  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 direct
formation of $e^+e^-\to X(3872)$ through the two-photon fusion process. No
enhancement of the cross section is observed at the $X(3872)$ peak and an upper
limit 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 to
the previous limit. Furthermore, using the latest result of
$\mathcal{B}(X(3872)\to\pi^+\pi^-J/\psi)$, an upper limit on the electronic
width $\Gamma_{ee}$ of $X(3872)$ is obtained to be $<0.32\,\text{eV}$ at the
$90\,\%$ confidence level.
