Charm-meson Triangle Singularity in ${e^+e^-}$ Annihilation into ${ D^{*0} \bar{D}^0 + \gamma }$

We calculate the cross section for $e^+ e^-$ annihilation into $D^{*0} \bar D^0 +\gamma$ at center-of-mass energies near the $D^{*0} \bar D^{*0}$ threshold under the assumption that $X(3872)$ is a weakly bound charm meson molecule. The Dalitz plot has a $\bar D^{*0}$ resonance band in the squared invariant mass $t$ of $\bar D^0 \gamma$. In the limit as the decay width of the $D^{*0}$ goes to 0, the Dalitz plot also has a narrow band in the squared invariant mass $u$ of $D^{*0} \bar D^0$ from a charm-meson triangle singularity. At the physical value of the $D^{*0}$ width, the narrow band reduces to a shoulder. Thus the triangle singularity cannot be observed directly as a peak in a differential cross section as a function of $u$. It may however be observed indirectly as a local minimum in the $t$ distribution for events with $u$ below the triangle singularity. The minimum is produced by the Schmid cancellation between triangle loop diagrams and a tree diagram. The observation of this minimum would support the identification of $X(3872)$ as a weakly bound charm meson molecule.