Non-equilibrium Dynamical Attractors and Thermalisation of Charm Quarks in Nuclear Collisions at the LHC Energy

This study demonstrates that while charm quarks in longitudinally expanding Quark-Gluon Plasma exhibit non-equilibrium dynamical attractors, their thermalisation is significantly delayed by temperature-dependent diffusion coefficients derived from lattice QCD, suggesting they may not fully equilibrate within the plasma's lifetime and challenging the applicability of viscous hydrodynamics to their dynamics.

The Gist

Imagine you drop a handful of heavy, glowing marbles (Charm Quarks) into a giant, expanding balloon filled with a super-hot, sticky soup (the Quark-Gluon Plasma or QGP). This happens when scientists smash heavy atoms together at the Large Hadron Collider (LHC).

The big question this paper asks is: How fast do these heavy marbles get "stirred" into the soup until they move exactly like the soup itself? And, do they ever forget where they started, or do they keep their own unique rhythm?

Here is the story of their journey, broken down into simple concepts:

1. The Two Types of Soup

The scientists ran two different simulations to see how the marbles behave, depending on how "sticky" the soup is.

• Scenario A: The Super-Sticky Soup (AdS/CFT)
  Imagine the soup is like thick honey. If you drop a marble in, it gets dragged along immediately. The scientists used a theoretical model where the soup is extremely sticky.

  • The Result: The marbles get "thermalized" (they start moving exactly like the soup) very quickly—within about 1 to 1.5 femtoseconds (a femtosecond is a quadrillionth of a second). They forget their original speed and direction almost instantly.

• Scenario B: The Realistic Soup (Lattice QCD)
  This is based on real data from supercomputers. The soup is still sticky, but not as sticky as the honey in Scenario A. It's more like warm maple syrup.

  • The Result: The marbles take much longer to catch up. It takes about 5 femtoseconds for them to start moving like the soup.
  • The Problem: The "soup" (the QGP) only exists for about 5 to 10 femtoseconds before it cools down and disappears. In this realistic scenario, the heavy marbles are still trying to catch up just as the party is ending. They never fully get to know the dance moves of the soup.

2. The "Universal Dance" (Dynamical Attractors)

The paper talks about something called "Dynamical Attractors." Think of this as a universal dance floor.

• The Setup: Imagine you have two groups of dancers. Group A starts dancing wildly and chaotically. Group B starts dancing in a slow, organized circle.
• The Magic: In the Super-Sticky Soup, no matter how they started, both groups quickly stop dancing their own weird dances and start doing the exact same "Universal Dance" together. They lose their individual memories and become one unit.
• The Twist: In the Realistic Soup, the dancers are still trying to learn the steps. They haven't reached the "Universal Dance" yet. They are still remembering how they started. This means that in smaller collisions (like smashing lighter atoms), the heavy marbles might never learn the dance at all.

3. The "Sticky" Math Problem

The scientists looked at how much the marbles' movement deviated from the "perfect" soup movement. They found a scary trend for the realistic scenario:

• Low Speed: At slow speeds, the marbles are okay.
• High Speed: As the marbles move faster, the difference between how they should move (if they were part of the soup) and how they actually move gets huge.
• The Analogy: Imagine trying to predict the path of a leaf in a stream. If the leaf is slow, it follows the water perfectly. But if the leaf is moving fast, it starts skipping over the water, jumping around, and doing its own thing.
• The Conclusion: For the realistic soup, at speeds around 3 GeV (a specific energy unit), the marbles are 100% out of sync with the soup. This breaks the rules of the "Hydrodynamics" math that scientists usually use to predict these collisions. It's like trying to use a map of a calm lake to navigate a raging waterfall; the map stops working.

4. Why Does This Matter?

For a long time, physicists thought heavy quarks (like Charm) were so heavy they would just sit there or move slowly, and that they would quickly join the "soup" and act like the lighter particles (like pions).

This paper says: "Wait a minute!"

