Dissociation-driven quarkonium spin alignment in Pb--Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV

This study proposes that medium vorticity induces spin-dependent dissociation of quarkonia via a modified color-singlet potential, thereby altering their spin alignment ($\rho_{00}$) in Pb--Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV and offering new insights into the microscopic dynamics of the vortical quark-gluon plasma.

The Gist

The Big Picture: A Cosmic Dance Floor

Imagine two massive lead balls (atomic nuclei) smashing into each other at nearly the speed of light. This collision is so violent that it creates a tiny, super-hot drop of "primordial soup" called the Quark-Gluon Plasma (QGP). Think of this soup as a swirling, boiling vortex of energy where the usual rules of matter don't apply.

In this chaotic dance floor, heavy particles called quarkonia (specifically charmonium and bottomonium) are born. These are like heavy-duty couples: a heavy quark and its partner (an antiquark) holding hands tightly.

The scientists in this paper are asking a specific question: When these couples spin, do they prefer to spin in a specific direction?

The Mystery: Why Do They Spin?

In a normal, calm world, these couples would spin randomly. Sometimes they spin "up," sometimes "down," and sometimes they stand straight up (spin 0). If there were no outside influence, the odds of them standing straight up would be exactly 1 in 3 (33.3%).

However, experiments at the Large Hadron Collider (LHC) have shown something weird. In these heavy-ion collisions, the couples aren't spinning randomly. They seem to be "aligned" or "polarized." Sometimes they stand up more often (more than 33%), and sometimes they lie down more often (less than 33%).

The paper tries to explain why this happens. The authors propose a new mechanism: The Spin-Disappearing Act.

The Core Idea: The "Spin-Sensitive" Melting Pot

The authors suggest that the swirling QGP isn't just hot; it's spinning (it has "vorticity," like a giant whirlpool).

Here is the analogy:
Imagine the QGP is a giant, spinning blender. Inside, you have three types of spinning tops (representing the three spin states: +1, 0, -1).

• The Old Theory: The blender just heats everything up equally, melting the tops at the same rate.
• The New Theory (This Paper): The blender's spin interacts differently with the tops depending on how they are spinning.
  • If a top spins in the same direction as the blender, the blender's force might make it wobble and break apart faster.
  • If a top spins the opposite way, the blender might actually help it stay together a bit longer.
  • If a top stands straight up (spin 0), the blender treats it differently again.

The Result: Because the "blender" (the QGP) destroys the tops at different rates depending on their spin, the ones that survive to be measured are the ones that were harder to break. This changes the ratio. If the "standing up" tops are harder to break, you see more of them (ρ00 > 1/3). If the "lying down" tops are harder to break, you see fewer standing ones (ρ00 < 1/3).

The Characters: The "Sturdy" vs. The "Fragile"

The paper looks at four specific types of couples (quarkonia), which behave very differently:

1. The Sturdy Couples (J/ψ and Υ(1S)):

   • These are the "tight hugs." They are very heavy and hold on very tightly.
   • What happens: Because they are so strong, the "spin-sensitivity" of the swirling blender matters a lot. The spin of the blender can tip the scales, making them align in a specific way (usually standing up more often).
   • Analogy: A heavy, well-built oak tree in a windstorm. The wind (vorticity) pushes it, and it leans in a specific direction because its roots are deep.

2. The Fragile Couples (ψ(2S) and Υ(2S)):

   • These are the "loose hugs." They are excited states, meaning they are larger and hold on less tightly.
   • What happens: They are so fragile that the sheer heat of the blender melts them instantly, regardless of how they are spinning. The "spin" effect is drowned out by the "heat" effect.
   • Analogy: A house of cards in a hurricane. It doesn't matter which way the wind blows; the heat and chaos melt it so fast that the specific direction of the wind doesn't change the outcome. They tend to lie down (ρ00 < 1/3) because the heat destroys the "standing up" ones first.

The Variables: Speed and Crowd Size

The paper also checks how this changes based on two things:

• Transverse Momentum (Speed): How fast the couple is moving across the room.
  • The Twist: The "effective temperature" the couple feels changes depending on how fast they are moving (like the Doppler effect with sound). If they move fast, they might feel cooler or hotter than the surrounding soup, which changes how quickly they melt. This creates a wavy, non-straight line in the data.
• Multiplicity (Crowd Size): How many particles are in the collision (how "busy" the room is).
  • The Finding: In a small, quiet room (low multiplicity), the spin of the blender is very obvious. In a huge, roaring crowd (high multiplicity), the heat is so intense that it overrides the spin effects, especially for the fragile couples.

