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Using optical theorem within sophisticated fits for exclusive hadronicproductions in $e^+e^-$ collisions the interference between leptonic andhadronic vacuum polarization functions is considered and applied forcalculation of $\mu-$pair productions below 3 GeV. Particularly, the dispersionrelation for the pion production form factor was obtained, confirming apossible evidence for the light $\rho(1250)$ resonance. The inflated error wasintroduced and used to discriminate the quality of various phenomenologicalfits. Resulting hadronic vacuum polarization was compared with otherconventional sources and compared with KLOE experiment for $\mu^{-}\mu^{+}$production at $\phi $ meson energy. Furthermore, the obtained running finestructure coupling is compared with KLOE2 experiment for radiative return$\mu^{-}\mu^{+}$ production at $ \omega/\rho $ meson energy and carefulanalysis of the error stemming form various fits of combined data was made anddiscussed.

Muon pair production with hadronic vacuum polarization re-evaluated using precise data fits