A 2026 evaluation framework by Ding and Sun (arXiv:2606.27541) found a strong linear correlation (Pearson coefficient 0.977) between theory–experiment binding-energy deviations of single-Λ\Lambda and double-Λ\Lambda hypernuclei computed with relativistic density functionals, anchored on the four double-Λ\Lambda systems with established binding-energy data (core nuclei 4{}^{4}He, 8{}^{8}Be, 9{}^{9}Be, 11{}^{11}B), and used it to propagate empirical single-Λ\Lambda constraints into evaluated ΛΛ\Lambda\Lambda interaction energies ΔBΛΛ\Delta B_{\Lambda\Lambda} for heavier nuclei. The evaluated values systematically exceed direct functional predictions calibrated only to the NAGARA event, implying absorbed beyond-mean-field physics of unknown origin. The open problem is twofold and joint: establish whether the correlation is robust, framework-independent, and valid across the mass range by reproducing or refuting it within independent theoretical frameworks, and quantitatively identify which microscopic mechanisms — few-body clustering, core polarization, coupled-channel ΛΛ\Lambda\Lambda–ΞN\Xi N mixing, or ΛΛ\Lambda\Lambda pairing — dominate the discrepancy between evaluated and direct predictions.

Background

A hypernucleus is an atomic nucleus containing one or more hyperons — baryons carrying strangeness, such as the Λ\Lambda (lambda, mass 1116 MeV, isospin zero) — in addition to protons and neutrons. Double-Λ\Lambda hypernuclei (strangeness S=−2S=-2) are the only existing window onto the ΛΛ\Lambda\Lambda hyperon–hyperon interaction inside nuclear matter, which is otherwise unconstrained because free ΛΛ\Lambda\Lambda scattering data do not exist; this interaction matters both for fundamental baryon–baryon physics and for the composition of neutron-star cores, where hyperons are expected to appear. The key observable is the ΛΛ\Lambda\Lambda interaction energy ΔBΛΛ(ΛΛA+2Z)=BΛΛ(ΛΛA+2Z)−2BΛ(ΛA+1Z)\Delta B_{\Lambda\Lambda}(^{A+2}_{\Lambda\Lambda}Z) = B_{\Lambda\Lambda}(^{A+2}_{\Lambda\Lambda}Z) - 2B_{\Lambda}(^{A+1}_{\Lambda}Z), where BΛB_{\Lambda} and BΛΛB_{\Lambda\Lambda} are the single- and double-Λ\Lambda separation energies: it isolates the extra binding due to the second Λ\Lambda beyond twice the single-Λ\Lambda binding, i.e. the effective ΛΛ\Lambda\Lambda attraction in the nuclear medium. Only a handful of double-Λ\Lambda events exist, all from nuclear-emulsion experiments: the NAGARA event (ΛΛ6{}^{6}_{\Lambda\Lambda}He, Takahashi et al. 2001; its interaction energy was revised to ΔBΛΛ=0.67±0.17\Delta B_{\Lambda\Lambda}=0.67\pm 0.17 MeV in the E373 analysis of Ahn et al. 2013), which established weak ΛΛ\Lambda\Lambda attraction and remains the sole calibration anchor for the ΛΛ\Lambda\Lambda coupling in most models; an E373 event interpreted as ΛΛ10{}^{10}_{\Lambda\Lambda}Be (Ahn et al. 2013) and a J-PARC E07 event interpreted as ΛΛ11{}^{11}_{\Lambda\Lambda}Be (Ekawa et al. 2019); and a new ΛΛ13{}^{13}_{\Lambda\Lambda}B candidate with an 11{}^{11}B core found by machine-learning-assisted emulsion analysis (He et al. 2025). On the theory side, relativistic density functionals (RDF — covariant mean-field models with meson-mediated interactions, here the relativistic Hartree–Fock functionals PKO1/PKO2 and the density-dependent relativistic mean-field functionals TW99, PKDD, DD-ME2, DD-MEX) describe both sectors uniformly; alternative frameworks include the Skyrme–Hartree–Fock approach (non-relativistic mean field; Zhou et al. 2007) and ab initio few-body/quantum-Monte-Carlo methods rooted in chiral effective field theory (the systematic low-energy expansion of nuclear forces; Tong et al. 2025). Ding and Sun (2026) computed theory–experiment deviations δEB\delta E_B of binding energies in both sectors and found a robust linear correlation, δEB(ΛΛA+2Z)=0.890 δEB(ΛA+1Z)+1.837\delta E_B(^{A+2}_{\Lambda\Lambda}Z) = 0.890\,\delta E_B(^{A+1}_{\Lambda}Z) + 1.837 MeV, from the four light systems with double-Λ\Lambda data (core nuclei 4{}^{4}He, 8{}^{8}Be, 9{}^{9}Be, 11{}^{11}B). Exploiting it, they transferred the abundant empirical single-Λ\Lambda constraints to the data-sparse S=−2S=-2 sector, obtaining evaluated BΛΛB_{\Lambda\Lambda} and ΔBΛΛ\Delta B_{\Lambda\Lambda} with quantified uncertainties for heavier nuclei. The evaluated ΔBΛΛ\Delta B_{\Lambda\Lambda} stay below about 5 MeV, peak in the medium-mass region, and are systematically larger than direct RDF predictions whose ΛΛ\Lambda\Lambda coupling is fitted only to NAGARA; the authors attribute the difference to correlations beyond the mean field — few-body cluster structure, core polarization (the distortion of the core nucleus by the hyperons), coupled-channel ΛΛ\Lambda\Lambda–ΞN\Xi N mixing (the mixing of double-Λ\Lambda states with states containing the Ξ\Xi hyperon, which share strangeness S=−2S=-2), and possible ΛΛ\Lambda\Lambda Cooper pairing — but explicitly do not disentangle these mechanisms. The correlation itself rests on only four data points from light nuclei, and its validity for medium-mass and heavy systems, or within other theoretical frameworks, is assumed but unverified.

