The hypernuclei Λ4He^4_\Lambda\mathrm{He} and Λ4H^4_\Lambda\mathrm{H} form a mirror pair whose different Λ separation energies in the 0+0^+ and 1+1^+ states measure charge symmetry breaking (CSB) in the Λ-nucleon interaction. A 2021 chiral-EFT analysis fitted the two leading CSB contact terms to the then-available emulsion-based splittings and predicted a strongly attractive Λn singlet scattering length (asΛn≈−3.2a_s^{\Lambda n}\approx -3.2 fm), but left open six issues: missing N²LO three-baryon forces, unevaluated two-pion-exchange CSB terms, superseded experimental inputs, the consequences for a possible Λnn resonance, extension to p-shell hypernuclei, and regulator/scheme consistency. In the subsequent literature, consistent chiral ΛNN+ΣNN three-body forces and an N²LO hyperon-nucleon potential now exist, and STAR has remeasured both splittings with substantially improved precision, but the central CSB redetermination with propagated uncertainties has not been performed. The problem remains open.

Background

A hypernucleus is a bound system of nucleons plus one or more hyperons — baryons carrying strangeness, such as the Λ (lambda) particle, which is electrically neutral and has the quark content udsuds. The key observable is the Λ separation energy BΛB_\Lambda, the energy needed to remove the hyperon; for the mirror pair Λ4He^4_\Lambda\mathrm{He} (Λ bound to a 3^3He core of two protons and a neutron) and Λ4H^4_\Lambda\mathrm{H} (Λ bound to a triton core of one proton and two neutrons), charge symmetry — the interchange of protons and neutrons — would force equal BΛB_\Lambda values in corresponding states if the strong interaction were exactly charge-symmetric. In reality the measured splittings differ by a few hundred keV in both the 0+0^+ ground states and the 1+1^+ excited states, which is direct evidence of charge symmetry breaking (CSB) in the ΛN interaction: the Λn and Λp forces are not identical. Because the Λ is neutral, no direct Λn scattering experiment exists, so hypernuclear mirror splittings are essentially the only handle on the Λn interaction at low energy.

The modern framework for this problem is chiral effective field theory (EFT), a low-energy expansion of baryon-baryon forces in powers of momentum over the breakdown scale, in which unknown short-range physics is encoded in contact terms whose coefficients, called low-energy constants (LECs), are fitted to data. Haidenbauer, Meißner, and Nogga (2021) added the two leading CSB contact terms, C1S0CSBC^{\mathrm{CSB}}_{1S0} and C3S1CSBC^{\mathrm{CSB}}_{\mathrm{3S1}} (one acting in the spin-singlet 1S0{}^1S_0 partial wave, one in the spin-triplet 3S1{}^3S_1), to their next-to-leading-order (NLO) chiral hyperon-nucleon potentials NLO13 and NLO19. Solving the exact four-body Faddeev–Yakubovsky equations (a system of coupled integral equations yielding essentially exact bound-state energies for the A = 4 system), they fitted the LECs to the then-accepted splittings ΔE(0+)=233±92\Delta E(0^+)=233\pm92 keV and ΔE(1+)=−83±94\Delta E(1^+)=-83\pm94 keV (their "CSB1" scenario, based on old emulsion measurements), obtaining CSB scattering-length shifts ΔaCSB(1S0)=0.62±0.08\Delta a_{\mathrm{CSB}}({}^1S_0)=0.62\pm0.08 fm and ΔaCSB(3S1)=−0.10±0.02\Delta a_{\mathrm{CSB}}({}^3S_1)=-0.10\pm0.02 fm and predicting a strongly attractive Λn singlet scattering length asΛn≈−3.2a_s^{\Lambda n}\approx-3.2 fm.

Several technical terms are needed to state what the 2021 analysis left open. Three-body forces (3BFs) — interactions among three baryons, such as ΛNN and ΣNN forces (where Σ denotes the sigma baryon, the isospin-1 partner of the Λ), which first enter the chiral expansion at next-to-next-to-leading order (N²LO) — were not included; their leading chiral structure was derived by Petschauer and coworkers. The similarity renormalization group (SRG) is a unitary transformation that softens two-body potentials for many-body use but induces compensating many-body forces, so consistency between the SRG-evolved potential and the 3BFs must be tracked; likewise the regulator — the momentum-space cutoff function that suppresses contributions above the EFT breakdown scale — must be applied uniformly to the two- and three-baryon forces. The no-core shell model (NCSM) is an ab initio method — one built directly from the constituent particles and their interactions, with no phenomenological core — that expands all nucleons (plus the hyperon) in a harmonic-oscillator basis without a core approximation, usable up to A≈7A\approx7. Irreducible two-pion-exchange CSB terms are N²LO contributions to the ΛN force that cannot be absorbed into the fitted contact terms and were neglected in 2021 as presumably small, without an explicit calculation. Finally, the Λnn system (a Λ with two neutrons) is a hypothetical three-body system that would become a resonance if the Λn singlet attraction were strong enough — a claim needing dedicated few-body resonance analysis.

