In Sec. II, in the un-numbered equation after Eq. (4), we reported the angular distribution of the proton momentum dN/d * as a function of the polarization vector 0. In our convention, which follows that of Ref. [10], this vector has a maximal magnitude of 1/2, i.e., the spin, whereas the usual convention in particle physics has as maximal magnitude 1, i.e., 100% polarization.

With a Yang-Mills flux-tube initial state and a high-resolution (3+1)D particle-in-cell relativistic (PICR) hydrodynamics simulation, we calculate the polarization for different energies. The origination of polarization in high energy collisions is discussed, and we find linear impact parameter dependence of the global polarization. Furthermore, the global polarization in our model decreases very quickly in the low energy domain, and the decline curve fits well the recent results of Beam Energy Scan (BES) program launched by the STAR Collaboration at the Relativistic Heavy Ion Collider (RHIC). The time evolution of polarization is also discussed.

The polarization of hyperons from relativistic flow vorticity is studied in peripheral heavy ion reactions at FAIR and NICA energies, just above the threshold of the transition to the quark-gluon plasma. Previous calculations at higher energies with larger initial angular momentum, predicted significant polarization based on the classical vorticity term in the polarization, while relativistic modifications decreased the polarization and changed its structure in the momentum space. At the lower energies studied here, we see the same effect namely that the relativistic modifications decrease the polarization arising from the initial shear flow vorticity.

We calculate the polarization in an exact analytical, rotating model based on parameters extracted from a high resolution (3+1)D particle-in-cell relativistic hydrodynamics calculation. The polarization is attributed to effects from thermal vorticity and for the first time the effects of the radial and axial acceleration are also studied separately.

With a Yang-Mills field, stratified shear flow initial state and a high resolution (3+1)D Particlein-Cell Relativistic (PICR) hydrodynamic model, we calculate the Λ polarization for peripheral Au+Au collisions at RHIC energy of √ SNN = 200 GeV. The obtained longitudinal polarization in our model agrees with the experimental signature and the quadrupole structure on transverse momentum plane. It is found that the relativistic correction (2nd term), arising from expansion and from the time component of the thermal vorticity, plays a crucial role in our results. This term is changing the signature and exceeds the first term, arising from the classical vorticity. Finally, the global polarization in our model shows no significant dependence on rapidity, which agrees with the experimental data. It is also found that the second term flattens the sharp peak arising from the classical vorticity (1st term). PACS