Effects of charged charm mesons on the line shapes of theX(3872)

Abstract

The quantum numbers ${J}^{PC}={1}^{++}$ of the $X(3872)$ and the proximity of its mass to the ${D}^{*0}{\overline{D}}^{0}$ threshold imply that it is either a loosely bound hadronic molecule whose constituents are a superposition of ${D}^{*0}{\overline{D}}^{0}$ and ${D}^{0}{\overline{D}}^{*0}$ or it is a virtual state of charm mesons. The line shapes of the $X(3872)$ can discriminate between these two possibilities. At energies within a few MeV of the ${D}^{*0}{\overline{D}}^{0}$ threshold, the lines shapes of the $X$ produced in $B\ensuremath{\rightarrow}K$ transitions are determined by its binding energy and its width. Their normalizations are determined by a short-distance constant that is different for ${B}^{+}\ensuremath{\rightarrow}{K}^{+}$ and ${B}^{0}\ensuremath{\rightarrow}{K}^{0}$. At energies comparable to the 8 MeV splitting between the ${D}^{*0}{\overline{D}}^{0}$ and ${D}^{*+}{D}^{\ensuremath{-}}$ thresholds, the charged meson channels ${D}^{*+}{D}^{\ensuremath{-}}$ and ${D}^{+}{D}^{*\ensuremath{-}}$ have a significant effect on the line shapes of the $X$. We calculate the line shapes taking into account the resonant coupling between the charged and neutral ${1}^{++}$ channels. The line shapes and their normalizations depend on one additional scattering parameter and two additional short-distance constants associated with the $B\ensuremath{\rightarrow}K$ transitions. The line shapes of the $X$ resonance depend on its decay channel; they are different for $J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, $J/\ensuremath{\psi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$, and ${D}^{0}{\overline{D}}^{0}{\ensuremath{\pi}}^{0}$. The line shapes are also different for $X$ produced in ${B}^{+}$ decays and in ${B}^{0}$ decays. Some conceptual errors in previous work on this problem are pointed out.