Thermalization at RHIC

Ideal hydroynamics provides an excellent description of all aspects of the single-particle spectra of all hadrons with transverse momenta below about 1.5-2 GeV/c at RHIC. This is shown to require rapid local thermalization at a time scale below 1 fm/c and at energy densities which exceed the critical value for color deconfinement by an order of magnitude. The only known thermalized state at such energy densities is the quark-gluon plasma (QGP). The rapid thermalization indicates that the QGP is a strongly interacting liquid rather than the weakly interacting gas of quarks and gluons that was previously expected. Collective flow as an indicator of thermalization Transverse flow in heavy-ion collisions is an unavoidable consequence of thermalization. Thermalization generates thermodynamic pressure in the matter created in the collision, which acts against the surrounding vacuum and causes rapid collective expansion (“flow”) of the reaction zone. Since the quark-gluon plasma (QGP) is a thermalized system of deconfined quarks, antiquarks, and gluons, collective flow is a necessary result of QGP formation in heavy-ion collisions, and its absence could be taken as proof that no such plasma was ever formed. Its presence, however, does not automatically signal QGP formation. Detailed studies of the observed final state flow pattern are necessary to convince oneself that the reflected time-integrated pressure history of the collision region indeed requires a thermalized state in the early collision stage whose pressure and energy density are so high that it can no longer be mistaken as consisting of conventional hadronic matter. Whereas radial flow (the azimuthally symmetric component of collective expansion transverse to the beam direction) integrates over the entire pressure history of the expanding fireball, anisotropic elliptic flow [1] (quantified by the second harmonic coefficient v2(y, p⊥;b) of a Fourier expansion in φp of the measured hadron spectrum dN/(dy p⊥dp⊥ dφp) [2]) is strongly weighted towards the very early stages of the expansion [3]. The higher the initial energy density, the less contributions it receives from the late, hadronic stage of the collision. At RHIC energies, elliptic flow almost saturates [4] before the energy density has dropped to the critical value ecr ≈ 1 GeV/fm3 (Tcr ≈ 170 MeV) where normal hadrons can begin to form [5]. At higher LHC energies, the elliptic flow is expected to peak even before the onset of hadronization (see curve c in Fig. 7 of [4]). At sufficiently high collision energies, elliptic flow is thus a QGP signature, probing the QGP equation of state p(e). Its advantage over other “early signatures” is that it affects the bulk of the hadrons and can thus be measured differentially with high statistical accuracy. The reason why elliptic flow must develop early in the collision is easy to understand. Since individual nucleon-nucleon collisions produce azimuthally symmetric momentum spectra, any final state momentum anisotropies must be generated dynamically during the nuclear reaction. They require the existence of an initial spatial anisotropy of the reaction zone, either by colliding deformed nuclei such as U+U [4, 6, 7], or by colliding spherical nuclei at non-zero impact parameter b 6=0 (the practical method of choice so far). Final state interactions within the produced matter transfer the initial spatial anisotropy onto a final momentum anisotropy. Microscopic transport calculations [8, 9] show a monotonic dependence of v2 on the opacity (density times scattering cross section) of the produced matter which is inversely related to its thermalization time. These studies strongly suggest that, for a given initial spatial anisotropy εx, the maximum momentum-space response v2 is obtained in the ideal hydrodynamic limit which assumes perfect local thermal equilibrium at every space-time point (i.e. a thermalization time which is much shorter than any macroscopic time scale in the system). Any significant delay of thermalization (modelled, for example, as an initial free-streaming stage) causes a decrease of the initial spatial anisotropy without concurrent build-up of momentum anisotropies, thereby reducing the finally observed elliptic flow signal [4]. In this talk I will present results from hydrodynamic simulations of hadronic spectra and elliptic flow at RHIC energies. We will see that the hydrodynamic approach provides an excellent quantitative description of the bulk of the data and fails only for very peripheral Au+Au collisions and/or at high p⊥>1.5−2 GeV/c. That the hydrodynamic approach fails if the initial nuclear overlap region becomes too small or the transverse momentum of the measured hadrons becomes too large is not unexpected. However, where exactly hydrodynamics begins to break down gives important information about the microscopic rescattering dynamics. What is really surprising is that the hydrodynamic approach works so well in central and semi-central