Local Spin Polarization in Anisotropic Gubser Flow: Suppression Mechanism and Formulation Dependence

Technical Summary: Local Spin Polarization in Anisotropic Gubser Flow

Problem Statement
The paper addresses the "sign puzzle" in local spin polarization observed in relativistic heavy-ion collisions. While global spin polarization of $\Lambda$ hyperons is well-described by thermal vorticity, theoretical calculations based solely on thermal vorticity yield a sign opposite to experimental measurements for local polarization along the beam direction. Recent developments suggest that including the thermal shear tensor in the modified Cooper-Frye formula can resolve this discrepancy, as shear-induced polarization often opposes vorticity-induced polarization. However, multiple theoretical formulations exist for incorporating thermal shear, differing primarily in the choice of the reference unit vector ($n^\mu$) aligned with the fluid velocity, laboratory time, or the freeze-out hypersurface normal. These formulations yield quantitatively different predictions, and the underlying mechanisms governing their differences, particularly regarding cancellation patterns, remain unclear from existing numerical studies.

Methodology
The authors employ an analytical approach using perturbed solutions of relativistic hydrodynamics to derive explicit expressions for longitudinal spin polarization.

1. Hydrodynamic Models: The study utilizes two analytical fluid solutions:
   • Rotating Hubble Flow: A simple model with spherical expansion and rigid rotation, serving as a toy model for global polarization.
   • Perturbed Gubser Flow: An anisotropic extension of the Gubser flow incorporating elliptic ($l=2$) and triangular ($l=3$) geometric deformations to model the transverse anisotropy and initial fluctuations relevant to local polarization. The analysis focuses on the large-system-size limit ($L \to \infty$), where $\tau/L \ll 1$.
2. Formulations Compared: The authors calculate the spin polarization vector $S^\mu(p)$ using three distinct formulations for the thermal shear contribution:
   • FLPSY: The unit vector $n^\mu$ is aligned with the fluid velocity $u^\mu$.
   • BBPIK: The unit vector $n^\mu$ is fixed along the laboratory time direction ($\delta^\mu_0$), with specific replacements for vorticity and shear tensors to remove explicit temperature gradients.
   • SBR: The unit vector $n^\mu$ is defined as the normal to the isothermal freeze-out hypersurface.
3. Calculation: The authors derive analytical expressions for the polarization components induced by thermal vorticity ($S_\varpi$) and thermal shear ($S_\xi$) by expanding the integrals over the freeze-out hypersurface to linear order in the perturbation parameters ($\epsilon_2, \epsilon_3$) and leading order in the large-$L$ limit.

Key Contributions and Results

• Analytical Derivation: The paper provides the first analytical derivation of local spin polarization directly from hydrodynamic solutions, avoiding the decoupling of dynamical variables found in modified blast-wave models.
• Sign of Polarization:
  • In all formulations, the thermal vorticity contribution alone yields a sign opposite to experimental data.
  • FLPSY: The total polarization recovers the desired experimental sign only at low transverse momenta ($p_T$) and only if the mass parameter is set to the constituent $s$-quark mass ($300$ MeV) rather than the $\Lambda$ hyperon mass ($1116$ MeV). At higher $p_T$, the sign reverts to the incorrect one.
  • BBPIK: This formulation successfully yields the desired sign for a wide range of transverse momenta ($p_T < 3$ GeV) using the $\Lambda$ mass, though the sign flips at very large $p_T$.
  • SBR: In the large-system-size limit, the contributions from thermal vorticity and thermal shear cancel exactly at leading order, resulting in a vanishing total polarization.
• Cancellation Mechanism (Acceleration Dominance): A central finding is the identification of a general cancellation pattern driven by acceleration terms.
  • In the BBPIK and SBR formulations, the acceleration components of the thermal vorticity tensor are exactly canceled by the acceleration components of the thermal shear tensor.
  • Consequently, the net longitudinal spin polarization in these formulations originates solely from non-acceleration effects.
  • This mechanism explains why the net polarization is significantly smaller than the individual contributions and why the SBR formulation leads to exact cancellation in the specific case of the perturbed Gubser flow (where non-acceleration terms also integrate to zero or are negligible).
• Dependence on Mass and Momentum: The ratio of vorticity to shear contributions depends on the particle mass and transverse momentum. Smaller masses (e.g., $s$-quark) enhance the vorticity-to-shear ratio, affecting the sign recovery in the FLPSY formulation.

Significance
The paper claims that its analytical approach clarifies the conditions under which the correct sign of local spin polarization is recovered and identifies the specific role of acceleration terms in the cancellation between thermal vorticity and thermal shear. By demonstrating that the total polarization in certain formulations is dominated by non-acceleration effects, the work suggests that the magnitude of the net polarization need not be substantial even when elliptic flow is finite. The authors emphasize that their study serves to deepen the analytical understanding of these mechanisms rather than to provide a phenomenological fit to experimental data, noting that relaxing the large-size limit and incorporating viscous terms are necessary steps for future phenomenological applications.