Production of $X(3872)$ in $e^+e^-$ annihilation*

The $X(3872)$ can be produced in $e^+e^-$ annihilation by the creation of $D^{*0}\bar D{}^{*0}$ from a virtual photon, followed by the rescattering of the charm mesons into $X(3872)$ and a photon. The cross section for $e^+e^-\to X(3872)\gamma$ has a narrow peak 2.2 MeV above the $D^{*0}\bar D {}^{*0}$ threshold. This peak is the result of a triangle singularity. We calculate the normalized cross section near the $D^{*0}\bar D{}^{*0}$ threshold. The calculated peak is large enough to be measured by the BESIII collaboration. The observation of this peak would provide strong evidence for the identification of the $X(3872)$ as a charm-meson molecule.