Inelastic heavy quarkonium photoproduction in $p$-$p$ and $Pb$-$Pb$ collisions at LHC energies

Abstract: We study the inelastic charmonium ($J/\psi$, $\psi(2S)$) and bottomonium ($\Upsilon(nS)$) photoproduction and fragmentation processes in $p$-$p$ and $Pb$-$Pb$ collisions at LHC energies, where the ultra-incoherent photon emission is included. In the framework of the NRQCD factorization approach, an exact treatment is developed which recovers Weizs\"{a}cker-Williams approximation (WWA) near the region $Q^{2}\sim0$, where the methods of Martin-Ryskin and BCCKL are used to avoid double counting. We calculate the $Q^{2}$, $y$, $z$, $\sqrt{s}$, $p_{T}$ dependent and the total cross sections. It turns out that the inelastic photoproduction and fragmentation processes provide valuable contributions to the heavy quarkonium production, especially in the large $p_{T}$ regions. While the relative contribution of ultra-incoherent photon channel is very important, which rapidly increases along with the growing quarkonium mass, and begins to dominate the photoproduction processes at large $p_{T}$ ranges. Moreover, we obtain the complete validity scopes of WWA in inelastic heavy quarkonium photoproduction in heavy-ion collisions. WWA has a much higher accuracy at high energies and in $Pb$-$Pb$ collisions. The existing photon spectra are generally derived beyond the applicable scopes of WWA, and the double counting exists when the different channels are considered simultaneously.