This paper studies why the direction of spin carried by particles produced in relativistic heavy‑ion collisions can be hard to predict. The authors use an analytic hydrodynamic model to follow how local spin polarization along the beam direction is produced. They find that two different fluid effects — thermal vorticity (a rotation‑like effect in the hot medium) and thermal shear (a stretching or velocity‑gradient effect) — have similar sizes but opposite tendencies. Which one wins depends on a technical choice in the calculation, and that choice can change the predicted sign of the polarization.

To reach these conclusions the researchers solved ideal relativistic hydrodynamics with a conformal equation of state (energy density e = 3p) and no conserved charges. They built a small anisotropic perturbation on a known analytical flow called the Gubser flow. That perturbed Gubser flow includes longitudinal expansion and transverse elliptic deformation, so it can represent the kind of flow seen in collisions. In the large‑system‑size limit they derived explicit formulas for the local spin polarization along the beam axis. For comparison they also discuss a rotating Hubble flow, a simpler model with rigid rotation.

At a conceptual level the paper separates two sources of spin polarization. Thermal vorticity is associated with local rotation of the fluid and tends to produce polarization with a particular sign. Thermal shear comes from gradients in the fluid velocity and produces polarization of the opposite sign. The authors show analytically that these two contributions are often comparable. Depending on how a unit vector in the thermal‑shear term is chosen, the shear can either partially cancel the vorticity or cancel it almost exactly at leading order.

The choice of unit vector in the shear term is important. In the Fu‑Liu‑Pang‑Song‑Yin (FLPSY) prescription, which aligns the vector with the local fluid velocity, the model gives the experimentally observed sign only at low transverse momentum. In the Becattini‑Buzzegoli‑Palermo‑Inghirami‑Karpenko (BBPIK) prescription, which fixes the vector along laboratory time, the desired sign appears over a wider range of transverse momentum. In a third recent prescription by Sheng, Becattini and Roselli (SBR), where the vector is taken normal to the freeze‑out surface, the authors find an exact cancellation between thermal vorticity and shear at leading order in the large‑size limit. From this they identify a general pattern of cancellation when acceleration effects dominate; the remaining polarization then comes from non‑acceleration effects and can be small even if elliptic flow is present.