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

Using optical theorem within sophisticated fits for exclusive hadronic productions in $e^+e^-$ collisions the interference between leptonic and hadronic vacuum polarization functions is considered and applied for calculation of $\mu-$pair productions below 3 GeV. Particularly, the dispersion relation for the pion production form factor was obtained, confirming a possible evidence for the light $\rho(1250)$ resonance. The inflated error was introduced and used to discriminate the quality of various phenomenological fits. Resulting hadronic vacuum polarization was compared with other conventional sources and compared with KLOE experiment for $\mu^{-}\mu^{+}$ production at $\phi$ meson energy. Furthermore, the obtained running fine structure coupling is compared with KLOE2 experiment for radiative return $\mu^{-}\mu^{+}$ production at $\omega/\rho$ meson energy and careful analysis of the error stemming form various fits of combined data was made and discussed.