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It was recently proposed that the $X(3872)$ binding energy, the differencebetween the $D^0\bar D^{*0}$ threshold and the $X(3872)$ mass, can be preciselydetermined by measuring the $\gamma X(3872)$ line shape from a short-distance$D^{*0}\bar D^{*0}$ source produced at high-energy experiments. Here, weinvestigate the feasibility of such a proposal by estimating the cross sectionsfor the $e^+e^-\to\pi^0\gamma X(3872)$ and $p\bar p\to\gamma X(3872)$ processesconsidering the $D^{*0}\bar D^{*0}D^0/\bar D^{*0}D^{*0}\bar D^0$ triangleloops. These loops can produce a triangle singularity slightly above the$D^{*0}\bar D^{*0}$ threshold. It is found that the peak structures originatingfrom the $D^{*0}\bar D^{*0}$ threshold cusp and the triangle singularity arenot altered much by the energy dependence introduced by the$e^+e^-\to\pi^0D^{*0}\bar D^{*0}$ and $p\bar p\to\bar D^{*0}D^{*0}$ productionparts or by considering a finite width for the $X(3872)$. We find that$\sigma(e^+e^-\to\pi^0\gamma X(3872)) \times {\rmBr}(X(3872)\to\pi^+\pi^-J/\psi)$ is $\mathcal{O}(0.1~{\rm fb})$ 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 $\sigma(p\barp\to\gamma X(3872))\times {\rm Br}(X(3872)\to\pi^+\pi^-J/\psi)$ is estimated tobe of $\mathcal{O}(10~{\rm pb})$. Our results suggest that a precisemeasurement of the $X(3872)$ binding energy can be done at PANDA.

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