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

The precise knowledge of the onset of the cross section $\sigma(e^+e^- \to \tau^+\tau^-)$ at the threshold is necessary for improving the accuracy of determination of the $\tau$ mass from the threshold measurements. The QED radiative corrections of relative order $\alpha$ and $\alpha^2$, additional to the well known Coulomb factor, are considered in the threshold region. The correction terms of order $\alpha^2$ are calculated, which contain coefficients enhanced by large parameters. As a result it is argued that the known $O(\alpha)$ 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.