⚛️ phenomenology

Prompt charmonia hadroproduction in High Energy Factorization with tree-level NLO corrections

This paper employs high-energy factorization with tree-level next-to-leading order corrections within the non-relativistic QCD framework to simultaneously extract long-distance matrix elements for prompt charmonium production and successfully describes LHC transverse momentum spectra and relative production rates, demonstrating that these higher-order corrections significantly improve the global agreement with experimental data.

Original authors: A. A. Prokhorov, S. P. Baranov, A. V. Lipatov, M. A. Malyshev

Published 2026-10-22

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CC BY 4.0

Original authors: A. A. Prokhorov, S. P. Baranov, A. V. Lipatov, M. A. Malyshev

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

Deep within the heart of modern particle physics lies a persistent mystery: how do the tiniest building blocks of matter, specifically heavy quarks, bind together to form stable particles? When protons smash into each other at nearly the speed of light inside massive machines like the Large Hadron Collider, they create a chaotic soup of energy. From this chaos, pairs of heavy charm quarks and their antimatter counterparts occasionally emerge. These pairs do not simply fly apart; instead, they must find a way to stick together, forming a family of particles known as charmonia. Understanding this process is like trying to figure out how a specific type of crystal forms in a storm; it requires knowing both the violent forces that create the ingredients and the gentle, invisible rules that guide them into a final shape. For decades, physicists have struggled to describe this transition perfectly, caught between the predictable laws of high-energy collisions and the fuzzy, complex rules of how particles hold hands.

A team of researchers has now taken a significant step toward solving this puzzle by refining the mathematical tools used to predict how these particles are born. In a new study, they focused on three specific members of the charmonium family: the J/psi, the chi-c, and the eta-c. These particles are essentially different arrangements of the same heavy quark pair, much like different ways of stacking the same set of bricks. The scientists applied a sophisticated method called high-energy factorization, which allows them to look at the collision not just as a straight-on hit, but as a complex interaction where the colliding particles carry sideways momentum. Crucially, they upgraded their calculations to include "next-to-leading order" corrections. In plain terms, this means they moved beyond the basic, first-pass estimate of what happens during a collision and added a second layer of detail that accounts for more subtle interactions, such as the emission of extra particles during the formation process. By doing this, they created a more complete picture of the event, one that avoids counting the same physical effect twice.

The researchers used this improved framework to analyze a vast amount of data collected from the Large Hadron Collider. They did not just look at one type of particle in isolation; instead, they performed a simultaneous fit, adjusting their theoretical model to match the experimental data for all three types of charmonia at once. This approach allowed them to extract precise values for the "long-distance matrix elements," which are essentially numbers that describe how likely a pair of quarks is to successfully transform into a specific bound state. The study found that by including these extra layers of calculation, their predictions matched the real-world data much better than previous attempts. They were able to accurately describe the speed and direction of the particles as they flew out of the collision zone, as well as the relative rates at which different types of charmonia were produced. For instance, their model correctly predicted how often a chi-c particle would appear compared to a J/psi particle, a ratio that had previously been difficult to get right.

However, the story is not entirely without its loose ends. While the new method worked beautifully for the J/psi and chi-c particles, the same approach faced challenges when applied to the eta-c particle. The team's calculations suggested that the eta-c should be produced more frequently than what was observed in the latest experiments, particularly in the forward direction of the detector. This discrepancy highlights a remaining gap in understanding, as the data from different experimental runs on the eta-c also showed significant differences from one another. The researchers noted that while their improved model reduced some of the theoretical inconsistencies that plagued earlier studies, the description of the eta-c remains an open problem. They also observed that the internal structure of the chi-c particles seemed to deviate slightly from the standard rules of symmetry that physicists expect to hold true, suggesting that the forces at play inside these particles are more complex than the simplest models predict.

Ultimately, this work represents a refinement of our theoretical map rather than a final destination. The authors demonstrated that adding tree-level next-to-leading order corrections significantly improves the global description of heavy quark production across a wide range of energies and angles. Their findings suggest that the current theoretical framework, when properly tuned with these higher-order details, is capable of providing a consistent explanation for most of the available data. The study confirms that the interplay between the violent creation of quark pairs and their subsequent soft transition into bound states can be modeled with increasing precision. While the eta-c particle continues to resist a perfect description, the success with the rest of the charmonium family provides a strong foundation for future investigations, bringing physicists one step closer to a complete understanding of how matter organizes itself from the chaos of high-energy collisions.

Technical Summary: Prompt Charmonia Hadroproduction in High Energy Factorization with Tree-Level NLO Corrections

Problem Statement
The production of charmonia (bound states of charmed quark-antiquark pairs) remains a complex challenge in high-energy particle physics, serving as a critical testbed for the interplay between perturbative and non-perturbative Quantum Chromodynamics (QCD). While the Non-Relativistic QCD (NRQCD) framework is the standard theoretical tool, previous calculations—particularly those relying on Leading Order (LO) approximations or standard collinear factorization—have struggled to resolve persistent issues such as the "polarization puzzle" and the "heavy quark spin symmetry (HQSS) puzzle." Furthermore, existing analyses often treat different charmonium states ( $J/\psi$ , $\chi_{cJ}$ , $\eta_c$ ) separately, lacking a unified description that simultaneously accounts for all available unpolarized Large Hadron Collider (LHC) data across the full kinematic range.

