Charmonium suppression in fixed target proton-nucleus collisions

This paper systematically investigates cold nuclear matter effects on charmonium production in fixed-target proton-nucleus collisions by analyzing the interplay of initial-state energy loss, nuclear shadowing, and final-state absorption using existing data to predict normal absorption levels for upcoming experiments at CERN and FAIR.

The Gist

Imagine you are trying to bake a very delicate, rare cake called a Charmonium Cake (specifically, a $J/\psi$ particle). This cake is made by smashing two heavy ingredients (charm quarks) together at high speeds.

Physicists have long been interested in these cakes because if they disappear or get "suppressed" (don't form properly) in a giant, hot oven (a heavy-ion collision creating Quark-Gluon Plasma), it proves the oven is hot enough to melt the very fabric of matter.

However, before you can blame the "hot oven," you need to make sure the cake didn't just get ruined by the kitchen itself (the cold, dense nuclear matter) before it even hit the hot oven. This paper is a detailed investigation of how the "kitchen" messes up the cake-making process in fixed-target collisions.

Here is a breakdown of the paper's story using everyday analogies:

1. The Setup: The "Kitchen" vs. The "Hot Oven"

In the world of particle physics, scientists smash protons into heavy metal targets (like Tungsten or Lead).

• The Goal: To see how the "kitchen" (Cold Nuclear Matter) affects the cake.
• The Problem: When you smash a proton into a heavy nucleus, the cake doesn't just appear instantly. It has to travel through a crowded room of other particles (nucleons) to get out. Along the way, it can get bumped, knocked over, or absorbed.

The authors wanted to figure out exactly how much the kitchen messes things up so that when they look at the "Hot Oven" experiments later, they know exactly what to subtract.

2. The Three Ways the Kitchen Ruins the Cake

The paper identifies three main ways the "kitchen" (the target nucleus) interferes with the cake-making:

• The Shadow Effect (Nuclear Shadowing): Imagine the ingredients (partons) inside the heavy metal target are hiding in the shadows. Because there are so many of them packed together, they block each other from being seen or used. This means fewer ingredients are available to make the cake, so you end up with fewer cakes than expected.
• The Energy Leak (Initial State Energy Loss): Imagine the delivery trucks (beam protons) carrying the ingredients have to drive through a thick, sticky mud (the nuclear matter) before they reach the mixing bowl. As they drive through the mud, they lose speed and energy. By the time they get to the bowl, they aren't moving fast enough to smash the ingredients together perfectly. This reduces the number of cakes made.
• The Bump-and-Run (Final State Absorption): Once the cake is baked (the $c\bar{c}$ pair is formed), it has to walk through the crowded room to get out. If the room is too crowded, the cake gets bumped into walls and destroyed before it can leave. This is called "absorption."

3. The Investigation: Who is to Blame?

The authors looked at data from old experiments (like NA50 and NA60 at CERN) where they smashed protons into different sized metal targets (from light Beryllium to heavy Uranium).

They built a computer model to simulate the cake-making process. They asked: "If we only account for the Shadow Effect, does it explain why we see fewer cakes? What if we add the Energy Leak?"

The Big Discovery:
Previously, scientists thought the "Bump-and-Run" (Final State Absorption) was the main culprit. They assumed the "Energy Leak" (Initial State Energy Loss) wasn't very important.

The authors found that the Energy Leak is actually a huge deal!
When they included the "Energy Leak" in their model, the amount of "Bump-and-Run" needed to explain the missing cakes dropped by about 50%.

• Analogy: Imagine you see a bakery with half the usual number of cakes. You might think, "Oh, the delivery trucks must have crashed into the walls (Absorption)." But this paper says, "Wait! The trucks actually got stuck in the mud and lost half their speed (Energy Loss) before they even started baking. So, the walls weren't the main problem; the mud was!"

4. The "Formation Length" Rule

The paper also explains when the cake gets ruined.

• If the cake forms quickly (short formation length) while still inside the crowded room, it gets bumped and destroyed (Absorption).
• If the cake forms slowly (long formation length) and only becomes a solid cake after it leaves the room, it doesn't get bumped.

The authors calculated that for the experiments they studied, the cakes formed quickly enough to get bumped. This confirmed that the "Absorption" model was the right one to use for these specific data sets.

