Effects of charged charm mesons on the line shapes of the $X$(3872)
Eric Braaten and Meng Lu

The quantum numbers $J^{PC} = 1^{++}$ of the $X(3872)$ and the proximity of its mass to the $D^{*0} \bar D^0$ threshold imply that it is either a loosely-bound hadronic molecule whose constituents are a superposition of $D^{*0} \bar D^0$ and $D^0 \bar D^{*0}$ 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 $D^{*0} \bar D^0$ threshold, the lines shapes of the $X$ produced in $B \to K$ transitions are determined by its binding energy and its width. Their normalizations are determined by a short-distance constant that is different for $B^+ \to K^+$ and $B^0 \to K^0$. At energies comparable to the 8 MeV splitting between the $D^{*0} \bar D^0$ and $D^{*+} D^-$ thresholds, the charged meson channels $D^{*+} D^-$ and $D^+ D^{*-}$ 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 $B \to K$ transitions. The line shapes of the $X$ resonance depend on its decay channel; they are different for $J/\psi \, \pi^+ \pi^-$, $J/\psi \, \pi^+ \pi^-\pi^0$, and $D^0 \bar D^0 \pi^0$. 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.

© 2008 The American Physical Society.