• If the soup is only as sticky as real-world data suggests, the heavy quarks are not fully thermalized. They are still "outliers."
• This is especially true in small collisions (like Oxygen-Oxygen or peripheral collisions). In these small systems, the soup disappears before the heavy marbles can even learn the dance.
• This means we can't just use the standard "fluid" math to predict what happens to heavy quarks in these collisions. We need new, more complex math to understand them.

The Bottom Line

The paper reveals that heavy quarks in nuclear collisions are like latecomers to a dance party.

• In a super-sticky world, they arrive, grab a drink, and immediately start dancing with everyone else.
• In the real world, they arrive, try to learn the steps, but the party ends before they can fully join in. They spend the whole time dancing to their own beat, never quite syncing up with the crowd.

This discovery forces scientists to rethink how they model these collisions, especially for smaller systems, because the heavy particles aren't behaving like the fluid everyone thought they were.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of whether charm quarks (Heavy Quarks, HQs) in the Quark-Gluon Plasma (QGP) created in ultra-relativistic heavy-ion collisions (at LHC energies) reach thermal equilibrium and exhibit dynamical attractors (universal behavior independent of initial conditions).

While hydrodynamics has successfully described light-flavor observables (suggesting rapid thermalization), the applicability of hydrodynamics to heavy quarks remains debated. Key issues include:

• Thermalization Timescales: Do charm quarks equilibrate fast enough to participate in the collective flow before the QGP decouples?
• Coupling Regimes: How do different interaction strengths—specifically the strong coupling limit (AdS/CFT) versus realistic temperature-dependent diffusion coefficients derived from Lattice QCD (lQCD)—affect the approach to equilibrium?
• Hydrodynamization: Does the charm sector exhibit "far-from-equilibrium" attractors similar to the bulk medium, and at what point does the deviation from equilibrium ($\delta f/f_{eq}$) become too large for viscous hydrodynamics to be applicable?

2. Methodology

The authors employ the Relativistic Boltzmann Transport (RBT) approach within a 1+1D Bjorken expansion framework.

• Simulation Setup:
  • Bulk Medium: Modeled as a fluid of massive light quarks and gluons ($m_i = 0.5$ GeV) with a fixed shear viscosity to entropy density ratio $\eta/s = 1/4\pi$. The Equation of State (EoS) is tuned to match lQCD data.
  • Charm Quarks: Treated as a Brownian motion component with mass $m_{HQ} = 1.3$ GeV. Their evolution is governed by coupled Boltzmann equations including elastic $2 \leftrightarrow 2$ collisions ($g+Q \to g+Q$ and $q+\bar{q}+Q \to q+\bar{q}+Q$).
  • Initial Conditions: The study tests a wide variety of initial momentum distributions to probe universality:
    • FONLL: Perturbative QCD calculation (standard for $p_T$ spectra).
    • EPOS4HQ: Monte Carlo generator including cascades.
    • Thermal (2D/3D): Boltzmann distributions at various initial temperatures ($T_0 = 0.5$ GeV and $1.0$ GeV).
• Coupling Regimes: Two distinct scenarios for the spatial diffusion coefficient $D_s$ are compared:
  1. Strong Coupling (AdS/CFT): Constant $2\pi T D_s = 1$.
  2. Realistic (lQCD): Temperature-dependent $D_s^{lQCD}(T)$ derived from recent unquenched lattice QCD data.
• Analysis Metrics:
  • Effective Temperature ($T_{eff}$): Derived from energy and particle density ratios.
  • Momentum Moments ($\mathcal{M}_{mn}$): Normalized moments of the distribution function to track isotropization and convergence.
  • Deviation from Equilibrium ($\delta f/f_{eq}$): Quantifying the non-equilibrium correction to the distribution function.
  • Scaled Time: Evolution analyzed as a function of $\tau/\tau_{eq}$, where $\tau_{eq}$ is the charm relaxation time.

