Analyzing $\Xi(1620)$ in the molecule picture in the Bethe-Salpeter equation approach

In this work, we assume that the observed state \(\Xi (1620)\) is a s-wave \(\Lambda {\bar{K}}\) or \(\Sigma {\bar{K}}\) bound state. Based on this molecule picture, we establish the Bethe–Salpeter equations for \(\Xi (1620)\) in the ladder and instantaneous approximations. We solve the Bethe–Salpeter equations for the \(\Lambda {\bar{K}}\) and \(\Sigma {\bar{K}}\) systems numerically and find that the \(\Xi (1620)\) can be explained as \(\Lambda {\bar{K}}\) and \(\Sigma {\bar{K}}\) bound states with \(J^P=1/2^-\), respectively. Then we calculate the decay widths of \(\Xi (1620)\rightarrow \Xi \pi \) in these two different molecule pictures systems, respectively.