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

Abstract: 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.