A Study of Degenerate Two-Body and Three-Body Coupled-Channel Systems -Renormalized Effective AGS Equations and Near-Threshold Resonances-

摘要

Motivated by the existence of candidates for exotic hadrons whose masses areclose to both of two-body and three-body hadronic thresholds lying close toeach other, we study degenerate two-body and three-body coupled-channelsystems. We first formulate the scattering problem of non-degenerate two-bodyand three-body coupled-channels as an effective three-body problem, i.e.\effective Alt-Grassberger-Sandhas (AGS) equations. We next investigate thebehavior of $S$-matrix poles near the threshold when two-body and three-bodythresholds are degenerate. We solve the eigenvalue equations of the kernel ofAGS equations instead of AGS equations themselves to obtain the $S$-matrix poleenergy. We then face a problem of unphysical singularity: though the physicaltransition amplitudes have physical singularities only, the kernel of AGSequations have unphysical singularities. We show, however, that theseunphysical singularities can be removed by appropriate reorganization of thescattering equations and mass renormalization. The behavior of $S$-matrix polesnear the degenerate threshold is found to be universal in the sense that thecomplex pole energy, $E$, is determined by a real parameter, $c$, as $c - E\log{\left( - E \right)} = 0$, or equivalently, ${\rm Im} E = - \pi {\rm Re} E/ \log{\mid {\rm Re} E \mid}$. This behavior is different from that of eithertwo-body or three-body system and is characteristic in the degenerate two-bodyand three-body coupled-channel system. We expect that this new class ofuniversal behavior might play a key role in understanding exotic hadrons.