GFIP: Enhanced Quarkonium-in-Jet Modeling

• The paper presents GFIP as a hybrid Monte Carlo/fragmentation method that replaces default Pythia quarkonium formation with explicit NRQCD functions after shower evolution.
• GFIP implements a modified shower cutoff at 1.6 GeV to evolve hard partons down to the charmonium scale, yielding improved z-distribution predictions for ψ(2S).
• GFIP serves as a phenomenological Monte Carlo realization of the fragmenting-jet picture, emphasizing sensitivity to NRQCD LDMEs and higher-order perturbative corrections.

Searching arXiv for papers on Gluon Fragmentation Improved Pythia and closely related quarkonium-in-jet work. Gluon Fragmentation Improved Pythia (GFIP) is a hybrid Monte Carlo/fragmentation approach for modeling quarkonium production inside jets more realistically than the default quarkonium treatment in Pythia. In the formulation described for $\psi(2S)$-in-jet production, the method separates the production of the energetic parton that initiates the jet from the nonperturbative hadronization of that parton into a quarkonium state, and then imposes the NRQCD fragmentation functions by hand after the shower has evolved down to the quarkonium formation scale (Copeland et al., 1 Aug 2025). GFIP is presented as a phenomenological implementation of the fragmenting-jet picture, in which a hard parton at the jet scale evolves down to the charmonium scale, where hadronization into quarkonium occurs (Copeland et al., 1 Aug 2025).

1. Definition and scope

GFIP was introduced in earlier work and is used as a phenomenological implementation of the “fragmenting jet” picture for quarkonium in jets (Copeland et al., 1 Aug 2025). The relevant physics picture is to produce a hard parton in the short-distance process, let it shower perturbatively in Pythia down to a scale near $2m_c$, and then replace Pythia’s default quarkonium hadronization with an explicit convolution with perturbative NRQCD fragmentation functions (Copeland et al., 1 Aug 2025).

The method is specifically motivated by the inadequacy of the default Pythia quarkonium model for LHCb $\psi(2S)$-in-jet data (Copeland et al., 1 Aug 2025). In that default treatment, a color-singlet $c\bar c$ pair is assumed to hadronize directly, with essentially no QCD radiation, while a color-octet pair is treated as a single colored particle that showers with a splitting function like $2P_{qq}(z)$, which strongly biases the distribution toward large momentum fraction $z\approx 1$ (Copeland et al., 1 Aug 2025). Within the paper’s terminology, this is not the same as the fragmenting-jet picture appropriate for quarkonium in jets, and the resulting $z$-distribution of $\psi(2S)$ in jets is described as being in “catastrophic” disagreement with the measured distribution (Copeland et al., 1 Aug 2025).

A plausible implication is that GFIP should be understood not as a generic modification of all hadronization in Pythia, but as a targeted replacement of the quarkonium formation stage in jet observables where gluon fragmentation is phenomenologically dominant. The paper’s discussion is confined to quarkonium inside jets, especially $\psi(2S)$, and the formalism is framed relative to NRQCD and the Fragmenting Jet Function (FJF) approach rather than as a universal hadronization model (Copeland et al., 1 Aug 2025).

2. Operational implementation

In the implementation described for proton-proton collisions at $\sqrt{s}=13$ TeV, GFIP proceeds through a concrete sequence (Copeland et al., 1 Aug 2025). Hard partonic events are generated with MadGraph. The analysis focuses on events where the hard process produces gluons and charm quarks that can seed jets. These events are then passed to Pythia for parton showering. The shower cutoff is modified by setting

$2m_c$0

instead of the default $2m_c$1 GeV, so that the shower stops near the physical charmonium scale $2m_c$2 (Copeland et al., 1 Aug 2025). The default hadronization module for quarkonium is disabled, the parton energy-fraction distribution after showering is extracted, and that distribution is convolved manually with LO NRQCD fragmentation functions at the scale $2m_c$3 to produce the $2m_c$4 spectrum (Copeland et al., 1 Aug 2025).

