⚛️ phenomenology

Forward $\chi_c$ production in $pp$ and $pA$ collisions within the CGC+NRQCD framework

This paper investigates forward $\chi_c$ production in $pp$ and $pA$ collisions using a CGC+NRQCD framework, finding that while predictions align with LHCb $p\mathrm{Pb}$ data and standard initial conditions, the model fails to simultaneously describe both $\chi_c$ and prompt $J/\psi$ production in $pp$ collisions, highlighting a persistent theoretical challenge.

Original authors: Zhi-Guo He, Hao-yu Liu

Published 2026-10-20

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhi-Guo He, Hao-yu Liu

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 matter, protons and neutrons are not solid, unchanging spheres but rather seething clouds of smaller particles called quarks and gluons. When these particles smash together at nearly the speed of light, they create a chaotic environment where the fundamental forces of nature are pushed to their limits. One of the most intriguing outcomes of these collisions is the birth of heavy quarkonium, a fleeting particle made of a heavy quark and its antimatter partner, the antiquark. These particles are like tiny, short-lived laboratories that allow physicists to test the rules governing how matter holds itself together. While scientists have long understood how these particles form at high speeds, a persistent mystery remains: how do they behave when moving more slowly, in a region where the density of gluons becomes so extreme that they begin to merge and interact in complex, non-linear ways?

A team of researchers at the Beijing University of Chemical Technology has taken a fresh look at this problem, focusing on a specific type of heavy quarkonium known as the $\chi_c$ particle. They investigated how these particles are produced when protons collide with other protons or with heavy lead nuclei, specifically looking at the particles that fly forward at low speeds. To do this, they combined two powerful theoretical tools. The first is a framework called non-relativistic quantum chromodynamics, which acts like a detailed instruction manual for how a heavy quark pair transforms into a stable particle. The second is the Color Glass Condensate, a theory that describes the dense, saturated cloud of gluons inside a fast-moving proton or nucleus. By weaving these two theories together, the researchers created a new model to predict exactly how often these $\chi_c$ particles should appear and how they should behave.

The team began by refining their model to handle the complex web of interactions between the quarks and the gluon clouds. They tested different ways of calculating how the color charges of the particles interact, eventually settling on a method that treats the particles with a high degree of mathematical precision while keeping the calculations manageable. Their simulations revealed a surprising detail: in the production of these specific $\chi_c$ particles, the "color-singlet" state—a configuration where the quarks are perfectly balanced and neutral—is far more dominant than previously thought. In fact, this balanced state accounts for nearly half to almost all of the $\chi_c$ particles produced, a stark contrast to other similar particles where this state plays a much smaller role. This finding suggests that the $\chi_c$ particle is an exceptionally sensitive probe for studying the intricate, multi-point connections within the gluon clouds of the target nucleus.

When the researchers compared their predictions with real-world data collected by the LHCb experiment at the Large Hadron Collider, the results were a mix of success and puzzling disagreement. For collisions involving a proton and a lead nucleus, their model predicted the ratio of different types of $\chi_c$ particles with remarkable accuracy, matching the experimental measurements almost perfectly. However, when they applied the same model to collisions between two protons, the theory consistently predicted more particles than were actually observed. This discrepancy suggests that while the model captures the physics of dense nuclear matter well, it struggles to fully describe the simpler proton-proton environment under these specific conditions.

The investigation went deeper, examining how these $\chi_c$ particles decay into a more common particle called the $J/\psi$ . The researchers tried two different approaches to compare their theory with data. In the first approach, they used the actual measured number of $J/\psi$ particles from experiments as a fixed baseline to check their $\chi_c$ predictions. In this scenario, their model worked reasonably well, provided they used a standard set of starting conditions for the gluon clouds. But when they switched to a purely theoretical approach—calculating both the $\chi_c$ and the $J/\psi$ production from scratch using only their model and no experimental data—the agreement vanished. The model failed to reproduce the observed relationship between the two particles, regardless of which set of theoretical parameters they chose.

This failure highlights a significant gap in our current understanding. The researchers found that no single combination of their theoretical inputs could simultaneously describe the production of both the $\chi_c$ and the $J/\psi$ particles correctly. While the model works well for some specific ratios in heavy-ion collisions, it cannot yet provide a unified picture that explains all the data across different collision types. The authors conclude that the current framework, while powerful, is missing crucial pieces of the puzzle, likely higher-order corrections that account for more subtle quantum effects. Until these missing pieces are added, the complete story of how heavy quarkonium forms in the most extreme environments remains just out of reach.

Technical Summary: Forward $\chi_{cJ}$ Production in $pp$ and $pA$ Collisions within the CGC+NRQCD Framework

Problem Statement
Heavy-quarkonium production serves as a critical testing ground for the interplay between perturbative and nonperturbative Quantum Chromodynamics (QCD). While the Color Glass Condensate (CGC) effective field theory successfully describes small- $x$ gluon dynamics and saturation effects, and Non-Relativistic QCD (NRQCD) factorization provides a systematic framework for quarkonium formation, their combined application has historically focused on the $S$ -wave vector state $J/\psi$ . The $P$ -wave charmonium states, specifically $\chi_{cJ}$ , have received less attention within the CGC+NRQCD framework. This gap is significant because $\chi_{cJ}$ production is a primary source of "feed-down" contributions to prompt $J/\psi$ yields and offers a unique probe of multipoint color correlations in the target color field, particularly through the color-singlet (CS) channel involving quadrupole Wilson-line correlators. Furthermore, achieving a unified description of both $\chi_{cJ}$ and prompt $J/\psi$ production within the same theoretical framework remains an open challenge.

