Cancellation of infrared divergence in inclusive production of heavy quarkonia

A scheme is presented to cancel out topologically unfactorized infrared divergences in the inclusive production of heavy quarkonia, which affect the nonrelativistic QCD (NRQCD) factorization of these processes. Heavy quarkonia are defined as resonance states of QCD instead of a color-singlet heavy quark pair. Thus the final heavy quark pair is not necessarily a color singlet. In addition, heavy quarkonia are reconstructed from their decay products. As a result, the transitions between states containing heavy quarks caused by exchanges of soft gluons are also taken into account here. Such cancellation is crucial for the NRQCD factorization of these processes.

Figure 1

Topology of diagrams which contribute to uncanceled infrared divergences that affect NRQCD factorization at NNLO. The summation over all cuts that produce a color-singlet heavy quark pair, which is produced as a color octet at the hard vertex, is understood implicitly.

Figure 2

Examples of diagrams where a color-singlet $c\overline{c}$ pair is produced by the evolution of a color-octet $c\overline{c}$ pair.

Figure 3

Diagrams with the Fock states $|c\overline{c}g⟩$ in the final states.

Figure 4

Definition of the vertex $\otimes$ in Fig. 3.

Figure 5

Diagrams that take the form shown in Fig. 3.

Figure 6

Diagrams that take the form shown in Fig. 3.

Figure 7

Diagrams that take the form shown in Fig. 3.

Figure 8

(a) Example of direct production of ${\mu }^{+}{\mu }^{-}$ pair in fragmentation functions $D_{i\to {\mu }^{+}{\mu }^{-}(n,p_H)}$ and $D_{Q\overline{Q}(\kappa )\to {\mu }^{+}{\mu }^{-}(n,p_H)}$. (b) Example of indirect production of ${\mu }^{+}{\mu }^{-}$ pair in fragmentation functions $D_{i\to {\mu }^{+}{\mu }^{-}(n,p_H)}$ and $D_{Q\overline{Q}(\kappa )\to {\mu }^{+}{\mu }^{-}(n,p_H)}$. (c) Example of interference terms between the direct and indirect production of a ${\mu }^{+}{\mu }^{-}$ pair.

Figure 9

(a) Example of topologically unfactorized QCD interactions. (b) Example of topologically factorized QCD interactions.

Figure 10

Typical diagram that contributes to the infrared divergences of topologically unfactorized diagrams.