Strange dibaryon resonance in the $\overline{K}\mathit{NN}$-$\pi YN$ system

Three-body resonances in the $\overline{K}\mathit{NN}$ system have been studied within a framework of the $\overline{K}\mathit{NN}$-$\pi YN$ coupled-channel Faddeev equation. By solving the three-body equation, the energy dependence of the resonant $\overline{K}N$ amplitude is fully taken into account. The $S$-matrix pole has been investigated from the eigenvalue of the kernel with the analytic continuation of the scattering amplitude on the unphysical Riemann sheet. The $\overline{K}N$ interaction is constructed from the leading order term of the chiral Lagrangian using relativistic kinematics. The $\Lambda (1405)$ resonance is dynamically generated in this model, where the $\overline{K}N$ interaction parameters are fitted to the data of scattering length. As a result we find a three-body resonance of the strange dibaryon system with binding energy $B~79$ MeV and width $\Gamma ~74$ MeV. The energy of the three-body resonance is found to be sensitive to the model of the $I=0$ $\overline{K}N$ interaction.

Figure 1

Graphical representation of (a) one particle exchange interaction $Z_{i,j}({\overrightarrow{p}}_i,{\overrightarrow{p}}_j,W)$ and (b) two-body $t$ matrix ${\tau }_i(W)$. The relative momentum of the interacting particles is given by ${\overrightarrow{q}}_i$ for spectator particle $i$.

Figure 2

Singularities of the one particle exchange interaction $Z(p_i,p_n,W)$ in the complex $p_n$ plane at $W=E+i\epsilon$ and the real $p_i$.

Figure 3

Integration contour $C$ and the singularity of $Z$ at $W=E-i\Gamma /2$ and real value of $p_i$.

Figure 4

Logarithmic singularities of the π exchange mechanism $Z(p_i,p_n,Z)$ at $W_{\mathrm{th}}=10-i35$ MeV in the complex $p_n$ plane. $C$ is the integration contour of $p_n$ and $p_i$.

Figure 5

Singularities due to the three-body resonance and the $\Lambda (1405)$ in (a) the complex energy plane and (b) the momentum plane.

Figure 6

Total cross section of (a) $K^{-}p\to K^{-}p$, (b) $K^{-}p\to {\pi }^{+}{\Sigma }^{-},$ (c) $K^{-}p\to {\pi }^{-}{\Sigma }^{+},$ (d) $K^{-}p\to {\pi }^0{\Sigma }^0,$ and (e) $K^{-}p\to {\pi }^0\Lambda$ reactions in the relativistic model. Data are from Refs. [29, 30, 31, 32, 33].

Figure 7

Phase shift of the $\pi N$ scattering for (a) $S_{11}$ and (b) $S_{31}$ partial waves. Data are from Ref. [35].

Figure 8

Phase shifts of $\mathit{NN}$ scattering for the ${}^1{S}_0$ state. The phase shifts calculated from the model of Ref. [36] are shown in triangles.

Figure 9

Pole trajectories of the $\overline{K}\mathit{NN}$-$\pi YN$ scattering amplitude for the $J^{\pi }=0^{-}$ and $I=1/2$ state. Filled circles (filled triangles) show the results of the relativistic (nonrelativistic) model (aA) for different steps in the analysis, as explained in the text. Here $W_{\mathit{KNN}}=m_K+2m_N$.

Figure 10

Pole trajectories of the three-body resonance and $\Lambda (1405)$ using the nonrelativistic model (aA).