Precise determination of the $ηΛ$ scattering length and effective range and relationship to the $Λ(1670)$ resonance

Abstract

We use the Belle data on the $K^- p$ mass distribution of the $Λ_c^+ \to p K^- π^+$ 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 $K^- p \to ηΛ$ 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 $Λ(1670)$.