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

Geoffrey T. Bodwin, Hee Sok Chung, U-Rae Kim, Jungil Lee

TL;DR

The paper addresses quark-initiated fragmentation into quarkonium through a heavy $Q\bar Q$ pair in the spin-triplet $^3S_1$ state, examining both color-octet and color-singlet channels for unequal and equal flavors. It employs the Collins-Soper fragmentation-function definition, explicit spin-polarization projectors, and NRQCD factorization to compute LO fragmentation functions at leading order in $\alpha_s$ and $v$, with full $d=4-2\epsilon$ expressions. The authors verify consistency with prior results for spin sums, correct specific longitudinal-polarization results, and provide new longitudinal-polarization functions, all cast in terms of NRQCD LDMEs. These results yield polarization-sensitive inputs for leading-power quarkonium production and highlight areas where higher-order ($\alpha_s^5$ and higher-$v$) corrections will be important in extending quark- and gluon-initiated fragmentation contributions.

Abstract

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.

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

TL;DR

The paper addresses quark-initiated fragmentation into quarkonium through a heavy pair in the spin-triplet state, examining both color-octet and color-singlet channels for unequal and equal flavors. It employs the Collins-Soper fragmentation-function definition, explicit spin-polarization projectors, and NRQCD factorization to compute LO fragmentation functions at leading order in and , with full expressions. The authors verify consistency with prior results for spin sums, correct specific longitudinal-polarization results, and provide new longitudinal-polarization functions, all cast in terms of NRQCD LDMEs. These results yield polarization-sensitive inputs for leading-power quarkonium production and highlight areas where higher-order ( and higher-) corrections will be important in extending quark- and gluon-initiated fragmentation contributions.

Abstract

We compute fragmentation functions for a quark to fragment to a quarkonium through an -wave spin-triplet heavy quark-antiquark pair. We consider both color-singlet and color-octet heavy quark-antiquark () 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 pairs confirm previous results. Our results for longitudinally polarized pairs agree with previous calculations for the same flavor cases and correct an error in a previous calculation for the different-flavor case.

Paper Structure

This paper contains 7 sections, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation, kinematics, and projectors
  3. Collins-Soper Definition of Fragmentation Function
  4. Color-octet fragmentation: different-flavor case
  5. Color-octet fragmentation: same-flavor case
  6. Color-singlet fragmentation
  7. Summary and discussion

Figures (3)

  • Figure 1: Feynman diagrams for quark fragmentation into a color-octet $Q\bar{Q}$ 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. (\ref{['unequal-frag']}).
  • Figure 2: Additional Feynman diagrams for quark fragmentation into a color-octet $Q\bar{Q}$ 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. \ref{['fig:unequal']} 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. (\ref{['exchange-frag']}).
  • Figure 3: Additional Feynman diagrams for quark fragmentation into a color-octet $Q\bar{Q}$ 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. \ref{['fig:unequal']} 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. (\ref{['interference-frag']}).