New Universality for Near-Threshold Three-Body Resonances

In the three-body system with one resonantly interacting pair, we study the behavior of the $S$-matrix pole near the threshold in the fourth quadrant of the unphysical complex energy plane. Our study is essentially based on the unitarity and analyticity of the $S$-matrix and employs the Alt-Grassberger-Sandhas (AGS) equations specifically for the three-body scattering problem and the dispersion relation for the inverse $T$-matrix. We find that the trajectory of the complex energy, $E$, of the $S$-matrix pole near the threshold is uniquely given by $c_0 + E \log{\left( - E \right)} \approx 0$ or $c_0 + E_R \log E_R \approx 0$, $E_I \approx \pi E_R/\log E_R$ in the fourth quadrant of the unphysical complex energy plane, in contrast to the non-unique trajectories with no resonantly interacting pair, $c_0 + c_1 E + E^2 \log{\left( - E \right)} \approx 0$ or $E_R \approx -c_0/c_1$, $E_I \approx -\pi E_R^2/c_1$ where $E_R$ and $E_I$ are the real and imaginary parts of $E$, respectively, and $c_0$ and $c_1$ are real constants. This is a new universal behavior of the $S$-matrix near the threshold. Also, we briefly discuss implications to exotic hadron candidates.