  We study the fragmentation function of the gluon to color-octet $^3S_1$ heavy
quark-antiquark pair using the soft gluon factorization (SGF) approach, which
expresses the fragmentation function in a form of perturbative short-distance
hard part convoluted with one-dimensional color-octet $^3S_1$ soft gluon
distribution (SGD). The short distance hard part is calculated to
next-to-leading order in $\alpha_s$ and a renormalization group equation for
the SGD is derived. By solving the renormalization group equation, threshold
logarithms are resummed to all orders in perturbation theory. The comparison
with gluon fragmentation function calculated in NRQCD factorization approach
indicates that the SGF formula resums a series of velocity corrections in NRQCD
which are important for phenomenological study.
