  We consider the electron-positron annihilation process into hadrons
$R_{e^+e^-}$ up to $\mathcal{O}(\alpha_{s}^{3})$ and we adopt the smearing
method suggest by Poggio, Quinn and Weinberg to confront the experimental data
with theory. As a theoretical model we use a QCD coupling constant frozen in
the low energy regime, where this coupling can be parameterised in terms of an
effective dynamical gluon mass ($m_g$) which is determined through
Schwinger-Dyson equations. In order to find the best fit between experimental
data and theory we perform a $\chi^2$ study, that, within the uncertainties of
the 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 phenomenological
determinations of this ratio and lead to an infrared effective charge
$\alpha_s(0) \approx 0.7$. We comment how this effective charge may affect the
global duality mass scale that indicates the frontier between perturbative and
non-perturbative physics.
