Energy dependence of $J/\psi$ production in pp collisions with the PACIAE model
In this work we investigate the $J/\psi$ production in proton-proton collisions at the center-of-mass energy ($\sqrt{s}$) equal to 2.76, 5.02, 7, 8 and 13 TeV with a parton and hadron cascade model PACIAE 2.2a. It is based on PYTHIA but extended considering the partonic and hadronic rescatterings before and after hadronization, respectively. In the PYTHIA sector the $J/\psi$ production quantum chromodynamics processes are selected specially and a bias factor is proposed correspondingly. The calculated total cross sections, the differential cross sections as a function of the transverse momentum and the rapidity of $J/\psi$ in the forward rapidity region reproduce the corresponding experimental measurements reasonably well. In the mid-rapidity region, the double differential cross sections at $\sqrt{s}=$ 5.02, 7 and 13 TeV are also in a good agreement with the experimental data. Moreover, we predict the double differential cross section as well as the total cross section of $J/\psi$ at $\sqrt{s}=$ 8 TeV, which could be validated when the experimental data is available.