Line shapes of theX(3872)
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Abstract
If the quantum numbers of the $X(3872)$ are ${J}^{PC}={1}^{++}$, the measurement of its mass implies that it is either a loosely bound hadronic molecule whose constituents are a superposition of the charm mesons pairs ${D}^{*0}{\overline{D}}^{0}$ and ${D}^{0}{\overline{D}}^{*0}$ or else it is a virtual state of these charm mesons. Its binding energy is small enough that the decay width of a constituent ${D}^{*0}$ or ${\overline{D}}^{*0}$ has a significant effect on the line shapes of the $X$ resonance. We develop a simple approximation to the line shapes that takes into account the effect of the ${D}^{*0}$ width as well as inelastic scattering channels of the charm mesons. We carry out a simultaneous fit to the line shapes in the $J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${D}^{0}{\overline{D}}^{0}{\ensuremath{\pi}}^{0}$ channels measured in the decays ${B}^{+}\ensuremath{\rightarrow}{K}^{+}+X$ by the Belle Collaboration. The best fit corresponds to the $X(3872)$ being a bound state just below the ${D}^{*0}{\overline{D}}^{0}$ threshold, but a virtual state just above the ${D}^{*0}{\overline{D}}^{0}$ threshold is not excluded.