S-matrix pole trajectory of the three-body system

Abstract

We have studied the behaviour of S-matrix poles near the thresholds for some three-body systems. In Amado's simple model for three bosons with an attractive S-wave two-body interaction and specetor angular momentum l=0, the S-matrix pole behaves as Im E ∞ −(Re E) 2 near the threshold in the fourth quadrant of the unphysical complex E-plane. This behaviour is formally the same as that in the two-body scattering with angular momentum l= 3 2 and a three-body resonance shows up near the threshold. In such a case, the Fredholm determinant of the Faddeev equation at zero energy, D F(0), is close to zero. However, it is shown that the value D F(0) does not definitely tell us whether the system is attractive or repulsive in contrast to the case of a two-body system. As physical applications, we have studied Λnn and Σ −nn systems. The S-matrix poles are found to be far away from the physical real axis and do not give rise to observable effects for the two-body interactions reproducing the low-energy scattering data, although the Λnn system could develop a three-body resonance if the Λn interaction were slightly more attractive.