collisions where it quantitatively reproduces the momenta of more than 99% of the particles. Below p⊥=1.5 GeV/c the elliptic flow data [11, 12, 13] actually exhaust the hydrodynamically predicted [4, 14, 15, 16, 17] upper limit. The significance of this agreement can hardly be overstressed, since it implies one of the biggest surprises so far at RHIC: the produced matter (which I call QGP since this is only known viable concept of thermalized matter at e ∼ (10−20)ecr) is not the originally expected weakly interacting gas of quarks and gluons (wQGP for “weakly interacting quark-gluon plasma”), but a strongly coupled liquid with extremely small viscosity (sQGP for “strongly interacting quarkgluon plasma” [18]). In fact, upper limits [19] on the dimensionless ratio of viscosity to entropy density, η/s, based on an analysis of the RHIC elliptic flow data, indicate that the sQGP is less viscous, by about an order of magnitude, than even liquid helium below the transition to superfluidity. The sQGP is therefore the most ideal fluid ever observed! Hydrodynamic expansion in heavy-ion collisions The natural language for describing collective flow phenomena is hydrodynamics. Its equations control the space-time evolution of the pressure, energy and particle densities and of the local fluid velocity. The system of hydrodynamic equations is closed by specifying an equation of state which gives the pressure as a function of the energy and particle densities. In the ideal fluid (non-viscous) limit, the approach assumes that the microscopic momentum distribution is thermal at every point in space and time (note that this does not require chemical equilibrium – chemically non-equilibrated situations can be treated by introducing into the equation of state non-equilibrium chemical potentials for each particle species [20, 21, 22]). Small deviations from local thermal equilibrium can in principle be dealt with by including viscosity, heat conduction and diffusion effects, but such a program is made difficult in practice by a number of technical and conceptual questions [23] and has so far not been successfully applied to relativistic fluids. Stronger deviations from local thermal equilibrium require a microscopic phasespace approach (kinetic transport theory), but in this case the concepts of equation of state and local fluid velocity field themselves become ambiguous, and the direct connection between flow observables and the equation of state of the expanding matter is lost. The assumption of local thermal equilibrium in hydrodynamics is an external input, and hydrodynamics offers no direct insights about the equilibration mechanisms. It is clearly invalid during the initial particle production and early recattering stage, and it again breaks down towards the end when the matter has become so dilute that rescattering ceases and the hadrons “freeze out”. The hydrodynamic approach thus requires a set of initial conditions for the dynamic variables at the earliest time at which the assumption of local thermal equilibrium is applicable, and a “freeze-out prescription” at the end. For the latter we use the Cooper-Frye algorithm [24] which implements an idealized sudden transition from perfect local thermal equilibrium to free-streaming. This is not unreasonable because freeze-out (of particle species i) is controlled by a competition between the local expansion rate ∂ · u(x) (where uμ(x) is the fluid velocity field) and the local scattering rate ∑ j〈σi jvi j〉ρ j(x) (where the sum goes over all particle species with densities ρ j(x) and 〈σi jvi j〉 is the momentum-averaged transport cross section for scattering between particle species i and j, weighted with their relative velocity); while the local expansion rate turns out to have a rather weak time-dependence, the scattering rate drops very steeply as a function of time, due to the rapid dilution of the particle densities ρ j [25], causing a rapid transition to free-streaming. – A better algorithm [26, 14] switches from a hydrodynamic description to a microscopic hadron cascade at or shortly after the quark-hadron transition, before the matter becomes too dilute, and lets the cascade handle the freeze-out kinetics. This also correctly reproduces the final chemical composition of the fireball, since the particle abundances already freeze out at hadronization, due to a lack of particle-number changing inelastic rescattering processes in the hadronic phase. The resulting radial flow patterns [14] from such an improved freeze-out algorithm don’t differ much from our simpler Cooper-Frye based approach. Hydrodynamic radial flow and RHIC particle spectra We have solved the relativistic equations for ideal hydrodynamics, as described in [4]. To simplify the numerical task, we imposed boost-invariant longitudinal expansion analytically [1, 27]. As long as we focus on the transverse expansion dynamics near midrapidity (the region which most RHIC experiments cover best), this does not give up any essential