Radiative decay of the $X(3872)$ as a mixed molecule-charmonium state in effective field theory

Assuming that $X(3872)$ is a mixture between $2P$ charmonium and $\bar{D}D^{*}$ molecular states with $J^{PC}=1^{++}$, an analysis of $X(3872)$ radiative decays into $J/\psi \gamma$ and $\psi(2S)\gamma$ is presented. The modification of the radiative branching ratio due to possible constructive or destructive interferences between the meson-loop and the short-distance contact term, which is modeled by a charm quark loop, is shown. The model predictions are shown to be compatible with the experimentally determined ratio of the mentioned branching fractions for a wide range of the $X(3872)$ charmonium content. In the case of the destructive interference, a strong restriction on the charmonium admixture is found.