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

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