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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 ($Q\bar Q$) 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 $Q\bar Q$ pairs confirm previous results. Our results for longitudinally polarized $Q\bar Q$ pairs agree with previous calculations for the same flavor cases and correct an error in a previous calculation for the different-flavor case.