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We study the electromagnetic Dalitz decay modes of $J/\psi \rightarrow Pl^+l^-$ ($P= \pi^0$, $\eta$ or $\eta^\prime$). In these decays, the leptonpairs are formed by internal conversion of an intermediate virtual photon withinvariant mass $m_{l^+l^-}$. Study of the effective-mass spectrum of the$l^+l^-$ will shed light on the dynamic transition form factor $F_{J/\psiP}(q^2)$ ($q^2 = m^2_{l^+l^-}$), which characterizes the electromagneticstructure arising at the vertex of the transition $J/\psi$ to pseudoscalars. Wealso discuss the direct productions of a GeV scale vector $U$ boson in theseprocesses $J/\psi \rightarrow PU$ ($U\rightarrow l^+l^-$). It is responsiblefor mediating a new $U(1)_d$ interaction, as recently exploited in the contextof weakly interacting massive particle dark matter.  In this paper, we firstly use the usual pole approximation for the formfactor to estimate the decay rate of $J/\psi \rightarrow P l^+l^-$ in thestandard 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.

Study of the electromagnetic transitions $J/\psi \rightarrow P l^+l^-$ and probe dark photon