Particle production in relativistic heavy-ion collisions: A consistent hydrodynamic approach

We derive relativistic viscous hydrodynamic equations invoking the generalized second law of thermodynamics for two different forms of the nonequilibrium single-particle distribution function. We find that the relaxation times in these two derivations are identical for shear viscosity but different for bulk viscosity. These equations are used to study thermal dilepton and hadron spectra within longitudinal scaling expansion of the matter formed in relativistic heavy-ion collisions. For consistency, the same nonequilibrium distribution function is used in the particle production prescription as in the derivation of the viscous evolution equations. Appreciable differences are found in the particle production rates corresponding to the two nonequilibrium distribution functions. We emphasize that an inconsistent treatment of the nonequilibrium effects influences the particle production significantly, which may affect the extraction of transport properties of quark-gluon plasma.

Figure 1

(a) Time evolution of shear $\Phi$ and bulk $\Pi$ viscous pressures, and (b) temperature dependence of the ratio of the longitudinal to transverse pressures $P_L/P_T$, for the various bulk relaxation times ${\tau }_{\Pi }$ defined in Eq. (20). Note that for ${\tau }_{\Pi }={\tau }_{\pi }$, cavitation ($P_L<0$) sets in.

Figure 2

Particle spectra as a function of the transverse momentum $p_T$, for ideal and viscous hydrodynamics with bulk relaxation times ${\tau }_{\Pi }$ defined in Eq. (20) for (a) dileptons of invariant mass $M=1,2,3$ GeV/$c^2$, and (b) hadrons.

Figure 3

Ratios of particle yields for viscous and ideal hydrodynamics as a function of $p_T$, for the two bulk relaxation times ${\tau }_{\Pi }$ defined in Eq. (20) for (a) dileptons of invariant mass $M=1,1.5,2$ GeV/$c^2$, and (b) pions. Inset: Pion yields in various evolution and production scenarios scaled by the consistent second-order calculation for Case 1 [blue (dark grey)] and Case 2 [red (light grey)]. Solid lines: second-order evolution with ideal production rate; dashed lines: second-order evolution with first-order correction to the production rate; dotted lines: ideal evolution with first-order correction to the production rate.

Figure 4

Dilepton yields as a function of the invariant mass $M$, in ideal and viscous hydrodynamics with the two bulk relaxation times ${\tau }_{\Pi }$ defined in Eq. (20). Inset: Same as Fig. 3 inset but for dileptons.