简并二体和三体耦合通道系统的研究-重整化有效AGS方程和近阈共振
A Study of Degenerate Two-Body and Three-Body Coupled-Channel Systems -Renormalized Effective AGS Equations and Near-Threshold Resonances-
摘要：
由于存在质量接近两体和三体强子阈值的奇异强子候选者，我们研究了简并的两体和三体耦合通道系统.本文首先将非简并二体和三体耦合道的散射问题表示为一个有效的三体问题，即有效Alt-Grassberger-Sandhas（AGS）方程。接下来，我们研究的行为$S$矩阵极点附近的阈值时，两体和三体阈值退化。通过求解AGS方程核的本征值方程而不是AGS方程本身来获得$S$ -矩阵极点能量。然后，我们面临一个非物理奇异性问题：虽然物理跃迁振幅只有物理奇异性，但AGS方程的核具有非物理奇异性。然而，我们表明，这些非物理的奇异性可以通过适当的重组的散射方程和质量重整化。退化阈值附近的$S$矩阵极点的行为被发现是普遍的意义上的复极点能量，$E$，是由一个真正的参数，$c$，作为$c - E \log{\left( - E \right)} = 0$，或等价地，${\rm Im} E = - π{\rm Re} E / \log{\mid {\rm Re} E \mid}$。这种行为不同于二体或三体系统，是简并二体和三体耦合通道系统的特征。我们期望这类新的普遍行为可能在理解奇异强子方面发挥关键作用。
英文摘要：
Motivated by the existence of candidates for exotic hadrons whose masses are close to both of two-body and three-body hadronic thresholds lying close to each other, we study degenerate two-body and three-body coupled-channel 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.\ 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: though the physical transition amplitudes have physical singularities only, the kernel 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 $c - E \log{\left( - E \right)} = 0$, or equivalently, ${\rm Im} E = - π{\rm Re} E / \log{\mid {\rm Re} E \mid}$. This behavior is different from that of either two-body or three-body system and is characteristic in the degenerate two-body and three-body coupled-channel system. We expect that this new class of universal behavior might play a key role in understanding exotic hadrons.
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