Global calibration and complete uncertainty quantification of higher-order chiral hyperon-nucleon interactions with YNN three-body forces against light hypernuclei

The hyperon-nucleon (YN) interaction from SU(3) chiral effective field theory is severely underconstrained: only about 35 low-energy YN scattering data points exist, and leading-order (LO) fits face a structural tension — an adjustment of two LO low-energy constants that reproduces p-shell hyperon separation energies $B_\Lambda$ leaves the hypertriton slightly unbound, while the hypertriton-constrained fits overbind the p shell. Knöll and Roth (Phys. Lett. B 846 (2023) 138258) showed that hypernuclear structure data can supplement scattering data in calibrating chiral YN interactions, but at LO the tension persists and neither chiral-truncation nor calibration uncertainties were quantified. The open problem is to construct and globally calibrate chiral YN interactions at next-to-leading order (NLO) and next-to-next-to-leading order (N2LO) together with the hyperon-nucleon-nucleon (YNN) three-body forces that formally appear at N2LO, against YN scattering and light-hypernuclear data, with complete uncertainty quantification — truncation errors, calibration uncertainties of the low-energy constants, and the correlated uncertainties of the nucleonic NN+3N Hamiltonian — propagated consistently to hypernuclear observables, using emulators or equivalent statistical tools to make the optimization scale to the larger number of constants at higher orders. Substantial partial progress exists — an N2LO YN potential with truncation-error estimates, Bayesian uncertainty estimates for light hypernuclei, and first consistent chiral-YNN calculations reproducing separation energies up to A = 7 — but no rigorous global higher-order calibration with complete correlated uncertainty propagation has been achieved, and the sparse experimental database (including the still-imprecise hypertriton binding energy) remains the binding constraint.

Background

A hypernucleus is an atomic nucleus in which one or more nucleons are replaced by a hyperon, a baryon containing a strange quark — most commonly the $\Lambda$. The $\Lambda$ separation energy $B_\Lambda$, the energy required to remove the hyperon, is the primary observable: it isolates the hyperon-nucleon (YN) interaction, since the nucleonic core is otherwise well described. The lightest hypernucleus, the hypertriton $^{3}_{\Lambda}\mathrm{H}$ (a proton, a neutron, and a $\Lambda$, bound by only about $0.13$ MeV), is uniquely sensitive to the $\Lambda N$ spin-singlet interaction. Direct scattering data are extremely scarce: roughly 35 low-energy cross-section points exist for $\Lambda p$, $\Sigma^- p$, and $\Sigma^+ p$, and there are no $\Lambda n$ data at all, so hypernuclear separation energies and spectra across the s shell ($A\le 4$) and the p shell ($A=5$–$16$) are the main additional constraint on the YN interaction.

The theoretical framework is SU(3) chiral effective field theory (chiral EFT), in which baryon-baryon interactions are organized by Weinberg power counting as an expansion in $Q/\Lambda_b$, the ratio of a typical low momentum to the breakdown scale. Each order — leading order (LO), next-to-leading order (NLO), next-to-next-to-leading order (N2LO), … — contributes fixed operator structures with unknown short-distance coefficients, the low-energy constants (LECs), which must be fitted to data. For the strangeness $S=-1$ YN system there are 5 LECs at LO and 23 at NLO and N2LO combined — far more than the available scattering data — so fits must impose SU(3) flavor symmetry assumptions and use hypernuclear binding energies. A consistent chiral treatment also requires three-baryon forces involving one hyperon and two nucleons (YNN forces), which first contribute at N2LO, exactly as three-nucleon (3N) forces do in the purely nucleonic sector. Because hypernuclear calculations take a full Hamiltonian as input, the nucleonic NN+3N interaction enters as well, and its own uncertainty propagates into $B_\Lambda$, increasingly so for heavier and neutron-rich hypernuclei.