5. Predicting the Future: The "Low Energy" Challenge

The paper ends with a crystal ball prediction. New experiments are coming up at lower energies (30 GeV, 50 GeV, 80 GeV).

• The Prediction: At these lower speeds, the delivery trucks will get stuck in the mud even more (more Energy Loss).
• The Result: Even though the "kitchen" is colder, the cakes will be suppressed more than we thought because the trucks are losing so much energy in the mud. The authors predict that the "Absorption" effect will look even stronger at these low energies because the "Energy Leak" is doing so much of the damage.

Summary

This paper is like a detective story in a bakery.

1. The Mystery: Why are so few "Charmonium Cakes" being made in heavy metal targets?
2. The Old Theory: It's because the cakes get bumped into walls (Absorption).
3. The New Clue: The delivery trucks are losing too much energy in the mud before they even start baking (Initial State Energy Loss).
4. The Conclusion: We need to account for the mud (Energy Loss) to understand how much the walls (Absorption) are actually hurting the cakes.

By understanding this "mud," scientists can now better interpret future experiments. If they see cakes disappearing in a "Hot Oven" (Quark-Gluon Plasma), they can be much more confident that it's the heat melting the cake, and not just the delivery trucks getting stuck in the mud.

Technical Summary

1. Problem Statement

The suppression of charmonium states (specifically $J/\psi$ and $\psi(2S)$) in relativistic heavy-ion collisions is a primary signature for the formation of Quark-Gluon Plasma (QGP). However, to isolate "anomalous" suppression caused by the hot QGP medium, one must first precisely understand the "normal" suppression caused by Cold Nuclear Matter (CNM) effects in proton-nucleus ($p+A$) collisions.

CNM effects arise from various physical mechanisms acting at different stages of resonance formation:

• Initial State Effects: Modifications of parton distribution functions (nuclear shadowing) and energy loss of projectile partons before the hard scattering.
• Final State Effects: Absorption of the pre-resonant or resonant $c\bar{c}$ pair by nucleons in the target nucleus.

Previous studies often treated these effects in isolation or used simplified parametrizations. A critical gap exists in systematically quantifying the interplay between initial-state parton energy loss and final-state nuclear absorption across a wide range of fixed-target beam energies, particularly to predict suppression levels for upcoming low-energy experiments (e.g., NA60+ at CERN SPS, CBM at FAIR).

2. Methodology

The authors perform a systematic phenomenological analysis using the following framework:

• Production Model: They adopt the Color Evaporation Model (CEM) to calculate the perturbative QCD production cross-section of $c\bar{c}$ pairs. This treats all quarkonium states as a constant fraction ($F_i$) of the total open heavy-flavor production, constrained by the invariant mass range $2m_c < m < 2m_D$.
• Initial State Effects:
  • Nuclear Shadowing: Implemented using the EPPS21 NLO nuclear parton distribution functions (nPDFs) combined with CT18ANLO free proton PDFs.
  • Parton Energy Loss: Two distinct formalisms are employed to model the energy loss ($\Delta E$) of beam partons traversing the target nucleus:
    1. Brodsky-Hoyer (BH): Assumes linear dependence on path length ($\Delta x \propto L$).
    2. BDMPS: Assumes quadratic dependence on path length ($\Delta x \propto L^2$), relevant for the Landau-Pomeranchuk-Migdal (LPM) regime.
  • The energy loss parameters ($\alpha$ and $\beta$) were previously constrained by the authors using Drell-Yan data from Fermilab E866 and E906 experiments.
• Final State Effects:
  • Modeled using the Glauber model framework.
  • The $c\bar{c}$ pair is assumed to undergo inelastic collisions with nucleons, characterized by an absorption cross-section $\sigma_{abs}$.
  • Data Selection: The authors carefully select datasets where the $J/\psi$ formation length ($d_0$) is smaller than the nuclear diameter, ensuring that color neutralization occurs inside the nucleus. This validates the use of the nuclear absorption scenario. Consequently, they exclude data from HERA-B, E866, and E906 (which probe forward rapidities where formation occurs outside the nucleus) and focus on NA50 and NA60 data from CERN SPS (beam energies 158–450 GeV).
• Analysis: They fit the ratio of per-nucleon production cross-sections ($\sigma_{pA}/A$ vs. $\sigma_{pBe}/9$) to extract the effective absorption cross-section ($\sigma_{abs}$) under three scenarios:
  1. nPDF only (No energy loss).
  2. nPDF + BH energy loss.
  3. nPDF + BDMPS energy loss.

