Vorticity in heavy-ion collisions

We study the event-by-event generation of flow vorticity in the BNL Relativistic Heavy Ion Collider $\mathrm{Au}+\mathrm{Au}$ collisions and CERN Large Hadron Collider $\mathrm{Pb}+\mathrm{Pb}$ collisions by using the hijing model. Different definitions of the vorticity field and velocity field are considered. A variety of properties of the vorticity are explored, including the impact parameter dependence, the collision energy dependence, the spatial distribution, the event-by-event fluctuation of the magnitude and azimuthal direction, and the time evolution. In addition, the spatial distribution of the flow helicity is also studied.

Figure 1

Illustration of the heavy-ion collisions with impact parameter $b$. Here “T” stands for target and “P” for projectile.

Figure 2

The impact parameter dependence of the angular momenta of the participants (black), the remnants (red), and the QGP (blue) with respect to the collision center $\mathit{x}=\mathit{0}$ for RHIC $\mathrm{Au}+\mathrm{Au}$ collisions (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions (b). The sum of the angular momenta of the remnants and the QGP is equal to the angular momentum of the participants.

Figure 3

The event-averaged longitudinal velocity profile at zero rapidity for RHIC (a) and LHC (b). Different curves correspond to different definitions of the event-averaged velocity; see Eqs. (3.10)–(3.13).

Figure 4

The double-averaged vorticity at $\tau ={\tau }_0$ and $\eta =0$ for RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s}=200$ GeV (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76$ TeV (b). Only the $y$ component of the vorticity is sizable; other components are negligibly small and are not drawn. Different curves correspond to different definitions of vorticity and velocity.

Figure 5

The vorticity squared of the energy flow at $\tau ={\tau }_0$ and $\eta =0$ for RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s}=200$ GeV (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76$ TeV (b). Different curves correspond to different definitions of the vorticity.

Figure 6

The collision energy dependence of the double-averaged vorticity $\langle {\overline{\omega }}_{2y}\rangle$ at fixed $\tau =0.4$ fm and $\eta =0$ computed by using the energy flow velocity ${\mathit{v}}_2$.

Figure 7

The event-averaged vorticity as a function of $x$ for RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s}=200$ GeV (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76$ TeV (b). Different curves represent different definitions of vorticity [see Eqs. (3.14) and (3.15)] based on different definitions of the velocity field [see Eqs. (3.1) and (3.2)].

Figure 8

The spatial distribution of the event-averaged vorticity, $\langle {\omega }_{2y}\rangle$, in the transverse plane for RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s}=200$ GeV (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76$ TeV (b).

Figure 9

The double-averaged vorticity $\langle {\overline{\omega }}_{2y}\rangle$ in $\mathrm{Au}+\mathrm{Au}$ collisions calculated based on ${\mathit{v}}_2$ as a function of spacetime rapidity at various collision energies. The proper time is fixed to be ${\tau }_0=0.4$ fm.

Figure 10

The event-averaged helicities $\mathit{v}·{\mathbf{\omega }}_1$ and $\mathit{v}·{\mathbf{\omega }}_2$ along the $y$ axis. Different curves correspond to different definitions of the vorticity and velocity fields.

Figure 11

The spatial distribution of the event-averaged helicity, $\langle {\mathbf{\omega }}_2·{\mathit{v}}_2\rangle$, in the transverse plane for RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s}=200$ GeV (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76$ TeV (b).

Figure 12

The spatial distribution of the event-averaged helicity, $\langle T^2{\mathbf{\omega }}_2·{\mathit{v}}_2\rangle$, in the transverse plane for RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s}=200$ GeV.

Figure 13

The histograms of ${\psi }_{\omega }-{\psi }_2$ at impact parameters $b=3$ and $10\mathrm{fm}$ for $\mathrm{Au}+\mathrm{Au}$ collisions at RHIC energy (a) and $\mathrm{Pb}+\mathrm{Pb}$ collisions at LHC energy (b). Here ${\psi }_{\omega }$ is the azimuthal direction of the space-averaged vorticity ${\overline{\mathbf{\omega }}}_2$ (at ${\tau }_0$ and $\eta =0)$ of ${\mathit{v}}_2$ and ${\psi }_2$ is the second harmonic participant plane.

Figure 14

The correlation $\langle cos\left[2({\psi }_{\omega }-{\psi }_2)\right]\rangle$ as a function of impact parameter for both RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $200\mathrm{GeV}$ (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $2.76\mathrm{TeV}$ (b). The vorticity field is calculated based on the energy flow velocity ${\mathit{v}}_2$.

Figure 15

The ${\mathbf{\omega }}^2$-weighted correlation $\langle {\mathbf{\omega }}^{2}cos\left[2({\psi }_{\omega }-{\psi }_2)\right]\rangle /\langle {\mathbf{\omega }}^2\rangle$ as a function of impact parameter for both RHIC $\mathrm{Au}+\mathrm{Au}$ collisions at $200\mathrm{GeV}$ (a) and LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $2.76\mathrm{TeV}$ (b). The vorticity field is calculated based on the energy flow velocity ${\mathit{v}}_2$.

Figure 16

The event-averaged longitudinal velocity profile at zero rapidity for RHIC (a) and LHC (b). Different curves correspond to different definitions of the event-averaged velocity; see Eqs. (3.10)–(3.13).