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A model for the $\bar K d \to \pi Y N$ reactions with $Y=\Lambda, \Sigma$ isdeveloped, aiming at establishing the low-lying $\Lambda$ and $\Sigma$ hyperonresonances through analyzing the forthcoming data from the J-PARC E31experiment. The off-shell amplitudes generated from the dynamicalcoupled-channels (DCC) model, which was developed in Kamano et al. [Phys. Rev.C 90, 065204 (2014)], are used as input to the calculations of the elementary$\bar K N \to \bar K N$ and $\bar K N \to \pi Y$ subprocesses in the $\bar K d\to \pi Y N$ reactions. It is shown that the cross sections for the J-PARC E31experiment with a rather high incoming-$\bar{K}$ momentum, $|\vec p_{\bar K}|=1$ GeV, can be predicted reliably only when the input $\bar K N \to \bar K N$amplitudes are generated from a $\bar KN$ model, such as the DCC model used inthis investigation, which describes the data of the $\bar K N$ reactions atenergies far beyond the $\bar K N$ threshold. We find that the data of thethreefold differential cross section $d\sigma/(dM_{\pi\Sigma}d\Omega_{p_n})$for the $K^- d \to \pi \Sigma n$ reaction below the $\bar K N$ threshold can beused to test the predictions of the resonance poles associated with$\Lambda(1405)$. We also find that the momentum dependence of the threefolddifferential cross sections for the $K^- d \to \pi^- \Lambda p$ reaction can beused to examine the existence of a low-lying $J^P=1/2^+$ $\Sigma$ resonancewith a pole mass $M_R = 1457 -i39$ MeV, which was found from analyzing the$K^-p$ reaction data within the employed DCC model.

Toward establishing low-lying $\Lambda$ and $\Sigma$ hyperon resonances with the $\bar K + d \to \pi + Y + N$ reaction