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The onset of e^+e^- -> tau^+ tau^- at threshold revisited

The precise knowledge of the onset of the cross section $σ(e^+e^- \to τ^+τ^-)$ at the threshold is necessary for improving the accuracy of determination of the $τ$ mass from the threshold measurements. The QED radiative corrections of relative order $α$ and $α^2$, additional to the well known Coulomb factor, are considered in the threshold region. The correction terms of order $α^2$ are calculated, which contain coefficients enhanced by large parameters. As a result it is argued that the known $O(α)$ corrections provide the accuracy of the description of the cross section close to $10^{-4}$, rather than $10^{-3}$ as claimed in a recent literature. Also analytical expressions are provided for some limiting cases of the corrections, previously calculated numerically.