Self-consistent conversion of a viscous fluid to particles

Comparison of hydrodynamic and “hybrid” $\text{hydrodynamics}+\text{transport}$ calculations with heavy-ion data inevitably requires the conversion of the fluid to particles. For dissipative fluids the conversion is ambiguous without additional theory input complementing hydrodynamics. We obtain self-consistent shear viscous phase-space corrections from linearized Boltzmann transport theory for a gas of hadrons. These corrections depend on the particle species, and incorporating them in Cooper–Frye freeze-out affects identified particle observables. For example, with additive quark model cross sections, proton elliptic flow is larger than pion elliptic flow at moderately high $p_T$ in $\text{Au}+\text{Au}$ collisions at the BNL Relativistic Heavy Ion Collider. This is in contrast to Cooper–Frye freeze-out with the commonly used “democratic Grad” ansatz that assumes no species dependence. Various analytic and numerical results are also presented for massless and massive two-component mixtures to better elucidate how species dependence arises. For convenient inclusion in pure hydrodynamic and hybrid calculations, Appendix G contains self-consistent viscous corrections for each species both in tabulated and parametrized form.

Figure 1

Ratio of dissipative corrections as a function of normalized proper time for a massless two-component system in a $0+1\text{D}$ Bjorken scenario, calculated from nonlinear $2\to 2$ covariant transport using MPC [18]. Five different scenarios (a)–(e) with various cross sections and densities are shown, labeled with the ratio of inverse Knudsen numbers $K_A/K_B$. See Table 1 for a detailed list of parameters. Thin, horizontal dotted lines and arrows on the right side of the plot correspond to the expectation from a self-consistent calculation based on linearized transport in the quadratic Grad approximation (“dynamical Grad” approach). Only four such lines and arrows are visible because scenarios (b) and (c) are identical except for the timescale of relaxation to Navier–Stokes regime; scenario (b) relaxes 5/3 times quicker than scenario (c).

Figure 2

Self-consistent dissipative corrections for shear stress as a function of temperature for a chemically equilibrated pion-nucleon gas, in the Grad approximation, with effective cross sections ${\sigma }_{\pi \pi }^{\mathrm{eff}}=30$ mb, ${\sigma }_{\pi N}^{\mathrm{eff}}=50$ mb, and ${\sigma }_{NN}^{\mathrm{eff}}=20$ mb. The ratio of coefficients $c_{\pi }/c_N$ is shown, where $c_i$ is the dissipative correction for species $i$ relative to the commonly used democratic ansatz (see text).

Figure 3

Differential elliptic flow $v_2\left(p_T\right)$ of pions and protons in $\text{Au}+\text{Au}$ at $\sqrt{s_{NN}}=200$ GeV at RHIC with impact parameter $b=7$ fm, using $2+1\text{D}$ boost invariant hydrodynamic solutions from azhydro [21, 22], and Cooper–Frye fluid-to-particle conversion at $T_{\text{conv}}=165$ MeV (left plot) or 140 MeV (right plot). Dashed lines are for pions, while solid curves are for protons. The standard democratic Grad approach (open boxes) is compared to self-consistent shear corrections (crosses) computed for a pion-nucleon gas from linearized kinetic theory (see text). In both cases, ${\eta }_s/s=0.1$ at conversion. Results with uncorrected, local equilibrium phase-space distributions ($\delta f=0$) are also shown (filled circles).

Figure 4

Differential elliptic flow $v_2\left(p_T\right)$ from a viscous generalization of the simple four-source model of Ref. [27] with parameters $v_x=0.5, v_y=0.45, T=140$ MeV, and $\kappa \equiv {\pi }_L/(e+p)=-0.06$. Dashed lines are for pions, while solid curves are for protons. The standard democratic Grad approach (open boxes, $c_{\pi }=c_p=1$) is compared with self-consistent shear corrections (crosses, $c_{\pi }=1.03$ and $c_p=0.55$) computed for a pion-nucleon gas from linearized kinetic theory (see Sec. 4b). Results with uncorrected, local equilibrium phase-space distributions ($c_{\pi }=c_p=0$) are also shown (filled circles).

Figure 5

Same as Fig. 3, except the self-consistent viscous corrections are computed for a gas of all hadron species up to $m=1.672$ GeV $\left[\mathrm{\Omega }\right(1672\left)\right]$, with members of each isospin multiplet (and antiparticles) combined together into a single effective species. There are 49 effective species this way. (left plot) All hadron species interacting with the same constant isotropic cross section ${\sigma }_{ij}=30$ mb. (right plot) Constant isotropic cross sections with additive quark model scaling ${\sigma }_{MM}:{\sigma }_{MB}:{\sigma }_{BB}=4:6:9$ and ${\sigma }_{MM}=30$ mb. Calculations with the democratic Grad ansatz for ${\eta }_s/s=0.1$ (open boxes) and with local equilibrium distribution (filled circles) are also shown. In all cases, and for both plots, the Cooper–Frye prescription is applied at $T_{\text{conv}}=165$ MeV.

Figure 6

Same as Fig. 5, except after feed down from resonance decays using the reso code in the azhydro package [21].

Figure 7

Differential elliptic flow $v_2\left(p_T\right)$ for pions and protons in $\text{Au}+\text{Au}$ at $\sqrt{s_{NN}}=200$ GeV at RHIC with impact parameter $b=7$ fm, using $2+1\text{D}$ boost invariant hydrodynamic solutions from azhydro [21, 22], followed by Cooper–Frye fluid-to-particle conversion at $T_{\text{conv}}=165$ MeV with either the standard democratic approach (open boxes) or self-consistent shear corrections (crosses) with momentum dependence $\delta f\propto p^{3/2}$ computed from kinetic theory for a gas of hadrons up to $m=1.672$ GeV. In both plots, dashed lines are for pions, while solid curves are for protons, and feed down from resonance decays is included. (left plot) Scenario with constant 30 mb hadronic cross sections. (right plot) Cross sections based on the additive quark model (AQM). Results with uncorrected, local equilibrium phase-space distributions ($\delta f=0$) are also shown (filled circles).

Figure 8

Same as Fig. 7, except with momentum dependence $\delta f\propto p$ for the curves from self-consistent fluid-to-particle conversion (crosses).

Figure 9

Same as the right plots in Figs. 6 and 7 for $\text{Au}+\text{Au}$ at RHIC with self-consistent viscous corrections in the additive quark model (AQM) scenario with $\delta f\propto p^2$ (left) and $\delta f\propto p^{3/2}$ (right) but with ${\eta }_s/s$ varied in the range $0.05\le {\eta }_s/s\le 0.15$. Shaded bands are shown together with curves at the lowest ${\eta }_s/s=0.05$ (squares) and highest ${\eta }_s/s=0.15$ (triangles) boundaries.