⚛️ nuclear theory

Rotational Brownian Motion and $\mathrm{J/\psi}$ Spin Alignment in Heavy-Ion Collisions

This paper proposes a microscopic mechanism for $\mathrm{J}/\psi$ spin alignment in heavy-ion collisions by deriving the polarization of charm quark pairs through rotational Brownian motion and spin-vorticity coupling in a strong magnetic field, ultimately linking these dynamics to the observed spin-density matrix element $\rho_{00}$ via hadronization.

Original authors: Bhagyarathi Sahoo, Captain R. Singh

Published 2026-09-16

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CC BY 4.0

Original authors: Bhagyarathi Sahoo, Captain R. Singh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the most violent collisions imaginable, where atomic nuclei smash together at nearly the speed of light, matter is forced into a state that has not existed since the first moments of the universe. This state, known as the quark-gluon plasma, is a seething, super-hot soup where the usual rules of atomic structure dissolve, and the fundamental building blocks of matter—quarks and gluons—roam freely. Scientists study this fleeting, extreme environment to understand how the universe cooled and organized itself after the Big Bang. One of the most intriguing questions in this field is how the tiny, spinning particles within this soup interact with their surroundings. Just as a spinning top might wobble or change direction when pushed by a breeze, these subatomic particles can have their orientation altered by the intense magnetic fields and the swirling, vortex-like motion of the plasma itself. Understanding how these particles spin and align offers a unique window into the invisible forces and fluid dynamics of this primordial matter.

A recent study by researchers at the Indian Institute of Technology Indore and the National Institute of Science Education and Research in India explores exactly this phenomenon, focusing on a specific type of heavy particle called the charm quark. When these heavy quarks are created in the initial crash of a heavy-ion collision, they are born with a specific spin direction, often aligned with the powerful magnetic fields generated by the passing protons. As they travel through the expanding quark-gluon plasma, they are buffeted by the hot, chaotic medium. The researchers wanted to know how much of that original spin direction survives this journey. They treated the movement of the quark's spin not as a smooth, predictable path, but as a random, jittery process similar to how a leaf might tumble in a turbulent stream, a concept known as rotational Brownian motion. By using mathematical models that describe this random tumbling under the influence of both magnetic fields and the swirling motion of the plasma, they calculated how the spin of the charm quark changes over time.

The team found that the amount of spin a quark retains depends heavily on how fast it is moving and how long it stays in the plasma. Faster-moving quarks, which zip through the medium more quickly, manage to keep a larger fraction of their original spin alignment because they have less time to be jostled out of place by the surrounding particles. Slower quarks, which linger longer in the soup, lose more of their initial orientation. This relationship between speed and spin retention provides a clear signature of how the plasma behaves, acting as a probe for the internal friction and magnetic properties of the medium. To make their theoretical predictions comparable to real-world observations, the researchers constrained their model by fitting it to experimental data on D∗+ meson spin alignment from Pb–Pb collisions. This fitting process yielded a spin relaxation time—the time it takes for the spin to lose its memory of the initial direction—of approximately 1.31 femtometers (a unit of length often used to represent time in this context), a scale so small it is measured in quadrillionths of a second.

The ultimate goal of tracking these spinning quarks is to understand what happens when they eventually stop and combine to form new particles. Specifically, the researchers looked at the formation of the J/ψ particle, a type of meson made of a charm quark and its antimatter partner, an anti-charm quark. When these two heavy particles meet at the edge of the plasma to form a J/ψ, their individual spins combine to determine the final orientation of the new particle. The study revealed that the way this combination happens is critical. If the particles merge directly from the soup, a process called coalescence, the resulting J/ψ tends to have a specific spin alignment that is stronger than random chance. However, if the particles form the J/ψ through a different mechanism known as fragmentation, the alignment is weaker and points in a different direction. This distinction is vital because it means that by measuring the spin of the J/ψ particles produced in collisions, scientists can deduce which formation process was dominant and, by extension, learn more about the conditions inside the plasma.

The researchers also noted that the spin alignment of the J/ψ is not a fixed value but changes depending on the momentum of the particle. At lower speeds, the interaction with the medium is so strong that the particles lose almost all memory of their initial spin, resulting in a random, unaligned state. But at higher speeds, the particles preserve more of their alignment, suggesting that the medium's ability to scramble their spin has limits. This finding challenges previous assumptions and suggests that the spin of heavy quarks is a sensitive tool for measuring the vortical, or swirling, structure of the quark-gluon plasma. While the current results are based on theoretical models and simulations rather than direct measurement of every variable, the study provides a detailed framework for interpreting future experimental data. It suggests that the spin of heavy quarks is not just a passive property but an active record of the extreme environment they traversed, offering a new way to map the invisible currents and magnetic forces of the early universe.

