Radiative decays of dynamically generated pentaquarks in the chiral unitary approach: the $P_c(4457)\to P_c(4312)\,γ$ transition
We study the radiative decay of dynamically generated pentaquarks and apply the formalism to the transition $P_c(4457)(3/2^-)\to P_c(4312)(1/2^-)γ$. Both states are treated as $S$-wave hadronic molecules generated in the chiral unitary approach with heavy-quark spin symmetry and the local hidden gauge interaction. The photon therefore couples to the meson-baryon components of the two poles. The calculation combines the strong coupling residues of the coupled-channel solution, heavy-quark spin symmetry for the electromagnetic vertices, and a transverse assembly of the $M1$ triangle loops. A complete calculation gives nineteen triangle loops. We reduce each loop to a single numerical quadrature and give the closed analytic form. The electromagnetic vertices that are not fixed by data are estimated with the naive-quark model and heavy-quark spin symmetry. We normalize the main $D^*Dγ$ coupling to $\bar D^{*0}\to\bar D^{0}γ$ and test the same convention with $J/ψ\toη_cγ$. The central width is $6.7\keV$, with a conservative range of about $2$ to $9\keV$. This radiative decay process is a pure $M1$ transition with photon energy $143\MeV$. The $\bar D^{*0}\to\bar D^{0}γ$ loop gives the leading contribution. The near-threshold $\bar D^{*}Λ_c$ loop gives the main correction. A soft Gaussian form-factor on the leading diagram reduces the width to about $2\keV$, compatible with earlier molecular results, and decreases the full width to about $4\keV$. The coherent result is sensitive to the relative residue phases in the coupled-channel convention. We also estimate the cascade rate for $Λ_b^{0}\to J/ψ\,p\,K^{-}γ$ and discuss how the line can be searched for. The pure $M1$ content, the ratio to the $P_c(4440)$ radiative decay, and the binding-energy dependence of the width are proposed as tests of the molecular nature.