Quark fragmentation into spin-triplet $S$-wave quarkonium

We compute fragmentation functions for a quark to fragment to a quarkonium through an $S$-wave spin-triplet heavy quark-antiquark pair. We consider both color-singlet and color-octet heavy quark-antiquark ($Q\overline{Q}$) pairs. We give results for the case in which the fragmenting quark and the quark that is a constituent of the quarkonium have different flavors and for the case in which these quarks have the same flavors. Our results for the sum over all spin polarizations of the $Q\overline{Q}$ pairs confirm previous results. Our results for longitudinally polarized $Q\overline{Q}$ pairs agree with previous calculations for the same flavor cases and correct an error in a previous calculation for the different-flavor case.

Figure 1

Feynman diagrams for quark fragmentation into a color-octet $Q\overline{Q}$ pair for the case in which the initial quark $q$ and the quark $Q$ that is a constituent of the quarkonium have different flavors. The diagram labels $d_i$ correspond to the quantities that appear in Eq. (17).

Figure 2

Additional Feynman diagrams for quark fragmentation into a color-octet $Q\overline{Q}$ pair for the case in which the initial quark $q$ and the quark $Q$ that is a constituent of the quarkonium have the same flavor. These diagrams differ from those in Fig. 1 in that the identical quarks have been interchanged in the amplitudes on both the left and the right sides of the final-state cut. The diagram labels $d_i$ correspond to the quantities that appear in Eq. (32).

Figure 3

Additional Feynman diagrams for quark fragmentation into a color-octet $Q\overline{Q}$ pair for the case in which the initial quark $q$ and the quark $Q$ that is a constituent of the quarkonium have the same flavor. These diagrams differ from those in Fig. 1 in that the identical quarks have been interchanged in an amplitude on only one side of the final-state cut. The diagram labels $d_i$ correspond to the quantities that appear in Eq. (34).