$K^- d\rightarrow \pi \Lambda N$ reaction for studying charge symmetry breaking in the $\Lambda N$ interaction

We discuss charge symmetry breaking in the $\Lambda N$ interaction. In order to investigate the $\Lambda p$ and $\Lambda n$ interactions at low energies, we propose to utilize the $K^-d\to \pi^-\Lambda p$ and $K^-d\to \pi^0\Lambda n$ reactions. They are symmetric under the exchange of both the pion and nucleon isospin partners in the final states. This advantage allows us to study charge symmetry breaking in the $\Lambda N$ interaction. We calculate the differential cross sections of these reactions with stopped kaons theoretically and discuss the possibility to extract the scattering length and effective range of the $\Lambda N$ scattering. With stopped kaons, the $\Lambda N$ interaction takes place dominantly in the spin-triplet state thanks to the deuteron spin and $s$-wave dominance of the scattering amplitudes at the low energy. We find that the ratio of the differential cross sections as a function of the $\Lambda N$ invariant mass between the two reactions is useful for revealing how charge symmetry breaking, or isospin symmetry breaking appears in the low-energy $\Lambda N$ scattering.