Analysis of $J/ψ$ and $ψ(2S)$ Charmonium Production in Ultraperipheral Lead-Lead and Proton-Lead Collisions at LHC Energies

This study utilizes the STARlight program and the two-gluon exchange model to analyze charmonium production in ultraperipheral PbPb and pPb collisions at LHC energies, introducing a phenomenological suppression factor to successfully reconcile theoretical predictions with experimental rapidity and transverse momentum distributions in PbPb collisions while confirming the model's validity for future UPC studies.

The Gist

Imagine the Large Hadron Collider (LHC) not just as a machine smashing particles together, but as a giant cosmic lighthouse. When two massive lead atoms (or a lead atom and a proton) zoom past each other at nearly the speed of light without actually crashing, they don't touch. Instead, their intense electromagnetic fields flash like powerful beams of light. These "beams" are actually streams of photons (particles of light) that can hit the other particle and create new, heavy particles called charmonium (specifically the $J/\psi$ and $\psi(2S)$).

This paper is like a team of physicists trying to predict exactly how many of these new particles will be created and where they will end up, and then checking if their predictions match what the LHC experiments actually see.

Here is a breakdown of their work using simple analogies:

1. The Blueprint: The "Two-Gluon Exchange" Model

To understand how a photon creates a heavy charmonium particle, the authors use a specific blueprint called the two-gluon exchange model.

• The Analogy: Imagine trying to build a heavy, complex Lego structure (the charmonium) using only a single, flimsy stick (a single gluon). It won't work. You need a sturdy, double-stick support (two gluons) to hold it together.
• What they did: They used this "double-stick" rule to calculate the basic probability of a photon hitting a proton and creating a charmonium particle. They checked this against existing data and found their blueprint was accurate for the basic building blocks.

2. The Simulation: The "STARlight" Program

Once they had the basic blueprint, they needed to simulate what happens when these collisions occur in the real world, involving massive lead nuclei. They used a computer program called STARlight.

• The Analogy: Think of STARlight as a flight simulator. It takes the basic rules of aerodynamics (the two-gluon model) and simulates a flight through a storm (the lead nucleus).
• The Problem: When they ran the simulation for Lead-Lead (Pb-Pb) collisions, the computer predicted too many particles, especially in the middle of the collision zone. It was like the flight simulator predicting the plane would fly straight through a mountain without slowing down. The real experiments (ALICE, CMS, LHCb) showed fewer particles than the computer said there should be.

3. The Fix: The "Suppression Factor"

To fix the over-prediction, the authors introduced a phenomenological suppression factor.

• The Analogy: Imagine you are baking a cake and your recipe says it will rise to the ceiling, but in reality, it only rises halfway. You realize you need to add a "dampener" to the recipe to account for the fact that the oven (the heavy nucleus) is denser than you thought.
• What they did: They added a mathematical "dampener" that gets stronger in the middle of the collision (where the density is highest) and weaker at the edges. This "dampener" represents the fact that the heavy lead nucleus gets in the way, blocking some of the light (photons) or making it harder for the particles to form.
• The Result: After adding this dampener, their predictions matched the real-world data perfectly. They could even reproduce a specific "double-hump" shape in the data, which looks like a rabbit's ears, a pattern caused by the way the particles are distributed.

4. The Asymmetry: Lead-Lead vs. Proton-Lead

The paper also looked at collisions between a Lead nucleus and a single Proton (p-Pb).

• The Analogy: Imagine a game of tennis.
  • Lead-Lead (Pb-Pb): Two giant, heavy players hitting the ball back and forth. Both sides are dense and block the ball heavily.
  • Proton-Lead (p-Pb): A giant player (Lead) vs. a tiny, light player (Proton).
• The Finding: In the Lead-Lead game, the "dampener" was needed because both sides were heavy and blocked the action. But in the Proton-Lead game, the authors found they didn't need a strong dampener.
• Why? Because when the tiny proton is the target, it's like hitting a light ping-pong ball; there's no heavy "shadow" to block the action. The heavy lead nucleus is only the source of the light, not the target being blocked. So, the simulation worked almost perfectly without needing to add extra "dampening."

5. The Conclusion

The authors conclude that:

1. Their "two-gluon" blueprint is a solid foundation for understanding these collisions.
2. When simulating heavy Lead-Lead collisions, you must account for the fact that the heavy nucleus gets in the way (suppresses the production), especially in the middle.
3. When simulating Proton-Lead collisions, the effect is much weaker because the proton is too small to cause the same kind of blockage.

In short: They built a better map for predicting how light creates heavy particles in high-speed collisions. They found that heavy nuclei act like thick fog that dims the light, and once they accounted for that fog, their map matched the real-world terrain perfectly.

