Precise calculation of the two-step process for K- d -> pi Sigma N in the Lambda (1405) resonance region

The reaction K- d -> pi Sigma N is investigated taking into account single scattering and the two-step process due to Kbar N -> pi Sigma rescattering. The influence of some common approximations are examined. It is found that the treatment of the kinematics in the Green's function that appears in the loop integral of the rescattering process has a rather strong impact on the resulting lineshape of the pi Sigma invariant mass spectrum. Specifically, a calculation with correct kinematics where the three-body unitarity cut due to the nK- p threshold occurs at the physical value yields a pronounced peak in the invariant mass spectrum at this threshold and, at the same time, suppresses the signal in the region of the Lambda(1405) resonance. On the other hand, an approximation applied in past calculations shifts that threshold down and, consequently, leads to an accidental and therefore erroneous enhancement of the signal of the Lambda(1405) in the pi Sigma invariant mass spectrum.