Thermal Vorticity-Induced Spin Polarization in QGP

• Thermal vorticity–induced spin polarization is the conversion of local fluid rotation into particle spin alignment in relativistic fluids.
• The phenomenon is modeled using relativistic hydrodynamics and kinetic theory, with measurable observables such as Λ hyperon polarization.
• It serves as a diagnostic tool for probing QGP properties, phase transitions, and transport coefficients, linking nuclear and condensed matter research.

Thermal vorticity–induced spin polarization refers to the phenomenon in which the intrinsic spins of particles produced in a relativistic fluid—such as the quark–gluon plasma (QGP) in noncentral heavy-ion collisions—develop a macroscopic alignment owing to the presence of local vorticity, i.e., collective fluid rotation. This effect is determined by gradients of the inverse temperature-weighted four-velocity (βμ = uμ/T), and has become a central probe of QGP properties, vorticity, phase transitions, and the dynamics of spin in both strongly and weakly coupled matter. The theoretical framework is based on relativistic hydrodynamics, kinetic theory, and statistical quantum field theory, with connections to observable phenomena in nuclear and condensed matter systems.

1. Theoretical Foundations and Mathematical Formalism

Noncentral heavy-ion collisions impart large orbital angular momentum to the QGP, creating one of the most vortical fluids observed in the laboratory (Becattini et al., 2016, Wei et al., 2018). In relativistic systems, the hydrodynamic vorticity is recast into the thermal vorticity tensor: $$\varpi_{\mu\nu} = \frac{1}{2} \left( \partial_\nu \beta_\mu - \partial_\mu \beta_\nu \right), \quad \text{with} \quad \beta_\mu = \frac{u_\mu}{T}$$ where uμ is the fluid four-velocity and T the local temperature.

For spin-½ particles of mass m, the mean spin polarization vector at position x and momentum p (in the linear regime) is given by (Becattini et al., 2016, Wei et al., 2018): $$S^\mu(x, p) = -\frac{1}{8m}(1-n_F)\,\epsilon^{\mu\rho\sigma\tau}\,p_\tau\,\varpi_{\rho\sigma}$$ with n_F the Fermi–Dirac distribution and ε the Levi–Civita symbol. More generally, for spin s, the leading order is

$$S^\mu(x, p) = -\frac{s(s+1)}{6m}(1-n_F)\,\epsilon^{\mu\nu\rho\sigma}p_\nu\,\varpi_{\rho\sigma}$$

Calculation of polarization observables typically proceeds through integration over the freezeout hypersurface (using the Cooper–Frye prescription) and Lorentz boost into the particle rest frame.

Extension to higher-order effects and arbitrary spin is provided by the all-orders formalism (Palermo, 2023), in which the spin density matrix and polarization vector are derived using an exact analytic continuation of the density operator including constant thermal vorticity. This leads to saturation effects at large vorticity and minor quantitative corrections at top RHIC and LHC energies.

2. Microscopic Origin: Connection to Hydrodynamics, Kinetic Theory, and Statistical Models

Thermal vorticity–induced spin polarization emerges naturally in the local equilibrium density operator, which includes, in addition to the conserved charges, angular momentum terms (βμ and ϖμν), and, in certain scenarios, a magnetic field coupling (Becattini et al., 2016, Kumar, 2018). In kinetic theory, the covariant Wigner function formalism provides a phase-space description with spin encoded via the antisymmetric tensor ωμν. The equilibrium Wigner function, expanded in the Clifford algebra, captures the coupling of spin degrees of freedom to fluid vorticity and additionally supports the derivation of hydrodynamic equations with spin (Kumar, 2018, Kumar, 2018).

One subtlety revealed by this approach is that in strict global equilibrium (βμ a Killing vector), the thermal vorticity and the spin polarization tensor coincide, but in general, especially for local or extended equilibrium, the spin polarization tensor can differ from the underlying vorticity field; thus, spin must be treated as an independent hydrodynamic variable (Kumar, 2018, Kumar, 2018).

Conversion of orbital to spin angular momentum is understood through nonlocal collision terms in the Boltzmann kinetic equation. The essential quantum correction, at order ℏ, involves “position shifts” in the collision integrals, allowing orbital angular momentum to be converted into spin, a mechanism manifest in the hydrodynamics of micropolar fluids (Weickgenannt et al., 2020).

3. Experimental Manifestations and Phenomenology

The primary observable is the global polarization of Λ hyperons (and their antiparticles), measured via parity-violating decay angular distributions: $$\frac{dN}{d\Omega^*} = \frac{1}{4\pi} \left[ 1 + \alpha_\Lambda \mathbf{P}_\Lambda^* \cdot \hat{\mathbf{p}}^* \right]$$ where α_Λ ≈ 0.642 and $\hat{\mathbf{p}}^*$ is the direction of the decay proton in the Λ rest frame (Becattini et al., 2016, Wei et al., 2018). Experimental determination of the global spin polarization provides a “vorticity thermometer” for the QGP (Wei et al., 2018).

Beyond the averaged polarization, differential and harmonic spin flow observables—such as azimuthal dependences decomposed into Fourier series (with coefficients f₁, f₂)—probe the spatial and temporal structure of the thermal vorticity field, including quadrupole and “smoke-loop” patterns (Wei et al., 2018). Comparative measurements of polarization among various hyperons (Λ, Ξ⁰, Ω⁻) provide additional constraints on the mass and spin dependence of the effect and help separate out magnetic-field-induced contributions.

Crucial corrections arise from polarization “feed-down” in secondary decays. The polarization transfer coefficient C (e.g., –1/3 for strong and electromagnetic Σ* decays, +0.9 to +0.93 for weak Ξ decays) must be accounted for to reliably extract the underlying vorticity and discern any splitting due to magnetic fields (Becattini et al., 2016).

