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

Abstract

We use the Belle data on the $K^{-}p$ mass distribution of the ${\Lambda }_c^{+}\to p{K}^{-}{\pi }^{+}$ reaction near the $\eta \Lambda$ threshold to determine the $\eta \Lambda$ 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 $K^{-}p\to \eta \Lambda$ 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 $\Lambda (1670)$.