physics. We use an EOS which is constructed by matching a free ideal quark-gluon gas above Tcr to a realistic hadron resonance gas [28, 29] below Tcr, using a Maxwell construction and fixing Tcr =165 MeV to reproduce lattice QCD results [5]. The Maxwell construction leads to an artificial first order transition, with a latent heat of 1.15 GeV/fm3, whereas the lattice QCD data indicate either a very weak first order transition or a rapid, but smooth, crossover. As long as this crossover is very steep (as the lattice data indicate [5]), the dynamical consequences of our first-order idealization of the EOS are not expected to be significant. We have used two variants of the hadron resonance gas below Tcr: In our first sets of calculations 1999-2002, we assumed the hadron resonance gas to be in chemical equilibrium until kinetic freeze-out. RHIC data on particle abundance ratios show, however, that hadronic particle yields freeze out directly at Tcr [30], due to inefficiency of particlechanging inelastic reactions in the relatively dilute and rapidly expanding hadron gas phase below Tcr. More recent hydrodynamic calculations [21, 31] therefore include nonequilibrium chemical potentials [20, 21, 22] ensuring number conservation for individual (stable) hadron species below Tcr. This does not affect the equation of state p(e) and therefore leads to the same flow pattern as the chemical equilibrium EOS [21]. What does change, however, is the relationship between energy density e and temperature T since in the chemical non-equilibrium case more of the energy is stored in the rest masses of heavy baryons and antibaryons (which are not allowed to annihilate as the temperature drops). The same decoupling energy density edec =0.075 GeV/fm3 obtained from a fit to RHIC data (see next paragraph) therefore corresponds to Tdec =130 MeV in the chemical equilibrium case, but drops to the much lower value Tdec =100 MeV if hadronic chemical freeze-out at Tcr =165 MeV is properly taken into account [21, 31]. The initial and final conditions for the hydrodynamic evolution are fixed by fitting the total charged multiplicity dNch/dy and the pion and proton spectra at midrapidity in central (b=0) collisions. (All calculated hadron spectra include feeddown from decays of unstable hadron resonances [32].) Since these two hadrons have quite different masses, their spectra allow for the independent extraction of the temperature and average radial flow velocity at freeze-out [33]. The freeze-out temperature together with the total charged multiplicity determines the total fireball entropy which in ideal hydrodynamics is preserved during the expansion. During the early, predominantly longitudinal expansion stage, where the fireball volume increases linearly with proper time τ , this constraint fixes the product τ · ∫ dr⊥ s(r⊥,τ) where s(r⊥,τ) is the entropy density distribution in the plane transverse to the beam. Since the initial shape of this distribution is fixed by the nuclear overlap geometry, using a Glauber model ansatz [17], this constraint determines the normalization of s(r⊥,τeq) (i.e. the central entropy density s0 =s(0,τeq)) as a function of the initial thermalization time τeq. The Glauber model contains one additional parameter: the ratio of “soft” and “hard” components of initial particle production. Since these scale with the transverse density of wounded nucleons and binary nucleon-nucleon collisions, respectively, their ratio can be determined from the collision centrality dependence of the produced charged multiplicity dNch/dy [17]. We use 25% hard and 75% soft contributions to the initial entropy production [34]. The initial thermalization time τeq, finally, is fixed by the need for the hydrodynamic evolution to have enough time to generate the finally observed radial flow. At RHIC energies, we find that this requires thermalization times τeq≤0.6 fm/c. In our calculations we take the upper limit of this interval. In Au+Au collisions at √ s=100A GeV, the initial central entropy density at this time is s0 =110 fm−3, corresponding to an initial central energy density1 e0 =30 GeV/fm≈30ecr and an initial central temperature T0 =360MeV≈2Tcr. FIGURE 1. (color online) Negative pion, kaon, antiproton, and Ω spectra from central Au+Au collisions at √ s=200A GeV, as measured by the four RHIC experiments [35, 36, 37, 38, 39]. The curves show hydrodynamical calculations as described in the text. In Fig. 1 I show the (absolutely normalized) single particle p⊥-spectra for negatively charged pions, kaons and antiprotons (left panel) as well as Ω baryons (right panel) measured in Au+Au collisions at RHIC together with hydrodynamical results [31]. In order to illustrate the effect of additional radial flow generated by elastic scattering in the late hadronic stage below Tcr, two sets of curves are shown: the lower (blue) bands correspond to kinetic decoupling at Tcr =165 MeV, whereas the upper (red) bands assume decoupling at Tdec =100 MeV. The width of the bands