Next-to-leading Order Calculation of the Color-Octet 3S1 Gluon Fragmentation Function for Heavy Quarkonium

TL;DR

This paper computes the short-distance coefficients for the color-octet ${}^3S_1$ gluon fragmentation function into heavy quarkonia at next-to-leading order in $oldsymbol{\a_s}$ using Collins–Soper’s gauge-invariant definition and MS-bar renormalization. The authors find that the longitudinal piece agrees with Beneke–Rothstein, while the transverse piece yields a different ${ m NLO}$ correction than Ma. The calculation carefully separates scales via NRQCD factorization, evaluates virtual and real corrections, and demonstrates ultraviolet renormalization and infrared safety after combination. The results are crucial for accurate predictions of quarkonium production at large transverse momentum, though the transverse ${}^3S_1$-channel coefficient requires independent verification due to discrepancies with prior work.

Abstract

The short-distance coefficients for the color-octet 3S1 term in the fragmentation function for a gluon to split into heavy quarkonium states is calculated to order alpha_s^2. The gauge-invariant definition of the fragmentation function by Collins and Soper is employed. Ultraviolet divergences are removed using the MS-bar renormalization procedure. The longitudinal term in the fragmentation function agrees with a previous calculation by Beneke and Rothstein. The next-to-leading order correction to the transverse term disagrees with a previous calculation.

Figures (4)

• Figure 1: Leading order Feynman diagram for $g \to Q\overline{Q}$.
• Figure 2: The Feynman diagrams of order $\alpha_s^2$ for $g \to Q\overline{Q}$ with $Q\overline{Q}$ final states. There are additional contributions from the complex-conjugate diagrams.
• Figure 3: One loop correction diagrams for $g^*\to Q\overline{Q}$.
• Figure 4: The Feynman diagrams of order $\alpha_s^2$ for $g \to Q\overline{Q}$ with $Q\overline{Q}g$ final states. There are a total of 25 diagrams, but only the left halves of the diagrams are shown.