Massive Quark Production in Electron Positron Annihilation to Order ${\alpha }_s^2$

Recent analytical and numerical results for the three-loop polarization function allow to present a phenomenological analysis of the cross section for massive quark production in electron positron annihilation to order ${\alpha }_s^2$. Numerical predictions based on fixed order perturbation theory are presented for charm and bottom production above 5 and 11.5 GeV, respectively. The contribution from these energy regions to $\alpha (M_Z^2)$, the running QED coupling constant at scale M_Z, are given. The dominant terms close to threshold, i.e. in an expansion for small quark velocity $\beta$, are presented.