Possible precise measurements of the $X(3872)$ mass with the $e^{+}{e}^{-}\to {\pi }^0\gamma X(3872)$ and $p\bar{p}\to \gamma X(3872)$ reactions

Abstract

It was recently proposed that the $X(3872)$ binding energy, the difference between the $D^0{\bar{D}}^{\ast 0}$ threshold and the $X(3872)$ mass, can be precisely determined by measuring the $\gamma X(3872)$ line shape from a short-distance $D^{\ast 0}{\bar{D}}^{\ast 0}$ source produced at high-energy experiments. Here, we investigate the feasibility of such a proposal by estimating the cross sections for the $e^{+}{e}^{-}\to {\pi }^0\gamma X(3872)$ and $p\bar{p}\to \gamma X(3872)$ processes considering the $D^{\ast 0}{\bar{D}}^{\ast 0}{D}^0/{\bar{D}}^{\ast 0}{D}^{\ast 0}{\bar{D}}^0$ triangle loops. These loops can produce a triangle singularity slightly above the $D^{\ast 0}{\bar{D}}^{\ast 0}$ threshold. It is found that the peak structures originating from the $D^{\ast 0}{\bar{D}}^{\ast 0}$ threshold cusp and the triangle singularity are not altered much by the energy dependence introduced by the $e^{+}{e}^{-}\to {\pi }^0{D}^{\ast 0}{\bar{D}}^{\ast 0}$ and $p\bar{p}\to {\bar{D}}^{\ast 0}{D}^{\ast 0}$ production parts or by considering a finite width for the $X(3872)$. We find that $\sigma (e^{+}{e}^{-}\to {\pi }^0\gamma X(3872))\times \mathrm{Br}(X(3872)\to {\pi }^{+}{\pi }^{-}J/\psi )$ is $\mathcal{O}(0.1\mathrm{fb})$ with the $\gamma X(3872)$ invariant mass integrated from 4.01 to 4.02 GeV and the c.m. energy of the $e^{+}{e}^{-}$ pair fixed at 4.23 GeV. The cross section $\sigma (p\bar{p}\to \gamma X(3872))\times \mathrm{Br}(X(3872)\to {\pi }^{+}{\pi }^{-}J/\psi )$ is estimated to be of $\mathcal{O}(10\mathrm{pb})$. Our results suggest that a precise measurement of the $X(3872)$ binding energy can be done at PANDA.

Comments: 29 pages, 11 figures