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 using the leading order (LO) non-relativistic QCD (NRQCD) model of heavy quarkonium production. We have included the contributions from $chi_{cJ}$ ($J$ = 0, 1, 2) and $psi(2S)$ decays to $J/psi$. The calculated values have been compared with 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 $B$ meson using Fixed-Order Next-to-Leading Logarithm (FONLL) formalism. It is found that all the experimental cross-sections are well reproduced for $p_T >$ 4 GeV within the theoretical uncertainties arising due to the choice of the factorization scale. We also predict the transverse momentum distributions of $J/psi$ and $psi(2S)$ both for the direct and feed-down processes at the upcoming LHC energies of $sqrt{s} =$ 5.1 TeV and 13 TeV for the year 2015.
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