Hyperon-nucleon interaction through the $K^-d\to\pi\Lambda N$ reaction

The hyperon-nucleon interaction is investigated through the final-state interaction in the $K^-d\to\pi^-\Lambda p$ reaction. We focus on the $\Lambda N$-$\Sigma N$ coupled-channel interaction, which produces characteristic structures around the $\Sigma N$ thresholds in the $\Lambda p$ invariant mass spectrum. The spin-triplet $\Sigma N\to\Lambda p$ conversion amplitude is constructed within the $K$-matrix formalism using scattering lengths in the isospin basis. We first examine the dependence of the conversion amplitude on the $\Sigma N$ scattering lengths and find that the threshold structure is particularly sensitive to the sign of the real part of the $I=1/2$ scattering length. We then calculate the $\Lambda p$ invariant mass spectrum of the $K^-d\to\pi^-\Lambda p$ reaction, including the contributions from the background diagrams. The resulting spectra show characteristic structures around the $\Sigma N$ thresholds, whose shapes depend on the choice of the interaction parameters. These results suggest that the $\Lambda p$ invariant mass spectrum can serve as a useful observable for constraining the $\Lambda N$-$\Sigma N$ coupled-channel interaction.