Technical Summary: Rotational Brownian Motion and J/ψ Spin Alignment in Heavy-Ion Collisions

Problem Statement
Heavy quarks (charm and bottom) produced in ultra-relativistic heavy-ion collisions (HICs) serve as probes for deconfined QCD matter (Quark-Gluon Plasma, QGP). While the translational Brownian motion of heavy quarks—governed by drag and diffusion coefficients—is well-established, the rotational Brownian motion and spin dynamics of these quarks in the presence of strong magnetic fields and fluid vorticity remain less explored. Recent observations of hadron spin polarization motivate the investigation of how heavy quarks propagate through the QGP, specifically how their spin orientation evolves under stochastic interactions. A critical open question is how the polarization of constituent charm ( $c$ ) and anti-charm ( $\bar{c}$ ) quarks translates into the spin alignment of the resulting $J/\psi$ meson, particularly distinguishing between different hadronization mechanisms.

Methodology
The authors employ a semiclassical stochastic approach to model the spin dynamics of heavy quarks in a rotating, magnetized medium.

Theoretical Framework: The study extends the Landau-Lifshitz-Gilbert (LLG) equation to describe the spin dynamics of a heavy quark in its rest frame under the influence of a vorticity field ( $\omega$ ) and a magnetic field ( $B$ ). This is formulated as a generalized multivariate Langevin equation.
Fokker-Planck Formalism: By applying the Kramers-Moyal expansion, the authors derive a Fokker-Planck equation governing the probability distribution $P(s, \tau)$ of the spin orientation. This equation incorporates a spin-diffusion coefficient ( $D_s$ ) and a damping coefficient ( $\lambda$ ), linked via a phenomenological Einstein-like relation ( $D_s = k_B T / \zeta_s$ ).
Perturbative Solution: The Fokker-Planck equation is solved perturbatively. The authors assume an exponentially decaying magnetic field profile and a constant vorticity profile. The solution is expanded using a Dyson series in terms of small perturbation parameters $\alpha$ (related to vorticity) and $\beta$ (related to the magnetic field).
Spin Alignment Calculation: The residual polarization of the heavy quark ( $\langle \cos \theta \rangle$ $⟨ cos θ ⟩$ ) is calculated after a characteristic interaction time $\tau$ $τ$ . This polarization is then used to determine the spin-density matrix element $\rho_{00}$ $ρ_{00}$ for the $J/\psi$ $J / ψ$ meson. The calculation considers two distinct hadronization mechanisms at the chemical freeze-out surface:
1. Coalescence (Recombination): Where the $J/\psi$ forms from the recombination of a polarized $c$ and $\bar{c}$ .
2. Fragmentation: Where the $J/\psi$ forms via the fragmentation of a high-momentum heavy quark.

Key Contributions and Results

Analytical Expression for Polarization: The paper derives an analytical expression for the time evolution of heavy-quark spin polarization, $\langle \cos \theta \rangle$ , as a function of transverse momentum ( $p_T$ ) and rapidity. The results indicate that $\langle \cos \theta \rangle$ increases systematically with $p_T$ , suggesting that faster-moving quarks retain a larger fraction of the initial vorticity-induced polarization due to reduced interaction time with the medium.
Dependence on Hadronization Mechanism: A primary finding is the strong dependence of the $J/\psi$ $J / ψ$ spin alignment ( $\rho_{00}$ $ρ_{00}$ ) on the hadronization mechanism. The authors find an inverted hierarchy for heavy-flavor mesons compared to light-flavor mesons:
- Coalescence: Yields $\rho_{00} > 1/3$ .
- Fragmentation: Yields $\rho_{00} < 1/3$ .
  This reversal is attributed to the opposite alignment of heavy quarks and antiquarks along the initial magnetic field direction (implemented via $\cos \theta_0 = +1$ for quarks and $-1$ for antiquarks).
Quantitative Estimates: Using a spin relaxation time $\tau_s$ fitted to prompt $D^{*+}$ data (and varied to account for Run 3 uncertainties), the study predicts $\rho_{00}$ values at mid-rapidity. At low $p_T$ ( $\le 4$ GeV/c), the model predicts $\rho_{00} \approx 1/3$ (no alignment), attributed to significant spin relaxation. At high $p_T$ , the model predicts stronger spin alignment.

Significance and Claims
The paper claims to provide a microscopic description of heavy-quark spin transport, linking the rotational diffusion of charm quarks to the observed spin alignment of $J/\psi$ mesons.

Mechanism Identification: It suggests rotational diffusion as a viable mechanism underlying $J/\psi$ spin alignment in the presence of strong magnetic and vortical fields.
Diagnostic Potential: The study posits that the $J/\psi$ spin alignment observable ( $\rho_{00}$ ) is sensitive to the hadronization mechanism (coalescence vs. fragmentation) and the spin relaxation time ( $\tau_s$ ) of the medium.
Modest Scope: The authors explicitly state that their results are qualitative due to the lack of first-principles determination for model parameters (such as $T$ , $R$ , $\Omega$ , $\tau_s$ ). They emphasize that a quantitative description would require a systematic Bayesian inference analysis combined with experimental data. Furthermore, the study is limited to directly produced $J/\psi$ mesons, excluding feed-down contributions from excited states like $\psi(2S)$ and $\chi_c$ .

The work concludes that establishing quarkonium spin observables as probes of the QGP's vortical structure and microscopic dynamics requires further theoretical development, including realistic hydrodynamic evolution and field-theoretical calculations of spin relaxation in rotating systems.

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