P c ( 4457 ) → P c ( 4312 ) π / γ in the molecular picture

The three pentaquark states, ${P}_{c}(4312)$, ${P}_{c}(4440)$, and ${P}_{c}(4457)$, discovered by the LHCb Collaboration in 2019, can be nicely arranged into a multiplet of ${\overline{D}}^{(*)}{\mathrm{\ensuremath{\Sigma}}}_{c}^{(*)}$ of seven molecules dictated by heavy quark spin symmetry. In this work we employ the effective Lagrangian approach to investigate the two decay modes of ${P}_{c}(4457)$, ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\pi}$, and ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\gamma}$, via the triangle mechanism, assuming that ${P}_{c}(4457)$ and ${P}_{c}(4312)$ are ${\overline{D}}^{*}{\mathrm{\ensuremath{\Sigma}}}_{c}$ and $\overline{D}{\mathrm{\ensuremath{\Sigma}}}_{c}$ bound states but the spin of ${P}_{c}(4457)$ can be either $1/2$ or $3/2$. Our results show that the spin of ${P}_{c}(4457)$ can not be discriminated through these two decay modes. The decay widths of ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\pi}$ and ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\gamma}$ are estimated to be of order of 100 keV and 1 keV, respectively. The ratio of the partial decay widths of ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\pi}$ to ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\gamma}$ is similar to the ratio of ${D}^{*}\ensuremath{\rightarrow}D\ensuremath{\pi}$ to ${D}^{*}\ensuremath{\rightarrow}D\ensuremath{\gamma}$, which could be used to check the molecular nature of ${P}_{c}(4457)$ and ${P}_{c}(4312)$ if they can be observed in the future.