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.