Roles of $a_0(980)$, $\Lambda(1670)$, and $\Sigma(1385)$ in the $\Lambda_c^+ \to \eta \Lambda \pi^+$ decay

Recently, the Belle Collaboration has measured the $\Lambda_c^+ \to \eta \Lambda \pi^+$ decay and reported the $\eta\Lambda$ and $\Lambda\pi^+ $ invariant mass distributions, which show the clear signals of the resonances $\Lambda(1670)$ and $\Sigma(1385)$, respectively. Based on our previous works [Eur. Phys. J. C 76 (2016) 496 and Phys. Rev. D 95 (2017) 074024], we re-analyze this process by considering the $S$-wave $\eta\Lambda$ and $\eta\pi^+$ final state interactions within the chiral unitary approach, which dynamically generate the $\Lambda(1670)$ and $a_0(980)$, respectively. Our results are in good agreement with the Belle measurements, which supports the molecular nature of the $\Lambda(1670)$ and $a_0(980)$. In addition, the $\eta \pi^+$ invariant mass distributions are also computed and a cusp structure of $a_0(980)$ is cleary shown around the $K\bar{K}$ mass threshold.