A short review on the compositeness of the $X(3872)$

The $X(3872)$ could be a shallow $D\bar D^*$ bound state, a compact four-quark state, or a partially composite particle, i.e. a superposition of the two. We will review how these hypotheses could be tested experimentally, examining especially the cases in which the $X$ is a pure bound state or a pure compact tetraquark. The comparison with data is done by comparing the results on the $X\to D\bar D\pi$ decay with the analysis of its lineshape. Looking at decay, the pure bound state hypothesis corresponds to a well-defined region in parameter space defined by the width of the $D^*$ versus the binding energy of the $X$. As far as the lineshape is concerned, the currently available experimental analysis is limited to showing compatibility with the purely compact hypothesis. We propose how to modify the analysis to test the molecular or the partially composite hypotheses. We also review the analysis on the radiative decays of the $X$, showing that it leaves little room for its purely molecular nature, even if pion corrections are included.
Comment: 54 pages, 4 figures