Abstract
Fluid dynamic approach is a workhorse for modelling collective dynamics in relativistic heavy-ion collisions. The approach has been successful in describing various features of the momentum distributions of hadrons produced in the heavy-ion collisions, such as \(p_T\) spectra and flow coefficients \(v_n\). As such, the description of the phenomenon of polarization of \(\Lambda \) hyperons in heavy-ion collisions has to be incorporated into the hydrodynamic approach. We start this chapter by introducing different definitions of vorticity in relativistic fluid dynamics. Then we present a derivation of the polarization of spin 1/2 fermions in the relativistic fluid. The latter is directly applied to compute the spin polarization of the \(\Lambda \) hyperons, which are produced from the hot and dense medium, described with fluid dynamics. It is followed by a review of the existing calculations of global or local polarization of \(\Lambda \) hyperons in different hydrodynamic models of relativistic heavy-ion collisions. We particularly focus on the explanations of the collision energy dependence of the global \(\Lambda \) polarization from the different hydrodynamic models, the polarization component in the beam direction as well as on the origins of the global and local \(\Lambda \) polarization.
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Notes
- 1.
Sometimes the vorticity is defined without the factor 1/2; we use the definition that gives the vorticity of the fluid rotating as a whole with a constant angular velocity \(\Omega \), to be \(\omega =\Omega \).
- 2.
Typical grid spacing used in the calculations: \(\Delta x=\Delta y=0.2\)Â fm, \(\Delta \eta =0.05-0.15\) and timestep \(\Delta \tau =0.05-0.1\)Â fm/c depending on the collision energy. A finer grid with \(\Delta x=\Delta y=0.125\)Â fm was taken to simulate peripheral collisions.
- 3.
The phenomenon of baryon transparency describes transporting the baryon charge of the colliding nuclei to the forward and backward rapidities. Opposite to that, baryon stopping implies that the baryon charge from the colliding nuclei is stopped around mi-rapidity.
- 4.
The projection is made on a plane orthogonal to the direction of the collective flow velocity: \(\varpi ^{\mu \nu }_\text {proj} u_\nu =0\).
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Karpenko, I. (2021). Vorticity and Polarization in Heavy-Ion Collisions: Hydrodynamic Models. In: Becattini, F., Liao, J., Lisa, M. (eds) Strongly Interacting Matter under Rotation. Lecture Notes in Physics, vol 987. Springer, Cham. https://doi.org/10.1007/978-3-030-71427-7_8
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