Measurement of Branching Fractions of $\Lambda_{c}^{+} \rightarrow \eta\Lambda\pi^{+}$, $\eta \Sigma^{0} \pi^{+}$, $\Lambda(1670) \pi^{+}$, and $\eta \Sigma(1385)^{+}$

We report branching fraction measurements of four decay modes of the $\Lambda_{c}^{+}$ baryon, each of which includes an $\eta$ meson and a $\Lambda$ baryon in the final state, and all of which are measured relative to the $\Lambda_{c}^{+} \rightarrow p K^{-} pi^{+}$ decay mode. The results are based on a $980~\mathrm{fb^{-1}}$ data sample collected by the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. Two decays, $\eta \Sigma^{0} \pi^{+}$ and $\Lambda(1670) \pi^{+}$, are observed for the first time, while the measurements of the other decay modes, $\Lambda_{c}^{+} \rightarrow \eta\Lambda\pi^{+}$ and $\eta\Sigma(1385)^{+}$, are more precise than those made previously. We obtain $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \eta \Lambda \pi^{+})/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $0.293 \pm 0.003 \pm 0.014$, $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \eta \Sigma^{0} \pi^{+})/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $0.120 \pm 0.006 \pm 0.006$, $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \Lambda(1670) \pi^{+}) \times \mathcal{B}(\Lambda(1670) \rightarrow \eta \Lambda)/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $(5.54 \pm 0.29 \pm 0.73 ) \times 10^{-2}$, and $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \eta \Sigma(1385)^{+})/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $0.192 \pm 0.006 \pm 0.016$. The mass and width of the $\Lambda(1670)$ are also precisely determined to be $1674.3 \pm 0.8 \pm 4.9~{\rm MeV}/c^{2}$ and $36.1 \pm 2.4 \pm 4.8~{\rm MeV}$, respectively, where the uncertainties are statistical and systematic, respectively.