Study of exotic hadrons in $S$-wave scatterings induced by chiral interaction in the flavor symmetric limit

We study $s$-wave bound states of a hadron and a light pseudoscalar meson induced by the Weinberg-Tomozawa (WT) interaction in the flavor SU(3) symmetric limit. The WT interaction is a driving force to generate quasibound states dynamically in the chiral unitary approaches. The strength and sign of the WT interaction are determined only by the group theoretical structure of the target hadrons, and we present a general expression of the strengths for the flavor SU(3) case. We show that, for the channels which are more exotic than the target, the interaction is repulsive in most cases, and the strength of the attractive interaction is universal for any possible target states. We demonstrate that the attractive coupling is not strong enough to generate an exotic state from the physically known masses of target hadrons. In addition, we also find a nontrivial $N_c$ dependence of the coupling strengths. We show that the channels which are attractive at $N_c=3$ change into repulsive ones for large $N_c$, and, therefore, no attractive interaction exists in exotic channels in the large-$N_c$ limit.

Figure 1

(a): Notation of the representations $\alpha$, Ad, and $T$ for the WT term. (b): The bound state pole diagram after unitarization of the amplitude.

Figure 2

The denominator of the scattering amplitude of $1-VG$ for light flavor baryons (top left panel: $T=\mathbf{8}$; bottom left panel: $T=\mathbf{10}$), for charmed baryon (top center panel: $T=\overline{\mathbf{3}}$; bottom center panel: $T=\mathbf{6}$) and for $D$ and $B$ mesons (top right panel: $T={\mathbf{3}}_c$; bottom right panel: $T={\mathbf{3}}_b$). Results for exotic channels are denoted by dashed lines.

Figure 3

Binding energies $E_b=M_T+m-M_b$ as functions of $M_T$ with coupling strengths $C_{\alpha ,T}=6$ (solid line), 5 (dash-dotted line), and 3 (dotted line). For reference, the upper limit of the binding energies is indicated as the dashed line. No bound state is found for $C_{\alpha ,T}=1$ in this energy region.

Figure 4

Critical coupling strength $C_{\mathrm{crit}}$ for $m=368\mathrm{MeV}$ (solid line). The universal strength of the attraction in exotic channels $C_{\mathrm{exotic}}=1$ is shown by the dashed line.

Figure 5

Lines for $C_{\mathrm{crit}}=1$ in the $(m,f)$ plane, with $M_T=1000\mathrm{MeV}$ (solid line), $M_T=3000\mathrm{MeV}$ (dotted line), and $M_T=6000\mathrm{MeV}$ (dashed line). In the region below (above) the lines, $C_{\mathrm{crit}}<1$ ($C_{\mathrm{crit}}>1$). The point of $(m,f)=(368,93)\mathrm{MeV}$ is indicated by the cross.