Triangle Singularity in $e^+e^-$ Annihilation into $D^{*0} \bar{D}{}^0 + \gamma$ with the Charm Mesons near the $X(3872)$ Resonance*

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 has a threshold enhancement in $D^{*0} \bar D{}^0$ from the $X(3872)$ resonance. The Dalitz plot has a $\bar D{}^{*0}$ resonance band in the $\bar D{}^0 \gamma$ invariant mass and a narrow band in the $D^{*0} \bar D{}^0$ invariant mass from a charm-meson triangle singularity. In the projection of the Dalitz plot onto the $D^{*0} \bar D{}^0$ invariant mass, the peak from the triangle singularity is suppressed by the Schmid cancellation. The measurement of the cross section for $D^{*0} \bar D{}^0 +\gamma$ would allow an accurate prediction of the height of the peak from the triangle singularity in the cross section for $X(3872)+\gamma$.