Variational calculations for $\overline{K}$–few-nucleon systems

Deeply bound $\overline{K}\mathit{NN}$, $\overline{K}\mathit{NNN}$, and $\overline{K}\mathit{NNNN}$ states are discussed. The effective force exerted by the $\overline{K}$ meson on the nucleons is calculated with static nucleons. Next the binding energies are obtained by solving the Schrödinger equation or by variational calculations. The dominant attraction comes from the $S$-wave $\Lambda (1405)$ and an additional contribution is due to $\Sigma (1385)$. The latter state is formed at the nuclear peripheries and absorbs a sizable piece of the binding energy. It also generates new branches of quasibound states. The lowest binding energies based on a phenomenological $\overline{K}\mathit{N}$ input fall into the 40- to 80-MeV range for $\overline{K}\mathit{NN}$, 90–150 MeV for $\overline{K}\mathit{NNN}$, and 120–220 MeV for $\overline{K}\alpha$ systems. The uncertainties are due to unknown $\overline{K}\mathit{N}$ interactions in the distant subthreshold energy region.

Figure 1

Binding energies $E_B$(r) of a $\overline{K}+\mathit{NN}$ pair fixed at a distance $r$ apart in the $I_{\mathit{NN}}=1$, $I_{\mathit{KNN}}=1/2$ state with the symmetric meson wave function. The $E_{B,S}$ line shows the binding for the $S$ wave $\overline{K}\mathit{N}$ interactions. The $E_{B,\mathit{SP}}$ one follows from the $S+P$-wave interactions. The upper curve $-\Gamma$ shows 2 $\mathrm{Im}E(r)$ for the $S$-wave case. These results are based on the A. Martin amplitudes [20].