Short-distance RG-analysis of $X(3872)$ radiative decays

We present a renormalization-group analysis of the $X(3872)$ radiative decays into $J/ \psi \gamma$ and $\psi(2S) \gamma$. We assume a $D\bar{D}^{*}$ molecule for the $X$ long-distance structure and parametrize its short-distance physics as contact interactions. Using effective field theory techniques and power-divergence subtraction scheme, we find that short- and long-distance physics are equally important in these decays. Our calculations set a lower limit to the corresponding decay widths, which can in principle be tested experimentally. Our results may be used as guide to build models for the $X$ short-distance.