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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 theisospin 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 molecularstructure of the $\Lambda(1405)$ resonance, and to test the existence of thepredicted 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 massdistribution, 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.

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