Structure of Near-Threshold $s$-Wave Resonances

We study the structure of two-body $s$-wave bound states as well as resonances in the threshold energy region. We focus on the single-channel scattering where the scattering length and the effective range are given by real numbers. It is shown that, in the energy region where the effective range expansion is valid, the properties of resonances are constrained only by the position of the pole. We find that the compositeness defined through the analytic continuation of the field renormalization constant is purely imaginary and normalized for resonances. We discuss the interpretation of this quantity by examining the structure of the hadron resonance ${\Lambda }_c(2595)$ in the $\pi {\Sigma }_c$ scattering. We show that the ${\Lambda }_c(2595)$ resonance requires an unnaturally large effective range and hence it is not likely a $\pi {\Sigma }_c$ molecule.

Figure 1

Trajectories of the pole positions $k^{-}$ and $k^{+}$ (a), and the field renormalization constant $Z$ (b) increasing the inverse scattering length $1/a$ for a fixed negative effective range $r_e<0$. Inverted triangles, squares, circles, crosses, and triangles correspond to $1/a=-\infty$, 0, $-2/r_e$, $-1/r_e$, and $+\infty$, respectively. Solid (dashed) line with filled (empty) symbols stands for $k^{-}$ ($k^{+}$).

Figure 2

Trajectories of the pole position $z_R$ with the interaction strength $C=4$ (a) and $C=1$ (b). Solid (dashed) lines with empty (filled) triangles represent the exact pole positions [pole positions calculated from ($a$, $r_e$)]. Symbols are plotted at $a=\pm 1$, $\pm 2$, and $\pm 10\mathrm{fm}$. Cross and circle in (a) represent the position of the physical ${\Lambda }_c(2595)$ and that of the natural renormalization scheme.