Perturbative $O\left({\alpha }_s^2\right)$ corrections to the hadronic cross section near heavy quark-antiquark thresholds in $e^{+}{e}^{-}$ annihilation

It is demonstrated how perturbative $O\left({\alpha }_s^2\right)$ relativistic corrections to the nonrelativistic stable heavy quark-antiquark production cross section in $e^{+}{e}^{-}$ annihilation based on a Coulombic QCD potential can be systematically calculated using the concept of effective field theories. The $O\left({\alpha }_s^2\right)$ corrections from the relativistic energy-momentum relation, the relativistic phase space corrections and the $O\left({\alpha }_s^2\right)$ corrections involving the group theoretical factors $C_F^2$ and $C_{F}T$ are determined explicitly. For the case of $t\overline{t}$ production the sum of these corrections amounts to 3%–7% over the whole threshold regime and is insensitive to variations in the top width. Perturbative $O\left({\alpha }_s^2\right)$ corrections to the leptonic decay width of heavy quark-antiquark ${}^3{S}_1$ vector resonances are extracted.