Production of $X(3872)$ Accompanied by a Soft Pion at Hadron Colliders

If the $X(3872)$ is a weakly bound charm-meson molecule, it can be produced by the creation of $D^{*0} \bar{D}^0$ or $D^{0} \bar{D}^{*0}$ at short distances followed by the formation of the bound state from the charm-meson pair. The $X$ can also be produced by the creation of $D^* \bar{D}^*$ at short distances followed by the rescattering of the charm-meson pair into $X \pi$. At a high-energy hadron collider, the prompt cross section from this mechanism has a narrow peak in the $X \pi$ invariant mass distribution near the $D^* \bar D^*$ threshold from a charm-meson triangle singularity. An order-of-magnitude estimate of the ratio of the cross section for producing the peak in the $X \pi^\pm$ distribution to the cross section for producing $X$ without an accompanying pion suggests that the peak may be observable at the LHC.