Precise determination of the Λ Scattering Length and Effective Range and Relationship to the Λ(1670) Resonance

We use the Belle data on the ${K}^{-}p$ mass distribution of the ${\Lambda }_{c}^{+}\to p{K}^{-}{\pi }^{+}$ reaction near the ηΛ threshold to determine the ηΛ 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 \Lambda $ data allows us to improve the precision of these magnitudes, with errors smaller than 15%. We also determine with high precision the pole position of the $\Lambda \left(1670\right)$.