Quark model estimate of hidden-charm pentaquark resonances

A quark model, which reproduces the ground-state mesons and baryons, i.e., the threshold energies, is applied to the $qqqc\overline{c}$ configurations, where $q$ is a light quark and $c$ the charmed quark. In the calculation, several open channels are explicitly included such as $J/\psi +N, {\eta }_c+N, {\mathrm{\Lambda }}_c+\overline{D}$, etc. To distinguish genuine resonances and estimate their width, we employ the Gaussian expansion method supplemented by the real scaling method (stabilization). No resonance is found at the energies of the $P_c\left(4380\right)$ and $P_c\left(4450\right)$ pentaquarks. On the other hand, there is a sharp resonant state at 4690 MeV with a $J=1/2^{-}$ state and another one at 4920 MeV with a $J=3/2^{-}$ state which have a compact structure.

Figure 1

Four sets of the Jacobi coordinate systems. The light ($u,d$) quarks, labeled by particles $1–3$, are to be antisymmetrized, while particles 4 and 5 stand for $c$ and $\overline{c}$, respectively. Scatterings of $qqq+c\overline{c}$ and $qqc+q\overline{c}$ are described in the coordinates $C=1$ and 2, respectively.

Figure 2

The calculated energy spectra for $J=1/2^{-}$ and $3/2^{-}$ states using the connected configurations of $C=3$ and 4. The dashed lines are thresholds. The numbers in parentheses are energies in MeV. Also the numbers in unit of MeV are listed.

Figure 3

The stabilization plots of the eigenenergies $E$ of the pseudostates for $J=1/2^{-}$ with the respect to the scaling factor $\alpha$ in the case of (a) only connected configuration of $C=3+4$, and ${\eta }_c+N$ configuration, and of (b) only connected configurations of $C=3+4$ and $J/\psi +N$.

Figure 4

The stabilization plots of the eigenenergies $E$ of the pseudostates with the respect to the scaling factor $\alpha$ in the case of (a) only scattering configurations of $C=1$ and 2, and of (b) full configurations of $C=1$–4. The Gaussian ranges $R_N$ for the coordinates ${\mathit{R}}_1$ and ${\mathit{R}}_2$ of $C=1$ and 2 configurations are scaled as $R_N\to \alpha R_N$ with $1.0$–1.5.

Figure 5

The stabilization plots of the eigenenergies $E$ of the pseudostates with the respect to the scaling factor $\alpha$ in the case of (a) only scattering configurations of $C=1$ and 2, and of (b) full configurations of $C=1$–4. The Gaussian ranges $R_N$ for the coordinates ${\mathit{R}}_1$ and ${\mathit{R}}_2$ of $C=1$ and 2 configurations are scaled as $R_N\to \alpha R_N$ with $1.0$–1.5.

Figure 6

The calculated $qq$ and $c\overline{c}$ correlation function for $J=1/2^{-}$ state with full configurations. The solid line is for $qq$ and the dashed line is for $c\overline{c}$.