Degenerate two-body and three-body coupled-channels systems: Renormalized effective Alt-Grassberger-Sandhas equations and near-threshold resonances

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

Figure 1

A diagrammatic representation of the AGS equations with three-body force.

Figure 2

Summary of Feynman rules. We denote the free Green function of ${\varphi }_i$ as solid lines and the unrenormalized free Green function of $\psi$ as dashed red lines.

Figure 3

Summary of Feynman rules. We denote the free Green function of ${\varphi }_i$ as solid lines and the unrenormalized free Green function of $\psi$ as dashed red lines.

Figure 4

The $S$-matrix pole trajectories near the thresholds in a degenerate two-body and three-body coupled-channels system: (a) $f_{\varphi \varphi }=-0.15$, (b) $-0.20$, (c) $-0.25$, (d) $-0.30$, (e) $-0.35$, and (f) $-0.40$. Each point in each figure corresponds to different $f_{\psi \varphi }$. The solid curve is a trajectory determined by the equation $c+Elog\left(-E\right)=0$, or equivalently $c+\mathrm{Re}Elog\left(\mathrm{Re}E\right)=0$ and $\mathrm{Im}E=\pi \mathrm{Re}E/log\left(\mathrm{Re}E\right)$. We can see that poles approach the thresholds from the fourth quadrant of the unphysical complex energy sheet. We can also see that when poles lie very close to the thresholds, they all lie on the solid curve irrespective of parameter sets.

Figure 5

The $S$-matrix pole trajectories near the thresholds in a degenerate two-body and three-body coupled-channels system: (a) $f_{\psi \varphi }=-1.33$, (b) $-1.35$, and (c) $-1.37$. Each point in each figure corresponds to different $f_{\varphi \varphi }$. The solid curve is a trajectory determined by the equation $c+Elog\left(-E\right)=0$, or equivalently $c+\mathrm{Re}Elog\left(\mathrm{Re}E\right)=0$ and $\mathrm{Im}E=\pi \mathrm{Re}E/log\left(\mathrm{Re}E\right)$. We can see that poles approach the thresholds from the fourth quadrant of the unphysical complex energy sheet. We can also see that when poles lie very close to the thresholds, they all lie on the solid curve irrespective of parameter sets.

Figure 6

The $S$-matrix pole trajectory: $s$-wave. (a) The unphysical complex energy sheet and (b) The physical complex energy sheet.

Figure 7

The $S$-matrix pole trajectory: higher partial wave. (a) The unphysical complex energy sheet and (b) The physical complex energy sheet.

Figure 8

A diagrammatic representation of the modified kernel.