Methodology
The authors apply the high-energy factorization (or $k_T$ -factorization) approach within the NRQCD framework. This method incorporates the leading-logarithmic parts of higher-order corrections by utilizing Transverse Momentum-Dependent (TMD) or unintegrated gluon densities in the proton, evolved via the Ciafaloni-Catani-Fiorani-Marchesini (CCFM) equation.

Key methodological components include:

NLO $^\dagger$ Scheme: The study implements a tree-level Next-to-Leading Order (NLO) correction scheme (denoted NLO $^\dagger$ ) for off-shell production amplitudes. This involves calculating $2 \to 1$ (LO) and $2 \to 2$ (NLO) subprocesses for gluon-gluon fusion ( $g^* + g^* \to c\bar{c}[n] + g$ ).
Merging Procedure: To avoid double-counting gluon emissions that are already accounted for in the parton evolution, the authors employ a Double-Counting-Exclusion (DCE) cut. This ensures that the hardest gluon emission in the $2 \to 2$ subprocess originates from the hard matrix element rather than the evolution.
Gluon Densities: Two sets of CCFM-evolved gluon densities are utilized: the A0 set and the JH'2013 set 2.
Fitting Strategy: A two-step global fitting procedure is developed to extract Long-Distance Matrix Elements (LDMEs):
1. Step 1: Fit inclusive prompt $J/\psi$ differential cross-sections ( $d\sigma/dp_T$ ) to determine linear combinations of color-octet (CO) LDMEs, addressing strong correlations between parameters.
2. Step 2: Incorporate direct $\chi_{cJ}$ production data and cross-section ratios ( $\sigma(\chi_{cJ})/\sigma(J/\psi)$ ) to refine LDMEs for $\chi_{cJ}$ and $J/\psi$ . The analysis explores scenarios where Heavy Quark Spin Symmetry (HQSS) relations are strictly enforced versus scenarios where the wave function derivatives $|R'_{\chi_{cJ}}(0)|^2$ are allowed to vary independently.
$\eta_c$ Prediction: The $\eta_c$ production is treated as a parameter-free prediction, deriving its CO LDMEs strictly from the fitted $J/\psi$ values via HQSS relations.

Key Contributions and Results

Simultaneous Description: The paper achieves a consistent, simultaneous description of unpolarized LHC data for the entire charmonium family ( $J/\psi$ , $\psi'$ , $\chi_{cJ}$ ) across a wide kinematic region. The inclusion of tree-level NLO corrections significantly improves the global description of data compared to pure LO calculations, particularly for the JH'2013 gluon density.
LDME Extraction: The authors extract updated LDMEs for $J/\psi$ and $\chi_{cJ}$ . The analysis reveals that while HQSS relations provide a useful constraint, the data favors a scenario where the wave function derivatives for $\chi_{c1}$ and $\chi_{c2}$ differ significantly (by up to 50%). Allowing these parameters to vary independently yields an excellent description of the $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ ratio, which is otherwise poorly reproduced under strict HQSS constraints.
Cross-Section Ratios: The model successfully predicts relative production rates, specifically $\sigma(\psi')/\sigma(J/\psi)$ and $\sigma(\chi_{cJ})/\sigma(J/\psi)$ , showing good agreement with ATLAS, CMS, and LHCb data in both central and forward rapidity regions.
$\eta_c$ Discrepancy: The parameter-free prediction for $\eta_c$ production overestimates the experimental cross-sections measured by LHCb in the forward rapidity region. The authors note a substantial discrepancy between two recent LHCb analyses at 13 TeV, suggesting that the accurate description of $\eta_c$ production remains an open problem.
Theoretical Uncertainties: The study provides detailed uncertainty bands, distinguishing between variations in soft gluon emission energies (inner bands) and renormalization scale uncertainties (outer bands).

Significance and Claims
The authors claim that their work demonstrates the efficacy of the NLO $^\dagger$ scheme within high-energy factorization. By properly merging LO and NLO contributions and utilizing CCFM-evolved gluon densities, the approach resolves several discrepancies present in previous LO-only studies. Specifically, the inclusion of tree-level NLO corrections reduces the violation of HQSS relations required to fit the data compared to LO calculations, although some deviation remains.

The paper concludes that while the framework provides a robust description for $J/\psi$ , $\psi'$ , and $\chi_{cJ}$ , the accurate description of $\eta_c$ production at the LHC remains elusive, likely due to unresolved issues in the experimental data or the need for further theoretical refinements regarding HQSS violations and relativistic corrections. The study does not propose new experimental setups but rather offers a refined theoretical tool for interpreting existing high-energy collider data.

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