The Conclusion: What Did We Learn?

The authors successfully built a mathematical model showing that the way these particles break apart (dissociate) depends on their spin and the rotation of the medium.

• For the strong couples: The rotation of the plasma (vorticity) is the main reason they align.
• For the weak couples: The heat of the plasma is the main reason they align (or rather, don't align with the rotation).

Why does this matter?
It's like finding a new way to measure the "spin" of the universe's most extreme environments. By watching how these heavy couples align, physicists can now "see" the rotation and temperature of the Quark-Gluon Plasma, giving us a deeper understanding of how the universe behaved just microseconds after the Big Bang.

In short: The paper explains that the "spin alignment" of heavy particles in a nuclear collision isn't magic; it's a survival game where the spinning, hot soup eats the particles at different rates depending on how they are spinning. The survivors tell us the story of the soup's rotation.

Technical Summary

1. Problem Statement

The paper addresses the recent experimental observation of quarkonium spin alignment in ultra-relativistic heavy-ion collisions (specifically Pb–Pb at $\sqrt{s_{NN}} = 5.02$ TeV).

• The Phenomenon: Experiments (ALICE) have measured the spin density matrix element $\rho_{00}$ for vector mesons ($J/\psi$, $\Upsilon$, etc.). In an unpolarized state, $\rho_{00} = 1/3$. Deviations indicate spin alignment.
• The Puzzle: Experimental data shows inconsistent results across different reference frames (helicity, Collins-Soper, event-plane). For instance, $\rho_{00} < 1/3$ in the helicity frame but $\rho_{00} > 1/3$ in the Collins-Soper frame.
• The Gap: While global vorticity in the Quark-Gluon Plasma (QGP) is known to polarize light hadrons (like hyperons), the mechanism driving quarkonium spin alignment remains unclear. Previous models often focused on spin transport or hadronization. This paper investigates whether dissociation mechanisms (the breaking apart of quarkonium) driven by the coupling between quarkonium spin and medium vorticity can explain the observed alignment.

2. Methodology

The authors employ a multi-stage theoretical framework combining relativistic hydrodynamics, quantum mechanics in a rotating frame, and in-medium dissociation physics.

A. Medium Evolution (Hydrodynamics)

• Framework: Second-order relativistic viscous hydrodynamics (Müller-Israel-Stewart formalism).
• Initial Conditions: Initial temperature ($T_0$) is estimated phenomenologically based on charged particle multiplicity ($dN_{ch}/d\eta$) using MC-Glauber model inputs for Pb-Pb collisions.
• Evolution: The temperature cooling profile $T(\tau)$ is solved using 1+1D boost-invariant hydrodynamics, incorporating shear viscosity ($\eta/s = 1/4\pi$).

B. Effective Temperature and Kinematics

• Relativistic Doppler Shift (RDS): Since heavy quarkonia do not thermalize with the medium, the authors calculate an effective temperature ($T_{eff}$) experienced by the moving quarkonium.
• Mechanism: $T_{eff}$ is derived from the relative velocity between the quarkonium and the local fluid cell, accounting for relativistic blue-shift (at low $p_T$) and red-shift (at high $p_T$). This $T_{eff}$ is crucial for calculating dissociation rates.

C. Spin-Vorticity Coupling (Quantum Mechanics)

• Hamiltonian: An effective Hamiltonian is constructed for a two-body system (heavy quark-antiquark) in a rotating medium.
• Coupling Term: The Hamiltonian includes a term $-\vec{\omega} \cdot \vec{S}$, where $\vec{\omega}$ is the vorticity and $\vec{S}$ is the total spin.
• Vorticity Parameterization: The angular velocity $\omega$ is related to a conserved circulation parameter $C$ via Stokes' theorem.
• Schrödinger Equation: The radial Schrödinger equation is modified to include a spin-dependent potential term proportional to $m_j C / r^2$ (where $m_j$ is the magnetic quantum number). This splits the energy levels and wavefunctions for different spin projections ($m_j = +1, 0, -1$).

D. Dissociation Mechanisms

The total decay width ($\Gamma_D$) is calculated as the sum of two spin-dependent processes:

1. Collisional Damping: Arises from the imaginary part of the in-medium complex potential (Landau damping). The width is calculated as the expectation value of $\text{Im}(V)$ using the spin-dependent wavefunctions.
2. Gluonic Dissociation: Arises from inelastic scattering with thermal gluons. The cross-section is calculated using the Peskin-Bhanot formalism, modified by the relativistic Doppler shift of the gluon distribution in the quarkonium rest frame.

