Next-to-leading-log renormalization-group running in heavy-quarkonium creation and annihilation

In the framework of potential nonrelativistic QCD, we obtain the next-to-leading-log renormalization-group running of the matching coefficients for the heavy-quarkonium production currents near threshold. This allows us to obtain S-wave heavy-quarkonium production or annihilation observables with next-to-leading-log accuracy within perturbative QCD. In particular, we give expressions for the decays of heavy quarkonium to $e^{+}{e}^{-}$ and to two photons. We also compute the $O\left(m{\alpha }^8{\ln }^3\alpha \right)$ corrections to the hydrogen spectrum.