The leading-order (LO) covariant chiral hyperon–nucleon (YN) interaction, propagated through relativistic Brueckner–Hartree–Fock nuclear-matter calculations into a parameter-free Skyrme–Hartree–Fock functional, reproduces the measured single-Λ binding energies of 29 hypernuclei from Λ5He^{5}_{\Lambda}\mathrm{He} to Λ208Pb^{208}_{\Lambda}\mathrm{Pb}, but with unquantified systematic uncertainty: predictions differ materially between regulator cutoffs of 550 MeV (systematic underbinding) and 700 MeV, discrepancies persist in light systems such as Λ7Li^{7}_{\Lambda}\mathrm{Li}, and the mapping neglects the Λ spin-orbit interaction, finite-range (gradient) terms, and the Λ-density dependence of the nucleon mean field. The covariant chiral YN force exists only at LO, higher chiral orders having been constructed only for the nucleon–nucleon sector and for non-relativistic chiral YN forces. The problem is to construct a quantified error budget for these hypernuclear predictions — attributing the residuals to chiral truncation, mapping approximations, and missing beyond-mean-field correlations — and to reduce them, for instance via a next-to-leading-order covariant chiral YN interaction carried through the same in-medium and finite-nucleus chain, so that single-Λ hypernuclear predictions become cutoff-stable and systematically improvable.

Background

A hypernucleus is a nucleus in which one nucleon is replaced by a hyperon — a baryon containing a strange quark — the lightest and best studied being the Λ. Single-Λ hypernuclei are the main source of experimental information on the hyperon–nucleon (YN) interaction, because direct YN scattering data are scarce: only about 36 low-energy data sets exist for the strangeness S=−1S=-1 sector (ΛN\Lambda N and the coupled ΣN\Sigma N channels) (Gal, Hungerford, and Millener, 2016). Most YN potentials used in hypernuclear structure have been phenomenological meson-exchange models of the Nijmegen type, mapped into Skyrme–Hartree–Fock (SHF) mean-field calculations through Brueckner–Hartree–Fock nuclear matter (Zhou et al., 2007). In SHF, the nucleus is described by an energy-density functional of the local nucleon (and hyperon) densities and their gradients, solved self-consistently for the single-particle orbitals; nucleon pairing is added in the BCS (Bardeen–Cooper–Schrieffer) approximation.

Chiral effective field theory (chiral EFT) replaces phenomenology with a systematic low-energy expansion of QCD: long-range meson exchanges plus short-range contact terms, organized by a power counting in small momenta, with unknown couplings — the low-energy constants (LECs) — fitted to data. In the Weinberg (non-relativistic) formulation, YN forces have been constructed up to next-to-next-to-leading order (N²LO) (Haidenbauer et al., 2013; Haidenbauer et al., 2023). A covariant alternative builds the baryon–meson interaction with manifest Lorentz invariance; for the YN system this program exists so far only at leading order (LO) (Li et al., 2016; Song et al., 2022), where 12 LECs are fixed by the 36 scattering data sets. The interaction is regularized with a smooth momentum-space cutoff ΛF\Lambda_{F}; residual dependence of observables on ΛF\Lambda_{F} (the regulator or cutoff) is a diagnostic of truncation of the chiral expansion at LO.

Carrying this force into medium, Zheng et al. (2025) computed the in-medium ΛN\Lambda N interaction in nuclear matter within the relativistic Brueckner–Hartree–Fock (RBHF) framework — a relativistic G-matrix resummation of two-body correlations — and showed that the LO covariant chiral force simultaneously reproduces the scattering data and the empirical Λ single-particle potential of about −30-30 MeV at saturation density, something non-relativistic LO forces fail to do. Zeng et al. (2026) then mapped these in-medium interactions, without any adjustable parameter, onto an SHF energy-density functional (parameter sets RCh-LO550 and RCh-LO700, for cutoffs ΛF=550\Lambda_{F}=550 and 700 MeV) and computed Λ binding energies of 29 hypernuclei from Λ5He^{5}_{\Lambda}\mathrm{He} to Λ208Pb^{208}_{\Lambda}\mathrm{Pb}. RCh-LO700 describes the data excellently, better than earlier microscopic interactions; RCh-LO550 systematically underbinds, because the in-medium attraction of the LO force grows with the cutoff. Residual discrepancies persist in light hypernuclei, most visibly Λ7Li^{7}_{\Lambda}\mathrm{Li}, where mean-field methods are known to be least reliable.

Three approximations of this mapping are unquantified. The Λ spin-orbit interaction is set to zero; the ΛN spin-orbit force is known to be small, producing an approximate spin symmetry — near-degeneracy of single-particle levels differing only in the Λ spin orientation — even at rather large spin-orbit strengths (Xue et al., 2023). Finite-range (gradient) corrections to the local ΛN functional are omitted. And the back-reaction of the hyperon density on the nucleon mean field is neglected. ΛΛ terms are set to zero. The stakes extend beyond hypernuclei: the hyperon puzzle — the tension between hyperon-induced softening of the neutron-star equation of state and the observed ∼2 M⊙\sim 2\,M_{\odot} pulsar masses — demands a YN interaction with controlled and quantifiable uncertainty.

Problem Statement

The problem is to attribute and reduce the systematic uncertainty of the parameter-free single-Λ hypernuclear predictions derived from the LO covariant chiral YN force, taking the RCh-LO550/RCh-LO700 results as the baseline. It has three linked parts.

(1) Cutoff/truncation error. Determine how much of the difference between the ΛF=550\Lambda_{F}=550 MeV and ΛF=700\Lambda_{F}=700 MeV predictions — including the systematic underbinding at 550 MeV across the 29 measured hypernuclei from Λ5He^{5}_{\Lambda}\mathrm{He} to Λ208Pb^{208}_{\Lambda}\mathrm{Pb} — is a leading-order truncation artifact. This could be pursued, for instance, by constructing the covariant chiral YN interaction at next-to-leading order (or another explicitly controlled higher order or improved regularization scheme), fitting its low-energy constants to the same scattering database, propagating it through the same in-medium (relativistic Brueckner–Hartree–Fock or equivalent) and finite-nucleus (Skyrme–Hartree–Fock or equivalent) chain, and establishing whether the hypernuclear binding-energy predictions become stable against the regulator cutoff within the estimated truncation uncertainty.

(2) Mapping approximations. Quantify, within the existing LO force, the effect on Λ binding energies and single-particle level splittings of the three neglected ingredients: the Λ spin-orbit interaction, finite-range/gradient terms in the ΛN functional, and the Λ-density dependence of the nucleon single-particle potential.

(3) Light hypernuclei. Determine whether the residual discrepancies in light systems, notably Λ7Li^{7}_{\Lambda}\mathrm{Li}, survive once (1) and (2) are accounted for, or whether they originate in beyond-mean-field (few-body, pairing, or symmetry) correlations; in the latter case, identify the mechanism.

These three parts are inseparable components of one objective: a validated error budget for the covariant-chiral hypernuclear predictions and a cutoff-stable, systematically improvable description of single-Λ hypernuclei. A submission that only constructs the NLO interaction without propagating it to finite hypernuclei, or that removes the light-nucleus discrepancy by refitting parameters rather than by identified physics, does not complete the problem. The source also lists extensions to ΛΛ/YY channels and the need for new scattering and hypernuclear-spectroscopy data as open items; these are genuine follow-on objectives but lie outside the completion boundary of this problem.

References

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