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Systematic study of the experimental measurements on $J/ψ$ cross section and kinematic distribution in $p+p$ collisions at different energies

The world experimental data on cross section and kinematic distribution in $p+p$ and $p+A$ collisions at $\sqrt{s}$ = 6.8 - 7000 GeV are examined in systematic way. The $\sqrt{s}$ dependence of the inclusive cross section, rapidity and transverse momentum distributions are studied phenomenologically. We explore empirical formulas to obtain the total cross section, rapidity and transverse momentum ($p_{T}$) distribution. This is crucial for the interpretation of A$+$A $J/ψ$ results at RHIC when the $p+p$ reference data are not available. In addition, the cross section at mid-rapidity and transverse momentum distributions in $p+p$ collisions at $\sqrt{s}$ = 39 and 62.4 GeV are evaluated.