Methodology
The authors investigate forward $\chi_{cJ}$ production at low transverse momentum ( $p_\perp$ ) in proton-proton ($pp$) and proton-lead ( $pPb$ ) collisions at LHC energies. The theoretical framework combines:

CGC Factorization: The dense target (nucleus or proton) is treated using CGC, where small- $x$ gluon evolution is governed by the running-coupling Balitsky-Kovchegov (rcBK) equation. The projectile is treated via collinear factorization.
NRQCD Factorization: The production cross-section is expressed as a sum of perturbatively calculable short-distance coefficients (SDCs) multiplied by long-distance matrix elements (LDMEs). The dominant intermediate states for $\chi_{cJ}$ are the color-octet $^3S_1^{[8]}$ and color-singlet $^3P_J^{[1]}$ states.
Multipoint Correlators: A key methodological component involves handling the target Wilson-line correlators. The authors compare three approaches for the color-singlet quadrupole correlator:
- A dipole-based ansatz (expressing the quadrupole entirely in terms of fundamental dipoles).
- A large- $N_c$ Gaussian approximation.
- A finite- $N_c$ Gaussian approximation.
  The study adopts the finite- $N_c$ Gaussian prescription as the default, as it retains the exact color structure and aligns with numerical JIMWLK results, unlike the dipole ansatz which showed significant deviations in spectral shapes.

The numerical analysis utilizes rcBK evolution with three distinct initial conditions (standard McLerran-Venugopalan and two $g1$ parametrizations fitted to HERA data). The study evaluates two primary observables:

The cross-section ratio $R_{\chi_c} = \sigma_{\chi_{c2}} / \sigma_{\chi_{c1}}$ .
The feed-down fractions $F_{\chi_{c1,2} \to J/\psi} = \sigma_{\chi_{c1,2} \to J/\psi} / \sigma_{J/\psi}$ .

These are compared against LHCb measurements using two schemes: a "data-driven" approach (normalizing theoretical $\chi_c$ predictions to experimental prompt $J/\psi$ cross-sections) and a "pure theoretical" approach (calculating both numerator and denominator within CGC+NRQCD using six representative $J/\psi$ LDME sets).

Key Results

Color-Singlet Dominance: Unlike $J/\psi$ production where the color-singlet contribution is small (~15%), the CS channel contributes substantially to $\chi_{cJ}$ production, accounting for approximately 45–90% of $\chi_{c2}$ and over 60% of $\chi_{c1}$ yields in the studied $p_\perp$ range. This highlights $\chi_{cJ}$ as a sensitive probe of multipoint color correlations.
Ratio $R_{\chi_c}$ : Theoretical predictions for the ratio $R_{\chi_c}$ $R_{χ_{c}}$ show weak dependence on the specific rcBK initial conditions due to the cancellation of common dipole dependencies.
- For $pPb$ collisions, the predictions align well with LHCb measurements within experimental uncertainties.
- For $pp$ collisions, the theoretical predictions systematically overshoot the measured ratio across the entire $p_\perp$ range.
Feed-Down Fractions:
- In the data-driven scheme, the predictions using the standard MV initial condition (CGC set 1) broadly match the experimental data for both $pp$ and $pPb$ collisions. Other initial conditions systematically underestimate the data.
- In the pure theoretical scheme, no combination of the three CGC initial conditions and the six tested $J/\psi$ LDME sets successfully describes the feed-down fractions simultaneously for both $\chi_c$ and prompt $J/\psi$ . Specifically, improving the description of prompt $J/\psi$ (e.g., by adjusting CGC inputs or LDMEs) leads to a deterioration in the description of $\chi_c$ production, and vice versa.

Significance and Claims
The paper concludes that while the CGC+NRQCD framework can describe certain aspects of $\chi_{cJ}$ production (such as the $R_{\chi_c}$ ratio in $pPb$ collisions or feed-down fractions when normalized to data), it faces a persistent challenge in providing a simultaneous, unified description of $\chi_{cJ}$ and prompt $J/\psi$ production at Leading Order (LO) in both $\alpha_s$ and $v^2$ .

The authors assert that the systematic discrepancies observed in the pure theoretical calculations—where the framework fails to reproduce the measured prompt $J/\psi$ spectra while simultaneously fitting $\chi_c$ data—suggest that higher-order corrections are necessary. The paper explicitly states that the tension between theory and data may be resolved by incorporating Next-to-Leading Order (NLO) QCD corrections and relativistic corrections in future studies, noting that such corrections are known to be crucial in collinear factorization calculations. The work underscores the necessity of moving beyond LO approximations to achieve a consistent phenomenological description of heavy quarkonium production in the saturation regime.

More like this