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Fragmentation is the dominant mechanism for heavy quarkonia production with large transverse momentum. The study of heavy quarkonium production is a powerful tool to understand the dynamics of strong interactions. In this work, we study the effect of Fermi motion of constituents on the direct fragmentation of the \( J/\psi\) and \( \Upsilon\) states from the gluon using a light-cone wave function. Following this study, we compute the process-independent fragmentation functions (FFs) for a gluon to fragment into these bound states. Consistent with such a wave function we set up the kinematics of a gluon fragmenting into a quarkonium such that the Fermi motion of the constituents splits into longitudinal as well as transverse direction. In all previous calculations of heavy quarkonia FFs, by ignoring the Fermi motion of constituents, a delta function form was approximated for the meson wave function. Here, we present our numerical results for the \( g\rightarrow J/\psi\) and \( g\rightarrow \Upsilon\) FFs and show how the proposed meson wave function improves the previous results.
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