Semi-analytical calculation of gluon fragmentation into ${^{1}\hspace{-0.6mm}S_{0}^{[1,8]}}$ quarkonia at next-to-leading order

TL;DR

This work delivers the next-to-leading order short-distance coefficients for gluon fragmentation into heavy quarkonia in the ${^{1}S_{0}^{[1]}}$ and ${^{1}S_{0}^{[8]}}$ channels within NRQCD. It combines integration-by-parts reduction to master integrals with differential-equation techniques, utilizing numerical boundary conditions and asymptotic expansions around the light-cone momentum fraction singularities to achieve high-precision results. The real and virtual NLO corrections are renormalized to yield finite SDCs, which exhibit large K-factors across much of the $z$-spectrum, underscoring their potential impact on quarkonium production at the LHC. The methodology, including careful treatment of rapidity divergences with a gluon-mass regulator and cross-checks against independent works, provides a robust framework for precise quarkonium fragmentation phenomenology, with detailed numerical data and ancillary high-precision coefficients for practical use.

Abstract

We calculate the NLO corrections for the gluon fragmentation functions to a heavy quark-antiquark pair in ${^{1}\hspace{-0.6mm}S_{0}^{[1]}}$ or ${^{1}\hspace{-0.6mm}S_{0}^{[8]}}$ state within NRQCD factorization. We use integration-by-parts reduction to reduce the original expression to simpler master integrals (MIs), and then set up differential equations for these MIs. After calculating the boundary conditions, MIs can be obtained by solving the differential equations numerically. Our results are expressed in terms of asymptotic expansions at singular points of $z$ (light-cone momentum fraction carried by the quark-antiquark pair), which can not only give FFs results with very high precision at any value of $z$, but also provide fully analytical structure at these singularities. We find that the NLO corrections are significant, with K-factors larger than 2 in most regions. The NLO corrections may have important impact on heavy quarkonia (e.g. $η_c$ and $J/ψ$) production at the LHC.

Figures (9)

• Figure 1: Feynman rules related to the gluon gauge link.
• Figure 2: One of the two Feynman diagrams of gluon fragmenting into ${^{1}\space S_{0}^{[1]}}$ or ${^{1}\space S_{0}^{[8]}}$$Q\bar{Q}$ at LO in $\alpha_s$. Another diagram can be obtained by permuting the heavy quark and anti-quark.
• Figure 3: Typical Feynman diagrams for $g\to Q\bar{Q}({^{1}\space S_{0}^{[1,8]}})+gg$. The other diagrams can be obtained by permuting the heavy quark and anti-quark or the two emitted gluons.
• Figure 4: One of the two Feynman diagrams for $g\to Q\bar{Q}({^{1}\space S_{0}^{[1,8]}})+q\bar{q}$. Another diagram can be obtained by permuting the heavy quark and anti-quark.
• Figure 5: Singularities of DEs of MIs for both $g\to Q\bar{Q} ({^{1}\space S_{0}^{[1]}}) +X$ and $g\to Q\bar{Q} ({^{1}\space S_{0}^{[8]}}) +X$. Plus signs denote singularities encountered in real corrections while multiplication signs denote singularities encountered in virtual corrections.