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.

Figure 1

(a) Tree-level diagram of one-gluon exchange. (b) The bubbles in the middle of (a) stand for the amplitude of $g^{*}{g}^{*}\to {\eta }_Q$.

Figure 2

(a) The diagram with two-gluon change. The middle bubble in the left part is the sum of diagrams given in (b) and (c). (b) The tree amplitude for $g^{*}{g}^{*}{g}^{*}\to {\eta }_Q$, in which all gluons are attached to the quark line. (c) The tree amplitude for $g^{*}{g}^{*}{g}^{*}\to {\eta }_Q$, in which a three-gluon vertex is involved.

Figure 3

Diagrams of the real part of the one-loop correction.

Figure 4

Diagrams of the gluon TMD distribution for the subtraction.

Figure 5

One-loop corrections for the soft factor. The two diagrams with their conjugated diagrams give the one-loop correction for the soft factor with $u^{+}=0$ and $v^{-}=0$.

Figure 6

Virtual corrections.