Estimates of the $X(3872)$ cross section at a hadron collider

The claim that the $X(3872)$ meson cannot be a charm-meson molecule because its prompt production cross section at hadron colliders is too large is based on an upper bound in terms of a cross section for producing charm-meson pairs. Assuming $X$ is sufficiently weakly bound, we derive an equality between the $X$ cross section and a charm-meson pair cross section that takes into account the threshold enhancement from the $X$ resonance. The cross section for producing $X$ is equal to that for producing $D^{*0}{\overline{D}}^0$ integrated up to a relative momentum $k_{\max }=7.7{\gamma }_X$, where ${\gamma }_X$ is the binding momentum of $X$. We also derive an order-of-magnitude estimate of the $X$ cross section in terms of a naive charm-meson pair cross section that does not take into account the threshold enhancement, such as that produced by a Monte Carlo event generator. The cross section for producing $X$ can be approximated by the naive cross section for producing $D^{*0}{\overline{D}}^0$ integrated up to a relative momentum $k_{\max }$ of order $(m_{\pi }^2{\gamma }_X{)}^{1/3}$. The estimates of the prompt $X$ cross section at hadron colliders are consistent with the cross sections observed at the Tevatron and the LHC.

Figure 1

Universal energy distribution $\mathrm{Im}[f_X(E+i\epsilon )]$ in Eq. (4) as a function of the energy $E$. The tall rectangle below the scattering threshold at $E=0$ represents the delta function from production of the bound state and should be multiplied by 10. The dashed curve is the naive energy distribution in the absence of the resonance given by Eq. (5) with $\mathrm{\Lambda }=5{\gamma }_X$.