E. Observable Calculation

• Spin Alignment ($\rho_{00}$): The probability of survival for each spin state $m$ is $P_m = \exp(-\int \Gamma_{D, m} d\tau)$.
• The observable is defined as:
  $$\rho_{00} = \frac{P_0}{P_{+1} + P_0 + P_{-1}}$$
  Deviations from $1/3$ indicate alignment.

3. Key Contributions

1. Dissociation-Driven Mechanism: The paper proposes that spin alignment is not solely a result of spin transport but is significantly driven by spin-dependent dissociation. The medium vorticity modifies the decay widths differently for different spin states ($m_j$), altering the relative survival probabilities.
2. State-Dependent Behavior: The study distinguishes between tightly bound 1S states ($J/\psi$, $\Upsilon(1S)$) and loosely bound 2S states ($\psi(2S)$, $\Upsilon(2S)$).
3. Integration of Hydrodynamics and Quantum Mechanics: It successfully couples the macroscopic evolution of the QGP (temperature and vorticity profiles) with the microscopic quantum dynamics of heavy quarkonia in a rotating frame.
4. Reference to Experimental Frames: The calculations are aligned with the event-plane frame (perpendicular to the reaction plane), which is relevant for interpreting recent ALICE data.

4. Key Results

A. Decay Widths ($\Gamma_D$)

• Spin Splitting: In a rotating medium ($C \neq 0$), the decay widths split based on $m_j$.
  • For $m_j = +1$, the effective potential barrier is lowered, increasing the wavefunction overlap at short distances and enhancing dissociation (larger $\Gamma_D$).
  • For $m_j = -1$, the potential becomes more confining, suppressing dissociation.
  • For $m_j = 0$, the behavior is intermediate but generally closer to $m_j = -1$ in terms of survival probability.
• Temperature Dependence: $\Gamma_D$ is highly sensitive to $T_{eff}$. At high temperatures, dissociation rates increase significantly.

B. Spin Alignment ($\rho_{00}$) vs. Multiplicity

• 1S States ($J/\psi, \Upsilon(1S)$):
  • $\rho_{00}$ generally exceeds 1/3 (longitudinal alignment) for small to intermediate circulation parameters ($C$).
  • This implies that the $m_j=0$ state survives better than the $m_j=\pm 1$ states due to rotational effects.
  • At very high multiplicities (high temperature), thermal dissociation dominates, and the effect of vorticity diminishes, causing $\rho_{00}$ to approach $1/3$.
• 2S States ($\psi(2S), \Upsilon(2S)$):
  • $\rho_{00}$ is consistently below 1/3 (transverse alignment) across the multiplicity range.
  • This is attributed to their weak binding energy, making them highly susceptible to thermal dissociation. The thermal environment preferentially destroys the $m_j=0$ component relative to the others, or the rotational effect is too weak to overcome thermal smearing.

C. Transverse Momentum ($p_T$) Dependence

• Non-Monotonic Behavior: $\rho_{00}$ exhibits a non-monotonic dependence on $p_T$.
  • At low $p_T$, $\rho_{00}$ is influenced by the initial effective temperature and vorticity.
  • As $p_T$ increases, $T_{eff}$ changes (due to Doppler shift), altering the dissociation rate.
  • For $J/\psi$, $\rho_{00}$ often peaks around $p_T \approx 15$ GeV/c before decreasing.
  • For $\psi(2S)$, $\rho_{00}$ remains $< 1/3$ and shows a distinct structure driven by the dominance of thermal dissociation over rotational effects.

5. Significance and Conclusion

• Unified Interpretation: The study provides a unified explanation for the state-dependent spin alignment observed in experiments. It suggests that the interplay between binding energy, medium temperature, and vorticity determines the final $\rho_{00}$.
• Microscopic Dynamics: It establishes spin-dependent dissociation as a viable microscopic mechanism for quarkonium spin alignment, complementing existing theories based on spin transport.
• Probe of QGP: The results suggest that quarkonium spin observables are sensitive probes not just of the thermal properties (temperature) but also of the vortical structure of the QGP.
• Future Directions: The authors recommend extending the model to 3+1D hydrodynamics to account for spatial fluctuations, applying it to small systems (p-Pb, p-p) to test QGP formation, and performing differential measurements in various polarization frames to constrain the circulation parameter $C$.

In summary, the paper demonstrates that the rotation of the QGP medium induces a spin-dependent modification of quarkonium decay widths, leading to measurable spin alignment that varies systematically with the quarkonium state (1S vs. 2S), collision centrality, and transverse momentum.