Quantization of exotic states in SU(3) soliton models: A solvable quantum mechanical analog

The distinction between the rigid-rotor and Callan-Klebanov approaches to the quantization of SU(3) solitons is considered in the context of exotic baryons. A numerically tractable quantum mechanical analog system is introduced to test the reliability of the two quantization schemes. We find that in the equivalent of the large $N_c$ limit of QCD, the Callan-Klebanov approach agrees with a numerical solution of the quantum mechanical analog. Rigid-rotor quantization generally does not. The implications for exotic baryons are briefly discussed.

Figure 1

Semiclassical solutions of the system based on the Callan-Klebanov approach as given in 8. The energy is measured in units of ${\omega }_r$ and the solutions are plotted as a function of the ratio ${\omega }_v/{\omega }_r$, with ${\omega }_r=qg/2M{R}^2$ and ${\omega }_v=\sqrt{2k/M}$. The plot shows $\Delta E=E-E_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}$. Each solution is labeled by the $n_1{n}_2{n}_3$ of Eq. (4.3), giving its decomposition in terms of the three nonzero positive normal modes of the system.

Figure 2

We plot the numerical simulation data (dots) against the semiclassical approximation solutions based on the Callan-Klebanov approach (solid lines) for each of the energy levels shown in Fig. 1. The plots are of $\Delta E=E-E_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}$. The numerical computation was done with $eg=40$ and $J_{\max }=L_{\max }=48$.

Figure 3

We plot the nonzero semiclassical eigenmodes of the system, ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$.