$\Lambda(1405)$ production in the process $\chi_{c0}(1P)\to \bar{\Lambda}\Sigma\pi$

We have performed a theoretical study on the process $\chi_{c0}(1P)\to \bar{\Lambda}\Sigma\pi$, by taking into account the final state interactions of $\pi\Sigma$ and $\pi\bar{\Lambda}$ based on the chiral unitary approach. As the isospin filters of $I=0$ in the $\pi\Sigma$ channel and $I=1$ in the $\pi\bar{\Lambda}$ channel, this process can be used to study the molecular structure of the $\Lambda(1405)$ resonance, and to test the existence of the predicted states $\Sigma(1380)$ and $\Sigma(1430)$ with spin-parity $J^P=1/2^-$. Our results show that there is a peak around $1350 \sim 1400$~MeV, and a cusp around the $\bar{K}N$ threshold in the $\pi\Sigma$ invariant mass distribution, which should be the important feature of the molecular state $\Lambda(1405)$. We also find a peak around $1380$~MeV, and a cusp around $\bar{K}N$ threshold in the $\pi\bar{\Lambda}$ invariant mass distribution, which are associated to the $\Sigma(1380)$ and $\Sigma(1430)$ resonances.