Charmonium production at SPS and FAIR energies

This study applies the Remler formalism within the Parton-Hadron-String Dynamics (PHSD) framework to investigate charmonium production and dissociation in baryon-rich matter at SPS and FAIR energies, demonstrating that an in-medium heavy quark potential successfully describes experimental data at SPS while providing estimates for future GSI/FAIR collisions.

The Gist

The Big Picture: Smashing Atoms to Find "Ghost" Particles

Imagine you are trying to understand how a specific type of rare, fragile object (let's call it a "Glass Vase") is made and how it survives when thrown into a chaotic, crowded mosh pit.

In the world of physics, these "Glass Vases" are called Charmonium (specifically the $J/\psi$ particle). They are made of a heavy "charm" quark and its anti-particle stuck together. Scientists smash heavy atoms (like Lead or Gold) together at incredibly high speeds to create a super-hot, super-dense soup of energy called a Quark-Gluon Plasma (QGP). This soup is like the "mosh pit."

The goal of this paper is to figure out:

1. How many of these "Glass Vases" are created in the crash?
2. How many survive the mosh pit?
3. How does the "crowd" (the dense matter) affect their ability to form or break apart?

The researchers looked at two different types of "mosh pits":

• SPS Energy: A very hot, dense crowd, but not too crowded with extra heavy people (baryons).
• FAIR Energy: A slightly cooler crowd, but packed with many more heavy people (high baryon density).

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The Tool: The "Remler Formalism" (The Coalescence Game)

To predict how these vases form, the authors used a mathematical tool called the Remler formalism.

The Analogy: Imagine you are throwing two magnets (a charm quark and an anti-charm quark) into a room. They are flying around wildly.

• The Old Way: You might just guess, "If they fly close enough, they stick."
• The Remler Way: This method is much more precise. It tracks the exact position and speed of every magnet. It asks: "At this exact moment, do their positions and speeds match the perfect pattern required to snap together and become a vase?"

The paper says this method works great for simple collisions (like hitting a single proton against another), but they had to tweak it to work for the chaotic, hot soup of heavy-ion collisions.

The Journey of the "Glass Vase"

The paper breaks the life of a Charmonium particle into three stages:

1. The Birth (The Crash)

When the atoms smash, the energy creates pairs of charm and anti-charm quarks.

• The Finding: At the lower energies (FAIR), the "crowd" is so dense with heavy particles that the quarks have a harder time finding each other to stick together. However, the authors found that the random jiggling of the heavy particles inside the nuclei (called Fermi motion) actually gives them an extra "kick." This kick helps them overcome the energy barrier to create the vases, making production much higher at these low energies than a simple guess would suggest.

2. The Mosh Pit (The Quark-Gluon Plasma)

Once the vases are formed (or trying to form), they are in the hot soup.

• The Problem: In a super-hot soup, the "glue" holding the vase together gets weaker. It's like trying to hold a snowball together in a blast furnace; it melts.
• The Discovery: The authors tried two scenarios:
  • Scenario A: The glue is constant. (This failed to match real-world data).
  • Scenario B: The glue gets weaker as the temperature rises. They found that the "vase" (the $J/\psi$) can survive up to a certain temperature (about 1.15 times the critical melting point), but right before it melts, it gets huge and floppy.
  • The Result: By accounting for this "melting glue," their calculations finally matched the experimental data from the SPS (European lab). This proves that the "glue" inside the plasma changes with temperature.

3. The Aftermath (The Hadronic Phase)

After the hot soup cools down, it turns back into normal particles (protons, neutrons, pions). The vases are now flying through a dense forest of these particles.

• Nuclear Absorption: Imagine the vase flying through a forest of trees (baryons). If it hits a tree, it shatters. The paper calculated how often this happens. They found that at lower energies, the vase is more likely to hit a tree and break.
• Comover Effects: Sometimes, the vase hits a flying rock (a meson) and breaks. But, interestingly, the reverse can happen too! Two broken pieces (open charm mesons) can fly together and rebuild the vase.
• The Surprise: The paper found that while the "rebuilding" process is important, the shattering (absorption by baryons) is the main reason fewer vases survive in heavy collisions.

