Different pole structures in line shapes of the $X(3872)$

Abstract: We introduce a near-threshold parameterization that is more general than the effective-range expansion up to and including the effective-range because it can also handle with a near-threshold zero in the $D<sup>0\bar{D}<sup>{*0}$ $S$-wave. In terms of it we analyze the CDF data on inclusive $p\bar{p}$ scattering to $J/\psi \pi<sup>+\pi<sup>-$, and the Belle and BaBar data on $B$ decays to $K\, J/\psi \pi<sup>+\pi<sup>-$ and $K D\bar{D}<sup>{*0}$ around the $D<sup>0\bar{D}<sup>{*0}$ threshold. It is shown that data can be reproduced with similar quality for the $X(3872)$ being a bound {\it and/or} a virtual state. We also find that the $X(3872)$ might be a higher-order virtual-state pole (double or triplet pole), in the limit in which the small $D<sup>{*0}$ width vanishes. Once the latter is restored the corrections to the pole position are non-analytic and much bigger than the $D<sup>{*0}$ width itself. The $X(3872)$ compositeness coefficient in $D<sup>0\bar{D}<sup>{*0}$ ranges from nearly 0 up to 1 in the different scenarios.