This procedure is summarized in the paper as “Pythia for the shower + NRQCD fragmentation by hand at the end” (Copeland et al., 1 Aug 2025). The purpose of the shower cutoff change is not merely technical. It encodes the factorization logic that perturbative evolution should proceed from the jet scale down to a scale near quarkonium formation, after which the nonperturbative quarkonium transition is imposed by the NRQCD fragmentation functions (Copeland et al., 1 Aug 2025).

The main observable is the fragmentation variable

$2m_c$5

measured in bins of jet transverse momentum $2m_c$6 (Copeland et al., 1 Aug 2025). The LHCb acceptance used in the comparison requires jet pseudorapidity $2m_c$7, jet radius $2m_c$8, jet transverse momentum $2m_c$9, and for muons and pions $\psi(2S)$0 and $\psi(2S)$1, together with $\psi(2S)$2 and $\psi(2S)$3 (Copeland et al., 1 Aug 2025). The comparison is restricted to $\psi(2S)$4 to avoid endpoint regions where fixed-order calculations are less reliable (Copeland et al., 1 Aug 2025).

Component	GFIP treatment	Default Pythia+NRQCD treatment
Hard process	MadGraph	Internal Pythia heavy-quarkonium model
Shower evolution	Pythia down to $\psi(2S)$5-scale via $\psi(2S)$6	Standard Pythia treatment
Quarkonium formation	Explicit NRQCD fragmentation functions inserted by hand	Direct internal hadronization model

3. Factorization logic and relation to FJF

The paper explains GFIP as the Monte Carlo realization of the same factorization idea used in the FJF formalism (Copeland et al., 1 Aug 2025). In the analytic SCET-based description, the factorization theorem is written schematically as

$\psi(2S)$7

where $\psi(2S)$8 and $\psi(2S)$9 are PDFs, $c\bar c$0 is the short-distance hard function, $c\bar c$1 is the jet function, $c\bar c$2 is the soft function, and $c\bar c$3 is the fragmenting jet function for a jet initiated by parton $c\bar c$4 containing the $c\bar c$5 (Copeland et al., 1 Aug 2025).

The FJF itself factorizes as

$c\bar c$6

and for quarkonium the fragmentation function is matched onto NRQCD through

$c\bar c$7

The fragmentation functions are then evolved from $c\bar c$8 to the jet scale $c\bar c$9 using DGLAP (Copeland et al., 1 Aug 2025).

Within this comparison, the paper states that, theoretically, FJF and GFIP should be equivalent, because in FJF the scale evolution is done analytically with DGLAP/RG evolution, while in GFIP the shower plays the role of that evolution (Copeland et al., 1 Aug 2025). At leading order, they are essentially the same; differences arise from higher-order perturbative corrections and from specific modeling details (Copeland et al., 1 Aug 2025). The paper further notes that FJF can include systematic higher-order perturbative corrections, especially important near $2P_{qq}(z)$0, whereas GFIP approximates the evolution numerically via the shower (Copeland et al., 1 Aug 2025).

This suggests that GFIP occupies an intermediate position between a purely analytic factorization calculation and an unmodified event-generator prediction. It retains the event-generator description of the hard process and shower while replacing the quarkonium-specific hadronization kernel with an NRQCD-based one.

4. NRQCD structure and gluon fragmentation content

NRQCD is the essential nonperturbative input in GFIP, because it provides the fragmentation functions and production probabilities for each $2P_{qq}(z)$1 quantum state (Copeland et al., 1 Aug 2025). The NRQCD factorization formula is written as

$2P_{qq}(z)$2

For $2P_{qq}(z)$3, the relevant channels are the color-singlet $2P_{qq}(z)$4 and the color-octet $2P_{qq}(z)$5, $2P_{qq}(z)$6, and $2P_{qq}(z)$7 channels (Copeland et al., 1 Aug 2025). The paper notes the standard velocity scaling that the singlet $2P_{qq}(z)$8 scales as $2P_{qq}(z)$9, while the octet channels scale as $z\approx 1$0 (Copeland et al., 1 Aug 2025).

