Precise determination of the $\eta \mathrm{\Lambda }$ scattering length and effective range and relationship to the $\mathrm{\Lambda }(1670)$ resonance

We use the Belle data on the $K^{-}p$ mass distribution of the ${\mathrm{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+}$ reaction near the $\eta \mathrm{\Lambda }$ threshold to determine the $\eta \mathrm{\Lambda }$ scattering length and effective range. We show that from these data alone we can determine the value of $a$ with better precision than so far determined, and the value of $r_0$ for the first time. The addition of the $K^{-}p\to \eta \mathrm{\Lambda }$ data allows us to improve the precision of these magnitudes, with errors smaller than $15\mathrm{\%}$. We also determine with high precision the pole position of the $\mathrm{\Lambda }(1670)$.