4. Magnetic and Shear Contributions; Extensions of the Framework

While thermal vorticity aligns the spins of particles and antiparticles in the same direction, a strong magnetic field introduces an additional polarization of opposite sign for particles and antiparticles, proportional to the magnetic moment μ: $$\langle S \rangle \sim \frac{1}{T}\left( \omega + \frac{\mu B}{S} \right)$$ This offers a means to disentangle vorticity and magnetic-field effects through splitting of Λ and Λ̄ polarizations (Becattini et al., 2016, Buzzegoli, 2022).

Recent developments have uncovered a symmetric-shear-induced spin polarization (SIP) mechanism, wherein the symmetric gradients of βμ, encoded in the thermal shear tensor ξμν = ½(∂μβν + ∂νβμ), contribute non-dissipatively to the spin polarization (Fu et al., 2021, Becattini et al., 2021, Lin et al., 2022, Buzzegoli, 15 May 2024). Quantum kinetic theory and linear response analyses demonstrate that SIP can compete with or even dominate vorticity effects in certain angular momentum distributions, resolving the so-called “local polarization puzzle” in experiment. Collisional contributions and redistribution effects due to shear further modulate both the magnitude and sign of the polarization, with nontrivial gauge dependence managed via Schwinger–Keldysh contour double gauge links (Lin et al., 2022).

5. Influence on Transport Properties and Critical Phenomena

Inclusion of thermal vorticity–induced spin polarization (VIP) in the phase-space distribution function modifies transport and thermodynamic coefficients—such as the speed of sound squared (c_s²), specific shear viscosity (η/s), specific bulk viscosity (ζ/s), and mean free path (λ)—in QGP and hadronic matter (Wei, 23 Apr 2025, Wei, 24 Jul 2025). Comprehensive kinetic theory frameworks show that VIP dominates changes in these coefficients relative to shear polarization, with the following consequences:

• η/s and λ are generally suppressed by VIP;
• ζ/s and c_s² exhibit nonmonotonic dependencies on collision energy, particularly notable as inflection points near √sₙₙ ≈ 19.6–27 GeV;
• These nonmonotonicities align with the expected location of the QCD critical point, suggesting that spin polarization and its impact on transport serve as qualitative diagnostic tools for phase structure (Singh et al., 2021, Wei, 23 Apr 2025, Wei, 24 Jul 2025).

Tables of calculated coefficients illustrate system-size insensitivity in the energy location of critical features and the parametric dominance of vorticity-driven polarization for scaling observables.

6. Model-Dependent Corrections, Higher-Order Effects, and Local Equilibrium

Thermal vorticity–induced polarization formulas are most robust at or near global equilibrium. In real heavy-ion collisions, non-equilibrium corrections are significant, especially for observables sensitive to the local angular momentum and dissipative processes.

• Projected (“magnetic-like”) components of the thermal vorticity are often required to reproduce certain angular modulations, as spatial rotations, not acceleration or “electric-like” vorticity, dominate the observable polarization (Florkowski et al., 2019, Banerjee et al., 8 May 2024).
• Higher-order (all-orders) analytic treatments incorporating strong vorticity predict saturation of the polarization and small quantitative corrections at top-energy collisions (Palermo, 2023).
• The spin relaxation time τ_s, empirically fitted to ∼5 fm/c, controls the build-up of spin polarization and its lag with respect to fluid vorticity, and is critical for matching theoretical models to differential polarization data (Banerjee et al., 8 May 2024).
• Inhomogeneous dynamical masses due to chiral condensates naturally generate spin polarization at O(ℏ), even in the absence of collisions, but in equilibrium the resulting distribution remains enslaved to the thermal vorticity field (Wang et al., 2021).

Quantum and radiative corrections, such as self-energy gradient corrections derived via hard-thermal-loop resummation, further modulate the axial current (chiral/vortical effect) and polarization spectrum, and need to be incorporated in precision modeling—especially relevant to the spin alignment of vector mesons (Fang et al., 17 Mar 2025).

7. Broader Applications and Extensions

The framework of thermal vorticity–induced spin polarization has been successfully translated to condensed matter systems, e.g., the relativistic regime of electron hydrodynamics in graphene, where it leads to experimentally accessible “thermovortical magnetization” (Jaiswal, 12 Sep 2024). In these (2 + 1)D systems, the spin-hydrodynamic model confirms that global equilibrium with constant thermal vorticity is a solution, and the resulting magnetization is directly proportional to the vorticity tensor and saturates with increasing chemical potential.

Statistical quantum field theory links the origin of thermal-vorticity-induced spin polarization to the fundamental spin–rotation coupling, which is quantified in-medium by the “gravitomagnetic” form factor g_Ω. This factor, unity in the vacuum by Lorentz invariance, is renormalized by finite-temperature effects and directly connects spin polarization observables to the underlying QCD matter properties (Buzzegoli, 15 May 2024).

The analogy to the Barnett effect is especially instructive: rotational motion in a fluid mimics the effect of a magnetic field on spin alignment, and the effective magnetic field responsible for spin polarization in rotating systems is B_eff = (ε_p/q) ω (Buzzegoli, 2022). This classical–quantum correspondence suggests that thermal vorticity–induced spin polarization is fundamentally non-anomalous in origin and robust to a range of model choices.

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In summary, thermal vorticity–induced spin polarization encompasses the conversion of macroscopic fluid vorticity into observable particle spin alignment in relativistic heavy-ion and condensed matter systems. This phenomenon links hydrodynamics, quantum kinetic theory, and statistical field theory with experimentally accessible spin observables, and has emerged as a sensitive diagnostic for rotational properties, transport coefficients, QCD phase transitions, and magnetic field effects in strongly correlated matter.