  In the framework of potential NRQCD, we obtain the next-to-leading-log
renormalization-group running of the matching coefficients for the heavy
quarkonium production currents near threshold. This allows to obtain S-wave
heavy-quarkonium production/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(m\alpha^8\ln^3\alpha) corrections to the Hydrogen spectrum.
