  Vacuum polarization is a part of the initial-state radiative correction for
the cross-section of $e^+e^-$ annihilation processes. In the energy region in
the vicinity of narrow resonances $J/\psi$ and $\psi(3686)$, the vacuum
polarization contribution from the resonant component has a significant effect
on the line-shape of the lepton pair production cross-section. This paper
discusses some basic concepts and describes an analytical calculation of the
cross-section of $e^+e^-\to\gamma^\ast/\psi\to\mu^+\mu^-$ considering the
single and double vacuum polarization effect of the virtual photon propagator.
Moreover, it presents some numerical comparisons with the traditional
treatments.
