aa r X i v : . [ nu c l - t h ] S e p Elliptic Flow at Large Viscosity
Volker Koch
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 USA
Abstract
In this contribution we present an alternative scenario for the large elliptic flow observed inrelativistic heavy ion collisions. Motivated by recent results from Lattice QCD on flavor o ff -diagonal susceptibilities we argue that the matter right above T c can be described by single-particle dynamics in a repulsive single-particle potential, which in turn gives rise to elliptic flow.These ideas can be tested experimentally by measuring elliptic flow of heavy quarks, preferablyvia the measurement of J / Ψ elliptic flow.
1. Introduction
One of the big surprises of the RHIC experimental program has been the large elliptic flow[1, 2], which, contrary to SPS energies, agreed more or less with predictions from ideal hydro-dynamics [3, 4]. This surprising agreement let to the conjecture that the matter at RHIC is astrongly coupled, nearly perfect fluid, with very small shear viscosity. Indeed using the AdS-CFT correspondence, is was shown that a large class of strongly coupled theories seem to have auniversal minimal shear viscosity of η/ s = / π [5]. Meanwhile, more refined calculations basedon relativistic viscous hydrodynamic [6, 7] seem to indicate that a finite but small value for theshear viscosity is required in order to reproduce the p t dependence of the measured transverseflow. On the theoretical side very little is known about the shear viscosity of high temperatureQCD. Perturbative calculations lead to a considerably larger value than the conjectured lowerbound. Extracting a value for the shear viscosity from Lattice QCD (LQCD), on the other hand,requires analytic continuation to real time, thus leading to substantial uncertainties [8]. It wasalso found that the elliptic flow of the observed hadrons scales with the number of quarks [2]as predicted by a quark coalescence picture of hadronization [9]. This in turn implies that thescaled v may be interpreted as that of the quarks prior to hadronization, and we will use thisinterpretation in the following where we always refer to the elliptic anisotropy of quarks.In this contribution we want to entertain an entirely di ff erent view and interpretation of theobserved elliptic flow. First, we note that Lattice QCD results [10] suggest a quasi-particlepicture, at least for the quarks. Both flavor-o ff -diagonal susceptibilities [11] and higher orderbaryon number susceptibilities [12] are consistent with vanishing correlations for temperaturesright above the transition, T & . T c . Estimating the strength of correlations in the gluon sectoris not so straightforward, due to the lack of any additional quantum numbers, such as flavor,which one can use to study correlations. Therefore, let us conjecture, that gluons behave likequasi-particles as well. In this case we have a single-particle description of the QGP right above T c . Preprint submitted to Elsevier December 15, 2018 ext we need to address the equation of state above T c , where LQCD finds the pressure tobe considerably ( ≈ repulsive , density dependent single-particle potential, i.e, p ∼ Z d p exp (cid:20) − E + UT (cid:21) < Z d p exp (cid:20) − E T (cid:21) ∼ p (1)for U >
0. Here p denotes the pressure of a free, non-interacting gas of partons.The presence of a repulsive single-particle potential has interesting consequences, especiallyfor the elliptic anisotropy, v . Given the almond shaped initial distribution of matter in the trans-verse plane in a semi-central heavy-ion collision, the (negative) gradient of the potential, andthus the force, is larger in the in-plane than in the out-of-plane direction. As a consequence themomentum kick due to the repulsive single-particle potential is larger in plane than out of plane,resulting in a deformation of the momentum distribution in qualitative agreement with the ob-served elliptic anisotropy, v ( p t ) (see also [13]). We note that this e ff ect does not require a shortmean free path or, equivalently, a small viscosity.Besides the positive v the single-particle dynamics leads to two additional, qualitative pre-dictions. First, the elliptic anisotropy should vanish for large transverse momenta, since theadditional momentum kick due to the potential becomes negligible. Thus, contrary to ideal hy-drodynamics we predict a maximum of the transverse momentum dependent elliptic anisotropy, v ( p t ), which is observed in experiment. At large p t , of course, v will be dominated by theattenuation of fast partons in the matter, and it needs to be determined at what momentum thistransition will take place [14]. Second, since for a given temperature the momentum distributionfor heavy quarks is titled towards higher momenta, T ∼ p m , the resulting elliptic anisotropy forheavy quarks should be considerably smaller than that for light quarks.
