Abstract
Analyses of the $J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay channel of the $X(3872)$ resonance by the CDF, Belle, and LHCb Collaborations have established its ${J}^{PC}$ quantum numbers as ${1}^{++}$. An analysis of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ invariant mass distribution in the $J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ decay channel by the BABAR Collaboration indicated a preference for ${2}^{\ensuremath{-}+}$ over ${1}^{++}$. We point out that a proper evaluation of the ${\ensuremath{\chi}}^{2}$ in that analysis increases the probability for ${1}^{++}$ from 7.1% to about 18.7%. In the case of quantum numbers ${1}^{++}$, where the $X$ has an $S$-wave coupling to $J/\ensuremath{\psi}\ensuremath{\omega}$, the proximity of the $J/\ensuremath{\psi}\ensuremath{\omega}$ threshold to ${D}^{*}\overline{D}$ thresholds and the narrow width of the $\ensuremath{\omega}$ suggest that the effects of scattering between $J/\ensuremath{\psi}\ensuremath{\omega}$ and charm meson pairs could be significant. We derive invariant mass distributions for $J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ and ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ that take into account $S$-wave scattering between the ${D}^{*0}{\overline{D}}^{0}$, ${D}^{*+}{D}^{\ensuremath{-}}$, and $J/\ensuremath{\psi}\ensuremath{\omega}$ channels. We also analyze the effects of scattering through the ${\ensuremath{\chi}}_{c1}(2P)$ charmonium resonance. We find that scattering effects are unable to produce significant changes in the shape of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ invariant mass distribution.
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