Strong decays of the latest LHCb pentaquark candidates in hadronic molecule pictures

We investigate the observed pentaquark candidates $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ from the latest LHCb measurement, as well as four possible spin partners in the ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_c^{*}$ system predicted from the heavy quark spin symmetry with the hadronic molecule scenarios. Similar to the previous calculation on $P_c(4380)$ and $P_c(4450)$, the partial widths of all the allowed decay channels for these $P_c$ states are estimated with the effective Lagrangian method. The cutoff dependence of our numerical results is also presented. Comparing with the experimental widths, our results show that $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ can be described well with the spin-parity-$1/2^{-}\text{−}\overline{D}{\mathrm{\Sigma }}_c$, $1/2^{-}\text{−}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$, and $3/2^{-}\text{−}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$ molecule pictures, respectively.

Figure 1

Three-body decays of the ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_c^{*}$ molecules.

Figure 2

The triangle diagram for the two-body decays of the $P_c$ states in the ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_c^{(*)}$ molecule scenarios, where $C1$, $C2$ denote the constituent particles of the ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_c^{(*)}$ composite system, $F1$, $F2$ denote the final states, $EP$ denotes the exchanged mesons.

Figure 3

The dependence of relative ratios $g_{RT}/g_{\mathrm{NR}}$, $g_0/g_{\mathrm{NR}}$ on binding energy, where $g_{\mathrm{NR}}$ and $g_{RT}$ denote the nonrelativistic and relativistic estimation for the effective couplings between $P_c(4312)$ and $\overline{D}{\mathrm{\Sigma }}_c$ composite system. And $g_0$ is the approximate $g_{\mathrm{NR}}$ as shown in Eq. (8). The red-circle, orange-square, and blue-diamond dots denote ${\mathrm{\Lambda }}_0=0.6$, 1.0, and 1.4 GeV, respectively. The upper dots are the values of $g_{RT}/g_{\mathrm{NR}}$, while the lower dots are $g_0/g_{\mathrm{NR}}$.

Figure 4

${\mathrm{\Lambda }}_1$ dependence of the total decay width and the branching fractions of the ${\overline{D}}^{*}{\mathrm{\Lambda }}_c$, $J/\psi p$, and $\overline{D}{\mathrm{\Lambda }}_c$ channels for the $P_c(4312)$ in the $J^P=1/2^{-}$ $\overline{D}{\mathrm{\Sigma }}_c$ molecule scenario. The form factor set is chosen as $(f_1,f_3)$ (denoted as RT) for the left panel, and it is $(f_2,f_3)$ (denoted as NR) for the right panel. The black solid line denotes the ${\mathrm{\Lambda }}_1$ dependence of total widths, while dashed line is the ${\mathrm{\Lambda }}_0$ dependence. And the blue-solid, origin-dashed, and red-dotted lines represent the ${\mathrm{\Lambda }}_1$ dependence of partial widths for the ${\overline{D}}^{*}{\mathrm{\Lambda }}_c$, $J/\psi p$, and $\overline{D}{\mathrm{\Lambda }}_c$ channels, respectively. The green bands in the upper half panels represent the measured widths with uncertainties, and the green-solid line denotes the central value.

Figure 5

${\mathrm{\Lambda }}_0$ dependence of the total decay widths for the $P_c(4440)$ in the left panel and $P_c(4457)$ in the right panel, where the blue-solid, blue-dashed, red-solid, and red-dashed lines denote that $1/2^{-}\text{−}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$ molecule with the form factor set $(f_2,f_3)$, $3/2^{-}\text{−}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$ with $(f_2,f_3)$, $1/2^{-}\text{−}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$ with $(f_1,f_3)$, and $3/2^{-}\text{−}{\overline{D}}^{*}{\mathrm{\Sigma }}_c$ with $(f_1,f_3)$, respectively. The green bands in the upper half panels represent the measured widths with uncertainties, and the green-solid line denotes the central value.

Figure 6

${\mathrm{\Lambda }}_1$ dependence of the total decay width and the branching fractions of the ${\overline{D}}^{*}{\mathrm{\Lambda }}_c$, $J/\psi p$, and $\overline{D}{\mathrm{\Lambda }}_c$ channels for the $P_c(4440)$ in the ${\overline{D}}^{*}{\mathrm{\Sigma }}_c$ molecule scenario with $J^P=1/2^{-}$ or $3/2^{-}$. The form factor set is chosen as $(f_1,f_3)$ (denoted as RT) for the left panel, and it is $(f_2,f_3)$ (denoted as NR) for the right panel. The black solid line denotes the ${\mathrm{\Lambda }}_1$ dependence of total widths. And the blue-solid, origin-dashed, and red-dotted lines represent the ${\mathrm{\Lambda }}_1$ dependence of partial widths for the ${\overline{D}}^{*}{\mathrm{\Lambda }}_c$, $J/\psi p$, and $\overline{D}{\mathrm{\Lambda }}_c$ channels, respectively. The green bands in the upper half panels represent the measured widths with uncertainties, and the green-solid line denotes the central value.

Figure 7

${\mathrm{\Lambda }}_1$ dependence of the total decay width and the branching fractions of the ${\overline{D}}^{*}{\mathrm{\Lambda }}_c$, $J/\psi p$, and $\overline{D}{\mathrm{\Lambda }}_c$ channels for the $P_c(4457)$ in the ${\overline{D}}^{*}{\mathrm{\Sigma }}_c$ molecule scenario with $J^P=1/2^{-}$ or $3/2^{-}$. The notations are same with Fig. 6.

Figure 8

$\mathrm{\Lambda }$ dependence of the total decay widths for the four spin partners predicted by Ref. [35] in the ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_c^{*}$ molecule pictures, where the blue-solid, blue-dashed, red-solid, and red-dashed lines denote that the dependence on ${\mathrm{\Lambda }}_0$ with form factor set $(f_2,f_3)$, dependence on ${\mathrm{\Lambda }}_1$ with $(f_2,f_3)$, dependence on ${\mathrm{\Lambda }}_0$ with $(f_1,f_3)$, and dependence on ${\mathrm{\Lambda }}_1$ with $(f_1,f_3)$, respectively.