Radiative decays of dynamically generated pentaquarks in the chiral unitary approach: the $P_c(4457)\to P_c(4312)\gamma$ transition

We study the radiative decay of dynamically generated pentaquarks and apply the formalism to the transition $P_c(4457)(3\mathrm{/}{2}^{-})\to P_c(4312)(1\mathrm{/}{2}^{-})\gamma$. 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 $D^{\ast }D\gamma$ coupling to ${\bar{D}}^{\ast 0}\to {\bar{D}}^0\gamma$ and test the same convention with $J\mathrm{/}\psi \to {\eta }_c\gamma$. The central width is $6.7\text{keV}$, with a conservative range of about $2$ to $9\text{keV}$. This radiative decay process is a pure $M1$ transition with photon energy $143\text{MeV}$. The ${\bar{D}}^{\ast 0}\to {\bar{D}}^0\gamma$ loop gives the leading contribution. The near-threshold ${\bar{D}}^{\ast }{\mathrm{\Lambda }}_c$ loop gives the main correction. A soft Gaussian form-factor on the leading diagram reduces the width to about $2\text{keV}$, compatible with earlier molecular results, and decreases the full width to about $4\text{keV}$. The coherent result is sensitive to the relative residue phases in the coupled-channel convention. We also estimate the cascade rate for ${\mathrm{\Lambda }}_b^0\to J\mathrm{/}\psi p{K}^{-}\gamma$ 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.