Key Takeaways for the General Audience

1. Temperature Matters: The "glue" holding these particles together isn't static; it weakens as the environment gets hotter. The paper successfully modeled this, showing that the particle survives just long enough to be detected before the heat destroys it.
2. The "Crowd" Effect: In the lower-energy experiments (FAIR), the environment is full of heavy particles. This density actually helps create more charm particles than expected because the heavy particles inside the nuclei are jiggling around, giving the quarks an extra boost to collide.
3. Survival of the Fittest: Most of the "Glass Vases" that disappear in heavy collisions aren't melting in the hot soup; they are getting smashed by other particles after the soup cools down.
4. Prediction for the Future: Using what they learned from the European lab (SPS), the authors made a prediction for the upcoming FAIR lab in Germany. They estimate that even though the energy is lower, the unique conditions there will still produce a significant number of these particles, perhaps even more than a simple calculation would predict.

Summary

The paper is like a detailed survival guide for a fragile object in a chaotic environment. By using a sophisticated tracking method (Remler) and understanding how the "glue" changes with heat, the authors successfully explained why we see the number of particles we do in current experiments and predicted what we should see in future, lower-energy experiments.

Technical Summary

Technical Summary: Charmonium production at SPS and FAIR energies

Problem Statement
Relativistic heavy-ion collisions offer a unique laboratory to study the transition from hadronic matter to a deconfined quark-gluon plasma (QGP). While lattice QCD establishes a crossover transition at small baryon chemical potential ($\mu_B$), the high-$\mu_B$ region remains challenging to access via first-principles calculations due to the fermion sign problem. Quarkonium suppression, specifically of the $J/\psi$, is a central probe for QGP formation, attributed to color screening and thermal dissociation. However, in the high-$\mu_B$ regime relevant to the Super Proton Synchrotron (SPS) and the Facility for Antiproton and Ion Research (FAIR), charm-anticharm ($c\bar{c}$) pair production is rare (typically one pair per collision). This scarcity allows for the tracing of individual charm quarks without the interference common at higher energies (RHIC/LHC), offering a unique opportunity to study correlations and interactions in dense baryonic matter. The specific challenge addressed is to estimate charmonium production and dissociation in these baryon-rich environments, accounting for both partonic (QGP) and hadronic interactions, where the properties of the heavy-quark potential and the nuclear absorption cross sections are critical but not fully established.

Methodology
The study employs the Parton-Hadron-String Dynamics (PHSD) transport approach, a non-equilibrium microscopic model that describes strongly interacting hadronic and partonic matter. The methodology proceeds in three stages:

1. Charm Production and Dynamics: Heavy quarks are produced via hard nucleon-nucleon scatterings simulated using the PYTHIA event generator, with kinematic distributions rescaled to match Fixed-Order-Next-to-Leading-Logarithm (FONLL) calculations. In heavy-ion collisions, nuclear parton distribution functions (PDFs) are modified using the EPS09 set to account for (anti)shadowing. Within the QGP phase, charm quarks interact with off-shell partons described by the Dynamical Quasi-Particle Model (DQPM), which incorporates medium-dependent masses and widths consistent with lattice QCD results.
2. Remler Formalism for Formation: The formation of charmonium is treated using the Remler formalism. This approach expresses the number of quarkonia as a projection of the charmonium Wigner function onto the phase-space distribution of charm and anticharm quarks. The rate of change in quarkonium number is determined by the difference between regeneration (gain) and thermal dissociation (loss) terms, updated whenever heavy quarks scatter in the QGP. The formalism is applied up to the hadronization temperature ($T_c$).
3. Hadronic Interactions: Post-hadronization, charmonia undergo interactions with the surrounding hadronic medium. These are categorized into:
   • Nuclear Absorption: Inelastic scattering with baryons (nucleons), dominant in $p+A$ collisions. The cross section is extracted from experimental $p+A$ data.
   • Comover Effects: Inelastic scattering with mesons (e.g., $J/\psi + \pi \to D + \bar{D}$) and regeneration from open-charm mesons ($D + \bar{D} \to J/\psi + \pi$). These cross sections are modeled using a two-body transition amplitude $|M_0|^2$, tuned to experimental data, with backward reactions determined by detailed balance.