Technical Summary

Technical Summary: Analysis of J/ψ and ψ(2S) Charmonium Production in Ultraperipheral Lead-Lead and Proton-Lead Collisions at LHC Energies

Problem Statement
The study addresses the challenge of accurately describing exclusive charmonium ($J/\psi$ and $\psi(2S)$) photoproduction in Ultraperipheral Collisions (UPCs) at Large Hadron Collider (LHC) energies. While the two-gluon exchange model provides a robust perturbative QCD (pQCD) framework for elementary photon-proton ($\gamma p \to Vp$) interactions, standard simulations (such as STARlight) based on the impulse approximation tend to overestimate coherent charmonium yields in heavy-ion collisions (Pb-Pb), particularly around midrapidity. This discrepancy suggests that nuclear effects, such as gluon shadowing or saturation, are not fully captured by standard phenomenological scaling. Furthermore, a systematic comparison between symmetric (Pb-Pb) and asymmetric (p-Pb) collision systems is required to isolate the specific impact of nuclear gluon modifications.

Methodology
The authors employ a hybrid theoretical approach combining perturbative QCD with Monte Carlo simulation:

1. Elementary Cross Sections: The study utilizes the two-gluon exchange model to calculate the elementary photoproduction cross sections ($\gamma p \to J/\psi, \psi(2S)$). This model treats the process as a photon fluctuating into a compact $c\bar{c}$ dipole, which scatters elastically via the exchange of two gluons in a color-singlet state. The scattering amplitude is factorized and linked to the generalized gluon distribution of the target.
2. UPC Simulation: These calculated cross sections serve as the fundamental input for the STARlight Monte Carlo generator. The simulation models the UPC environment by treating the electromagnetic fields of relativistic ions as quasi-real photon fluxes (Weizsäcker-Williams approximation).
3. Phenomenological Correction: To reconcile theoretical predictions with experimental data, the authors introduce a phenomenological, rapidity-dependent suppression factor, $F(y) = e^{-\frac{1}{a|y|+b}}$. This factor is applied specifically to Pb-Pb simulations to account for the observed reduction in yields, particularly at midrapidity where small Bjorken-$x$ values are probed.
4. System Comparison: The framework is applied to both symmetric Pb-Pb collisions and asymmetric p-Pb collisions at LHC energies ($\sqrt{s_{NN}} = 2.76, 5.02, 8.16$ TeV) to analyze rapidity and transverse momentum ($p_T$) distributions.

Key Results

• Pb-Pb Collisions: The unmodified STARlight baseline significantly overestimates the experimental cross sections for coherent $J/\psi$ and $\psi(2S)$ production. After applying the phenomenological suppression factor $F(y)$, the corrected predictions successfully reproduce the experimental data from ALICE, CMS, and LHCb. The corrected model accurately captures the characteristic "rabbit-ear" (double-peak) structure of the rapidity distributions.
• p-Pb Collisions: In contrast to Pb-Pb, the p-Pb results show no significant need for strong rapidity-dependent suppression. The theoretical predictions align well with experimental data without the heavy correction applied to Pb-Pb. This is attributed to the asymmetric photon fluxes and the dominance of the photon-proton ($\gamma p$) interaction branch, which is not subject to the same nuclear gluon modifications as the photon-nucleus ($\gamma A$) branch.
• Transverse Momentum ($p_T$): The STARlight simulations successfully reproduce the low-$p_T$ diffractive pattern characteristic of coherent production off the entire nucleus. However, the overall yield remains slightly overpredicted even after rapidity corrections, necessitating a global, rapidity-independent normalization factor (approximately 0.90 for $J/\psi$ and 0.60 for $\psi(2S)$) to match experimental $p_T$ spectra.
• Nuclear Effects: When the fitted suppression factor is mapped to Bjorken-$x$, the reduction becomes stronger as $x$ decreases. This trend is qualitatively consistent with leading-twist nuclear shadowing and saturation-based descriptions, though the authors do not claim to isolate a specific microscopic mechanism.

Significance and Claims
The paper claims to validate the applicability of the two-gluon exchange model for charmonium UPC studies when combined with phenomenological nuclear suppression. Its primary contributions are:

1. Systematic Validation: It demonstrates that the two-gluon exchange model, when integrated into STARlight, provides a solid baseline for describing the kinematic features (rapidity shape and $p_T$ diffractive pattern) of charmonium production.
2. Quantification of Nuclear Suppression: By introducing and fitting the suppression factor, the work quantifies the magnitude of the discrepancy between impulse-approximation models and experimental data in Pb-Pb collisions.
3. Differentiation of Collision Systems: The study highlights the distinct behavior of symmetric versus asymmetric systems, showing that the strong suppression observed in Pb-Pb is not present in p-Pb due to the dominance of the $\gamma p$ channel.
4. Predictive Baseline: The authors provide predicted total cross sections for $J/\psi$ and $\psi(2S)$ production in UPCs at various LHC energies, offering reference values for future experimental measurements.

The authors maintain a modest stance, explicitly stating that the suppression factor is a phenomenological parametrization of the data's requirements rather than a first-principles derivation of a unique microscopic mechanism (e.g., distinguishing between shadowing and saturation). The work serves as a guide for future experimental design and data analysis in UPC measurements.