We study the fragmentation function of the gluon to color-octet 3S1 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 3S1 soft gluon distribution (SGD). The short distance hard part is calculated to the next-to-leading order in αs and all orders in velocity expansion. By deriving and solving the renormalization group equation of the SGD, 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.
We have studied color-octet contributions for J/ψ inclusive production at B factories, i.e., e+e− → J/ψ(3$$ {P}_J^{\left[8\right]} $$ P J 8 ,1$$ {S}_0^{\left[8\right]} $$ S 0 8 ) + $$ {X}_{\mathrm{non}-c\overline{c}} $$ X non − c c ¯ , using the soft gluon factorization (SGF) approach, in which the J/ψ energy spectrum is expressed in a form of perturbatively calculable short-distance hard parts convoluted with one-dimensional soft gluon distributions (SGDs). The series of velocity corrections originated from kinematic effect can be naturally resummed in this approach. Short-distance hard parts have been calculated analytically to next-to-leading order in αs. Renormalization group equations for SGDs have been derived and solved, which resums Sudakov logarithms originated from soft gluon emissions. Our final result gives a upper bound for color-octet matrix elements consistent with that extracted from hadron colliders. This may relieve the well-known universality problem in the NRQCD factorization.As a comparison, we also analytically calculated short-distance hard parts in the NRQCD factorization, with Sudakov logarithms resummed by using soft collinear effective theory. The comparison shows that velocity corrections from kinematic effect, which have been resummed in SGF, are significant for phenomenological study. Furthermore, it is found that Sudakov logarithms originated from soft gluon emissions are very important, while it is not the case for Sudakov logarithms originated from jet function. Therefore, the partial Sudakov resummation in SGF has already captured the main physics.
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