On the computational side, ab initio methods — chiefly the no-core shell model (NCSM), a variational diagonalization of the full $A$-body Hamiltonian in a harmonic-oscillator basis that treats all particles explicitly — made p-shell hypernuclei accessible starting with Wirth et al. (Phys. Rev. Lett. 113 (2014) 192502). A long-standing problem was that chiral YN interactions fitted to scattering data plus the hypertriton systematically overestimate p-shell $B_\Lambda$. Knöll and Roth (Phys. Lett. B 846 (2023) 138258) addressed this by adjusting the two dominant LO YN LECs to p-shell separation energies within the NCSM: the overestimation is remedied and the excitation spectra are well described, but the hypertriton becomes slightly unbound, demonstrating that LO alone cannot simultaneously describe the hypertriton and the heavier p-shell $B_\Lambda$. They also introduced an artificial-neural-network (ANN) emulator — a machine-learning surrogate trained on nucleonic systems — to extrapolate finite-model-space energies to the infinite basis with quantified many-body uncertainties. Their conclusions define the problem: full theoretical uncertainty estimation requires YN interactions at higher chiral orders with the consistently appearing YNN three-body forces, a framework that captures the dependence on the nucleonic NN+3N interaction consistently, and a calibration procedure that can handle the growing number of LECs with rigorous optimization and complete uncertainty quantification, supported by new, more precise hypernuclear data.

Several ingredients have since become available and define the current frontier: an N2LO YN potential in the semi-local momentum-space (SMS) regularization scheme with truncation-error estimates for scattering observables (Haidenbauer et al., Eur. Phys. J. A 59 (2023) 63); the leading chiral YNN three-body forces, whose LECs can be estimated by decuplet saturation — i.e. by saturating intermediate states with the lowest decuplet baryons ($\Delta$, $\Sigma^*$, $\Xi^*$) — but which remain essentially unconstrained by data; and first NCSM calculations combining N2LO YN and chiral YNN forces for hypernuclei up to $A=7$. The YN interaction also matters beyond hypernuclei: the appearance of hyperons in neutron-star matter (the so-called hyperon puzzle, the tension between hyperon-induced softening of the equation of state and observed two-solar-mass pulsars) makes a YNN force with quantified uncertainties one of the candidate mechanisms for stiffening the equation of state.

Problem Statement

Construct and calibrate chiral hyperon-nucleon interactions beyond leading order — at next-to-leading order (NLO) and next-to-next-to-leading order (N2LO) in SU(3) chiral effective field theory — together with the hyperon-nucleon-nucleon (YNN) three-body forces that formally appear at N2LO, so that all low-energy constants are fixed in a single consistent procedure: the YN low-energy constants fitted to the combined available YN scattering data (about 35 low-energy $\Lambda p$, $\Sigma^- p$, and $\Sigma^+ p$ cross-section points) and measured hypernuclear separation energies and spectra from the hypertriton through the p shell, and the YNN low-energy constants constrained within the same global procedure by the available hypernuclear data sensitivity together with theory priors (decuplet saturation / SU(3) symmetry relations), with the prior uncertainties quantified and propagated; the inclusion of additional subleading YNN operator terms is permitted where needed for a consistent description (e.g. of the $A=4$ $0^+$/$1^+$ states). The calibrated interaction must achieve, with quantified total uncertainties: (1) simultaneous reproduction of the hypertriton binding and the heavier p-shell $B_\Lambda$ values, resolving the tension that a two-parameter LO adjustment cannot; (2) a quantified truncation uncertainty of the chiral expansion at each order, together with the calibration (parameter) uncertainty of the fitted LECs, propagated through the many-body calculation to predicted observables; (3) consistent treatment of the nucleonic Hamiltonian — the nucleon-nucleon (NN) plus three-nucleon (3N) interaction — as part of the calibration — varying it within its own uncertainty and propagating its uncertainty and its correlations with the YN sector, whose importance grows for heavier and neutron-rich hypernuclei; and (4) an optimization procedure that scales to the much larger number of LECs at higher chiral orders, for example through emulators or other statistical/algorithmic strategies, making a rigorous global fit with complete uncertainty propagation computationally feasible. Requirements (1)–(3) are mandatory components of the calibrated interaction and its uncertainty budget; requirement (4) is an enabling means rather than a separate deliverable, and “simultaneous reproduction” in (1) means reproducing the hypertriton binding and the heavier p-shell $B_\Lambda$ values within the propagated $1\sigma$ total uncertainty. Predictions for hypernuclei and observables not included in the calibration must validate the interaction. Improved experimental constraints (a more precise hypertriton binding energy, $\Lambda p$ angular distributions, spectroscopy of neutron-rich hypernuclei) may be incorporated as they become available; the scientific objective is the calibrated, uncertainty-quantified interaction itself, with all the listed components jointly supporting that single objective.

References

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