$\mathrm{\Lambda }$ spin polarization in event-by-event relativistic heavy-ion collisions

We present a systematic study of $\mathrm{\Lambda }$ hyperon's polarization observables using event-by-event ($3+1$) dimensional relativistic hydrodynamics. The effects of initial hot spot size and quark gluon plasma's specific shear viscosity on the polarization observables are quantified. We examine the effects of the two formulations of the thermal shear tensor on the polarization observables using the same hydrodynamic background. With event-by-event simulations, we make predictions for the Fourier coefficients of ${\mathrm{\Lambda }}^{\prime }\mathrm{s}$ longitudinal polarization $P^z$ with respect to the event planes of different orders of anisotropic flow. We propose new correlations among the Fourier coefficients of $P^z$ and charged hadron anisotropic flow coefficients to further test the mapping from fluid velocity gradients to hyperon's polarization. Finally, we present a system size scan with $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Ru}+\mathrm{Ru}$, and $\mathrm{O}+\mathrm{O}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$ to study the system size dependence of polarization observables at the BNL Relativistic Heavy-ion Collider.

Figure 1

(a)–(e) Charged hadron pseudo-rapidity distributions of $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV in different centrality bins compared to the PHOBOS measurements [63]. (f) Charged hadron anisotropic flow coefficients compared with the STAR measurements [64]. Model simulations are performed with three values of specific shear viscosity in the hydrodynamic phase.

Figure 2

Similar comparisons as in Fig. 1 but for model simulations with three initial hot spot sizes $w$.

Figure 3

(a) The hyperon's global polarization $P^y$ as a function of the collision centrality in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV with four combinations of the axial-vector ${\mathcal{A}}^{\mu }$ in Eq. (16). (b) The $p_T$-differential $P^y$ with $\left|\eta \right|<1$ in 20–60% $\mathrm{Au}+\mathrm{Au}$ collisions. (c) The pseudorapidity distribution of $P^y$ in 20–60% $\mathrm{Au}+\mathrm{Au}$ collisions. Simulations are performed with the initial hot spot size $w=0.4$ fm, a specific shear viscosity $\eta T/(e+P)=0.16$ in the hydrodynamic phase, and a switching energy density $e_{\mathrm{sw}}=0.5\mathrm{GeV}/{\mathrm{fm}}^3$. Comparisons are made with the STAR measurements [4]. The STAR measurements are scaled by 0.877 because the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ from Ref. [67].

Figure 4

The azimuthal dependence of the global polarization $P^y$ with respect to the elliptic flow event plane for four combinations of the axial-vector ${\mathcal{A}}^{\mu }$.

Figure 5

The sensitivity of hyperon's global polarization on the QGP's specific shear viscosity (a), initial hot spot size (b), and switching energy density (c). The STAR measurements [4] are scaled by 0.877 because the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ from Ref. [67].

Figure 6

The averaged cosine of the daughter proton's polar angle in the ${\mathrm{\Lambda }}^{\prime }\mathrm{s}$ rest frame computed from ${\mathrm{\Lambda }}^{\prime }\mathrm{s}$ longitudinal polarization with four combinations of the axial-vector ${\mathcal{A}}^{\mu }$ in 20–60% $\mathrm{Au}+\mathrm{Au}$ collisions. Model calculations are compared with the STAR measurements [5].

Figure 7

The centrality dependence of the second-order Fourier coefficients of the azimuthal dependent $P^z$ with respect to the elliptic flow event plane angle ${\mathrm{\Psi }}_2$ for four combinations of the axial-vector ${\mathcal{A}}^{\mu }$. Results are compared with the STAR data [5].

Figure 8

The centrality dependence of the second-order Fourier coefficients of the longitudinal polarization $P^z$ with respect to the elliptic flow event-plane angle ${\mathrm{\Psi }}_2$ for different values of specific shear viscosity (a), initial hot spot size (b), and switching energy density (c). Results are compared with the STAR data [5].

Figure 9

The $\langle cos\left({\theta }_p^{*}\right)\rangle$ in Eq. (24) with respect to the third-order event plane angle computed from the ${\mathrm{\Lambda }}^{\prime }\mathrm{s}$ longitudinal polarization $P^z\left(\varphi \right)$ using four combinations of the axial vector ${\mathcal{A}}^{\mu }$ in 20–60% $\mathrm{Au}+\mathrm{Au}$ collisions.

Figure 10

The centrality dependence of the $n\mathrm{th}$ order Fourier coefficients of $P^z\left(\varphi \right)$ with respect to $n\mathrm{th}$ order event-plane determined by charged hadron anisotropic flow in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV for $n=1\text{–}5$.

Figure 11

The centrality dependence of the Pearson correlation between charged hadron anisotropic flow $v_n$ and the Fourier coefficients of the longitudinal polarization $P_n^z$ in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV.

Figure 12

The centrality dependence of ${\mathrm{\Lambda }}^{\prime }\mathrm{s}$ global polarization in $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Ru}+\mathrm{Ru}$, and $\mathrm{O}+\mathrm{O}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$.

Figure 13

The Fourier coefficients of ${\mathrm{\Lambda }}^{\prime }\mathrm{s}$ longitudinal polarization as functions of charged hadron multiplicity for $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Ru}+\mathrm{Ru}$, and $\mathrm{O}+\mathrm{O}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$.