Estimating Longitudinal Polarization of $\Lambda$ and $\bar{\Lambda}$ Hyperons at Relativistic Energies using Hydrodynamic and Transport models

The global and local spin polarization measurements of $\Lambda$ ($\bar{\Lambda}$) hyperons by STAR and ALICE Collaborations open up an immense interest in investigating the spin polarization dynamics in heavy-ion collisions. Recent studies suggest the transverse component of the vorticity field is responsible for the global spin polarization. In contrast, the longitudinal component of the vorticity field accounts for the local spin polarization. The local (longitudinal) spin polarization of $\Lambda$-hyperons arises due to the anisotropic flows in the transverse plane, indicating a quadrupole pattern of the longitudinal vorticity along the beam direction. In this study, we derive a simple solution relating the longitudinal mean spin vector with the second-order anisotropic flow coefficient due to the thermal shear tensor for an ideal uncharged fluid in a longitudinal boost invariant scenario. The present study focuses on the local spin polarization of $\Lambda$ and $\bar{\Lambda}$ in Au$+$Au and Pb$+$Pb collisions at $\sqrt{s_{NN}}$ = 200 GeV and 5.02 TeV, respectively. Further, we explore the azimuthal angle, centrality, and transverse momentum ($p_{\rm T}$) dependence study of longitudinal spin polarization using hydrodynamic and transport models. All these models predict a maximum longitudinal spin polarization in mid-central collisions around 30-50 \% centrality at $p_{\rm T} \approx$ 2.0 - 3.0 GeV/c. These findings on longitudinal spin polarization advocate the existence of a thermal medium in non-central heavy-ion collisions.