Isospin breaking decays as a diagnosis of the hadronic molecular structure of the $P_c(4457)$
The LHCb Collaboration announced the observation of three narrow structures consistent with hidden-charm pentaquark states. They are candidates of hadronic molecules formed of a pair of a charmed baryon and an anticharmed meson. Among them, the $P_c(4457)$ mass is consistent with earlier predictions of a $Σ_c\bar D^*$ molecule with $I=1/2$. We point out that if such a picture were true, one would have $\mathcal{B}(P_c(4457)\to J/ψΔ^+)/\mathcal{B}(P_c(4457)\to J/ψp)$ at the level ranging from a few percent to about 30%. Such a large isospin breaking decay ratio is two to three orders of magnitude larger than that for normal hadron resonances. It is a unique feature of the $Σ_c\bar D^*$ molecular model, and can be checked by LHCb.
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