Pseudoscalar heavy-quarkonium hadroproduction from nonrelativistic fragmentation at NLL/NLO$^+$

TL;DR

This work develops NRFF1.0, a set of collinear fragmentation functions for pseudoscalar quarkonia $η_c$ and $η_b$, constructed from NRQCD inputs at NLO and evolved through a two-stage DGLAP procedure in a variable-flavor-number scheme to include heavy-quark thresholds.The analysis is embedded in the NLL/NLO$^+$ hybrid factorization (HyF) framework with JETHAD, enabling robust predictions for forward hadroproduction observables at $ extsqrt{s}=13$ TeV, such as $η_Q$ in association with a heavy-flavor hadron or a jet.Key findings show natural stabilization of high-energy cross sections due to heavy-flavor fragmentation, a dominant color-singlet mechanism for pseudoscalar quarkonia, and predictable, perturbatively stable azimuthal and transverse-momentum observables across the explored kinematics.The results provide a solid phenomenological toolkit for quarkonium fragmentation at large $p_T$, and set the stage for future extensions to vector quarkonia and quarkonium-in-jet studies, as well as exploration of intrinsic heavy-flavor effects.

Abstract

We investigate the inclusive hadroproduction of pseudoscalar heavy quarkonia, $η_c$ and $η_b$ mesons, in high-energy proton collisions. Our framework bases on the single-parton collinear fragmentation within a variable-flavor number scheme, tailored to describe the moderate to large transverse momentum regime. To this end, we construct a new set of collinear fragmentation functions, denoted as NRFF1.0, which evolve via standard DGLAP equations with a consistent treatment of flavor thresholds. Initial conditions for all parton-induced channels are computed using next-to-leading-order nonrelativistic QCD. We perform our analysis within the NLL/NLO$^+$ hybrid factorization framework, employing the JETHAD numerical interface together with the symJETHAD symbolic engine. These tools allow us to deliver predictions for high-energy observables sensitive to quarkonium final states at 13 TeV LHC. To the best of our knowledge, the NRFF1.0 sets represent the first-ever release of collinear fragmentation functions for heavy quarkonia that consistently includes all partonic channels within collinear factorization.

Figures (18)

• Figure 1: Left panel: A leading diagram for the fragmentation of a gluon to ${\@fontswitch\mathcal{S}}^-_Q \, [^1S_0^{(1)}]$ at ${\@fontswitch\mathcal{O}}(\alpha_s^2)$. Central panel: One of the leading diagrams for the fragmentation of a constituent heavy quark ($Q$) to a $S$-wave color-singlet pseudoscalar quarkonium ${\@fontswitch\mathcal{S}}^-_Q \, [^1S_0^{(1)}]$ at ${\@fontswitch\mathcal{O}}(\alpha_s^2)$. Right panel: A leading diagram for the fragmentation of a nonconstituent light ($\tilde{q}$) or heavy ($\tilde{Q}$) quark to ${\@fontswitch\mathcal{S}}^-_Q \, [^1S_0^{(1)}]$ at ${\@fontswitch\mathcal{O}}(\alpha_s^3)$. Green blobs refer to the ${\@fontswitch\mathcal{S}}^-_Q \, [^1S_0^{(1)}]$ nonperturbative NRQCD LDME.
• Figure 2: Left panel: A leading diagram for the fragmentation of a gluon to ${\@fontswitch\mathcal{V}}_Q \, [^3S_1^{(1)}]$ at ${\@fontswitch\mathcal{O}}(\alpha_s^3)$. Central panel: One of the leading diagrams for the fragmentation of a constituent heavy quark ($Q$) to a $S$-wave color-singlet vector quarkonium ${\@fontswitch\mathcal{V}}_Q \, [^3S_1^{(1)}]$ at ${\@fontswitch\mathcal{O}}(\alpha_s^2)$. Right panel: A leading diagram for the fragmentation of a nonconstituent light ($\tilde{q}$) or heavy ($\tilde{Q}$) quark to ${\@fontswitch\mathcal{V}}_Q \, [^3S_1^{(1)}]$ at ${\@fontswitch\mathcal{O}}(\alpha_s^4)$. Violet blobs refer to the ${\@fontswitch\mathcal{V}}_Q \, [^3S_1^{(1)}]$ nonperturbative NRQCD LDME.
• Figure 3: NRQCD [$g \to \eta_b$] fragmentation channel at LO and NLO, for $\mu_{R,0}=\mu_{F,0}=2m_b$. Left plot: $\mu_R$ runs from $\mu_{R,0}/2$ to $2\mu_{R,0}$, while $\mu_F$ is fixed at $\mu_{F,0}$. Right plot: $\mu_R$ is fixed at $\mu_{R,0}$, while $\mu_F$ runs from $\mu_{F,0}/2$ to $2\mu_{F,0}$.
• Figure 4: NRQCD [$c \to \eta_c$, upper] and [$b \to \eta_b$, lower] fragmentation channels at LO and NLO, for $\mu_{R,0}=2m_Q$ and $\mu_{F,0}=3m_Q$. Left plot: $\mu_R$ runs from $\mu_{R,0}/2$ to $2\mu_{R,0}$, while $\mu_F$ is fixed at $\mu_{F,0}$. Right plot: $\mu_R$ is fixed at $\mu_{R,0}$, while $\mu_F$ runs from $\mu_{F,0}/2$ to $2\mu_{F,0}$.
• Figure 5: NRQCD [$b \to \eta_c$, lower] fragmentation channel at NLO for $\mu_{R,0}=2m_c$ and $\mu_{F,0}=m_b+2m_c$. Left plot: $\mu_R$ runs from $\mu_{R,0}/2$ to $2\mu_{R,0}$, while $\mu_F$ is fixed at $\mu_{F,0}$. Right plot: $\mu_R$ is fixed at $\mu_{R,0}$, while $\mu_F$ runs from $\mu_{F,0}/2$ to $2\mu_{F,0}$.