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

A discussion on vacuum polarization correction to the cross-section of $e^+e^-\to\gamma^\ast/\psi\to\mu^+\mu^-$