⚛️ nuclear theory

$\psi(2S)$ production in jets using NRQCD

This paper utilizes recent LHCb data to demonstrate that both the Fragmenting Jet Function (FJF) and Gluon Fragmentation Improved Pythia (GFIP) formalisms significantly outperform default Pythia+NRQCD predictions in describing $\psi(2S)$ production in jets, while highlighting the distribution's utility as a discriminator to reveal large discrepancies among different $\psi(2S)$ Long-Distance Matrix Element (LDME) extractions.

Original authors: Marston Copeland, Lin Dai, Yu Fu, Jyotirmoy Roy

Published 2026-06-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Marston Copeland, Lin Dai, Yu Fu, Jyotirmoy Roy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is a giant, high-speed particle collider, like a cosmic pinball machine where tiny particles smash together at nearly the speed of light. When these particles collide, they don't just bounce off; they shatter and create new, heavier particles. One of these heavy particles is called $\psi(2S)$ (pronounced "psi-two-S"). Think of it as a heavy, excited version of a "charmed" particle family.

Physicists want to understand exactly how these particles are born. To do this, they look at them inside "jets." You can imagine a jet as a high-speed spray of debris shooting out from the collision point, like water spraying from a broken fire hose. The $\psi(2S)$ is a rare gem hidden inside this chaotic spray.

The Problem: The "Naive" Prediction Failed

For a long time, scientists used a standard computer program called Pythia to simulate these collisions. It's like using a basic weather app to predict a hurricane. For simple things, it works fine. But when they tried to use this basic program to predict how the $\psi(2S)$ particles were distributed inside the jets, it failed miserably. The computer said one thing, but the actual data from the LHCb experiment (a giant detector at CERN) said something completely different. It was like the weather app predicting sunshine while a tornado was actually happening.

The Solution: Two New "Maps"

To fix this, the authors of this paper used two advanced, more sophisticated "maps" to describe how the heavy particle forms inside the jet.

The FJF Map (The Analytical Route): This method uses heavy mathematics (specifically something called "Non-Relativistic QCD" or NRQCD) to calculate the process step-by-step. It's like a master architect drawing a perfect blueprint of how the particle forms, accounting for every tiny force and interaction using complex equations.
The GFIP Map (The Simulation Route): This method also uses the basic Pythia program but gives it a major upgrade. Imagine taking that basic weather app and manually forcing it to pay attention to the specific, messy details of how the heavy particle "sticks together" (hadronizes) at the very end of the process. It's a "glue-fix" for the simulation.

The Result: Both of these new maps worked! They predicted the distribution of the $\psi(2S)$ particles inside the jets much better than the old, naive method. It turns out that to understand these heavy particles, you can't just use a simple simulation; you need to account for the specific, complex way they form inside the jet spray.

The Mystery: Which "Recipe" is Correct?

Here is where it gets interesting. To use these maps, the scientists need to know the "recipe" for the $\psi(2S)$ particle. This recipe is a set of numbers called LDMEs (Long-Distance Matrix Elements). These numbers tell us how likely the particle is to form in different ways.

The problem is, different groups of scientists have calculated these numbers differently, and they don't agree with each other. It's like having three different chefs who all claim to know the exact recipe for a cake, but they use different amounts of sugar and flour.

The authors tested these three different "recipes" against the real data:

Recipe A (Bodwin et al.): This recipe worked perfectly with both new maps. The predictions matched the real data almost exactly.
Recipe B (B&K) and Recipe C (Brambilla et al.): These recipes worked okay in some situations, but when the particles were moving very fast (at the "end" of the jet spray), the predictions went wild and didn't match the data.

The Conclusion

The paper concludes that the $\psi(2S)$ particle is a great "test subject" for figuring out the correct recipe. Because the data is so precise, it can tell us which group of scientists has the right numbers.

Currently, the data suggests that the Bodwin et al. recipe is the most accurate. The other recipes need to be tweaked because they predict too many particles at the very edge of the jet spray. The authors argue that we need to do more experiments to pin down these numbers with perfect precision, because until we do, our "maps" of how the universe builds these heavy particles will have some blurry spots.

In short: The old way of simulating these particles was broken. The authors fixed it with two advanced methods. Now, by comparing their results to real data, they are helping the scientific community figure out exactly which "recipe" nature uses to build these heavy particles.

Technical Summary: $\psi(2S)$ Production in Jets using NRQCD

Problem Statement
Quarkonium production, particularly for charmonium states like the $J/\psi$ and $\psi(2S)$ , serves as a critical probe for understanding both perturbative and non-perturbative Quantum Chromodynamics (QCD). While Non-Relativistic QCD (NRQCD) factorization has been successful in describing inclusive production, the long-distance matrix elements (LDMEs) governing the hadronization of heavy quark-antiquark pairs remain poorly constrained, often exhibiting uncertainties of 100–200%. Recent measurements by the LHCb collaboration of $\psi(2S)$ production inside jets revealed a significant discrepancy: standard theoretical predictions combining NRQCD with the default Pythia event generator failed catastrophically to describe the data. This failure mirrors previous issues observed with $J/\psi$ production in jets, suggesting that the standard modeling of quarkonium fragmentation within the parton shower is insufficient. Furthermore, the $\psi(2S)$ meson offers a unique testing ground compared to the $J/\psi$ , as its production is less contaminated by feed-down from higher charmonium resonances. However, existing extractions of $\psi(2S)$ LDMEs from various collaborations (e.g., Bodwin et al., Brambilla et al., Butenschoen & Kniel) show large disparities, necessitating new observables to constrain these parameters.

