Dynamical equation for quark spin polarization in the rotating medium

TL;DR

This work addresses how heavy-quark spins polarize in a rotating quark-gluon plasma produced in non-central relativistic heavy-ion collisions. It derives a nonrelativistic heavy-quark Hamiltonian from the Dirac equation in a rotating frame, $H_{\rm NR}=\frac{\boldsymbol{p}^{2}}{2m}-\boldsymbol{\omega}\cdot\boldsymbol{j}$ with $\boldsymbol{j}=\boldsymbol{l}+\boldsymbol{s}$, via a Foldy–Wouthuysen transformation, establishing rotation-induced spin–orbit coupling as the basis for spin dynamics. The authors then formulate a Landau-Lifshitz–like equation by introducing a non-Hermitian Hamiltonian and a stochastic torque $\boldsymbol{\omega}_{\mathrm{th}}$, yielding $\frac{d}{dt}\langle \mathbf{S}\rangle = \langle \mathbf{S}\rangle \times (\boldsymbol{\omega}+\boldsymbol{\omega}_{\mathrm{th}}) - \lambda \langle \mathbf{S}\rangle \times (\langle \mathbf{S}\rangle \times (\boldsymbol{\omega}+\boldsymbol{\omega}_{\mathrm{th}}))$ with a fluctuation-dissipation relation $\langle \omega_{\mathrm{th},\mu}(t)\omega_{\mathrm{th},\nu}(t')\rangle = C_Q 2\lambda k_B T \delta_{\mu\nu}\delta(t-t')$ and $C_Q=1/3$. Numerical results in a static medium demonstrate that polarization grows with the damping parameter and with the rotation rate, approaching an equilibrium set by $\omega/T$, consistent with detailed-balance expectations, and imply possible impacts on $J/\psi$ spin alignment. This framework provides a quantitative tool to study heavy-quark spin polarization and its phenomenology in relativistic heavy-ion collisions.

Abstract

In non-central relativistic heavy-ion collisions, the produced quark-gluon plasma (QGP) behaves approximately as a rotating fluid due to the system's initial angular momentum. In this rotating fluid, the spins of quarks become polarized due to the coupling between spin and angular momentum, as well as random spin-spin interactions. Since the Landau-Lifshitz (LL) equation effectively describes the spin polarization of fermions in a medium with a magnetic field, we derive a phenomenological equation analogous to the LL equation for heavy quark spin dynamics in the rotating medium. The spin-angular momentum coupling and random spin-spin interactions are incorporated, leading to a detailed balance of heavy quark spin distributions. This equation provides insight into the spin dynamics of heavy quarks and quarkonium in relativistic heavy-ion collisions.

Figures (3)

• Figure 1: The precession and polarization of the heavy quark spin vector around the angular momentum $\boldsymbol{\omega}$.
• Figure 2: The time evolution of the mean value of the normalized heavy quark spin vector $\langle S_z \rangle / |{\bf S}|$ along the $z$-direction in rotating quark matter with a constant temperature $T = 0.2 \, \text{GeV}$ and a constant angular momentum $\omega = 20 \, \text{MeV}$, with the angular momentum along the $z$-direction. The damping factor is taken as $\lambda = 0.03, 0.05, 0.1$, respectively.
• Figure 3: The spin polarization of quarks plotted over time, with the parameters $\lambda=0.1$, $T=0.2$ GeV, and $\omega=10,20,30$ MeV.