$\Sigma^*_{1/2^-}(1380)$ in the $\Lambda^+_c \to \eta \pi^+ \Lambda$ decay

A $\Sigma^*$ state with spin-parity $J^P = 1/2^-$ with mass and width around $1380$ MeV and $120$ MeV, refereed to as the $\Sigma^*_{1/2^-}(1380)$, has been predicted in several pentaquark models and inferred from the analysis of CLAS $\gamma p$ data. In the present work, we discuss how one can employ the $\Lambda^+_c \to \eta \pi^+ \Lambda$ decay to test its existence, as well as to study the $\Sigma^*(1385)$ state with $J^P = 3/2^+$. Because the final $\pi^+ \Lambda$ system is in a pure isospin $I = 1$ combination, the $\Lambda^+_c \to \eta \pi^+ \Lambda$ decay can be an ideal process to study these $\Sigma^*$ resonances. In particular, we show that the decay angle and energy distributions of the $\pi^+$ are very different for $\Sigma^*(1385)$ and $\Sigma^*_{1/2^-}(1380)$. The proposed decay mechanism as well as the existence of the $\Sigma^*_{1/2^-}(1380)$ state can be checked by future BESIII and Belle experiments.