Fragmentation of charm to charmonium in e^+e^- and pp collisions

We perform numerical comparison of the fragmentation mechanism of charmonium production ($g\,g\to c\,\bar{c}$ followed by $c\to\psi\,c$) with the full leading order calculation ($g\,g\to\psi\,c\,\bar{c}$ at ${\O} (\alpha_s^4)$). We conclude that the non-fragmentation contributions remain important up to $J/\psi$ transverse momenta about as large as 50 GeV, thus making questionable the applicability of the fragmentation approximation at smaller transverse momenta.