  It has been a long-standing problem that the rare electromagnetic decay
process $J/\psi\to 3\gamma$ is plagued with both large and negative radiative
and relativistic corrections. To date it remains futile to make a definite
prediction to confront with the branching fraction of $J/\psi\to 3\gamma$
recently measured by the \textsf{CLEO-c} and \textsf{BESIII} Collaborations. In
this work, we investigate the joint perturbative and relativistic correction
(i.e. the ${\mathcal O}(\alpha_s v^2)$ correction, where $v$ denotes the
characteristic velocity of the charm quark inside the $J/\psi$) for this decay
process, which turns out to be very significant. After incorporating the
contribution from this new ingredient, with the reasonable choice of the input
parameters, we are able to account for the measured decay rates in a
satisfactory degree.
