Thermal vorticity and spin polarization in heavy-ion collisions

The hot and dense matter generated in heavy-ion collisions contains intricate vortical structure in which the local fluid vorticity can be very large. Such vorticity can polarize the spin of the produced particles. We study the event-by-event generation of the so-called thermal vorticity in Au+Au collisions at energy region $\sqrt{s}=7.7–200$ GeV and calculate its time evolution, spatial distribution, etc., in a multiphase transport model. We then compute the spin polarization of the $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ hyperons as a function of $\sqrt{s}$, transverse momentum $p_T$, rapidity, and azimuthal angle. Furthermore, we study the harmonic flow of the spin, in a manner analogous to the harmonic flow of the particle number. The measurement of the spin harmonic flow may provide a way to probe the vortical structure in heavy-ion collisions. We also discuss the spin polarization of ${\mathrm{\Xi }}^0$ and ${\mathrm{\Omega }}^{-}$ hyperons, which may provide further information about the spin polarization mechanism of hadrons.

Figure 1

The $zx$ component of the thermal vorticity averaged over transverse plane and over colliding events, $\langle {\overline{\varpi }}_{zx}\rangle$, as a function of time at $\sqrt{s}=19.6,62.4$, and 200 GeV for fixed impact parameter $b=9$ fm and rapidity $\eta =0$.

Figure 2

The distribution of the $yz$ (top-left panel) and $zx$ (top-right panel) components of the event averaged thermal vorticity in the transverse plane at very early time, $t=0.6$ fm, and rapidity $\eta =0$ for Au+Au collisions at $\sqrt{s}=19.6$ GeV averaged over the centrality region 20–50%. It reflects a vortical structure as displayed by the arrows in the lower-right panel with the color representing the magnitude $|{\mathbf{\varpi }}_{\perp }|={({\varpi }_{zx}^2+{\varpi }_{yz}^2)}^{1/2}$. The bottom-left panel shows the spatial distribution of the radial thermal vorticity $\widehat{\mathit{r}}·{\mathbf{\varpi }}_{\perp }$.

Figure 3

The longitudinal component of the event-averaged thermal vorticity distributed in the transverse plane at very early time, $t=0.6$ fm, and rapidity $\eta =0$ for Au+Au collisions at $\sqrt{s}=19.6$ GeV averaged over the centrality region 20–50%.

Figure 4

The vector plot for the thermal vorticity projected to the transverse plane at space-time rapidity $\left|\eta \right|=2$ for Au+Au collisions at 19.6 and 200 GeV averaged over events in 20–50% centrality range at fixed time $t=3$ and 9 fm, respectively. The background color represents the distribution of the ${\varpi }_{zx}$ component.

Figure 5

The distribution of event-averaged thermal vorticity in the $x\text{−}\eta$ plane for Au+Au collisions at 19.6 and 200 GeV, respectively. Other parameters are the same as Fig. 4.

Figure 6

(Left) The averaged $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ spin polarization along $y$ direction in 20–50% centrality range of Au+Au collisions as a function of collision energy. The rapidity window for $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ is $\left|Y\right|<1$. Open points: STAR data [5, 6]. Red solid points: this work. (Right) The spin polarization $P_y$ for ${\mathrm{\Xi }}^0$ and ${\mathrm{\Omega }}^{-}$. Other parameters are the same as the left panel.

Figure 7

The $p_T$ and rapidity dependence of the global polarization at different collision energies. Open points: STAR data [6]. Dotted lines: this work.

Figure 8

The rapidity-azimuth distribution of the event-averaged spin polarization of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ for Au+Au collisions at 20–50% centrality range at 19.6 and 200 GeV, respectively.

Figure 9

The azimuthal angle dependence of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ polarization in rapidity region $\left|Y\right|<1$ for Au+Au collisions at 19.6, 62.4, and 200 GeV. The experimental data [6] is also shown.

Figure 10

The directed and elliptic spin harmonic coefficients, $f_1$ and $f_2$, versus rapidity for Au+Au collisions with fixed impact parameter $b=9$ fm for $\sqrt{s}$ from 19.6–200 GeV.