The A = 4 mirror hypernuclei ⁴ΛH and ⁴ΛHe exhibit the largest known charge-symmetry breaking (CSB) in the hyperon–nucleon interaction: their Λ binding-energy differences, ΔBΛ(0+)\Delta B_\Lambda(0^+) in the ground state and ΔBΛ(1+)\Delta B_\Lambda(1^+) in the excited state, lie at the 150–350 keV level, several times the nuclear (NN) CSB scale and far above the ≈70 keV produced by standard microscopic calculations. γ-ray spectroscopy at J-PARC established that this CSB is strongly spin dependent, and recent precision measurements at RHIC revised the empirical pattern to comparable magnitudes with opposite signs, ΔBΛ(0+)≈+0.15\Delta B_\Lambda(0^+)\approx+0.15 MeV and ΔBΛ(1+)≈−0.17\Delta B_\Lambda(1^+)\approx-0.17 MeV. The open problem is the dynamical origin of this spin-dependent CSB: the ab initio no-core-shell-model calculation of Gazda and Gal (2016) with a Λ–Σ⁰ mixing vertex whose amplitude is fixed by baryon mass splittings — independently of the A = 4 data — does describe the revised pattern quantitatively at a chosen momentum cutoff (STAR 2026), but with a strong cutoff dependence (ΔBΛ(0+)\Delta B_\Lambda(0^+) ranging over roughly 30–300 keV across cutoffs 550–700 MeV), while its updated Dalitz–von-Hippel one-pion-exchange variant underestimates ΔBΛ(1+)\Delta B_\Lambda(1^+); no cutoff-robust, independently anchored microscopic derivation of both channels currently exists. Λ–Σ⁰ mixing (the Dalitz–von Hippel mechanism, supported by lattice QCD+QED — the strong and electromagnetic interactions simulated numerically on a discrete space-time grid) is the leading candidate, and chiral effective-field-theory (chiral-EFT) fits to the A = 4 differences yield untested predictions such as the Λn scattering lengths. Resolving the problem requires a quantitative microscopic account of the sign, magnitude, and spin dependence of the ΛN CSB; the precision remeasurements raised by the source work and needed to test it (the ⁴ΛH 1⁺ spacing, keV-scale ground-state binding energies) form the experimental side of the same research program.

Background

Charge symmetry is the invariance of the strong interaction under interchange of proton and neutron; it is broken mildly by the up–down quark mass difference and by electromagnetism, as seen for example in the binding-energy difference of the mirror nuclei ³H and ³He (about 67 keV after Coulomb corrections) and in the difference of the neutron–neutron and proton–proton scattering lengths. A hypernucleus is a nucleus in which one nucleon is replaced by a hyperon, i.e. a baryon containing a strange quark. The lightest and most studied hyperon is the Λ (quark content uds, isospin 0, electrically neutral), so under charge symmetry the ΛN force acts identically in mirror hypernuclei — pairs related by p↔n such as ⁴ΛH (a Λ bound to a triton core) and ⁴ΛHe (a Λ bound to ³He). Their Λ separation energies BΛB_\Lambda, the energy needed to remove the Λ, should therefore be equal up to small, calculable Coulomb and nuclear-mass effects, and any residual difference ΔBΛ=BΛ(Λ4He)−BΛ(Λ4H)\Delta B_\Lambda = B_\Lambda(^4_\Lambda\mathrm{He}) - B_\Lambda(^4_\Lambda\mathrm{H}) measures charge-symmetry breaking (CSB) in the ΛN interaction. Each of these A = 4 systems has a spin-parity 0+0^+ ground state and a 1+1^+ excited state, forming a spin doublet whose two members probe different spin states of the ΛN pair.

