Production rates of hidden-charm pentaquark molecules in $\Lambda_b$ decays

The partial decay widths and production mechanism of the three pentaquark states, $P_{\psi}^{N}(4312)$, $P_{\psi}^{N}(4440)$, and $P_{\psi}^{N}(4457)$, discovered by the LHCb Collaboration in 2019, are still under debate. In this work, we employ the contact-range effective field theory approach to construct the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$, $\bar{D}^{*}\Lambda_c$, $\bar{D}\Lambda_c$, $J/\psi p$, and $\eta_c p$ coupled-channel interactions to dynamically generate the multiplet of hidde-charm pentaquark molecules by reproducing the masses and widths of $P_{\psi}^{N}(4312)$, $P_{\psi}^{N}(4440)$, and $P_{\psi}^{N}(4457)$. Assuming that the pentaquark molecules are produced in the $\Lambda_b$ decay via the triangle diagrams, where $\Lambda_{b}$ firstly decays into $D_{s}^{(\ast)}\Lambda_{c}$, then $D_{s}^{(\ast)}$ scatters into $\bar{D}^{(\ast)}K$, and finally the molecules are dynamically generated by the $\bar{D}^{(\ast)}\Lambda_{c}$ interactions, we calculate the branching fractions of the decays $\Lambda_b \to {P_{\psi}^{N}}K$ using the effective Lagrangian approach. With the partial decay widths of these pentaquark molecules, we further estimate the branching fraction of the decays $\Lambda_b \to ( P_{\psi}^{N} \to J/\psi p )K$ and $\Lambda_b \to ( P_{\psi}^{N}\to \bar{D}^* \Lambda_c )K$. Our results show that the pentaquark states $P_{\psi}^{N}(4312)$, $P_{\psi}^{N}(4440)$, and $P_{\psi}^{N}(4457)$ as hadronic molecules can be produced in the $\Lambda_b$ decay, and on the other hand their heavy quark spin symmetry partners are invisible in the $J/\psi p$ invariant mass distribution because of the small production rates. Our studies show that is possible to observe some of the pentaquark states in the $\Lambda_b\to \bar{D}^*\Lambda_c K$ decays.