QCD factorization for quarkonium production in hadron collisions at low transverse momentum

Inclusive production of a quarkonium eta(c,b) in hadron collisions at low transverse momentum can be used to extract various transverse-momentum-dependent (TMD) gluon distributions of hadrons, provided the TMD factorization for the process holds. The factorization involving unpolarized TMD gluon distributions of unpolarized hadrons has been examined with on-shell gluons at the one-loop level. In this work, we study the factorization at the one-loop level with a diagram approach in the most general case, where all TMD gluon distributions at leading twist are involved. We find that the factorization holds and the perturbative effects are represented by one perturbative coefficient. Since the initial gluons from hadrons are off shell, in general, there exists the so-called superleading region found recently. We find that the contributions from this region can come from individual diagrams at the one-loop level, but they are canceled in the sum. Our factorized result for the differential cross section is explicitly gauge invariant.