In the GFIP implementation, the focus is on gluon fragmentation, because for quarkonium inside jets the gluon channel is phenomenologically dominant and corresponds most naturally to the fragmenting-jet picture (Copeland et al., 1 Aug 2025). For gluon fragmentation, the included channels are $z\approx 1$1 at $z\approx 1$2, $z\approx 1$3 at leading order in $z\approx 1$4, and $z\approx 1$5 and $z\approx 1$6 at $z\approx 1$7 (Copeland et al., 1 Aug 2025). For charm-quark fragmentation, only the color-singlet $z\approx 1$8 contribution is retained, since that is the dominant one for the analysis presented (Copeland et al., 1 Aug 2025).

The general NRQCD fragmentation matching at the scale $z\approx 1$9 is

$z$0

The relative rates are assembled channel by channel, with weighting factor

$z$1

The paper states that this ensures the different mechanisms contribute in the correct relative proportions (Copeland et al., 1 Aug 2025).

The dependence on long-distance matrix elements is a central phenomenological feature. The paper gives three $z$2 LDME extractions—Bodwin et al., Butenschoen–Kniehl, and Brambilla et al.—and emphasizes that the octet LDMEs are very poorly constrained, often with $z$3 uncertainties (Copeland et al., 1 Aug 2025). This matters because the jet distribution is sensitive to the relative sizes of the different NRQCD channels (Copeland et al., 1 Aug 2025).

5. Phenomenological performance and comparison with data

The principal phenomenological conclusion is that GFIP and FJF both describe the LHCb $z$4-in-jet data much better than default Pythia+NRQCD (Copeland et al., 1 Aug 2025). GFIP central values generally lie close to the data, especially in most $z$5 bins, though the highest $z$6 bin has scattered data and is not well described (Copeland et al., 1 Aug 2025). The comparison also shows that FJF and GFIP agree well at small to moderate $z$7, but can differ noticeably at large $z$8 (Copeland et al., 1 Aug 2025).

The origin of the large-$z$9 difference is identified with the NLO fragmenting jet function corrections in the FJF calculation, which contain logarithms behaving like

$\psi(2S)$0

and become increasingly important as $\psi(2S)$1, especially at lower jet $\psi(2S)$2 (Copeland et al., 1 Aug 2025). The paper states that if only LO FJFs were used, the FJF results at large $\psi(2S)$3 would look much closer to GFIP (Copeland et al., 1 Aug 2025). It further finds that the FJF formalism often gives slightly better agreement with data, particularly in lower $\psi(2S)$4 bins, because it systematically includes these perturbative corrections (Copeland et al., 1 Aug 2025).

A second major conclusion is that $\psi(2S)$5-in-jet production is a powerful discriminator among different LDME fits (Copeland et al., 1 Aug 2025). Bodwin et al. generally give the best overall agreement with LHCb data; GFIP and FJF with these LDMEs track the measured $\psi(2S)$6-shape fairly well (Copeland et al., 1 Aug 2025). Butenschoen–Kniehl give small uncertainty bands; GFIP predictions are close to data except in the highest $\psi(2S)$7 bin, while FJF predictions tend to rise in the large-$\psi(2S)$8 region (Copeland et al., 1 Aug 2025). Brambilla et al. have very large uncertainty bands, making the comparison less decisive (Copeland et al., 1 Aug 2025).

These results position GFIP as an effective phenomenological tool for quarkonium-in-jet observables. The article’s own comparison does not claim that GFIP supersedes the analytic FJF framework; rather, it shows that the Monte Carlo realization captures much of the same physics and substantially improves on the default generator prediction (Copeland et al., 1 Aug 2025).

6. Relation to broader fragmentation modeling and common misconceptions

GFIP is closely related in spirit to other efforts to improve gluon fragmentation modeling, but it is not identical to them. A common misconception would be to treat GFIP as any generic enhancement of gluon fragmentation in Pythia. The explicit definition in the quarkonium-in-jet study is narrower: GFIP is a hybrid gluon-fragmentation Monte Carlo implementation in which MadGraph and Pythia generate and shower hard partons, the shower is stopped at the charmonium scale $\psi(2S)$9, and NRQCD fragmentation functions are inserted to model quarkonium formation (Copeland et al., 1 Aug 2025).

