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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. Themodification of the radiative branching ratio due to possible constructive ordestructive interferences between the meson-loop and the short-distance contactterm, which is modeled by a charm quark loop, is shown. The model predictionsare shown to be compatible with the experimentally determined ratio of thementioned branching fractions for a wide range of the $X(3872)$ charmoniumcontent. In the case of the destructive interference, a strong restriction onthe charmonium admixture is found.

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