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The $\Lambda(1405)$ resonance production reaction is investigated within theframework of the coupled-channels Alt-Grassberger-Sandhas (AGS) equations. Weperform full three-body calculations for the $\bar{K}NN-\pi YN$ amplitudes onthe physical real energy axis and investigate how the signature of the$\Lambda(1405)$ appears in the cross sections of the $K^-d\rightarrow \pi\Sigman$ reactions, also in view of the planned E31 experiment at J-PARC. Two typesof meson-baryon interaction models are considered: an energy-dependentinteraction based on chiral $SU(3)$ effective field theory, and anenergy-independent version that has been used repeatedly in phenomenologicalapproaches. These two models have different off-shell properties that implycorrespondingly different behavior in the three-body system. We investigate howthese features show up in differential cross sections of $K^- d\rightarrow\pi\Sigma n$ reactions. Characteristic patterns distinguishing between the twomodels are found in the invariant mass spectrum of the final $\pi\Sigma$ state.The $K^-d\rightarrow \pi\Sigma n$ reaction, with different($\pi^{\pm}\Sigma^{\mp}$ and $\pi^{0}\Sigma^{0}$) charge combinations in thefinal state, is thus demonstrated to be a useful tool for investigating thesubthreshold behavior of the $\bar{K}N$ interaction.

Structure of the $\Lambda(1405)$ and the $K^-d\rightarrow\pi\Sigma n$ reaction