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We study heavy quarkonium production associated with gluons in $\rm e^+e^-$annihilation as an illustration of the perturbative QCD (pQCD) factorizationapproach, which incorporates the first nonleading power in the energy of theproduced heavy quark pair. We show how the renormalization of the four-quarkoperators that define the heavy quark pair fragmentation functions usingdimensional regularization induces "evanescent" operators that are absent infour dimensions. We derive closed forms for short-distance coefficients forquark pair production to next-to-leading order ($\alpha_s^2$) in the relevantcolor singlet and octet channels. Using non-relativistic QCD (NRQCD) tocalculate the heavy quark pair fragmentation functions up to $v^4$ in thevelocity expansion, we derive analytical results for the differential energyfraction distribution of the heavy quarkonium. Calculations for ${}^3S_1^{[1]}$and ${}^1S_0^{[8]}$ channels agree with analogous NRQCD analytical resultsavailable in the literature, while several color-octet calculations of energyfraction distributions are new. We show that the remaining corrections due tothe heavy quark mass fall off rapidly in the energy of the produced state. Toexplore the importance of evolution at energies much larger than the mass ofthe heavy quark, we solve the renormalization group equation perturbatively totwo-loop order for the ${}^3S_1^{[1]}$ case.

Power expansion for heavy quarkonium production at next-to-leading order in $\rm e^+e^-$ annihilation