3. Key Contributions

• Systematic Interplay Analysis: The study is one of the first to rigorously quantify how initial-state parton energy loss modifies the extracted value of the final-state absorption cross-section.
• Re-evaluation of Absorption Cross-Sections: By incorporating energy loss, the authors demonstrate that previous extractions of $\sigma_{abs}$ (which often ignored energy loss or used older nPDFs) were overestimated.
• Kinematic Filtering: They explicitly filter experimental data based on formation length calculations to ensure the validity of the nuclear absorption hypothesis, avoiding the mixing of regimes where final-state energy loss (rather than absorption) would dominate.
• Predictions for Future Facilities: They provide quantitative predictions for $J/\psi$ suppression at lower beam energies (30, 50, and 80 GeV) relevant for the NA60+, J-PARC, and CBM experiments.

4. Key Results

• Impact of Energy Loss on Shadowing:
  • Without energy loss, the EPPS21 nPDFs predict an anti-shadowing effect (enhancement of $c\bar{c}$ production) in the SPS energy range ($\sqrt{s_{NN}} \approx 13-100$ GeV).
  • When parton energy loss is included, this anti-shadowing is converted into a net shadowing (suppression) effect. The energy loss reduces the effective center-of-mass energy available for hard scattering, significantly suppressing production, especially at lower beam energies.
• Extraction of $\sigma_{abs}$:
  • Reduction in $\sigma_{abs}$: When initial-state energy loss is accounted for, the extracted final-state absorption cross-section ($\sigma_{abs}$) decreases by approximately 50% compared to the "no energy loss" scenario.
  • Values: For $J/\psi$ at 400 GeV, $\sigma_{abs}$ drops from $\sim 5.1$ mb (no energy loss) to $\sim 2.5$ mb (with energy loss).
  • Model Independence: The extracted $\sigma_{abs}$ values are remarkably similar for both BH and BDMPS energy loss models, suggesting that current $p+A$ target mass dependence data cannot distinguish between linear and quadratic path-length dependencies of energy loss.
  • Energy Dependence: A strong beam-energy dependence is confirmed: $\sigma_{abs}$ increases as beam energy decreases (e.g., $\sigma_{abs} \approx 7.3$ mb at 158 GeV without energy loss).
• $\psi(2S)$ Analysis: Similar trends are observed for $\psi(2S)$, with an even larger absorption cross-section ($\sim 8-9$ mb without energy loss, $\sim 4-6$ mb with energy loss), consistent with its larger size and weaker binding energy.
• Predictions for Low Energies:
  • At lower energies (30–80 GeV), the initial-state energy loss effect becomes dominant.
  • Even with a reduced final-state absorption cross-section, the total suppression is predicted to be stronger at lower energies due to the combined effect of strong initial-state shadowing/energy loss and increased final-state absorption.
  • The nuclear modification factor ($R_{pA}$) is predicted to drop significantly as beam energy decreases.

5. Significance

• Baseline for QGP Search: The study provides a refined baseline for "normal" nuclear suppression. By disentangling initial-state energy loss from final-state absorption, it offers a more accurate determination of the final-state absorption cross-section, which is crucial for interpreting heavy-ion collision data.
• Implications for QGP Interpretation: The finding that initial-state energy loss significantly reduces the required final-state absorption cross-section implies that previous attributions of suppression solely to final-state interactions may have been overestimated. This necessitates a re-evaluation of QGP-induced suppression signals in heavy-ion collisions.
• Guidance for Future Experiments: The predictions for NA60+, J-PARC, and CBM highlight that low-energy collisions will probe a regime where initial-state energy loss is the dominant CNM effect. This guides experimentalists on what to expect and underscores the need for precise measurements of $p+p$ and $p+A$ collisions at these energies to disentangle CNM effects from potential QGP signals in future heavy-ion runs.
• Methodological Rigor: The paper sets a standard for combining modern nPDFs (EPPS21), specific energy loss formalisms, and kinematic filtering (formation length) to analyze charmonium production.