Methodology
The authors investigate $\psi(2S)$ production in jets using two distinct theoretical frameworks that improve upon the default Pythia approach, both grounded in NRQCD:

Fragmenting Jet Function (FJF) Formalism: This approach utilizes Soft-Collinear Effective Theory (SCET) to rigorously factorize the cross-section. The calculation involves convolving parton distribution functions, hard scattering coefficients, and a specific Fragmenting Jet Function ( $G_i^{\psi(2S)}$ ) that describes the probability of a parton $i$ fragmenting into a jet containing a $\psi(2S)$ . The FJF is calculated by matching perturbative partonic cross-sections to NRQCD LDMEs at the scale $2m_c$ . The resulting fragmentation functions are then evolved analytically from the scale $2m_c$ to the jet scale $\mu_J \approx 2E \tan(R/2)$ using DGLAP evolution equations. This evolution is performed in Mellin space to handle the convolution integrals.
Gluon Fragmentation Improved Pythia (GFIP): This semi-analytic Monte Carlo approach modifies the standard Pythia event generator. Instead of allowing the default hadronization module to handle the transition from partons to quarkonium, the authors disable this module. They generate hard scattering events (charm quarks and gluons) using MadGraph, then allow the parton shower to evolve down to a cutoff scale of approximately $2m_c$ (achieved by setting TimeShower:pTmin to 1.6 GeV). At this scale, the gluon energy fraction distribution is extracted and manually convolved with leading-order NRQCD fragmentation functions. This method aims to mimic the physical picture where a hard gluon showers until it reaches the virtuality scale required for quarkonium formation.

The study compares the predictions of both formalisms against LHCb data for $\psi(2S)$ production in jets at $\sqrt{s} = 13$ TeV. The analysis focuses on the transverse momentum fraction $z = p_T(\psi(2S))/p_T(\text{jet})$ across various jet $p_T$ bins. Three different sets of $\psi(2S)$ LDMEs (Bodwin et al., Brambilla et al., and B&K) are tested to assess their consistency with the data.

Key Results

Improvement over Default Models: Both the FJF and GFIP formalisms provide a significantly better description of the LHCb $\psi(2S)$ in-jet data compared to the default Pythia+NRQCD prediction, which fails to reproduce the observed distributions.
LDME Sensitivity: The distribution of $\psi(2S)$ $ψ (2 S)$ in jets acts as a powerful discriminator for different LDME extractions.
- The GFIP predictions generally agree well with experimental data across all $p_T$ bins for all three LDME sets, although discrepancies appear in the highest $p_T$ bin where experimental scatter is large.
- The FJF predictions show a more complex behavior. While the Bodwin et al. LDME set yields results consistent with both the data and the GFIP predictions, the Brambilla et al. and B&K LDME sets exhibit significant deviations from the data and GFIP predictions as $z \to 1$ .
Endpoint Behavior: The authors identify that the FJF formalism predicts a significant enhancement of the cross-section near the endpoint ( $z \to 1$ ) for the Brambilla and B&K LDME sets, whereas the GFIP approach (which relies on Pythia's parton shower) shows an artificial suppression in this region. This divergence is attributed to the different evolution mechanisms: FJF uses DGLAP evolution (akin to angular ordering) which accesses the $z \to 1$ region, while Pythia uses $k_T$ ordering. The Bodwin set avoids this large divergence because the contributions from the $^3P_J^{[8]}$ and $^3S_1^{[8]}$ channels approximately cancel, a phenomenon also observed in $J/\psi$ production.
Consistency: The results suggest that the FJF calculation is most consistent with the Bodwin et al. LDME set, while the other sets lead to unphysical endpoint divergences in the FJF framework.

Significance and Claims
The paper claims that the study of $\psi(2S)$ production in jets provides a robust and necessary tool for constraining $\psi(2S)$ LDMEs, which are currently poorly determined. The work demonstrates that the "fragmenting jet" picture, implemented via either FJF or GFIP, is the correct theoretical framework for describing quarkonium production within jets, resolving the failures of naive NRQCD+Pythia combinations.

Crucially, the authors highlight a large disparity in predictions arising from different LDME extractions, particularly in the endpoint region of the $z$ distribution when using the FJF formalism. They conclude that a more precise extraction of $\psi(2S)$ LDMEs is necessary, as none of the currently available sets provide perfect agreement with in-jet data across all kinematic bins. The paper posits that future global fits could incorporate LHCb in-jet production data alongside inclusive production data to further constrain these non-perturbative parameters. The authors also note that while their FJF results show endpoint singularities, these might be regulated by threshold resummation, a feature included in a concurrent preprint (Ref. [51]) but outside the scope of the current work.

The Problem: The "Naive" Prediction Failed

The Solution: Two New "Maps"

The Mystery: Which "Recipe" is Correct?

The Conclusion

More like this