Flow harmonics $v_n$ at finite density

We investigate the Gubser solution of viscous hydrodynamics at finite density and analytically compute the flow harmonics $v_n$. We explicitly show how $v_n$ and their viscous corrections depend on the chemical potential. The difference in $v_n$ between particles and antiparticles is also analytically computed and shown to be proportional to various chemical potentials and the viscosity. Excellent agreement is obtained between the results and the available experimental data from the SPS, RHIC and the LHC.

Figure 1

${\mu }_B/T$ at freeze-out as a function of the collision energy $\sqrt{s}$ (per nucleon).

Figure 2

$v_n$ versus different values of $n$ from 2 to 7 measured by the CMS collaboration (0%–0.2% centrality) [37]. The black solid curve represents the ideal hydrodynamic result, while the blue dotted curve and red dashed curve correspond to the viscous results with $\eta /s=0.08$ and $\eta /s=0.2$, respectively. See, also, Ref. [32].

Figure 3

The energy dependence of $v_2$ compared with the SPS data [36] for midcentral collisions. We used (78) and (80) in the left and right plots, respectively.

Figure 4

$\mathrm{\Delta }{v}_2^X\equiv v_2(X)-v_2(\overline{X})$ as a function of $\sqrt{s}$ compared with the STAR data for five different species of hadrons [13, 14]. We use open symbols for baryons and filled symbols for mesons. The color of data points is chosen to match the color of the corresponding line for each hadron species. We have used $\eta /s=0.2$, ${\mu }_S=0.23{\mu }_B$ and ${\mu }_I=-0.15{\mu }_B$.