$X(3872)$ as virtual companion pole of the charm-anticharm state $\chi_{c1}(2P)$

Abstract

We study the spectral function of the axial-vector charmonium state $\chi_{c1}(2P)$ coupled to $DD^{\ast}$ mesons, by employing a quantum field theoretical approach: a pronounced enhancement close to the $D^{0}% D^{\ast0}$ threshold, to be identified with the $X(3872)$, emerges. In the complex plane we find two poles: a pole for the broad seed state $\chi_{c1}(2P)$, and -- in the easiest scenario -- a virtual pole for the $X(3872)$. Thus, our approach describes both the seed state and the dynamically generated $X(3872)$ simultaneously. In particular, it explains the most prominent, both molecular-like and quarkonium-like, features of the $X(3872)$: its very small width (the decay into $D^{0}D^{\ast0}$ is predicted to be about 0.5 MeV), the enhanced radiative decay into $\psi(2S)\gamma$ w.r.t. $\psi(1S)\gamma$, and the isospin breaking decay into $J/\psi\rho$ (thanks to $DD^{\ast}$ loops mediating this decay channel). At the same time, we aim to determine the pole position and the properties of the charmonium seed state: quite interestingly, even if a pole is always present, it is possible that there is no peak corresponding to this state in the spectral function, thus potentially explaining why the corresponding resonance could not yet be seen in experiments.