  Based on the factorization formalism of nonrelativistic quantum
chromodynamics (NRQCD), we calculate the production cross section for the
charmonium and the bottomonium produced by the hard photoproduction processes
and fragmentation processes in relativistic heavy ion collisions. It is shown
that the existing experimental data on heavy quarkonium production at the Large
Hadron Collider (LHC) can be described in the framework of the NRQCD formalism,
and the phenomenological values of matrix elements for color-singlet and
color-octet components give the main contribution. The numerical results of
photoproduction processes and fragmentation processes for the heavy quarkonium
production become prominent in p-p collisions and Pb-Pb collisions at LHC
energies.