3. Key Contributions

• First Study of HQ Attractors: This is the first investigation identifying dynamical attractors specifically for a Brownian component (charm quarks) embedded in an evolving medium, distinct from the bulk attractor.
• Comparison of Coupling Regimes: A rigorous comparison showing that while $D_s^{lQCD}(T)$ approaches the AdS/CFT limit near the critical temperature ($T_c$), its temperature dependence leads to drastically different dynamical evolution at higher temperatures.
• Quantification of Hydrodynamics Validity: The paper provides a quantitative threshold for the validity of viscous hydrodynamics for charm quarks by analyzing the power-law growth of deviations from equilibrium.

4. Key Results

A. Thermalization Timescales and Attractors

• Strong Coupling ($2\pi T D_s = 1$): Charm quarks reach the bulk temperature and exhibit dynamical attractors rapidly, within $\tau \sim 1 - 1.5$ fm. Different initial conditions (FONLL, EPOS, Thermal) converge to a universal behavior quickly.
• Realistic lQCD Coupling ($D_s^{lQCD}(T)$): The relaxation is significantly slower. The system takes $\tau \sim 5$ fm to approach the attractor.
  • Implication: In small systems (peripheral collisions, light-ion collisions like O-O) where the QGP lifetime is short ($< 5$ fm), charm quarks do not fully thermalize and retain memory of their initial conditions. Even in central Pb-Pb collisions, full thermalization is marginal.

B. Momentum Moments and Universality

• Far-from-Equilibrium Attractors: When plotted against scaled time $\tau/\tau_{eq}$, different interaction regimes converge to universal curves for specific initial distributions.
• Initial Condition Dependence: Unlike the bulk, the charm attractor depends on the initial shape of the distribution (e.g., FONLL vs. Boltzmann). These distinct attractor curves only merge at late times when the system reaches the Navier-Stokes limit.
• Stress Tensor Anisotropy: The anisotropy $2T_{zz}/(T_{xx}+T_{yy})$ reaches the attractor behavior faster than lower-order moments in the strong coupling case, but lags significantly in the lQCD case.

C. Deviation from Equilibrium ($\delta f/f_{eq}$)

• Strong Coupling: The deviation remains small ($\delta f/f_{eq} \lesssim 0.25$) up to $p_T \approx 2$ GeV. The deviation grows quadratically ($\beta \approx 2.2$).
• Realistic lQCD: The deviation is much larger.
  • At $p_T \approx 1.5$ GeV, the deviation is already $\sim 10\%$.
  • At $p_T \approx 3$ GeV, the deviation reaches $\sim 100\%$ ($\delta f/f_{eq} \sim 1$).
  • The growth follows a power law with $\beta \approx 4.5$.
• Hydrodynamics Applicability: The large deviation and high power index ($\beta > 4$) in the lQCD scenario strongly suggest that viscous hydrodynamics is not applicable to charm quark dynamics at intermediate $p_T$ in realistic scenarios, challenging previous assumptions based on strong coupling limits.

5. Significance and Outlook

• System Size Dependence: The findings imply that the success of hydrodynamic models for charm quarks is highly system-dependent. It may hold for large, central collisions (where lifetimes are long) but fails for small systems (peripheral or light-ion collisions) under realistic lQCD coupling.
• Theoretical Challenge: The large deviation from equilibrium ($\delta f/f_{eq} \sim 1$) questions the standard approach of using hydrodynamics to describe heavy-flavor flow ($v_2$) without explicit kinetic corrections.
• Future Directions: The authors note that while 1+1D results are robust, a full 3+1D study is required to account for transverse expansion effects and to solidify conclusions regarding small systems. They also highlight the need to test these findings against the recently derived Kolmogorov equation for HQ dynamics in AdS/CFT.

In summary, the paper demonstrates that while charm quarks exhibit dynamical attractors, the temperature dependence of the diffusion coefficient in realistic QCD scenarios leads to significantly delayed thermalization and large non-equilibrium deviations, limiting the applicability of hydrodynamic descriptions for charm quarks in many collision systems.