  Within the NRQCD factorization framework, we compute the
next-to-leading-order QCD corrections to the gluon fragmentation into the
${}^1S_0^{(1,8)}$ Fock components of a quarkonium, at the lowest order in
velocity expansion. We follow the operator definition of the fragmentation
function advanced by Collins and Soper. The key technique underpinning our
calculation is the sector decomposition method widely used in the area of
multi-loop computation. It is found that the NLO QCD corrections have
significant effects, and qualitatively modify the profiles of the corresponding
leading-order fragmentation functions.
