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

We perform numerical comparison of the fragmentation mechanism of charmonium production ($g,gto c,bar{c}$ followed by $ctopsi,c$) with the full leading order calculation ($g,gtopsi,c,bar{c}$ 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.