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.