This paper shows that a specific pattern of Lambda hyperon spin, called longitudinal polarization, can tell us how much of the initial sideways shape of a heavy-ion collision survives until particles stop interacting. The authors ran 48 hydrodynamic simulations of lead–lead (Pb–Pb) collisions at a center-of-mass energy of 5.36 TeV per nucleon pair and used a local-equilibrium spin formula to compute the polarization. They find a compact quantitative relation that links the measured polarization harmonic to the leftover spatial eccentricity at freeze-out and to elliptic flow, the momentum-space anisotropy seen in particle spectra.

The polarization they study splits naturally into two parts. A positive contribution comes from kinematic shear, which correlates with the freeze-out eccentricity ε2,fo — a measure of how elongated the matter is when particles decouple. A negative contribution comes from kinematic vorticity, whose magnitude depends on both the elliptic flow v2 and the eccentricity survival fraction S_ε = ε2,fo/ε2,init (the fraction of the original eccentricity that survives to freeze-out). The competition between these two pieces follows R ≃ 0.84 (ε2,fo / v2)^0.47, and the total longitudinal harmonic in the isothermal local-equilibrium (ILE) prescription is compactly written as P_{z,s2}^{ILE} ≃ 0.0404 v2 S_ε [0.850 (ε2,fo / v2)^{1/2} − 1].

Those simple formulas are not just formal. The relation explains about 87% of the variation in the calculated polarization across the 48 initial-state scan. After the authors fixed the coefficients, the same relation described about 86% of the variation across 12 independently calculated initial states, including cases at a previously unsampled impact parameter. The formulas also predict the correct sign of the polarization in most cases; when they fail, the states lie very close to the boundary where shear and vorticity cancel and a small change can flip the sign.