We perform a numerical comparison between the fragmentation approach and the full $\mathcal{O}({\alpha }_s^5)$ calculation for the production of $J/\psi$ mesons in the partonic subprocess $q\overline{q}\to \psi ggg$. We make a similar comparison for the color-octet channel by considering the subprocess $gg\to {}^3{S}_1^{[8]}g$. We find that the nonfragmentation contributions in the color-singlet channel remain important up to $J/\psi$ transverse momenta as large as approximately 30 or 40 GeV, while the convergence of the factorized and full calculations in the color-octet channel is seen already at $p_{\psi T}\approx 10\mathrm{GeV}$.

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• Feynman diagrams
• Particle interactions
• Perturbation theory
• Phenomenology
• Scattering amplitudes
• Total cross sections