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The production of heavy quarkonia is a powerful tool to test our understanding of strong interaction dynamics. It is well-known that the dominant production mechanism for heavy quarkonia with large transverse momentum is fragmentation. In this work we, analytically, calculate the QCD leading order contribution to the process-independent fragmentation functions (FFs) for a gluon to split into the vector ($J/\psi$) and pseudoscalar ($\eta_c$) $S$-wave charmonium states. The analyses of this paper differ in which we present, for the first time, an analytical form of the
$g\rightarrow J/\psi$ FF using a different approach (Suzuki's model) in comparison with other results presented in literatures, where the Braaten's scheme was used and
the two-dimensional integrals were presented for the gluon FFs which must be evaluated numerically.
The universal fragmentation probability for the $g\rightarrow J/\psi$ is about $10^{-6}$ which is in good consistency with the result obtained in the Braaten's model. %We show that our analytical result for the $g\rightarrow J/\psi$ FF is in good consistency
%with the numerical results obtained through the Braaten's scheme.
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