Since 2021 the landscape has moved on three fronts. On the theory side, the Jülich–Bonn group derived the NLO-to-N²LO chiral hyperon-nucleon potential, benchmarked the partial-wave decomposition of the leading ΛNN 3BF, and performed the first NCSM calculations of A≤7A\le7 Λ hypernuclei with consistent chiral ΛNN and ΣNN three-body forces, finding that experimental separation energies are reproduced fairly well once 3BFs are included. On the experimental side, the STAR Collaboration (Solenoid Tracker at RHIC, the Relativistic Heavy Ion Collider at Brookhaven National Laboratory) measured BΛB_\Lambda for both Λ4H^4_\Lambda\mathrm{H} and Λ4He^4_\Lambda\mathrm{He} in Au+Au collisions, first in 2022 and then far more precisely in 2026, giving ground-state CSB ΔBΛ4(0+)=0.15±0.05±0.04\Delta B_\Lambda^4(0^+)=0.15\pm0.05\pm0.04 MeV and excited-state CSB ΔBΛ4(1+)=−0.17±0.05±0.04\Delta B_\Lambda^4(1^+)=-0.17\pm0.05\pm0.04 MeV — values with central magnitudes comparable to the old emulsion numbers but with smaller uncertainties, superseding the CSB1 input scenario. On the phenomenology side, the possible Λnn resonance and the extension of CSB predictions to p-shell hypernuclei — those whose valence nucleons occupy the 0p harmonic-oscillator shell, roughly A = 5 to 16 — have been pursued by several groups with other inputs, but not with the CSB-strengthened Λn singlet interaction. A review by the same Jülich–Bonn authors (2025 preprint, published 2026) states explicitly that the 2021 CSB analysis should be repeated with the new data, the updated interactions, and propagated experimental uncertainties, and reports the N²LO CSB analysis with three-body forces as work in progress.

Problem Statement

Within chiral effective field theory for the ΛN system, determine — consistently at next-to-next-to-leading order (N²LO), with chiral ΛNN and ΣNN three-body forces included and with regulator and SRG treatment held consistent — the charge-symmetry-breaking part of the Λ-neutron interaction from the mirror hypernuclei Λ4He^4_\Lambda\mathrm{He} and Λ4H^4_\Lambda\mathrm{H}, using the modern experimental inputs ΔBΛ4(0+)=0.15±0.05±0.04\Delta B_\Lambda^4(0^+)=0.15\pm0.05\pm0.04 MeV and ΔBΛ4(1+)=−0.17±0.05±0.04\Delta B_\Lambda^4(1^+)=-0.17\pm0.05\pm0.04 MeV (STAR, 2026) in place of the superseded emulsion-based values. Concretely, the open problem as posed by the source work and confirmed open by later literature comprises: (1) refit the two leading CSB contact LECs C1S0CSBC^{\mathrm{CSB}}_{1S0} and C3S1CSBC^{\mathrm{CSB}}_{\mathrm{3S1}} and redetermine the Λn scattering lengths, with experimental and truncation uncertainties propagated to the final numbers; (2) evaluate the neglected irreducible two-pion-exchange CSB contributions arising at N²LO and quantify their effect on the fitted LECs and observables; (3) analyze, with the resulting CSB-strengthened Λn ¹S₀ interaction (for which asΛn≈−3.2a_s^{\Lambda n}\approx-3.2 fm at NLO), whether the Λnn system supports a bound state or resonance and, if such a state exists, its energy and width; (4) test whether the same CSB contact terms, fitted only at A = 4, describe charge symmetry breaking in p-shell Λ hypernuclei — for completion, the A = 7 and A = 8 mirror pairs — without refit, that is, within the propagated uncertainties and without re-adjusting the LECs; and (5) establish regulator- and scheme-independent conclusions by controlling the interplay of the ΛN–ΣN transition strength, SRG-induced many-body terms, and consistently defined three-baryon interactions, demonstrated by the stability of the fitted CSB LECs and Λn scattering lengths under variation of the regulator and of the scheme-dependent inputs. The single unifying objective is a quantitative, uncertainty-controlled determination of charge symmetry breaking in the Λn interaction from hypernuclear data, and its few-body phenomenological consequences. For completion, all five sub-questions are jointly required, and a complete answer must report them together: the refitted CSB LECs and Λn scattering lengths with propagated uncertainties (1), the quantified effect of the two-pion-exchange CSB contributions (2), the stated Λnn outcome (3), the A = 7 and A = 8 p-shell test (4), and the regulator- and scheme-variation stability of the fitted quantities (5). Broader extensions, such as heavier p-shell hypernuclei or additional interaction variants, are optional.

References

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