Evolution of compact states to molecular ones with coupled channels: The case of the $X(3872)$

We study the molecular probability of the $X(3872)$ in the $D^0 \bar D^{*0}$ and $D^+ D^{*-}$ channels in several scenarios. One of them assumes that the state is purely due to a genuine nonmolecular component. However, it gets unavoidably dressed by the meson components to the point that in the limit of zero binding of the $D^0 \bar D^{*0}$ component becomes purely molecular. Yet, the small but finite binding allows for a nonmolecular state when the bare mass of the genuine state approaches the $D^0 \bar D^{*0}$ threshold, but, in this case the system develops a small scattering length and a huge effective range for this channel in flagrant disagreement with present values of these magnitudes. Next we discuss the possibility to have hybrid states stemming from the combined effect of a genuine state and a reasonable direct interaction between the meson components, where we find cases in which the scattering length and effective range are still compatible with data, but even then the molecular probability is as big as $95 \%$. Finally, we perform the calculations when the binding stems purely from the direct interaction between the meson-meson components. In summary we conclude, that while present data definitely rule out the possibility of a dominant nonmolecular component, the precise value of the molecular probability requires a more precise determination of the scattering length and effective range of the $D^0 \bar D^{*0}$ channel, as well as the measurement of these magnitudes for the $D^+ D^{*-}$ channel which have not been determined experimentally so far.