Elliptic flow of charm quarks produced in the early stage of pA collisions

This paper demonstrates that charm quarks produced in the early pre-equilibrium glasma stage of high-energy proton-nucleus collisions acquire significant elliptic flow ($v_2$) through interactions with non-Abelian fields, suggesting that pre-hydrodynamic dynamics play a crucial role in the observed heavy-flavor collectivity in small systems.

The Gist

Imagine two cars crashing into each other at nearly the speed of light. In the world of particle physics, these "cars" are protons and heavy atomic nuclei (like lead). When they smash together, they don't just bounce off; they create a tiny, super-hot explosion that lasts for a fraction of a second. Scientists call this the "early stage" of the collision.

This paper is about what happens to the heaviest particles in that explosion—specifically, charm quarks (which are like the "heavyweights" of the particle world)—during the very first split-second, before the explosion settles into a smooth, flowing liquid.

Here is the story of the paper, broken down with some everyday analogies:

1. The Setup: A Storm of Invisible Energy

Usually, when scientists study these collisions, they focus on what happens after the crash, when the debris has turned into a hot, flowing soup called a "quark-gluon plasma." But this paper looks at the very first moment (less than 0.0000000000000004 seconds).

At this instant, the collision isn't a smooth soup yet. It's a chaotic storm of intense, tangled energy fields. The authors call this the "Glasma."

• The Analogy: Imagine two giant fans blowing air at each other. Before the air mixes into a smooth breeze, there is a chaotic zone where the air streams crash, creating swirling eddies and strong gusts. The Glasma is that chaotic zone of invisible "color" forces (the strong nuclear force) before it smooths out.

2. The Players: The Heavyweights

Inside this storm, there are heavy particles called charm quarks.

• The Analogy: Think of the Glasma as a raging river with strong currents. The charm quarks are like heavy, dense rocks dropped into that river. Because they are heavy, they don't get swept away easily, but they do get pushed around by the strongest currents.

3. The Experiment: Simulating the Crash

The authors used a supercomputer to simulate these crashes. They didn't just make the proton and the nucleus look like smooth balls; they gave them "bumps" and "hotspots" (like the uneven surface of a real fruit) to make the simulation more realistic.

They tracked how these heavy rocks (charm quarks) moved through the storm (Glasma) using a set of rules called the Wong equations.

• The Analogy: It's like running a video game where you drop a bowling ball into a hurricane and watch exactly how the wind pushes it. They wanted to see: Does the wind push the ball in a specific direction?

4. The Discovery: The "Elliptic Flow"

In physics, "elliptic flow" is a fancy way of saying: "Do the particles fly out in a circle, or do they prefer to fly out in a football shape?"

• The Surprise: Scientists thought that this "football shape" (anisotropy) only happened later, when the hot soup formed and started flowing like a fluid.
• The Finding: The authors found that the Glasma storm itself is already shaped like a football. Because the storm is stronger in some directions than others, it pushes the heavy charm quarks in that same direction.
• The Result: Within a tiny fraction of a second (about 0.4 femtoseconds), the heavy quarks have already "learned" to fly in that football shape, just by interacting with the initial storm.

5. The "Aha!" Moment: Explaining Real Experiments

For years, experiments at the Large Hadron Collider (LHC) have seen that particles like the J/ψ (which is made of a charm quark and its anti-particle) fly out in this football shape. Scientists have been trying to figure out why.

• The Old Theory: "It must be the hot liquid soup pushing them later."
• The New Insight from this Paper: "Wait a minute! The initial storm (Glasma) alone is strong enough to give them that shape."

The authors calculated that the "kick" the charm quarks get from the initial Glasma storm accounts for a huge chunk of the shape we see in real experiments.

• The Analogy: Imagine a runner starting a race. We used to think they only started running fast after the starting gun and the first few steps. This paper says, "Actually, the wind at the starting line was so strong it already pushed them forward significantly before they even took their first step."

6. Why Does This Matter?

This changes how we understand the universe's earliest moments.

