  A search for the invisible decays of $\omega$ and $\phi$ mesons in $J/\psi\to
\omega (\phi)\eta$ transitions has been performed by the BESIII Collaboration
very recently. Inspired by this experimental study, we compute the lowest order
contribution to branching ratios of ${\cal B}(V\to\bar{\nu}\nu)$ with $V$
denoting $\rho,\;\omega,\;\phi$, as the standard model background to these
invisible decays. Our predictions are far below the upper bounds given by the
BESIII experiment. We also analyze the $J/\psi\to
\eta(\eta^\prime)\bar{\nu}\nu$ processes, and estimate their decay rates.
Furthermore, the invisible decays of light pseudoscalar mesons $P$ including
$\pi^0$, $\eta$, and $\eta^\prime$ are reexamined in the present note. It is
shown that, due to the helicity suppression of the two-neutrino final state,
the standard model contributions to $P\to{\rm invisible}$ decays are dominated
by $P\to\bar{\nu}\nu \bar{\nu}\nu$ processes.