2. Schematic Model
In order to study the qualitative features of the proposed single-particle dynamics in a trans-parent fashion let us start with a simple schematic model. If we ignore collisions among thepartons, the dynamics of the phasespace distribution follows as Vlasov equation. To allow for ananalytical treatment of the expansion, we assume a Gaussian distribution in configuration spaceand non-relativistic kinematics, resulting in a Gaussian momentum space distribution, f ( x , y , v x , v y , t = = N π σ x σ y ( T / m ) exp( − x σ x ) exp( − mv x T ) exp( − y σ y ) exp( − mv y T ) . (2)Here, v x , v y are the velocities and σ x , σ y denote the widths of the distribution in the transverse x , y direction, respectively. Assuming, for simplicity, that the single particle potential U isproportional to the density of the light degrees of freedom, U (cid:0) ~ x , t (cid:1) = g ρ (cid:0) ~ x , t (cid:1) = g Z d ~ v f ( ~ x ,~ v , t ) , the Vlasov equation leads to the following expression for the velocity space distribution, n (cid:0) ~ v (cid:1) , ∂∂ t n ( ~ v ) = gm Z d x ~ ∇ v f ( ~ x ,~ v , t ) ~ ∇ x ρ ( ~ x , t ) (3)2hich can be solved analytically under the assumption that the time dependence of the densityfollows free streaming, i.e. we ignore the e ff ect of the potential on the density distribution. Afully consistent solution will require a numerical treatment, which will be briefly discussed be-low. Since the heavy quarks are rare, we ignore their contribution to the potential, and propagatethem in the potential generated by the light degrees of freedom.To leading order in the initial spatial eccentricity ǫ ≡ σ y − σ x σ x + σ y we then obtain the followingresult for the elliptic anisotropy of the light quarks v , light ( u ) = ǫ U T u h − exp (cid:16) − u (cid:17) (cid:16) u + (cid:17)i (4)which only depends on the kinetic energy u = mv T . Here, U = U ( ~ r = , t =
0) is the initialstrength of the potential at the center. We note that v is (a) proportional to the initial spatialeccentricity, (b) proportional to the initial transverse density (via U ) and (c) a function of thekinetic energy only. This is precisely what is seen in experiment.At a given velocity the heavy quark elliptic anisotropy, v , heavy , is related to that of the lightquarks by v , heavy ( v ) = + γ v . light ( v ). Both, v , light and v , heavy are plotted as a function of thekinetic energy in the left panel of Fig.1 for an initial temperature of T =
300 MeV. As expected v exhibits a maximum, the position of which depends on the choice of initial temperature andis shifted to larger values of the kinetic energy for heavy quarks. As a result the momentumaveraged, or integrated anisotropy, ¯ v , for the heavy quarks is much smaller. This is shown in leftpanel of Fig.1, where we plot the ratio of heavy over light quark v as a function of the ratio ofthe quark masses, γ = m light / M heavy . Thus, a unique prediction of the single-particle dynamics isthat the integrate v for heavy quarks, specifically the J / Ψ should be considerably smaller thanthat for the pions, contrary to hydrodynamics where they should be about equal. γ = m light / M heavy v , h ea vy / v , li gh t Charm (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Figure 1: Left panel: Schematic model results for the elliptic anisotropy as a function of the kinetic energy, v ( E kinetic )for light quarks (full line) and heavy quarks (dashed line). The dotted line represents the kinetic energy spectrum forboth heavy and light quarks. Right panel: Ratio of integrated v for heavy over light quarks as a function of the ratio ofheavy and light quark mass.
3. More realistic (transport) model
A realistic treatment of the proposed single-particle dynamics requires relativistic kinematicsas well as a fully consistent treatment of the Vlasov (Boltzmann) equation. In order to ensureenergy momentum conservation it is best to start from an e ff ective energy functional which is3 p t [GeV] v ( p t ) ( % ) Light partonsheavy quarks
Figure 2: Result for elliptic anisotropy from transport calculation without parton scattering. tuned to reproduce the Lattice equation of state and from which one derives the single-particleVlasov equation. The results from such an exercise (for details see [15]) is shown in Fig. 2.Again we see the same features as in the schematic model, and we find that the intergrated v forheavy quarks is considerably smaller than that of light quarks.
4. Conclusions
In this contribution we have shown that the qualitative features of the observed ellipticanisotropy can be obtained from single-particle dynamics motivated by recent Lattice QCD re-sults. In this description there is no need for a very short mean free path or, equivalently, smallviscosities. This model can be tested in experiment by measuring the elliptic anisotropy of heavyquarkonium, which is predicted to be considerably smaller than that of pions, in contradistinctionto the predictions from hydrodynamics.
Acknowledgments
This work is supported by the Director, O ffi ce of Energy Research, O ffi ce of High Energyand Nuclear Physics, Divisions of Nuclear Physics, of the U.S. Department of Energy underContract No. DE-AC02-05CH11231. References [1] K. H. Ackermann, et al.,
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