K−d→πΛN reaction for studying charge symmetry breaking in the ΛN interaction

We propose to utilize the ${K}^{\ensuremath{-}}d\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}\mathrm{\ensuremath{\Lambda}}p$ and ${K}^{\ensuremath{-}}d\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}\mathrm{\ensuremath{\Lambda}}n$ reactions in order to investigate the difference between the low-energy $\mathrm{\ensuremath{\Lambda}}p$ and $\mathrm{\ensuremath{\Lambda}}n$ interactions. In these reactions, the $\mathrm{\ensuremath{\Lambda}}p$ and $\mathrm{\ensuremath{\Lambda}}n$ scatterings appear as final state interactions. We calculate the differential cross sections of these reactions with stopped kaons theoretically. We introduce isospin breaking by using the physical masses for the participating hadrons and through the chiral unitary amplitudes for ${K}^{\ensuremath{-}}N\ensuremath{\rightarrow}\ensuremath{\pi}N$ and ${K}^{\ensuremath{-}}N\ensuremath{\rightarrow}\overline{K}N$. With stopped kaons, the $\mathrm{\ensuremath{\Lambda}}N$ interaction takes place dominantly in the spin-triplet state thanks to the deuteron spin and $s$-wave dominance of the scattering amplitudes at low energy. We find that the ratio of the $\mathrm{\ensuremath{\Lambda}}N$ invariant mass spectra of these reactions is useful for revealing qualitative properties of charge symmetry breaking in the low-energy $\mathrm{\ensuremath{\Lambda}}N$ interaction. We also find that the contributions coming from the $\ensuremath{\pi}$ and $\mathrm{\ensuremath{\Sigma}}$ exchange diagrams which have $\ensuremath{\pi}N$ final state interaction and $\mathrm{\ensuremath{\Sigma}}N\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}N$ conversion, respectively, are negligibly small around the $\mathrm{\ensuremath{\Lambda}}N$ threshold, while the contributions from the impulse diagram without final state interactions and the $\overline{K}$ exchange diagram containing the $\ensuremath{\pi}\mathrm{\ensuremath{\Lambda}}$ final state interaction are main sources of the background.