Virtual State and Two-Pole Structure: Ξ (1690) and Ξ (1820) from BESIII data on ψ (3686) \to K^\,-Λ \barΞ⁺

We investigate the nature of the Ξ (1690) and Ξ (1820) resonances by analyzing the recent BESIII data on ψ (3686) K^\,\bf -Λ \barΞ⁺ within a coupled-channel rescattering framework. The calculation includes the channels \barKΛ, \barKΣ ^(*), πΞ ^(*), ηΞ ^(*), and KΩ. A direct fit to the data reveals that the Ξ(1690) is best described as a virtual state located right below the \barKΣ threshold, a feature that naturally explains its very sharp peak observed in the K^\,-Λ invariant mass spectrum. In contrast, the Ξ(1820) exhibits a two-pole structure originating mainly from the coupled contributions of the KΣ ^* and πΞ ^* channels, showing a two-pole pattern similar to that of the well-known Λ(1405) and accounting for the unexpectedly broad width measured by BESIII. These findings provide new insights into the dynamical generation of hyperon resonances and highlight the importance of high-precision data from charmonium decays.