The broader literature illustrates adjacent but distinct ideas. A calculation of gluon fragmentation functions in the Nambu–Jona-Lasinio model addresses the absence of explicit gluon degrees of freedom by treating a gluon as a pair of color lines formed by fictitious quark and anti-quark and constructing effective gluon fragmentation functions for mesons (Yang et al., 2016). That work is described as conceptually in the spirit of GFIP-style improvements, because it supplies a physically motivated effective description of gluon hadronization so that evolution and hadron-production predictions become more realistic (Yang et al., 2016). However, its object is low-scale nonperturbative gluon fragmentation into pions and kaons, not quarkonium in jets (Yang et al., 2016).

Likewise, the first NLO calculation of a genuinely $\psi(2S)$0-dependent gluon fragmentation function into quarkonium, namely $\psi(2S)$1, supplies a perturbatively calculable kernel directly relevant to GFIP-style treatments (Artoisenet et al., 2014). The paper emphasizes that NLO corrections have a dramatic effect on the shape of the fragmentation function and significantly increase the fragmentation probability (Artoisenet et al., 2014). This suggests that GFIP implementations based only on LO fragmentation functions may miss important shape information, particularly if extended to channels where NLO fragmentation kernels are available (Artoisenet et al., 2014).

A further neighboring development is the implementation of NLO DGLAP evolution in parton showers in Pythia and Sherpa (Höche et al., 2017). That framework improves the perturbative evolution of PDFs and fragmentation functions at next-to-leading order precision and is described as conceptually aligned with GFIP-type improvements, although it is not a dedicated gluon-fragmentation-only model (Höche et al., 2017). A plausible implication is that such shower improvements would affect the perturbative evolution stage that GFIP delegates to Pythia before the explicit NRQCD fragmentation step.

7. Limitations and prospective directions

The limitations discussed in the source material are specific and technically important. GFIP differs from FJF in that the shower approximates the RG evolution numerically, whereas FJF performs it analytically with explicit matching coefficients and systematic higher-order perturbative corrections (Copeland et al., 1 Aug 2025). This becomes especially relevant at large $\psi(2S)$2, where the NLO fragmenting jet function corrections generate logarithms of the form $\psi(2S)$3 that are increasingly important as $\psi(2S)$4 (Copeland et al., 1 Aug 2025). Accordingly, the agreement between GFIP and FJF is best at small to moderate $\psi(2S)$5, while the large-$\psi(2S)$6 region exposes the limitations of a leading-order fragmentation treatment in the Monte Carlo realization (Copeland et al., 1 Aug 2025).

A second limitation is the uncertainty in the $\psi(2S)$7 LDMEs, particularly the color-octet matrix elements (Copeland et al., 1 Aug 2025). Because the jet observable is highly sensitive to the relative sizes of the NRQCD channels, poorly constrained LDMEs propagate directly into the GFIP predictions (Copeland et al., 1 Aug 2025). The paper therefore concludes that the $\psi(2S)$8 LDMEs are still too poorly constrained and that in-jet $\psi(2S)$9 data should be included in future global LDME extractions (Copeland et al., 1 Aug 2025).

The supplied material also indicates a broader methodological direction. Since GFIP is intended to mimic the fragmenting-jet picture by replacing Pythia’s internal quarkonium hadronization with explicit fragmentation functions, improved perturbative fragmentation inputs and improved shower evolution are natural extensions. The NLO quarkonium fragmentation calculation demonstrates that higher-order corrections can significantly reshape the kernel (Artoisenet et al., 2014), and the NLO DGLAP shower framework shows how the timelike evolution underlying fragmentation can be made more accurate in a Pythia-like environment (Höche et al., 2017). This suggests that a systematically upgraded GFIP would require simultaneous control over the perturbative shower stage and the fragmentation-function stage.

In its present documented form, however, GFIP is defined by a simpler and more concrete statement: it keeps the parton shower machinery of Pythia but replaces its quarkonium hadronization model with the NRQCD fragmentation treatment appropriate to quarkonium inside jets (Copeland et al., 1 Aug 2025). Within that scope, it serves as an intuitive Monte Carlo realization of the fragmenting-jet picture and yields a substantially improved description of $\sqrt{s}=13$0-in-jet data relative to default Pythia+NRQCD (Copeland et al., 1 Aug 2025).