Faddeev calculations of the $\overline{K}NN$ system with a chirally motivated $\overline{K}N$ interaction. II. The $K{}^{-}pp$ quasibound state

New calculations of the quasibound state in the $K{}^{-}pp$ system using Faddeev-type equations in Alt-Grassberger-Sandhas (AGS) form with coupled $\overline{K}NN$ and $\pi \Sigma N$ channels were performed. A chirally motivated $\overline{K}N$ potential together with phenomenological models of $\overline{K}N$ interaction with one- and two-pole structure of the $\Lambda \left(1405\right)$ resonance were used. All three potentials reproduce experimental data on low-energy $K{}^{-}p$ scattering and kaonic hydrogen with the same level of accuracy. A new method of calculating the subthreshold resonance position and width in a three-body system was proposed and used together with the direct search of the resonance pole. We obtained binding energy of the $K{}^{-}pp$ quasibound state $\approx 32$ MeV with the chirally motivated and $47–54$ MeV with the phenomenological $\overline{K}N$ potentials. The width is about 50 MeV for the two-pole models of the interaction, while the one-pole potential gives $\approx 65$ MeV width. The question of using an energy-dependent potential in few-body calculations is discussed in detail. It is shown that “self-consistent” variational calculations using such a potential are unable to produce a reasonable approximation to the exact result.

Figure 1

Function $1/{\left|\mathrm{Det}\left(z\right)\right|}^2$ calculated with the one-pole $V_{\overline{K}N\text{−}\pi \Sigma }^{1,\mathrm{SIDD}}$ (black triangles), two-pole $V_{\overline{K}N\text{−}\pi \Sigma }^{2,\mathrm{SIDD}}$ (blue [gray] circles) phenomenological potentials, and the chirally motivated $V_{\overline{K}N\text{−}\pi \Sigma \text{−}\pi \Lambda }^{\mathrm{Chiral}}$ potential (red [gray] squares). Breit-Wigner fits for all three functions are plotted by the lines of the corresponding color.

Figure 2

Series of the quasibound system pole positions in the $K{}^{-}pp$ system calculated with fixed energy $z_{\overline{K}N}$ of the chirally motivated potential $V_{\overline{K}N\text{−}\pi \Sigma \text{−}\pi \Lambda }^{\mathrm{Chiral}}$. Each line contains a set of results obtained with the same $\mathrm{Im}{z}_{\overline{K}N}$ and $|\mathrm{Re}{z}_{\overline{K}N}|$ changing up to 100 MeV (numbers near the lines). The lines with $\mathrm{Im}{z}_{\overline{K}N}=-20$ MeV (black upward-pointing triangles), $\mathrm{Im}{z}_{\overline{K}N}=-10$ MeV (red [gray] diamonds), $\mathrm{Im}{z}_{\overline{K}N}=0$ MeV (blue [gray] squares), and $\mathrm{Im}{z}_{\overline{K}N}=+10$ MeV (green [gray] downward-pointing triangles) are shown. The exact result of Faddeev calculation with coupled channels (black circle) and two results of variational calculations (crosses), BGL [10] and type I result with HNJH $\overline{K}N$ potential DHW [9], are also shown. Due to the substantial difference between the potentials used in Refs. [9, 10] and ours, one should be careful in comparing our numbers with those of Refs. [9, 10]. However, since all three calculations are aimed at getting numbers for the same physical observables, their comparison might be informative.