J/psi + $\gamma$ production at the Tevatron energy

We study the process \bar p p \rightarrow J/\psi + \gamma +X at the Tevatron energy (\sqrt{s} = 1.8~TeV). The perturbative QCD contributions to this process from the gluon-fusion and the fragmentation mechanisms are computed. For the entire range of p_T that can be probed at the Tevatron, the fusion contribution is found to be dominant. Consequently the QCD prediction for this process has better precision than the purely hadronic production of J/\psi. An experimental study of this process at the Tevatron will help to clarify several aspects of quarkonium production at large p_T.