Hidden-charm pentaquarks as a meson-baryon molecule with coupled channels for ${\overline{D}}^{(*)}{\mathrm{\Lambda }}_{\mathrm{c}}$ and ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_{\mathrm{c}}^{(*)}$

The recent observation of two hidden-charm pentaquark states by LHCb collaborations prompted us to investigate the exotic states close to the $\overline{D}{\mathrm{\Lambda }}_{\mathrm{c}}$, ${\overline{D}}^{*}{\mathrm{\Lambda }}_{\mathrm{c}}$, $\overline{D}{\mathrm{\Sigma }}_{\mathrm{c}}$, $\overline{D}{\mathrm{\Sigma }}_{\mathrm{c}}^{*}$, ${\overline{D}}^{*}{\mathrm{\Sigma }}_{\mathrm{c}}$ and ${\overline{D}}^{*}{\mathrm{\Sigma }}_{\mathrm{c}}^{*}$ thresholds. We therefore studied the hadronic molecules that form the coupled-channel system of ${\overline{D}}^{(*)}{\mathrm{\Lambda }}_{\mathrm{c}}$ and ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_{\mathrm{c}}^{(*)}$. As the heavy quark spin symmetry manifests the mass degenerations of $\overline{D}$ and ${\overline{D}}^{*}$ mesons, and of ${\mathrm{\Sigma }}_{\mathrm{c}}$ and ${\mathrm{\Sigma }}_{\mathrm{c}}^{*}$ baryons, the coupled channels of ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_{\mathrm{c}}^{(*)}$ are important in these molecules. In addition, we consider the coupling to the ${\overline{D}}^{(*)}{\mathrm{\Lambda }}_{\mathrm{c}}$ channel whose thresholds are near the ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_{\mathrm{c}}^{(*)}$ thresholds, and the coupling to the state with nonzero orbital angular momentum mixed by the tensor force. This full coupled-channel analysis of ${\overline{D}}^{(*)}{\mathrm{\Lambda }}_{\mathrm{c}}-{\overline{D}}^{(*)}{\mathrm{\Sigma }}_{\mathrm{c}}^{(*)}$ with larger orbital angular momentum has never been performed before. By solving the coupled-channel Schrödinger equations with the one meson exchange potentials with respect to the heavy quark spin and chiral symmetries, we studied the hidden-charm hadronic molecules with $I(J^P)=1/2(3/2^{\pm })$ and $1/2(5/2^{\pm })$. We conclude that the $J^P$ assignment of the observed pentaquarks is $3/2^{+}$ for $P_{\mathrm{c}}^{+}(4380)$ and $5/2^{-}$ for $P_{\mathrm{c}}^{+}(4450)$, which is in agreement with the results of the LHCb analysis. In addition, we give predictions for other $J^P=3/2^{\pm }$ states at 4136.0, 4307.9 and 4348.7 MeV in $J^P=3/2^{-}$, and 4206.7 MeV in $J^P=3/2^{+}$, which can be further investigated by means of experiment.

Figure 1

Cutoff $\mathrm{\Lambda }$ dependence of the obtained energies $E$ in $I(J^P)=1/2(3/2^{-})$. The solid, dashed and dot-dashed lines correspond to the obtained energy spectra of first, second and third states, respectively. The explicit values are shown in Table 3. The black horizontal lines show the meson-baryon thresholds.

Figure 2

The same as Fig. 1 for $I(J^P)=1/2(3/2^{+})$.

Figure 3

The same as Fig. 1 for $I(J^P)=1/2(5/2^{-})$.

Figure 4

The same as Fig. 1 for $I(J^P)=1/2(5/2^{+})$.