Longitudinal $\Lambda$ polarization as a competition between freeze-out geometry and elliptic flow in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.36\mathrm{TeV}$

Abstract

The second sine harmonic $\Pz$ of longitudinal $\Lambda$ polarization is sensitive to the balance between the kinematic-vorticity contribution $\Pz^ω$ and the kinematic-shear contribution $\Pz^Ξ$ at decoupling. We quantify this competition in a scan of 48 Pb-Pb initial states evolved with the (3+1)-dimensional Particle-In-Cell Relativistic (PICR) ideal-hydrodynamic code. With the isothermal local-equilibrium (ILE) prescription, the total harmonic follows $\Pz\simeq a\,\efo-b\,v_2$, with $a\simeq 0.010$ and $b\simeq 0.014$, where $\efo$ is the freeze-out eccentricity and $v_2$ the elliptic flow of $\Lambda$ hyperons. With coefficients fixed on these 48 states, the relation describes 86\% of the variation of $\Pz$ in 12 states outside the fit. To identify the origin of the two terms, we decompose the ILE integrand exactly into an emission-gradient term and a curl term. On the PICR decoupling surfaces, the tangential emission gradient is positive on average, correlates most strongly with $\efo$, and admits a surface-curvature representation, whereas the normal emission gradient and the curl term are negative on average and correlate most strongly with $v_2$. A blast-wave calculation with the same integrand gives coefficients of comparable magnitude and shows how the cancellation point $\efo/v_2=b/a$ changes with the flow and the surface geometry; both PICR and blast-wave calculations predict that $b/a$ rises with hadron mass from the proton to the $\Omega$ when the decoupling surface is held fixed. Although the geometry inferred from ALICE data depends on the background, the measured polarization constrains the displacement of $\efo$ from the cancellation boundary $\efo=(b/a)v_2$. For the central ALICE measurement values at $5.36\mathrm{TeV}$, the backgrounds examined give $\efo\simeq0.2$--$0.4$ in 30--50\% Pb-Pb collisions.