$K^{-}$ nuclear states: Binding energies and widths

$K^{-}$ optical potentials relevant to calculations of $K^{-}$ nuclear quasibound states were developed within several chiral meson-baryon coupled-channels interaction models. The applied models yield quite different $K^{-}$ binding energies and widths. Then the $K^{-}$ multinucleon interactions were incorporated by a phenomenological optical potential fitted recently to kaonic atom data. Though the applied $K^{-}$ interaction models differ significantly in the $K^{-}N$ subthreshold region, our self-consistent calculations of kaonic nuclei across the periodic table lead to conclusions valid quite generally. Due to $K^{-}$ multinucleon absorption in the nuclear medium, the calculated widths of $K^{-}$ nuclear states are sizable, ${\mathrm{\Gamma }}_{K^{-}}\ge 90$ MeV, and exceed substantially their binding energies in all considered nuclei.

Figure 1

Energy dependence of real (left) and imaginary (right) parts of free-space $K^{-}p$ (top) and $K^{-}n$ (bottom) amplitudes in considered chiral models (see text for details). Thin vertical lines mark threshold energies.

Figure 2

Energy dependence of real (left) and imaginary (right) parts of WRW modified $K^{-}p$ (top) and $K^{-}n$ (bottom) amplitudes at ${\rho }_0=0.17{\mathrm{fm}}^{-3}$ in considered models. Thin vertical lines mark threshold energies.

Figure 3

Energy dependence of free-space (dotted line) amplitude $f_{K^{-}N}=\frac{1}{2}(f_{K^{-}p}+f_{K^{-}n})$ compared with WRW modified amplitude (solid line), Pauli (dashed line), and Pauli + SE (dot-dashed line) modified amplitude for ${\rho }_0=0.17{\mathrm{fm}}^{-3}$ in the P model (left: real parts, right: imaginary parts). The thin vertical line indicates the $K^{-}N$ threshold.

Figure 4

Subthreshold energies probed in the ${}^{16}\mathrm{O}+K^{-}$ nucleus as a function of relative density $\rho /{\rho }_0$, calculated self-consistently using $K^{-}N$ amplitudes in the P (dot-dashed line), KM (solid line), M1 (dashed line), and M2 (dotted line) models.

Figure 5

$K^{-}$ nuclear potential $V_{K^{-}}^{\left(1\right)}$ in ${}^{40}\mathrm{Ca}$ calculated using chiral $K^{-}N$ P amplitudes at threshold (dashed lines) and with $\sqrt{s}$ (Eq. (8) of Ref. [15]) (solid lines) in two in-medium versions: WRW (left panel) and Pauli+SE (right panel). The Pauli version (right panel, dotted line) for $\sqrt{s}$ from [15] is shown as well (see text for details).

Figure 6

Real (left) and imaginary (right) parts of the $K^{-}$ nuclear potential $V_{K^{-}}^{\left(1\right)}$ in ${}^{40}\mathrm{Ca}$ calculated self-consistently using chiral P (dot-dashed line), KM (solid line), M1 (dashed line), and M2 (dotted line) amplitudes.

Figure 7

1s $K^{-}$ binding energies (left) and corresponding widths (right) in various nuclei calculated self-consistently in P (circles), KM (squares), M1 (diamonds), and M2 (triangles) models. $K^{-}$-multinucleon interactions are not considered.

Figure 8

$K^{-}$ binding energies (left) and widths (right) in $s,p$, and $d$ levels in ${}^{40}\mathrm{Ca}$ calculated self-consistently in P, KM, M1, and M2 models. $K^{-}$-multinucleon interactions are not considered.

Figure 9

Subthreshold energies probed in the ${}^{208}\mathrm{Pb}+K^{-}$ nucleus as a function of relative density $\rho /{\rho }_0$, calculated self-consistently in all $K^{-}N$ amplitude models considered, supplemented by the FD variant of $V_{K^{-}}^{\left(2\right)}$. The dashed and dotted areas denote the uncertainty bands calculated in the KM1 and KM2 models and the shaded gray band represents their overlap.

Figure 10

The respective contributions from $K^{-}N$ (dashed dotted line) and $K^{-}NN$ (dashed line) potentials to the total real (left) and imaginary (right) $K^{-}$ optical potential in the ${}^{208}\mathrm{Pb}+K^{-}$ nucleus, calculated self-consistently in the FD version of KM1 (top) and KM2 (bottom) models. The shaded areas denote the uncertainty bands. The $K^{-}$ single-nucleon potential (KN, blue solid line) calculated in the KM model (i.e., without multinucleon interactions) is shown for comparison.

Figure 11

The ratio of $\mathrm{Im}{V}_{K^{-}}^{\left(1\right)}$ (dashed line) and $\mathrm{Im}{V}_{K^{-}}^{\left(2\right)}$ (solid line) potentials to the total $K^{-}$ imaginary potential $\mathrm{Im}{V}_{K^{-}}$ as a function of radius, calculated self-consistently for the $1s{K}^{-}$ state in ${}^{208}\mathrm{Pb}$ using the KM1 model and different options for the $K^{-}$ multinucleon potential. The relative nuclear density $\rho /{\rho }_0$ (dotted line) is shown for comparison.