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 $D^0{\bar{D}}^{\ast 0}$ $S$-wave. In terms of it we analyze the CDF data on inclusive $p\bar{p}$ scattering to $J/\psi {\pi }^{+}{\pi }^{-}$, and the Belle and BaBar data on $B$ decays to $KJ/\psi {\pi }^{+}{\pi }^{-}$ and $KD{\bar{D}}^{\ast 0}$ around the $D^0{\bar{D}}^{\ast 0}$ 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 $D^{\ast 0}$ width vanishes. Once the latter is restored the corrections to the pole position are non-analytic and much bigger than the $D^{\ast 0}$ width itself. The $X(3872)$ compositeness coefficient in $D^0{\bar{D}}^{\ast 0}$ ranges from nearly 0 up to 1 in the different scenarios.

43 pages, 9 figures, 3 tables. The paper has been much extended. Extra algebraic derivations and matching with QFT models are also included

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