The scattering lengths of the $\pi \Sigma$ systems are key quantities in understanding the structure of the $\Lambda \left(1405\right)$ resonance and the subthreshold extrapolation of the $\overline{K}N$ interaction. We demonstrate that the $\pi \Sigma$ scattering lengths can be extracted from the threshold cusp phenomena in the weak ${\Lambda }_c\to \pi \pi \Sigma$ decays, analogously with Cabibbo’s method for determination of the $\pi \pi$ scattering length. We show that the substantial cusp effect should be observed in the spectrum, when the $\pi \Sigma$ interaction in $I=0$ is strongly attractive to generate a near-threshold singularity, such as a bound state or a virtual state.

• Received 27 May 2011

DOI:https://doi.org/10.1103/PhysRevC.84.035201