Initial angular momentum and flow in high energy nuclear collisions

We study the transfer of angular momentum in high energy nuclear collisions from the colliding nuclei to the region around midrapidity, using the classical approximation of the color glass condensate (CGC) picture. We find that the angular momentum shortly after the collision (up to times $\sim 1/Q_s$, where $Q_s$ is the saturation scale) is carried by the “$\beta$-type” flow of the initial classical gluon field, introduced by some of us earlier. ${\beta }^i\sim {\mu }_1{\mathbf{\nabla }}^i{\mu }_2-{\mu }_2{\mathbf{\nabla }}^i{\mu }_1\left(i=1,2\right)$ describes the rapidity-odd transverse energy flow and emerges from Gauss's law for gluon fields. Here ${\mu }_1$ and ${\mu }_2$ are the averaged color charge fluctuation densities in the two nuclei, respectively. Interestingly, strong coupling calculations using anti-de Sitter/conformal field theory (AdS/CFT) techniques also find an energy flow term featuring this particular combination of nuclear densities. In classical CGC the order of magnitude of the initial angular momentum per rapidity in the reaction plane, at a time $1/Q_s$, is $|d{L}_2/d\eta |\approx R_A{Q}_s^{-3}{\overline{\varepsilon }}_0/2$ at midrapidity, where $R_A$ is the nuclear radius, and ${\overline{\varepsilon }}_0$ is the average initial energy density. This result emerges as a cancellation between a vortex of energy flow in the reaction plane aligned with the total angular momentum, and energy shear flow opposed to it. We discuss in detail the process of matching classical Yang-Mills results to fluid dynamics. We will argue that dissipative corrections should not be discarded to ensure that macroscopic conservation laws, e.g., for angular momentum, hold. Viscous fluid dynamics tends to dissipate the shear flow contribution that carries angular momentum in boost-invariant fluid systems. This leads to small residual angular momentum around midrapidity at late times for collisions at high energies.

Figure 1

The initial energy density ${\varepsilon }_0$ (dashed), directed flow in the $x$ direction ${\beta }^1$ (dash-dotted), and the divergence of directed flow $x{\mathbf{\nabla }}^i{\beta }^i$ weighted by $x$ (solid line) as functions of coordinate $x$ at $y=0$ for Pb+Pb collisions at impact parameter $b=6$ fm.

Figure 2

Energy flow components ${(T^{01},T^{03})}_{\mathrm{odd}}\sim (\tau {\beta }^1,-{\tau }^2{\mathbf{\nabla }}^i{\beta }^i/4)$ contributing to local angular momentum $d{L}_2/d\eta$, up to second order in time, plotted at $\tau =0.25$ fm/$c$ with the initial energy density ${\varepsilon }_0$, indicated by shading in the background, for Pb+Pb collisions at impact parameter $b=6$ fm.

Figure 3

The angular momentum per rapidity $d{L}_2/d\eta$ at $\eta =0$ at a time $\tau =0.25$ fm/$c$ as a function of impact parameter $b$ for Pb + Pb collisions as given by the leading terms in the power series in time.

Figure 4

Fluid flow vector field in the reaction plane, $(v_1,v_3)$, together with the local fluid energy density $e$, indicated by shading in the background, obtained from the matching at $\tau =0.1$ fm/$c$ as described in the text for Pb+Pb collisions at impact parameter $b=6$ fm. The flow is dominated by the Bjorken expansion.

Figure 5

Fluid flow vector components $v_1$ (dashed blue line) and $v_3$ (solid black line) as functions of $x$ at $\eta =0,y=0$ obtained from the matching at $\tau =0.1$ fm/$c$ as described in the text for Pb+Pb collisions at impact parameter $b=6$ fm. Because of the quadratic time dependence of the longitudinal shear flow, the buildup of $v_3$ is lagging behind $v_1$.

Figure 6

The flow of energy in the reaction plane for Pb+Pb collisions at impact parameter $b=6$ fm at time $\tau =0.1$ fm/$c$. The local energy density $e$ at the same time is indicated by background shading. Left panel (a): the part of the ideal energy momentum tensor contributing to angular momentum, ${(T_{\text{id}}^{01},T_{\text{id}}^{03})}_{\text{odd}}$. Right panel (b): the same for the shear stress tensor, ${({\pi }^{01},{\pi }^{03})}_{\text{odd}}$. Only the rapidity-even part of longitudinal flow and the rapidity-odd part of transverse flow contribute to angular momentum. The ideal part exhibits longitudinal shear flow aligned with the motion of the nuclei. The dissipative part carries directed flow aligned with, and longitudinal shear flow opposite to, the motion of the nuclei.

Figure 7

The longitudinal energy flow $T^{03}$ as a function of $x$ at $\eta =0$ and $y=0$. As in Fig. 6, the ideal contribution (dashed blue line) and the dissipative contribution (dot-dashed red line) are shown separately. The total energy flow (solid black line), with the two additional nodes as discussed in the text, emerges from a nontrivial cancellation of the ideal and dissipative contributions.

Figure 8

The time evolution of energy density (solid black), transverse pressure (dashed blue), and longitudinal pressure (dash-dotted red line) as a function of time $\tau$ at the center $(x,y,\eta )=(0,0,0)$ of a Pb+Pb collision with $b=6$ fm. For the evolution before ${\tau }_0=0.1$ fm/$c$, we use our classical Yang-Mills formalism. After that time, the system is evolved with the relativistic viscous hydro code viral in two configurations: (a) dissipative stress is carried over from the classical Yang-Mills simulation to viscous fluid dynamics (thick lines), and (b) dissipative stress is discarded at the beginning of the fluid dynamic simulation. In the latter case, longitudinal and transverse pressure are discontinuous at the matching time and the system cools much faster at the center of the collision.

Figure 9

For the same system as shown in Fig. 8, we plot the ratio of energy densities in cases (b) and (a) discussed in the text (without and with initial dissipative stress; solid black line), and the ratio of transverse pressures in cases (b) and (a) (dashed blue line), as functions of time $\tau$. The energy density and pressure are calculated in viral fluid dynamics and taken at the center of the collision $(x,y,\eta )=(0,0,0)$. We also show the ratio of cases (b) and (a) for the transverse velocity component $v_1$ measured at a point $(x,y,\eta )=(5\text{fm},0,0)$ away from the center (dash-dotted purple line). Discarding the initial dissipative stress leads to a 30–40% decrease in energy density and transverse pressure at a given time over the entire time evolution. At the same time, the transverse velocity is reduced by 5–10%.

Figure 10

The time evolution of the ratio $T^{03}/T^{00}$ as a function of time $\tau$ at the center point $(x,y,\eta )=(-4\text{fm},0,0)$ in a Pb+Pb collision with $b=6$ fm. Again we have used our classical Yang-Mills formalism before time ${\tau }_0=0.1$ fm/$c$ and after that time we use the relativistic viscous hydro code viral with initial dissipative stress.

Figure 11

Analytic results (black solid lines) and results obtained with the viral code (red circles) for the cold plasma limit problem described in Ref. [59]. We show the temperature profile along the $x$ axis [upper panel (a)] and the shear stress component ${\pi }^{xy}$ along a line $x=y$ [lower panel (b)] at midrapidity for times 4 and 6 fm/$c$. The system is initialized at time 1 fm/$c$.