Violations of Quark-Hadron Duality in Low-Energy Determinations of $\alpha_s$

Using the spectral functions measured in $\tau$ decays, we investigate the actual numerical impact of duality violations on the extraction of the strong coupling. These effects are tiny in the standard $\alpha_s(m_\tau^2)$ determinations from integrated distributions of the hadronic spectrum with pinched weights, or from the total $\tau$ hadronic width. The pinched-weight factors suppress very efficiently the violations of duality, making their numerical effects negligible in comparison with the larger perturbative uncertainties. However, combined fits of $\alpha_s$ and duality-violation parameters, performed with non-protected weights, are subject to large systematic errors associated with the assumed modelling of duality-violation effects. These uncertainties have not been taken into account in the published analyses, based on specific models of quark-hadron duality.