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The reaction $K^-d \to \pi \Sigma n$ is studied within a Faddeev-typeapproach, with emphasis on the specific kinematics of the E31 experiment atJ-PARC, i.e. $K^-$ beam momentum of $p_K = 1$ GeV/c and neutron angle of$\theta_n=0^\circ$. The employed Faddeev approach requires as main inputamplitudes for the two-body subsystems $\bar KN \to \bar KN$ and $\bar KN \to\pi\Sigma$. For the latter results from recently published chiral unitarymodels of the $\bar KN$ interaction are utilized. The $\bar KN \to \bar KN$amplitude itself, however, is taken from a recent partial-wave analysis. Due tothe large incoming momentum of the $K^-$, the $\bar KN$ interaction is probedin a kinematical regime where those chiral potentials are no longer applicable.A comparison of the predicted spectrum for various $\pi \Sigma$ charge channelswith preliminary data is made and reveals a remarkable agreement as far as themagnitude and the line shape in general is concerned. Noticeable differencesobserved in the $\pi\Sigma$ spectrum around the $\bar KN$ threshold, i.e. inthe region of the $\Lambda$(1405) resonance, indicate a sensitivity to thedetails of the employed $\bar KN \to \pi\Sigma$ amplitudes and suggest thatpertinent high-precision data could indeed provide substantial constraints onthe structure of the $\Lambda$(1405).

Faddeev approach to the reaction $K^- d \to \pi \Sigma n$ at $p_{K} = 1$ GeV/c