On the Application of Gluon to Heavy Quarkonium Fragmentation Functions

We analyze the uncertainties induced by different definitions of the momentum fraction $z$ in the application of gluon to heavy quarkonium fragmentation function. We numerically calculate the initial $g \to J / \psi$ fragmentation functions by using the non-covariant definitions of $z$ with finite gluon momentum and find that these fragmentation functions have strong dependence on the gluon momentum $\vec{k}$. As $| \vec{k} | \to \infty$, these fragmentation functions approach to the fragmentation function in the light-cone definition. Our numerical results show that large uncertainties remains while the non-covariant definitions of $z$ are employed in the application of the fragmentation functions. We present for the first time the polarized gluon to $J/\psi$ fragmentation functions, which are fitted by the scheme exploited in this work.