Probing early-time longitudinal dynamics with the $\mathrm{\Lambda }$ hyperon's spin polarization in relativistic heavy-ion collisions

We systematically study the hyperon global polarization's sensitivity to a collision system's initial longitudinal flow velocity in hydrodynamic simulations. By explicitly imposing local energy-momentum conservation when mapping the initial collision geometry to macroscopic hydrodynamic fields, we study the evolution of the system's orbital angular momentum (OAM) and fluid vorticity. We find that a simultaneous description of the $\mathrm{\Lambda }$ hyperon's global polarization and the slope of the pion's directed flow can strongly constrain the size of longitudinal flow at the beginning of hydrodynamic evolution. We extract the size of the initial longitudinal flow and the fraction of orbital angular momentum in the produced quark-gluon plasma fluid as a function of collision energy with the STAR measurements in the Beam Energy Scan program at the BNL Relativistic Heavy-Ion Collider. We find that there is about 100–200 $\hslash$ OAM that remains in the mid-rapidity fluid at the beginning of hydrodynamic evolution. We further examine the effects of different hydrodynamic gradients on the spin polarizations of $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$. The gradients of ${\mu }_B/T$ can change the ordering between $\mathrm{\Lambda }$'s and $\overline{\mathrm{\Lambda }}$'s polarizations.

Figure 1

Color contours show the initial energy density distributions in the $x\text{−}{\eta }_s$ plane for 20–30% $\mathrm{Au}+\mathrm{Au}$ collisions at 19.6 GeV with the longitudinal rapidity fractions $f=0$ (a) and $f=1$ (b). The grey arrows in panel (b) indicate the nonzero initial longitudinal flow $u^{\eta }$ with $y_L=y_{\mathrm{CM}}$ in Eqs. (13) and (14). $u^{\eta }=0$ in panel (a).

Figure 2

(a) Time evolution of the averaged thermal vorticity of fluid with different longitudinal rapidity fraction $f$ in mid-rapidity 20–30% $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV. (b) Time evolution of the averaged thermal vorticity of fluid for four centrality bins in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV with $f=0.2$.

Figure 3

(a) The hyperon's global polarization as a function of hydrodynamic proper time. (b) The hyperon production as a function of $\tau$. (c) The time development of $\mathrm{\Lambda }$'s global polarization with different fluid gradients. (d) The comparison of $\mathrm{\Lambda }$'s and $\overline{\mathrm{\Lambda }}$'s global polarization developments. The results are for $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ with $p_T\in [0.5,3.0]$ GeV and $\left|y\right|<1$ in 30–40% $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV with the longitudinal rapidity fraction $f=0.2$.

Figure 4

The global $\mathrm{\Lambda }$ polarization's dependence on the initial-state longitudinal rapidity fraction in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV compared with the STAR measurements [53]. The $\mathrm{\Lambda }$'s global polarization is computed with all the gradient terms in Eq. (37). Panel (a) shows the $P_{\mathrm{\Lambda }}^y$'s centrality dependence. Panel (b) presents the $p_T$ differential $P_{\mathrm{\Lambda }}^y$ in 20–60% $\mathrm{Au}+\mathrm{Au}$ collisions. Panel (c) shows the pseudorapidity dependence of $P_{\mathrm{\Lambda }}^y$.

Figure 5

The directed flow of ${\pi }^{+}$ as a function of rapidity with different initial-state longitudinal rapidity fraction $f$ for 10–40% $\mathrm{Au}+\mathrm{Au}$ collisions at 7.7 GeV compared with the STAR measurement [54].

Figure 6

(a) The value of longitudinal rapidity fraction $f$ as a function of collision energy. (b) The slope of ${\pi }^{+}$ directed flow at $y=0, d{v}_1{/dy|}_{y=0}$, compared with the STAR measurements [54]. (c),(d) The global $\mathrm{\Lambda }$ polarization in 20–50% $\mathrm{Au}+\mathrm{Au}$ collisions as a function of collision energy. Calculations including different gradient terms are compared with the STAR measurements [1]. The STAR polarization data points are rescaled by 0.877 because of the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ [55].

Figure 7

(a) The percentage fraction of orbital angular momentum (OAM) in the mid-rapidity region $|{\eta }_s|<0.5$ relative to the system's total OAM from the participant nucleons for 20–30% $\mathrm{Au}+\mathrm{Au}$ collisions at the RHIC BES energies. (b) The system's total and mid-rapidity OAM as a function of the collision energy.

Figure 8

(a),(b) The centrality dependence of the global $\mathrm{\Lambda }$ polarization with different gradient terms in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV compared with the STAR measurements [53]. (c) Model prediction for $P_{\mathrm{\Lambda }}^y$ at 7.7 GeV. The STAR polarization data points are rescaled by 0.877 because of the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ [55].

Figure 9

(a),(b) The $p_T$-differential polarization for $\mathrm{\Lambda }$ with different gradient terms in 20–60% $\mathrm{Au}+\mathrm{Au}$ collisions compared with the STAR measurements [53]. (c) Model prediction for $P_{\mathrm{\Lambda }}^y\left(p_T\right)$ at 7.7 GeV. The STAR polarization data points are rescaled by 0.877 because of the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ [55].

Figure 10

(a),(b) The pseudorapidity dependence of the $\mathrm{\Lambda }$ polarization with different gradient terms in 20–60% $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV compared with the STAR measurements [53]. (c) Model prediction for $P_{\mathrm{\Lambda }}^y\left(\eta \right)$ at 7.7 GeV. The STAR polarization data points are rescaled by 0.877 because of the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ [55].

Figure 11

The global $\mathrm{\Lambda }$ polarization computed with different vorticity tensors with $f=0.15$ in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV compared with the STAR measurements [53]. The STAR polarization data points are rescaled by 0.877 because of the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ [55].

Figure 12

The global $\mathrm{\Lambda }$ polarization's dependence on the switching energy density in $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV compared with the STAR measurements [53]. The STAR polarization data points are rescaled by 0.877 because of the latest hyperon decay parameter ${\alpha }_{\mathrm{\Lambda }}$ [55].