1. Small Systems Matter: Even in small collisions (proton hitting a nucleus), the initial chaos is powerful enough to create organized movement.
2. Heavy Quarks are Probes: Because charm quarks are heavy, they act like perfect sensors. They tell us exactly how strong the initial "wind" (Glasma) was.
3. Pre-Hydrodynamics: It proves that "order" (like the football shape) can emerge from "chaos" (the Glasma) before the system even becomes a fluid.

Summary

In simple terms: This paper shows that when a proton and a nucleus crash, the initial explosion isn't just random noise. It has a specific shape and strong winds. These winds are so powerful that they immediately push the heavy charm particles into a specific pattern. This pattern is so strong that it explains a lot of what we see in real-world experiments, proving that the "pre-explosion" phase is just as important as the "post-explosion" phase in shaping the universe.

Technical Summary

1. Problem Statement

The paper addresses the origin of collective behavior (specifically elliptic flow, $v_2$) observed in small collision systems like proton-nucleus (pA) collisions at the Large Hadron Collider (LHC). While hydrodynamic models successfully describe heavy-ion (AA) collisions, the mechanism generating flow in small systems (pA, pp) remains debated.

• Key Question: Can the pre-equilibrium stage, dominated by strong coherent gluon fields (the "Glasma"), generate sufficient momentum anisotropy in heavy quarks (charm) to explain the experimentally observed elliptic flow of $J/\psi$ mesons, without relying solely on later-stage hydrodynamic evolution?
• Gap in Literature: Previous studies modeled the Glasma with homogeneous color charge distributions or focused on bulk gluon anisotropies. Few studies have investigated how these early-stage anisotropies are transmitted to heavy quarks, particularly incorporating realistic subnucleonic fluctuations (hotspots) in both the proton and the nucleus.

2. Methodology

The authors employ a multi-stage theoretical framework combining Classical Yang-Mills (CYM) evolution with relativistic kinetic theory for heavy quarks.

A. Glasma Initialization (Color Glass Condensate Framework)

• Model: The initial state is modeled using the McLerran-Venugopalan (MV) model within the Color Glass Condensate (CGC) effective theory.
• Subnucleonic Fluctuations: A key improvement over previous works is the implementation of constituent-quark hotspots.
  • Proton: Modeled as three constituent quarks with Gaussian spatial distributions.
  • Nucleus: Modeled as $N_{part}$ nucleons, each containing three hotspots.
  • Color Sheets: The MV model is extended to include $N_s = 50$ stacked color sheets to smooth statistical fluctuations.
• Field Evolution: The initial color charges generate gauge fields ($A_\mu$) via the Poisson equation. The system evolves in real-time using the classical Yang-Mills equations on a 3+1D lattice with a gauge-invariant formulation.
• Numerical Scheme: A leapfrog algorithm is used to evolve the gauge links and electric fields in proper time ($\tau$), ensuring unitarity and preserving Casimir invariants.

B. Heavy Quark Dynamics

• Equations of Motion: Charm quarks are treated as test particles propagating through the evolving Glasma background. Their dynamics are governed by the relativistic Wong equations, which describe the evolution of position ($x$), momentum ($p$), and non-Abelian color charge ($Q^a$).
  • The equations include the Lorentz force term $g Q^a F^{\mu\nu} p_\nu$ and the precession of color charge.
• Initialization:
  • Momentum: Initialized according to FONLL (Fixed Order + Next-to-Leading Log) pQCD distributions, isotropic in azimuth.
  • Position: Extracted event-by-event based on the transverse energy density profile.
  • Color Charge: Generated to satisfy SU(3) Casimir invariants ($q^2, q^3$).
• Neglect of Back-reaction: The influence of heavy quarks on the Glasma fields is neglected, as previous studies suggest this effect is negligible during the short pre-equilibrium lifetime.

C. Analysis Methods

• Elliptic Flow ($v_2$): Calculated using two standard methods to ensure robustness:
  1. Event-Plane (EP) method: Based on the weighted average of the particle spectrum.
  2. Two-Particle Correlation (2PC) method: Based on azimuthal correlations between particles in a specific bin and a reference region.
• Comparison to Data: The calculated charm quark $v_2$ is converted to $J/\psi$ $v_2$ using a naive coalescence model ($v_2^{J/\psi} \approx 2 v_2^c$) and compared to CMS experimental data for p-Pb collisions at $\sqrt{s_{NN}} = 8.16$ TeV.

