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We have performed a systematic study of $J/\psi$ and $\psi(2S)$ production in$p-p$ collisions at different LHC energies and at different rapidities usingthe leading order (LO) non-relativistic QCD (NRQCD) model of heavy quarkoniumproduction. We have included the contributions from $\chi_{cJ}$ ($J$ = 0, 1, 2)and $\psi(2S)$ decays to $J/\psi$. The calculated values have been comparedwith the available data from the four experiments at LHC namely, ALICE, ATLAS,CMS and LHCb. In case of ALICE, inclusive $J/\psi$ and $\psi(2S)$cross-sections have been calculated by including the feed-down from $B$ mesonusing Fixed-Order Next-to-Leading Logarithm (FONLL) formalism. It is found thatall the experimental cross-sections are well reproduced for $p_T >$ 4 GeVwithin the theoretical uncertainties arising due to the choice of thefactorization scale. We also predict the transverse momentum distributions of$J/\psi$ and $\psi(2S)$ both for the direct and feed-down processes at theupcoming LHC energies of $\sqrt{s} =$ 5.1 TeV and 13 TeV for the year 2015.

Systematic study of Charmonium production in pp collisions at the LHC energies