The role of electromagnetic interaction in the $X(3872)$ and its analogs

We investigate the role of the electromagnetic interaction in the formation and decay of the $X(3872)$. The binding properties of the $X(3872)$ are studied by assuming the molecular nature and considering the $S$-$D$ wave mixing, isospin breaking, and coupled channel effects, and in particular the correction from the electromagnetic interaction. The radiative decays can better reflect the difference between the charged and neutral $D{\bar{D}}^{\ast }$ components, since the electromagnetic interaction explicitly breaks the isospin symmetry. We further study the radiative decay widths with the obtained wave functions for different $D{\bar{D}}^{\ast }$ channels. We also explore other similar hidden-charm molecular states. The electromagnetic interaction can make the molecule tighter. Our result of the radiative decay width for $X(3872)\to \gamma J\mathrm{/}\psi$ is in agreement with the experiment. The branching ratio $R_{\gamma \psi }$ is less than 1 in our framework, which supports the Belle and BESIII measurements.