  We search for $J/\psi$ radiative decays into a weakly interacting neutral
particle, namely an invisible particle, using the $J/\psi$ produced through the
process $\psi(3686)\to\pi^+\pi^-J/\psi$ in a data sample of
$(448.1\pm2.9)\times 10^6$ $\psi(3686)$ decays collected by the BESIII detector
at BEPCII. No significant signal is observed. Using a modified frequentist
method, upper limits on the branching fractions are set under different
assumptions of invisible particle masses up to 1.2 $\mathrm{\ Ge\kern -0.1em
V}/c^2$. The upper limit corresponding to an invisible particle with zero mass
is 7.0$\times 10^{-7}$ at the 90\% confidence level.
