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We consider the electron-positron annihilation process into hadrons$R_{e^+e^-}$ up to $\mathcal{O}(\alpha_{s}^{3})$ and we adopt the smearingmethod suggest by Poggio, Quinn and Weinberg to confront the experimental datawith theory. As a theoretical model we use a QCD coupling constant frozen inthe low energy regime, where this coupling can be parameterised in terms of aneffective dynamical gluon mass ($m_g$) which is determined throughSchwinger-Dyson equations. In order to find the best fit between experimentaldata and theory we perform a $\chi^2$ study, that, within the uncertainties ofthe approach, has a minimum value when $m_g/\Lambda_{QCD}$ is in the range $1.2\, - \, 1.4$. These values are in agreement with other phenomenologicaldeterminations of this ratio and lead to an infrared effective charge$\alpha_s(0) \approx 0.7$. We comment how this effective charge may affect theglobal duality mass scale that indicates the frontier between perturbative andnon-perturbative physics.

$R_{e^+ e^-}$ and an effective QCD charge