Precision Λ-separation energies of light hypernuclei (A ≤ 16): settling the hypertriton binding energy and mapping charge-symmetry breaking of the ΛN interaction

The Λ separation energies $B_\Lambda$ of the lightest hypernuclei are the primary empirical anchors of the hyperon–nucleon interaction, yet they are not known at the accuracy needed to map charge-symmetry breaking (CSB) of the $\Lambda N$ interaction. Botta, Bressani and Feliciello (2017) showed that the historical emulsion data carry unquantified nucleus-dependent systematics of order 100 keV, that the KEK SKS spectrometer scale required a +600±80 keV renormalization, and that key values — above all $B_\Lambda({}^3_\Lambda\mathrm{H})$ and $B_\Lambda({}^4_\Lambda\mathrm{He})$ — lacked modern, absolutely calibrated determinations. A decade of subsequent work has sharpened rather than settled the problem: STAR (2020), ALICE (2023), MAMI decay-pion spectroscopy (2026), and a J-PARC $(K^-,\pi^0)$ evaluation (2026) now disagree on $B_\Lambda({}^3_\Lambda\mathrm{H})$ at the level of 0.1–0.5 MeV, with the MAMI line assignment itself publicly contested; $B_\Lambda({}^4_\Lambda\mathrm{He})$ still has no direct counter measurement; and the absolutely calibrated ≲100 keV data set needed to compare the p-shell CSB pattern with modern ab initio predictions does not yet exist. The problem is to deliver and validate this data set.

Background

A hypernucleus is a bound nuclear system in which one or more nucleons are replaced by a hyperon, a baryon containing a strange quark — most commonly the $\Lambda$, which carries isospin zero. The $\Lambda$ separation energy $B_\Lambda$ is the energy required to remove the $\Lambda$ from the hypernuclear ground state; because direct $\Lambda p$ scattering data are scarce and no $\Lambda n$ scattering measurement exists, hypernuclear $B_\Lambda$ values are the main empirical constraints on the hyperon–nucleon ($YN$) interaction. Charge symmetry is the invariance of the strong interaction under interchange of proton and neutron; its breaking (CSB) can be isolated in $\Lambda$ hypernuclei by comparing $B_\Lambda$ within isomultiplets of mirror hypernuclei, since the $\Lambda$ itself contributes no isospin. The archetype is the $A=4$ isodoublet ${}^4_\Lambda\mathrm{He}$–${}^4_\Lambda\mathrm{H}$, for which emulsion data gave a ground-state splitting $\Delta B_\Lambda = B_\Lambda({}^4_\Lambda\mathrm{He}) - B_\Lambda({}^4_\Lambda\mathrm{H}) = 0.35\pm0.06$ MeV, an anomalously large CSB that has resisted first-principles explanation and serves as the calibration point for CSB terms in hyperon–nucleon potentials. Until the 2000s essentially all $B_\Lambda$ values came from nuclear-emulsion experiments of the 1960s–70s, with percent-level statistics and never fully quantified systematics. Modern magnetic spectrometers measured $B_\Lambda$ for p-shell hypernuclei by missing-mass spectroscopy, but the KEK Superconducting Kaon Spectrometer (SKS) normalized its energy scale to the emulsion value of $B_\Lambda({}^{12}_\Lambda\mathrm{C})$, so those results were not absolutely calibrated. Botta, Bressani and Feliciello reorganized the data from FINUDA (at the DA$\Phi$NE collider) and from electroproduction experiments at JLab and MAMI — the only magnetic-spectrometer data with an absolute energy-scale calibration — inferred that the SKS scale needed a $+600\pm80$ keV offset, and showed that no common correction reconciles emulsion with spectrometer data (reduced $\chi^2$ of 2.8 and 3.8 when a single common emulsion systematic is fitted), implying nucleus-dependent emulsion systematics of order 100 keV whose origin and magnitude remain unquantified. From the absolutely calibrated data they found the $A=7$, 12 and 16 isomultiplet splittings consistent with zero within 100–200 keV — CSB far smaller than in the $A=4$ system, in agreement with a prediction by Gal that p-shell CSB should be small. They also identified the missing pieces: $B_\Lambda({}^4_\Lambda\mathrm{He})$ rested only on emulsion data, the hypertriton value $B_\Lambda({}^3_\Lambda\mathrm{H}) = 0.13\pm0.05\pm0.04$ MeV came only from emulsions, and achieving ~100 keV absolute accuracy for meson-produced hypernuclei would require a new spectrometer generation. In the same period a new technique was established at the Mainz Microtron (MAMI): hypernuclear decay-pion spectroscopy, which measures the monochromatic momentum of pions from the two-body weak decay of hyperfragments stopped in an electroproduction target, and which delivered the first high-precision counter value of $B_\Lambda({}^4_\Lambda\mathrm{H})$.

Problem Statement

Provide absolutely calibrated, systematically controlled determinations of the Λ separation energies $B_\Lambda$ of the light ($A\le16$) hypernuclei, with total uncertainties at the $\lesssim100$ keV level, sufficient to fix the charge-symmetry-breaking pattern of the $\Lambda N$ interaction across the s and p shells. Concretely, the required data set is currently missing the following components: (i) a settled value of $B_\Lambda({}^3_\Lambda\mathrm{H})$ — present determinations disagree, spanning roughly 0.10 to 0.52 MeV among the emulsion average, heavy-ion invariant-mass measurements (STAR, ALICE), MAMI A1 decay-pion spectroscopy, and the J-PARC $(K^-,\pi^0)$ evaluation, with the MAMI line assignment itself contested and acknowledged by its authors to need higher-statistics dedicated experiments; (ii) a modern direct determination of $B_\Lambda({}^4_\Lambda\mathrm{He})$, whose accepted value still rests on emulsion data alone, complementing the precision value already achieved for the mirror ${}^4_\Lambda\mathrm{H}$; (iii) a quantification of the nucleus-dependent systematic errors of the historical emulsion $B_\Lambda$ values (known to be of order 100 keV and demonstrably non-uniform across nuclei), so that emulsion and spectrometric data can be combined coherently; and (iv) absolutely calibrated $B_\Lambda$ values for the p-shell isomultiplet members — including the still-unmeasured proton-rich partners such as ${}^7_\Lambda\mathrm{Be}$ — with uncertainties small enough that the p-shell CSB splittings $\Delta B_\Lambda(A,Z)$ can be compared with modern ab initio predictions at the level of the predicted effects (tens of keV). The single overarching objective is a mutually consistent, absolutely calibrated $B_\Lambda$ data set for $A\le16$ hypernuclei at the ~100 keV level; the listed items are precisely the components of that data set that are missing today. Resolution consists in producing this data set as the best mutually consistent, calibration-corrected $B_\Lambda$ evaluation attainable from the existing published measurements — each value carrying a stated uncertainty and its dominant systematic, and the combined set carrying a stated residual systematic floor against the $\lesssim100$ keV target — so that a component lacking an independent calibrated measurement is delivered at the precision the available data support.

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