclear matter, three-body correlations can be treated consistently only by solving the Bethe–Faddeev equation, a technically demanding step that the literature commonly avoids. Charge-symmetry breaking (CSB, the difference of the $\Lambda p$ and $\Lambda n$ interactions under $p\leftrightarrow n$), visible in the measured separation-energy splitting between the mirror hypernuclei ${}^{4}_{\Lambda}\mathrm{He}$ and ${}^{4}_{\Lambda}\mathrm{H}$, was not included; contributions such as $\pi^{0}$ exchange combined with $\Lambda$–$\Sigma^{0}$ mixing were explicitly ignored. The paper closes by identifying these unresolved issues: the $YNN$ forces (size, operator structure, LECs) undetermined; truncation at NLO and noticeable regulator (cutoff) dependence of the $G$-matrix results; insufficient empirical constraints (no spin-dependent or differential observables to pin down P-wave LECs and the relative singlet/triplet strengths; needed inputs being more precise hypertriton and $A=4$ separation energies, planned $\Sigma^{-}p$ scattering, and correlation measurements in heavy-ion or $pp$ collisions); missing CSB; and the computational challenge of observables that fully include three-body correlations.

Problem Statement

Complete the SU(3) chiral effective-field-theory determination of the strangeness $S=-1$ hyperon-nucleon interaction at N²LO, to the point where one consistent set of low-energy constants — including the $YNN$ ($\Lambda NN$/$\Sigma NN$) three-baryon forces that first contribute at N²LO and the charge-symmetry-breaking $\Lambda N$ component — predicts, within quantified truncation and regulator uncertainties: (i) the available $\Lambda p$, $\Sigma^{\pm}p$ and new differential/correlation data; (ii) the separation energies of light hypernuclei (${}^{3}_{\Lambda}$H, the $A=4$ mirror pair, ${}^{5}_{\Lambda}$He and heavier systems); and (iii) the $\Lambda$ and $\Sigma$ single-particle potentials in nuclear matter, with three-body correlations treated by Bethe–Faddeev-type calculations rather than density-dependent approximations. As of 2026 the two-body N²LO potential and the leading $YNN$ forces have been constructed and applied, so the surviving open core is: (a) an empirical determination of the $YNN$ low-energy constants that is independent of the hypernuclear observables used to validate them, and a settling of their interplay with the $\Lambda N$–$\Sigma N$ transition-potential strength; (b) inclusion of chiral $YNN$ forces in nuclear-matter calculations beyond the density-dependent effective-two-body approximation; (c) incorporation of CSB into the N²LO interaction, constrained by the best available empirical information on the $A=4$ CSB splittings and the hypertriton separation energy; and (d) empirical constraints on the P-wave contact terms from the currently available data, to be refined as $\Lambda p$ angular distributions such as those from the J-PARC E86 program become available. These components share one objective — a reliable, regulator-independent chiral $YN$+$YNN$ interaction — and the problem is complete exactly when the four surviving-core items (a)–(d) are each resolved and the resulting single LEC set predicts all observable classes (i)–(iii) within quantified truncation and regulator uncertainties.

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