3. Key Contributions

1. Realistic Initial Conditions: First implementation of subnucleonic (constituent quark) fluctuations for both the proton and the nuclear participants in a Glasma heavy-quark study.
2. Transmission Mechanism: Demonstrated quantitatively how momentum anisotropies intrinsic to the Glasma (due to domain structures in the transverse plane) are efficiently transferred to heavy quarks via the color Lorentz force.
3. Time Scale: Established that significant $v_2$ is generated within the very early pre-equilibrium stage ($\tau \sim 0.4$ fm/c), well before hydrodynamic thermalization.
4. Quantitative Agreement: Showed that the early-stage contribution alone can account for a substantial fraction of the experimentally observed $J/\psi$ elliptic flow in p-Pb collisions.

4. Key Results

A. Nuclear Modification Factor ($R_{pA}$)

• The Glasma fields induce a net energy injection into the charm quark spectrum.
• Result: $R_{pA} < 1$ for low $p_T$ ($k_T \lesssim 2.5$ GeV) and $R_{pA} > 1$ for high $p_T$.
• Magnitude: The modification is mild (max deviation $\sim 5\%$), suggesting that the initial stage has a minor effect on the suppression of the spectrum compared to later stages, but a major effect on anisotropy.

B. Glasma Elliptic Flow ($v_2$)

• The Glasma fields themselves possess a non-zero $v_2$ due to the finite transverse correlation length of color domains.
• Trend: The integrated $v_2$ of the bulk Glasma decreases as the number of participants ($N_{part}$) increases.
  • Reasoning: Higher $N_{part}$ leads to a more isotropic distribution of correlation domains, washing out the anisotropy.
  • Maximum: $v_2 \sim 0.11$ for pp collisions.

C. Charm Quark Elliptic Flow ($v_2$)

• Transmission: The Glasma anisotropy is efficiently transmitted to charm quarks within $\tau \sim 0.4$ fm/c.
• Trend (Opposite to Bulk): Unlike the bulk Glasma, the charm quark $v_2$ increases with the number of participants ($N_{part}$) and the saturation scale ($Q_s$).
  • Reasoning: Higher $N_{part}$ implies stronger Glasma fields (higher energy density). Stronger fields exert a larger color Lorentz force, imprinting the anisotropy onto the heavy quarks more effectively.
• Magnitude:
  • For pA (0-5% centrality), $v_2$ peaks around $k_T \sim 2$ GeV with values up to $\sim 0.025$.
  • The results from the Event-Plane and 2PC methods are consistent.

D. Comparison with Experiment

• Using the coalescence ansatz, the calculated $v_2$ for $J/\psi$ derived from early-stage charm quarks matches the order of magnitude of the CMS experimental data for p-Pb collisions.
• This suggests that pre-hydrodynamic dynamics play a non-negligible role in the final-state heavy-flavor collectivity.

5. Significance

• Paradigm Shift: The paper challenges the notion that elliptic flow in small systems is exclusively a hydrodynamic phenomenon. It provides strong evidence that pre-hydrodynamic (Glasma) dynamics can generate significant heavy-flavor collectivity.
• Heavy Quarks as Probes: It reinforces the utility of heavy quarks as clean probes of the earliest stages of collisions, as they are produced early and interact with the strong coherent fields before thermalization.
• Future Directions: The authors suggest that future work should analyze how much of this early-stage $v_2$ is retained or modified during subsequent QGP evolution (e.g., via relativistic Boltzmann transport), which is currently underway.

In summary, the study demonstrates that the strong, coherent color fields of the Glasma, evolving on a timescale of $\sim 0.4$ fm/c, are sufficient to generate a sizeable elliptic flow in charm quarks, offering a compelling explanation for the observed $J/\psi$ flow in small collision systems.