Analysis of the $\Xi \left(1620\right)$ resonance and $\overline{K}\Lambda$ scattering length with a chiral unitary approach

We study the $\Xi \left(1620\right)$ resonance near the $\overline{K}\Lambda$ threshold in the light of recent experimental constraints. The Belle Collaboration have found a resonance peak of $\Xi \left(1620\right)$ slightly below the ${\overline{K}}^0\Lambda$ threshold in the ${\pi }^{+}{\Xi }^{-}$ invariant mass spectrum, and the ALICE Collaboration have determined the $K^{-}\Lambda$ scattering length from the measurement of the momentum correlation functions in heavy ion collisions. Using the effective range expansion, we classify the nature of the pole of the near-threshold eigenstate in terms of the scattering length, in the presence of the decay channel. It is shown that the quasibound state below the threshold can be described by only the scattering length, while the description of the resonance above the threshold requires the contribution from the effective range. Based on the chiral unitary approach, we construct a theoretical model which generates the pole of $\Xi \left(1620\right)$ below the $\overline{K}\Lambda$ threshold with relatively narrow width, as reported by the Belle Collaboration. It is quantitatively demonstrated that the spectrum of the $\Xi \left(1620\right)$ quasibound state is distorted by the effect of the nearby $\overline{K}\Lambda$ threshold. We then construct another model which reproduces the $K^{-}\Lambda$ scattering length by the ALICE Collaboration. In this case, the eigenstate pole does not appear in the physically relevant Riemann sheet, and the spectrum shows a cusp structure at the $\overline{K}\Lambda$ threshold. We finally examine the compatibility of the value of the $\overline{K}\Lambda$ scattering length and the subthreshold pole of $\Xi \left(1620\right)$ including the experimental uncertainties.