Faddeev calculations of the $\overline{K}NN$ system with a chirally motivated $\overline{K}N$ interaction. I. Low-energy $K^{-}d$ scattering and antikaonic deuterium

A chirally-motivated coupled-channel $\overline{K}N$ potential, reproducing all low-energy experimental data on $K^{-}p$ scattering and kaonic hydrogen and suitable for using in accurate few-body calculations, was constructed. The potential was used for calculations of low-energy amplitudes of elastic $K^{-}d$ scattering using Faddeev-type Alt-Grassberger-Sandhas (AGS) equations with coupled $\overline{K}NN$ and $\pi \Sigma N$ channels. A complex $K^{-}-d$ potential reproducing the three-body $K^{-}d$ amplitudes was constructed and used for calculation of the $1s$ level shift and the width of kaonic deuterium. The predicted shift $\Delta E_{1s}^{K^{-}d}\sim -830$ eV and width ${\Gamma }_{1s}^{K^{-}d}\approx 1055$ eV are close to our previous results obtained with phenomenological $\overline{K}N$ potentials. No quasi-bound states in the $K^{-}d$ system were found.

Figure 1

Comparison of the elastic and inelastic $K^{-}p$ cross sections for the chirally-motivated potential $V_{\overline{K}N\text{−}\pi \Sigma \text{−}\pi \Lambda }^{\mathrm{Chiral}}$ (solid lines) with the one-pole $V_{\overline{K}N\text{−}\pi \Sigma }^{1,\mathrm{SIDD}}$ (dash-dotted lines) and two-pole $V_{\overline{K}N\text{−}\pi \Sigma }^{1,\mathrm{SIDD}}$ (dotted lines) phenomenological potentials from [2]. The experimental data are taken from [18, 19, 20, 21, 22] (data points).

Figure 2

Trajectories of the strong poles when the coupling between the $\overline{K}N$, $\pi \Sigma$, and $\pi \Lambda$ channels is gradually being switched off (empty symbols). The filled symbols denote the original values with coupled channels.

Figure 3

The same as Fig. 2 for the $1s$ level of kaonic hydrogen.

Figure 4

Elastic ${\pi }^0{\Sigma }^0$ cross sections of the chirally-motivated potential $V_{\overline{K}N\text{−}\pi \Sigma \text{−}\pi \Lambda }^{\mathrm{Chiral}}$. The Particle Data Group value for the mass of the $\Lambda \left(1405\right)$ resonance and the $K^{-}p$ threshold are also shown.

Figure 5

The elastic near-threshold $K^{-}d$ amplitudes presented in the form of a $kcot\delta \left(k\right)$ function. Real (filled circles) and imaginary (empty circles) parts of the functions obtained from the coupled-channel three-body AGS equations using the chirally motivated potential $V_{\overline{K}N\text{−}\pi \Sigma \text{−}\pi \Lambda }^{\mathrm{Chiral}}$ are shown. The two-body results obtained with the complex $K^{-}-d$ potential are demonstrated as well (by a solid line for the real part and a dotted line for the imaginary part of the function).

Figure 6

Pole trajectories in the $K^{-}d$ system for increasing absolute value of the ${\lambda }_{I=0}^{\overline{K}\overline{K}}$ strength constant of the phenomenological $V_{\overline{K}N\text{−}\pi \Sigma }^{1,\mathrm{SIDD}}$ and $V_{\overline{K}N\text{−}\pi \Sigma }^{2,\mathrm{SIDD}}$ potentials. The numbers along the trajectories indicate the multiplication factors. The position of the $K^{-}d$ threshold is also shown; quasi-bound states are the poles to the left of it.