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Heavy quarkonium production is a powerful implement to study the stronginteraction dynamics and QCD theory. Fragmentation is the dominant productionmechanism for heavy quarkonia with large transverse momentum. With the largeheavy quark mass, the relative motion of the heavy quark pair inside a heavyquarkonium is effectively nonrelativistic and it is also well-known that theirfragmentation functions can be calculated in the perturbative QCD framework.Here, we analytically calculate the process-independent fragmentation functionsfor a gluon to split into the spin-singlet and -triplet $S$-wave heavyquarkonia using three different scenarios. We will show that the fragmentationprobability of the gluon into the spin-triplet bound-state is the biggest one.

Spin-dependent fragmentation functions of Gluon splitting into heavy quarkonia considering three different scenarios