Abstract
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 $c+Elog\left(\ensuremath{-}E\right)=0$, or equivalently, $c+\mathrm{Re}\phantom{\rule{0.16em}{0ex}}Elog\left(\mathrm{Re}\phantom{\rule{0.16em}{0ex}}E\right)=0$ and $\mathrm{Im}\phantom{\rule{0.16em}{0ex}}E=\ensuremath{\pi}\mathrm{Re}\phantom{\rule{0.16em}{0ex}}E/log\left(\mathrm{Re}\phantom{\rule{0.16em}{0ex}}E\right)$. 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.
Highlights
The X(3872) was first observed in 2003 [1] and is considered not to be a simple charmonium [2,3,4] and is a candidate for the exotic hadron
II, we present basic setups, namely, the Hamiltonian we consider, effective interactions constructed by the Feshbach projection, the Alt-Grassberger-Sandhas (AGS) equations which three-body transition amplitudes satisfy, a problem of unphysical singularity, and its solution with the mass renormalization plus an appropriate reorganization of the Feynman diagrams
We show that the S-matrix pole behavior is characteristic in the system and universal in a sense that it is determined by the equation c + E log (−E) = 0 or, equivalently, c + Re E log (Re E) = 0 and Im E = π Re E/ log (Re E), where E is the S-matrix pole energy, while c a real parameter
Summary
The X(3872) was first observed in 2003 [1] and is considered not to be a simple charmonium [2,3,4] and is a candidate for the exotic hadron. In this paper, motivated by such circumstances, we develop two-body and three-body coupled-channels scattering equations and investigate the S-matrix pole behavior near the thresholds in the case of a degenerate two-body and three-body coupledchannels system. It is known that the S-matrix pole behavior near the threshold in a singlechannel two-body and three-body system has a universal property [47]. The behavior we are going to discuss is interesting on its own and might play a key role in understanding those observed candidates for the exotic hadrons lying in the energy regions where two-body and three-body hadronic thresholds rest close to each other. IV, we summarize the results and discuss their physical applications
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