Relativistic corrections to gluon fragmentation into spin-triplet S-wave quarkonium

TL;DR

The study computes relativistic (v^2) corrections to the gluon fragmentation function into spin-triplet S-wave quarkonia within NRQCD, using the Collins-Soper gauge-invariant definition. Short-distance coefficients for both color-octet and color-singlet channels are derived at LO in α_s with v^2 corrections, revealing large opposite-signed corrections: a negative shift for color-octet and a positive shift for color-singlet. The octet correction implies a need to enlarge the leading color-octet matrix element to maintain Tevatron J/ψ data fits, while the singlet correction modestly enhances the fragmentation probability. These findings impact quarkonium production phenomenology, test NRQCD universality, and motivate further higher-order and lattice studies to assess the velocity expansion's validity.

Abstract

We use the NRQCD factorization formalism to calculate the relativistic corrections to the fragmentation function for a gluon fragmenting into a spin-triplet S-wave heavy quarkonium. We make use of the gauge-invariant formulation of the fragmentation function of Collins and Soper. The color-octet contribution receives a large, negative relativistic correction, while the color-singlet contribution receives a large, positive relativistic correction. The considerable decrease in the color-octet contribution requires a corresponding increase in the phenomenological value of the leading color-octet matrix element in order to maintain a fit to the Fermilab Tevatron data.

Figures (3)

• Figure 1: Feynman diagram for the color-octet contribution at leading order in $\alpha_s$ to the fragmentation function for a gluon fragmenting into a color-octet spin-triplet $Q\overline{Q}$ pair.
• Figure 2: One of the Feynman diagrams for the color-singlet contribution at leading order in $\alpha_s$ to the fragmentation function for a gluon fragmenting into a color-singlet spin-triplet $Q\overline{Q}$ pair. The other diagrams are obtained by permuting the connections of the gluons to the heavy-quark lines.
• Figure 3: The color-singlet short-distance coefficients $d_1(z)$ and $d_1'(z)$, which are defined in Eq. (\ref{['singlet-frag-QQ']}). The scaling factors in this figure are $c_1=10^{-4}\times\alpha_s^3/m^3$ and $c_1'=10^{-3}\times\alpha_s^3/m^3$.