Color SU(3) symmetry, confinement, stability, clustering and quark mass dependence in the $q^4\overline{q}$ system

We examine the color SU(3) dynamics of $q^4\overline{q}$ system, i.e. of the pentaquark. First we study this system in the model with two-body interaction proportional to the color charges. We construct the potential matrix and show (1) Confinement: the color singlet $q\overline{q}$ potential energy rises infinitely with the separation distance, (2) Stability: All colorless states’ energies are bounded from below, (3) Color singlet clustering: the pentaquark color-singlet state Hamiltonian turns into a sum of a three-quark (baryon) and a quark-antiquark (meson) cluster Hamiltonian, in the limit of asymptotically large separations. We evaluate the four excitation eigenfrequencies of pentaquarks in the harmonic oscillator two-body confining potential and the ground states’ dependence on the quark-antiquark mass ratio. We show that the pentaquark is unlikely to bind even with a top antiquark, in contrast to tetraquarks.

Figure 1

Pauli allowed color singlet pentaquark states in the harmonic oscillator potential. The excitation energies $\hslash {\omega }_{\alpha }$, where $\alpha =\lambda ,\rho ,\sigma ,\tau$ denotes the eigenmodes of the pentaquark (not necessarily identical to the Jacobi coordinates of five particles) depends on the color singlet state the pentaquark is in. The numbers denote the dimensions of the ${\mathrm{SU}}_{FS}(6)$ irreducible representations allowed by the Pauli principle in the corresponding state, see Ref. 13.

Figure 2

Ratios of pentaquark energies $E_{|{\overline{\mathbf{3}}}_{12}{\overline{\mathbf{3}}}_{34}{\overline{\mathbf{3}}}_5⟩}/E_{|{\mathbf{1}}_{123}{\mathbf{1}}_{45}⟩}-1$ (solid), and $E_{|{\overline{\mathbf{3}}}_{12}{\mathbf{6}}_{34}{\overline{\mathbf{3}}}_5⟩}/E_{|{\mathbf{1}}_{123}{\mathbf{1}}_{45}⟩}-1$ (dashed) in the nonrelativistic chromo-harmonic quark model as a function of the antiquark-quark mass ratio $m_5/m$.

Figure 3

Mixing angle $\Theta$ (degrees) as a function of the cluster separation $\Delta R$ (in units of meson mean square radii ${\sqrt{⟨r^2⟩}}_{\mathrm{meson}}={\omega }_{\mathrm{meson}}^{-1}\sim 1\mathrm{fm}$) in the nonrelativistic chromo-harmonic quark model.