  We study the electromagnetic Dalitz decay modes of $J/\psi \rightarrow P
l^+l^-$ ($P= \pi^0$, $\eta$ or $\eta^\prime$). In these decays, the lepton
pairs are formed by internal conversion of an intermediate virtual photon with
invariant mass $m_{l^+l^-}$. Study of the effective-mass spectrum of the
$l^+l^-$ will shed light on the dynamic transition form factor $F_{J/\psi
P}(q^2)$ ($q^2 = m^2_{l^+l^-}$), which characterizes the electromagnetic
structure arising at the vertex of the transition $J/\psi$ to pseudoscalars. We
also discuss the direct productions of a GeV scale vector $U$ boson in these
processes $J/\psi \rightarrow PU$ ($U\rightarrow l^+l^-$). It is responsible
for mediating a new $U(1)_d$ interaction, as recently exploited in the context
of weakly interacting massive particle dark matter.
  In this paper, we firstly use the usual pole approximation for the form
factor to estimate the decay rate of $J/\psi \rightarrow P l^+l^-$ in the
standard model. Then the reach of searching for the dark photon is estimated.
We suggest that these Dalitz decays can be used to search for the light $U$
boson in the BESIII experiment with a huge $J/\psi$ data set.
