Chiral coupled-channel models of the KˉN\bar K N interaction, combined with a phenomenological multi-nucleon absorption term fitted to kaonic atoms, predict that K−K^- nuclear quasi-bound states in medium and heavy nuclei would have widths of order 100 MeV or would not bind at all — but this verdict rests on extrapolating the fitted multi-nucleon term into the nuclear interior (ρ≳0.5ρ0\rho\gtrsim0.5\rho_0), where no data constrain it. The open problem is to derive the multi-nucleon K−K^- absorption (K−NNK^-NN and higher) microscopically from chiral meson-baryon dynamics with validated in-medium corrections, demonstrate its consistency with kaonic-atom shifts/widths and measured absorption branching ratios, and apply it in self-consistent calculations to determine the widths — and hence the observability — of K−K^- nuclear quasi-bound states. Microscopic K−NNK^-NN absorption models now describe kaonic atoms well (2020-2025) but have not yet been applied to quasi-bound states, and the interior density dependence remains experimentally unvalidated, so the source question is open.

Background

An antikaon K−K^- is the negatively charged strange meson; at low energies the K−NK^-N interaction in the isospin-0 channel is strongly attractive because it dynamically generates the Λ(1405)\Lambda(1405) resonance, a quasi-bound KˉN\bar KN-dominated state just below the K−pK^-p threshold. This attraction suggested that a K−K^- could bind inside a nucleus, forming a kaonic nuclear quasi-bound state characterized by a binding energy BK−B_{K^-} and a width ΓK−\Gamma_{K^-} that is dominated by strong absorption of the kaon inside the medium.

Two experimental anchors constrain the K−K^--nucleus interaction. First, kaonic atoms — exotic atoms in which a K−K^- replaces an electron — show strong-interaction shifts and widths of the lowest atomic levels, which are described by a complex optical potential VK−=Re VK−+i Im VK−V_{K^-}=\mathrm{Re}\,V_{K^-}+i\,\mathrm{Im}\,V_{K^-}; the imaginary part encodes K−K^- absorption. Global fits show these atomic data probe the potential only up to about 50% of nuclear saturation density ρ0≈0.17\rho_0\approx0.17 fm−3^{-3}, because the atomic wavefunction stays outside the nuclear interior. Second, old bubble-chamber data and modern measurements by the AMADEUS experiment at the DAΦ\PhiNE electron–positron collider (Frascati) give branching ratios of K−K^- absorption at rest on one nucleon (K−N→πYK^-N\to\pi Y, with Y=Λ,ΣY=\Lambda,\Sigma) versus two and more nucleons (non-mesonic absorption K−NN→YNK^-NN\to YN).

Hrtánková and Mareš (2017, 2018) performed self-consistent calculations of K−K^- quasi-bound states, solving the Klein-Gordon equation with an optical potential built from in-medium K−NK^-N scattering amplitudes of chiral SU(3) coupled-channel models — chiral meson-baryon interaction models in which the KˉN\bar KN, πΣ\pi\Sigma and πΛ\pi\Lambda channels are coupled and the near-threshold amplitudes are strongly energy dependent; they used the 'KM' model of Ikeda, Hyodo and Weise and the 'P' model of Cieplý and Smejkal, the only two amplitudes that reproduce both kaonic-atom data and the measured absorption fractions. To this single-nucleon potential they added a phenomenological multi-nucleon term VK−(2)∝B(ρ/ρ0)αρV^{(2)}_{K^-}\propto B(\rho/\rho_0)^\alpha\rho whose parameters (B,α)(B,\alpha) had been fitted by Friedman and Gal to kaonic atoms. Because the atom data probe only ρ≲0.5ρ0\rho\lesssim0.5\rho_0, they bracketed the interior extrapolation with a 'half-density' prescription (HD: V(2)V^{(2)} frozen at its 0.5ρ00.5\rho_0 value inside) and a 'full-density' prescription (FD: the formula applied at all densities). Their result: the multi-nucleon absorption drastically enlarges the widths — HD gives ΓK−∼100\Gamma_{K^-}\sim100 MeV, generally much larger than the binding energies, and FD removes binding in most nuclei altogether — so identification of such states in experiment 'seems highly unlikely'. The width prediction, not the binding energy, is what controls observability, and it is precisely the quantity left undetermined by the uncontrolled interior extrapolation.

Since then, the adjacent few-body sector has advanced: the J-PARC E15 experiment (at the J-PARC proton accelerator facility in Japan) observed a KˉNN\bar KN N bound state in the 3He(K−,Λp)n{}^3\mathrm{He}(K^-,\Lambda p)n reaction with B≈42B\approx42 MeV and Γ≈100\Gamma\approx100 MeV, confirming that the lightest kaonic nuclear system exists but is very broad; and the SIDDHARTA-2 experiment (an X-ray spectrometer at the DAΦ\PhiNE collider) has measured kaonic deuterium for the first time (2026), which together with kaonic hydrogen pins the isospin-dependent K−NK^-N scattering lengths at threshold. On the theory side, a microscopic model of K−NNK^-NN absorption in nuclear matter derived from chiral amplitudes has been developed (Hrtánková and Ramos, 2020) and applied to kaonic atoms (Óbertová et al., 2022, 2025), where it works well but still requires a residual phenomenological multi-nucleon component. What has not been done is to carry this microscopic absorption dynamics into the nuclear interior and through self-consistent quasi-bound-state calculations — the step needed to settle whether any K−K^- nuclear states beyond the lightest few-body system are observable.

Problem Statement

Determine the multi-nucleon absorption of K−K^- mesons in the nuclear interior from the underlying dynamics, and thereby decide whether K−K^- nuclear quasi-bound states are observable. Concretely: (1) derive the K−NNK^-NN (and, where relevant, K−3NK^-3N) absorption contribution to the K−K^--nucleus optical potential microscopically — from chiral coupled-channel KˉN\bar KN amplitudes with controlled in-medium modifications (Pauli correlations, hadron self-energies, meson exchange between the nucleon pair) — including its real and imaginary parts as functions of density ρ\rho up to and above ρ0\rho_0, replacing the phenomenological ansatz VK−(2)∝B(ρ/ρ0)αρV^{(2)}_{K^-}\propto B(\rho/\rho_0)^\alpha\rho whose interior behavior is only bracketed by the half-density and full-density prescriptions; (2) demonstrate that the resulting total potential reproduces the measured kaonic-atom shifts and widths across the periodic table and the measured one- vs multi-nucleon absorption branching ratios, the interior density dependence — which these surface-weighted data alone do not determine — being thereby fixed by the microscopic derivation rather than by fitted extrapolation; and (3) use it in self-consistent calculations of K−K^- quasi-bound states in nuclei across the periodic table to determine their binding energies and, decisively, their widths, settling whether any states beyond the light few-body KˉNN\bar KN N system have widths small enough relative to binding energies to be identifiable as resonances in experiments such as in-flight (K−,N)(K^-,N) spectroscopy. These three components jointly serve one objective: replacing the uncontrolled interior extrapolation of multi-nucleon absorption with a microscopic, data-validated determination, which is the missing link in the source papers' conclusion that K−K^- nuclear widths are 'excessively large to allow observation'.

References

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