Two scenarios for the heavy-quark potential in the QGP are investigated:

• Scenario A: A temperature-independent potential (radii fixed as in vacuum).
• Scenario B: A temperature-dependent potential derived from the free energy of a heavy-quark pair, where the $J/\psi$ radius increases with temperature and diverges near the dissociation temperature ($T_{diss}$).

Key Results

• Validation in $p+p$ and $p+A$: The Remler formalism successfully reproduces charmonium production cross sections in $p+p$ collisions across a wide energy range (SPS to LHC) using a single set of radii. In $p+A$ collisions at SPS energies ($E_{kin} = 400$ GeV and $158$ GeV), the model extracts nuclear absorption cross sections of $\sigma_{abs}(J/\psi) \approx 7$ mb (at 400 GeV) and $\approx 10$ mb (at 158 GeV), with $\psi'$ absorption roughly twice that of $J/\psi$.
• Heavy-Ion Collisions at SPS ($E/A = 158$ GeV):
  • Using a temperature-independent potential, the model underestimates the suppression observed in NA50 and NA60 data, even when assuming the maximum possible interaction rate (twice the charm-quark rate).
  • Implementing the temperature-dependent potential (Scenario B) yields successful reproduction of experimental data. In this scenario, the initial Wigner projection is delayed until the temperature drops to $\approx 1.15 T_c$, where the $J/\psi$ radius is maximized before shrinking. This delay, combined with hadronic interactions (nuclear absorption and comover effects), accounts for the observed suppression. The results suggest the $J/\psi$ dissociation temperature is close to $T_c$.
  • The study finds that the final $J/\psi$ yield is primarily suppressed by nuclear absorption and the dissociation of excited states ($\chi_c, \psi'$), while regeneration from $D\bar{D}$ mesons contributes significantly to the final $J/\psi$ yield but negligibly to $\psi'$.
• Estimates for FAIR Energies:
  • For $p+Au$ collisions at $E/A = 29$ GeV, the integrated production cross sections are estimated at $210-310$ nb for $J/\psi$ and $22-37$ nb for $\psi'$, depending on the assumed nuclear absorption cross section.
  • A critical finding for FAIR energies ($\sqrt{s_{NN}} \approx 5.2$ GeV) is the significant enhancement of charm production due to the Fermi motion of nucleons. Because the nominal collision energy lies very close to the charm production threshold, the broadening of the nucleon-nucleon center-of-mass energy distribution increases the average charm production cross section by a factor of approximately 800 compared to a naive $p+p$ estimate at the nominal energy.

Significance and Claims
The paper claims that the Remler formalism, when coupled with the PHSD transport approach and an in-medium heavy-quark potential, provides a consistent description of charmonium production from $p+p$ to central heavy-ion collisions at SPS energies. The study emphasizes that:

1. In-Medium Potential is Essential: A temperature-independent potential fails to describe the data; the inclusion of a temperature-dependent potential, where the $J/\psi$ radius expands near $T_c$, is necessary to reproduce the observed suppression.
2. Hadronic Phase Importance: At SPS energies, the hadronic phase (specifically nuclear absorption and comover interactions) plays a crucial role in the final suppression, particularly due to the large hadronic corona surrounding the QGP fireball.
3. FAIR Prospects: While direct charmonium production at FAIR energies is rare, the Fermi motion of nucleons significantly enhances the probability of $c\bar{c}$ production near the threshold, suggesting that charmonium signals may be more accessible than previously estimated based solely on nominal beam energies.

The authors conclude that their framework successfully extracts interaction rates and cross sections at SPS energies, providing a baseline for future experimental investigations at FAIR, where the high baryon density offers a distinct regime for studying QCD matter.