indicates the sensitivity of the calculated spectra to an initial transverse flow of the fireball already at the time of thermalization: The lower end assumes no initial transverse flow whereas the upper end implements an initial radial flow profile vr(r⊥,τeq)= tanh(αr⊥) with α =0.02 fm−1 at τeq =0.6 fm/c (which seems to be slightly preferred by the data). The hydrodynamic model output shows [4] that it takes about 9-10 fm/c until the fireball has become sufficiently dilute to completely convert to hadronic matter and another 7-8 fm/c to completely decouple. Figure 1 shows clearly that by the time of hadronization hydrodynamics has not yet generated enough radial flow to reproduce the measured proton and Ω spectra; these heavy hadrons, which are particularly sensitive to radial flow effects, require the additional collective “push” created by resonant (quasi)elastic interactions during the fairly long-lived hadronic rescattering stage between Tcr and Tdec. Even though independent flow fits to π,K, p spectra on the one hand and multistrange hyperon spectra on the other [40] suggest (within some systematic uncertainty related to the strong 1 When averaged over the transverse profile, this corresponds to 〈e〉=13 GeV/fm3 which is still a good order of magnitude above the critical value for deconfinement, ecr =0.6−1 GeV/fm3 [5]. 2 If the initial QGP is not chemically equilibrated, but rather dominated by gluons, this initial temperature could be as high as 460 MeV. anticorrelation between the extracted flow and temperature values) that Ξ and Ω hyperons decouple slightly earlier (at somewhat higher temperature and with less radial flow) than pions, kaons, and protons, their immediate decoupling directly at hadronization is obviously not dynamically consistent with the successful hydrodynamic approach. The strong flattening of the (anti)proton spectra by radial flow provides a natural explanation for the (initially puzzling) experimental observation that for p⊥>2 GeV/c antiprotons become more abundant than pions. For a hydrodynamically expanding thermalized fireball, at relativistic transverse momenta p⊥≫m0 all hadron spectra have the same slope [33], and at fixed m⊥≫m0 their relative normalization is given by (giλi)/(g jλ j) (where gi, j is the spin-isospin degeneracy factor and λi, j = eμi, j/T is the fugacity of hadron species i, j). Due to the large antiproton chemical potential at kinetic freeze-out which is required to maintain the p̄ abundance at temperatures below chemical freeze-out, this asymptotic p̄/π− ratio is predicted to be about 28 [41]. Within the hydrodynamic approach the surprising fact is therefore not that p̄/π >1 at p⊥>2 GeV/c, but that this ratio seems to saturate around 1 and never much exceeds this value [35]. As we will see below, this is a signature of the beginning breakdown of the hydrodynamic model which stops working for baryons with transverse momenta above about 2.5 GeV/c. As shown elsewhere (see Fig. 1 in [42]), once the hydrodynamic model parameters have been fixed to describe pions, protons, and total multiplicity in central Au+Au collisions, the model describes these and all other hadron spectra not only in central, but also in peripheral collisions, up to impact parameters of about 10 fm and with similar quality. No additional parameters enter at non-zero impact parameter – only the initial conditions change due to the changing overlap geometry, but this is completely accounted for by the Glauber model. Therefore, hydrodynamic results for the elliptic flow, discussed in the next Section, are parameter-free predictions of the model. Hydrodynamic elliptic flow and RHIC data Figure 2 shows the predictions for the elliptic flow coefficient v2 from Au+Au collisions at RHIC, together with the data [11, 12, 43]. For impact parameters b≤7 fm (corresponding to nch/nmax≥0.5) and transverse momenta p⊥≤1.5−2 GeV/c the data are seen to exhaust the upper limit for v2 obtained from the hydrodynamic calculations. For larger impact parameters b>7 fm the p⊥-averaged elliptic flow v2 increasingly lags behind the hydrodynamic prediction. It is tempting to attribute this to a lack of early thermalization when the initial overlap region becomes too small [34]. However, the work by Teaney [14], who coupled hydrodynamic evolution of the quark-gluon plasma above Tcr with a microscopic kinetic evolution of a hadronic resonance gas using RQMD below Tcr, showed that at least a strong contributing factor to the lack of elliptic flow in peripheral collisions (as well as in central collisions at lower energy) is the large viscosity in the late hadronic stage. If the initial energy density is not high enough, the initial spatial deformation does not completely disappear until hadronization. Whereas ideal hydrodynamics then predicts a continued growth of the elliptic flow during the hadronic stage, driven by the still existing anisotropies in the pressure gradients, realistic hadron 0 0.25 0.5 0.75 1 0 2 4 6 8 10 n ch /n max v 2 ( % ) eWN