Emulsion experiments — hypernuclear production and decay recorded as microscopic tracks in photographic emulsion — found in the early 1970s ΔBΛ(0+)=0.35±0.05\Delta B_\Lambda(0^+) = 0.35\pm0.05 MeV (Jurić et al. 1973), anomalously large — several times the NN CSB scale. The classic candidate mechanism, due to Dalitz and von Hippel (1964), is Λ–Σ⁰ mixing: the physical Λ acquires a small isospin-1 admixture (amplitude about 1.5%, tied to the electromagnetic Λ–Σ⁰ mass-mixing matrix element), which permits a charge-asymmetric one-pion-exchange contribution to the ΛN force. Yet quantitative four-body Faddeev–Yakubovsky calculations (an exact few-body method for the hyperon–three-nucleon YNNN system) with coupled ΛN–ΣN channels — the strong interaction mixes the two-baryon ΛN and ΣN states — built on the Nijmegen NSC97e/f soft-core hyperon–nucleon interaction models gave at most about 70 keV (Nogga, Kamada and Glöckle 2002), and no microscopic calculation reached the emulsion value — a gap that stood for decades. γ-ray spectroscopy sharpened the problem: the 1+→0+1^+\to0^+ transition energy of ⁴ΛH averages 1.09±0.021.09\pm0.02 MeV from 1970s measurements (Bedjedian et al.), while the J-PARC E13 experiment (at the Japan Proton Accelerator Research Complex hadron hall) measured the ⁴ΛHe spacing to be 1406±2±21406\pm2\pm2 keV (Yamamoto et al. 2015). Combined with the emulsion ground-state energies, this implied ΔBΛ(1+)≈0.03\Delta B_\Lambda(1^+)\approx0.03 MeV — an order of magnitude below ΔBΛ(0+)\Delta B_\Lambda(0^+) — establishing that the ΛN CSB is strongly spin dependent and leaving open what mechanism produces a large effect in one spin channel together with a much smaller one in the other. The empirical input has itself moved since: decay-pion spectroscopy at MAMI, the Mainz electron accelerator (measuring the monoenergetic pions from the two-body weak decay of stopped hyperfragments; Esser et al. 2015, Schulz et al. 2016) revised BΛ(Λ4H)B_\Lambda(^4_\Lambda\mathrm{H}) upward, and heavy-ion measurements at RHIC, the Relativistic Heavy Ion Collider at Brookhaven (STAR 2022, 2026), now give ΔBΛ(0+)≈+0.15\Delta B_\Lambda(0^+)\approx+0.15 MeV and ΔBΛ(1+)≈−0.17\Delta B_\Lambda(1^+)\approx-0.17 MeV — comparable magnitudes with opposite signs — so any mechanism must in fact explain an approximately antisymmetric pattern across the two spin channels.

Problem Statement

What is the dynamical origin of the spin-dependent charge-symmetry breaking in the ΛN interaction, as quantified by the Λ binding-energy differences of the A = 4 mirror hypernuclei, ΔBΛ(0+)=BΛ(Λ4He;0+)−BΛ(Λ4H;0+)\Delta B_\Lambda(0^+) = B_\Lambda(^4_\Lambda\mathrm{He};0^+) - B_\Lambda(^4_\Lambda\mathrm{H};0^+) and ΔBΛ(1+)\Delta B_\Lambda(1^+), and can it be derived microscopically rather than fitted? A complete answer must: (i) identify the mechanism — for example Λ–Σ⁰ mixing of the Dalitz–von Hippel type, charge-symmetry-breaking contact terms in SU(3) chiral effective field theory (the low-energy theory of the baryon octet organized by the approximate up–down–strange flavor symmetry), or another source; (ii) demonstrate it within a microscopic calculation of the A = 4 mirror hypernuclei — for example, an ab initio four-body calculation with coupled-channel ΛN–ΣN interactions (the four-body bound-state equation solved directly from the input baryon–baryon forces) — that reproduces both the ground-state and excited-state differences, in sign and magnitude, at their current values ΔBΛ(0+)≈+0.15\Delta B_\Lambda(0^+)\approx+0.15 MeV and ΔBΛ(1+)≈−0.17\Delta B_\Lambda(1^+)\approx-0.17 MeV, robustly rather than only at a single tuned momentum cutoff, and with the charge-symmetry-breaking content of the interaction anchored independently of those A = 4 differences rather than fitted to them; and (iii) expose the account to test, for example through predictions of Λn scattering observables. These components serve one overarching objective — a quantitative microscopic determination of the spin-dependent ΛN charge-symmetry breaking — and the companion experimental task raised by the source work, independent high-precision determinations of the A = 4 binding energies (in particular the ⁴ΛH 1+1^+ spacing, still based on 1970s γ-ray data, and keV-scale ground-state BΛB_\Lambda values), forms the experimental side of the same research program rather than a deliverable of a theoretical answer, because it fixes the values any mechanism must reproduce and provides the discriminating tests.

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