<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>P</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>4457</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">?</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>4312</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mi>?</mml:mi><mml:mo>/</mml:mo><mml:mi>?</mml:mi></mml:math> 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 ${\overline{D}}^{(*)}{\mathrm{\ensuremath{\Sigma}}}_{c}^{(*)}$ seven molecules dictated heavy quark spin symmetry. In this work we employ effective Lagrangian approach to investigate two decay modes ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\pi}$, ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\gamma}$, via triangle mechanism, assuming that ${P}_{c}(4457)$ ${P}_{c}(4312)$ are ${\overline{D}}^{*}{\mathrm{\ensuremath{\Sigma}}}_{c}$ $\overline{D}{\mathrm{\ensuremath{\Sigma}}}_{c}$ bound states but either $1/2$ or $3/2$. Our results show not discriminated through these modes. widths ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\pi}$ ${P}_{c}(4457)\ensuremath{\rightarrow}{P}_{c}(4312)\ensuremath{\gamma}$ estimated order 100 keV 1 keV, respectively. ratio partial is similar ${D}^{*}\ensuremath{\rightarrow}D\ensuremath{\pi}$ ${D}^{*}\ensuremath{\rightarrow}D\ensuremath{\gamma}$, which could used check molecular nature if they observed future.