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

TL;DR

This paper investigates the molecular versus nonmolecular nature of the $X(3872)$ state, analyzing how coupled channels and interactions influence its structure and scattering parameters, concluding that current data favor a predominantly molecular state.

Contribution

It provides a detailed analysis of the $X(3872)$ structure considering various scenarios, including pure molecular, nonmolecular, and hybrid states, and highlights the importance of scattering parameters.

Findings

Pure nonmolecular component is incompatible with data.

Hybrid states can have high molecular probability (~95%).

Precise scattering data are needed for definitive conclusions.

Abstract

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.