Hidden-charm and bottom meson-baryon molecules coupled with five-quark states

In this paper, we investigate the hidden-charm pentaquarks as ${\overline{D}}^{(*)}{\mathrm{\Lambda }}_{\mathrm{c}}$ and ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_{\mathrm{c}}^{(*)}$ molecules coupled to the five-quark states. Furthermore, we extend our calculations to the hidden-bottom sector. The coupling to the five-quark states is treated as the short range potential, where the relative strength for the meson-baryon channels is determined by the structure of the five-quark states. We found that resonant and/or bound states appear in both the charm and bottom sectors. The five-quark state potential turned out to be attractive and, for this reason, it plays an important role to produce these states. In the charm sector, we need the five-quark potential in addition to the pion exchange potential in producing bound and resonant states, whereas, in the bottom sector, the pion exchange interaction is strong enough to produce states. Thus, from this investigation, it emerges that the hidden-bottom pentaquarks are more likely to form than their hidden-charm counterparts; for this reason, we suggest that the experimentalists should look for states in the bottom sector.

Figure 1

One pion exchange potential (left) and the effective interaction due to the coupling to the $5q$ channel (right). The meson-baryon channels are generally represented by $\overline{D}$ and $Y_c$, respectively, and $i$ is for the initial and $j$ the final channels. A $5q$ channel is denoted by $\alpha$.

Figure 2

Jacobi coordinates of “$\overline{D}$ meson” and “$Y_c$ baryon” in the $5q$ configuration. $q_i$ ($i=1$, 2, 3) stands for the light quark, and $c_4$ (${\overline{c}}_5$) stands for the (anti)charm quark. The coordinate $\overrightarrow{\rho }$ is the relative coordinate of $q_1{q}_2$, $\overrightarrow{\lambda }$ the relative coordinate between the center of mass of $q_1{q}_2$ and $c_4$, $\overrightarrow{r}$ the relative coordinate of $q_3{\overline{c}}_5$, and $\overrightarrow{x}$ the relative coordinate between the centers of mass of $q_1{q}_2{c}_4$ and ${\overline{c}}_5{q}_3$. Though we do not use the total center-of-mass coordinate $\overrightarrow{X}$ in the present paper explicitly, it is also shown in the figure.

Figure 3

The plot of the $5q$ potential, $V^{5q}$, (solid line) and the central force of the OPEP in the diagonal ${\overline{D}}^{*}{\mathrm{\Sigma }}_{\mathrm{c}}-{\overline{D}}^{*}{\mathrm{\Sigma }}_{\mathrm{c}}$ term, $V^{\pi }$, (dashed line).

Figure 4

Bound and resonant state energies of the hidden-charm molecules (solid lines) with various coupling constants $f$ for $J^P=1/2^{-}$, using the OPEP and one of the three $5q$ potentials derived from the configuration (a) $(S_{c\overline{c}},S_{3q})=(0,1/2)$, (b) $(1,1/2)$, or (c) $(1,3/2)$. The horizontal axis shows the ratio $f/f_0$, where $f_0$ is the reference value defined in Sec. 3a. Filled circle is the starting point where the states appear. Dashed lines are the $\overline{D}{\mathrm{\Sigma }}_{\mathrm{c}}$, $\overline{D}{\mathrm{\Sigma }}_{\mathrm{c}}^{*}$, ${\overline{D}}^{*}{\mathrm{\Sigma }}_{\mathrm{c}}$, and ${\overline{D}}^{*}{\mathrm{\Sigma }}_{\mathrm{c}}^{*}$ thresholds. Dot-dashed lines are the $\overline{D}{\mathrm{\Lambda }}_{\mathrm{c}}$ and ${\overline{D}}^{*}{\mathrm{\Lambda }}_{\mathrm{c}}$ thresholds.

Figure 5

The same as Fig. 4 for the bound and resonant states of the hidden-charm molecules for $J^P=1/2^{-}$ using the OPEP and the sum of the three $5q$ potentials.

Figure 6

The same as Fig. 4 for the resonant states of the hidden-charm molecules for $J^P=3/2^{-}$ using the OPEP and one of the three $5q$ potentials derived from the configuration (a) $(S_{c\overline{c}},S_{3q})=(0,3/2)$, (b) $(1,1/2)$, or (c) $(1,3/2)$.

Figure 7

The same as Fig. 4 for the resonant states of the hidden-charm molecules for $J^P=3/2^{-}$ using the OPEP and the sum of the three $5q$ potentials.

Figure 8

The same as Fig. 4 for the resonant states of the hidden-charm molecules for $J^P=5/2^{-}$ using the OPEP and the $5q$ potential from the configuration $(S_{c\overline{c}},S_{3q})=(1,\frac{3}{2})$.

Figure 9

Bound and resonant states of the hidden-bottom molecules with various coupling constants $f$ for $J^P=1/2^{-}$, using the OPEP and one of the three $5q$ potentials derived from the configuration (a) $(S_{b\overline{b}},S_{3q})=(0,1/2)$, (b) $(1,1/2)$ or (c) $(1,3/2)$. The horizontal axis shows the ratio $f/f_0$, where $f_0$ is the reference value defined in Sec. 3a. Solid line shows the obtained state. Filled circle is the starting point where the states appear. Dashed lines are the $B{\mathrm{\Sigma }}_{\mathrm{b}}$, $B{\mathrm{\Sigma }}_{\mathrm{b}}^{*}$, $B^{*}{\mathrm{\Sigma }}_{\mathrm{b}}$, and $B^{*}{\mathrm{\Sigma }}_{\mathrm{b}}^{*}$ thresholds. Dot-dashed lines are the $B{\mathrm{\Lambda }}_{\mathrm{b}}$ and $B^{*}{\mathrm{\Lambda }}_{\mathrm{b}}$ thresholds.

Figure 10

The same as Fig. 9 for the bound and resonant states of the hidden-bottom molecules for $J^P=1/2^{-}$ using the OPEP and the sum of the three $5q$ potentials.

Figure 11

The same as Fig. 9 for the bound and resonant states of the hidden-bottom molecules for $J^P=3/2^{-}$ using the OPEP and one of the three $5q$ potentials derived from the configuration (a) $(S_{b\overline{b}},S_{3q})=(0,3/2)$, (b) $(1,1/2)$, or (c) $(1,3/2)$.

Figure 12

The same as Fig. 9 for the bound and resonant states of the hidden-bottom molecules for $J^P=3/2^{-}$ using the OPEP and the sum of the three $5q$ potentials.

Figure 13

The same as Fig. 9 for the resonant states of the hidden-bottom molecules for $J^P=5/2^{-}$ using the OPEP and the $5q$ potential from the configuration $(S_{b\overline{b}},S_{3q})=(1,\frac{3}{2})$.