Analytical calculation for the gluon fragmentation into spin-triplet $S$-wave quarkonium

Fragmentation is the dominant mechanism for hadron production with high transverse momentum. For spin-triplet $S$-wave heavy quarkonium production, contribution of gluon fragmenting to color-singlet channel has been numerically calculated since 1993. However, there is still no analytic expression available up to now because of its complexity. In this paper, we calculate both polarization-summed and polarized fragmentation functions of gluon fragmenting to a heavy quark-antiquark pair with quantum number ${{}^{3}S}_1^{[1]}$. Our calculations are performed in two different frameworks. One is the widely used nonrelativistic QCD factorization, and the other is the newly proposed soft gluon factorization. In either case, we calculate at both leading order and next-to-leading order in velocity expansion. All of our final results are presented in terms of compact analytic expressions.

Figure 1

One typical diagram for the gluon fragmenting into ${{}^{3}S}_1^{[1]}Q\overline{Q}$ pair in the light-cone gauge at LO order in ${\alpha }_s$. The other diagrams are obtained by permutation.

Figure 2

Polarization-summed SDCs as functions of $z$. Solid curve corresponds to lowest order in $v^2$ expansion in either SGF [with superscript “(0)”] or NRQCD (with superscript “$O$”), based on the relation in Eq. (32). Dashed curve corresponds to order $v^2$ expansion in SGF. Dash-dotted curve corresponds to order $v^2$ expansion in NRQCD.

Figure 3

Transversely polarized SDCs as functions of $z$. Meaning of each curve is similar to that in Fig. 2.

Figure 4

SDCs at lowest order in $v^2$ expansion in either SGF [with superscript “(0)”] or NRQCD (with superscript “$O$”), based on relations in Eqs. (32) and (34). Solid curve represents polarization summed SDCs, dashed curve represents transversely polarized SDCs, and dash-dotted curve represents longitudinally polarized SDCs.

Figure 5

SDCs at order $v^2$ expansion in SGF framework. Meaning of each curve is similar to that in Fig. 4.

Figure 6

SDCs at order $v^2$ expansion in NRQCD framework. Meaning of each curve is similar to that in Fig. 4.