Problem Statement

Determine whether the linear correlation between single- and double-Λ\Lambda binding-energy deviations found by Ding and Sun (2026) — δEB(ΛΛA+2Z)=0.890 δEB(ΛA+1Z)+1.837\delta E_B(^{A+2}_{\Lambda\Lambda}Z) = 0.890\,\delta E_B(^{A+1}_{\Lambda}Z) + 1.837 MeV, established within relativistic density functionals from the four light systems with double-Λ\Lambda data — is a robust, framework-independent property of hypernuclear many-body theory, by reproducing, refining, or refuting it within independent theoretical frameworks (e.g. Skyrme–Hartree–Fock or ab initio methods), where a refutation in any single framework is decisive whereas a determination of framework-independence requires concordance across more than one such framework, and characterizing its slope, intercept, statistical significance, and possible mass dependence toward medium-mass and heavy double-Λ\Lambda systems; and identify, quantitatively and in controlled model studies, which of these microscopic mechanisms — few-body clustering, core polarization, coupled-channel ΛΛ\Lambda\Lambda–ΞN\Xi N mixing, or ΛΛ\Lambda\Lambda pairing — dominate the systematic difference between the correlation-evaluated ΔBΛΛ\Delta B_{\Lambda\Lambda} and direct functional predictions calibrated only to the NAGARA event. These two legs serve one objective: turning the correlation-based evaluation into a validated, physically understood predictor of ΛΛ\Lambda\Lambda interaction energies across the nuclear chart, with predictions concrete enough to be confronted with upcoming double-Λ\Lambda measurements at J-PARC (Japan Proton Accelerator Research Complex) and HIAF (China's High Intensity heavy-ion Accelerator Facility) (the measurements themselves are the ultimate empirical test but are not part of what a theoretical answer must supply).

References

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