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We report a measurement of the cross section for the process$e^+e^-\to\pi^+\pi^-J/\psi$ around the $X(3872)$ mass in search for the directformation of $e^+e^-\to X(3872)$ through the two-photon fusion process. Noenhancement of the cross section is observed at the $X(3872)$ peak and an upperlimit on the product of electronic width and branching fraction of$X(3872)\to\pi^+\pi^-J/\psi$ is determined to be$\Gamma_{ee}\times\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)<7.5\times10^{-3}\,\text{eV}$at $90\,\%$ confidence level under an assumption of total width of$1.19\pm0.21$ MeV. This is an improvement of a factor of about $17$ compared tothe previous limit. Furthermore, using the latest result of$\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)$, an upper limit on the electronicwidth $\Gamma_{ee}$ of $X(3872)$ is obtained to be $<0.32\,\text{eV}$ at the$90\,\%$ confidence level.

Measurement of the $e^+e^-\to\pi^+\pi^-J/\psi$ cross section in the vicinity of 3.872 GeV