The viscosity of quark-gluon plasma at RHIC and the LHC

Abstract

The specific shear viscosity (eta/s)_QGP of quark-gluon plasma (QGP) can be
extracted from elliptic flow data in heavy-ion collisions by comparing them
with the dynamical model VISHNU which couples a viscous fluid dynamic
description of the QGP with a microscopic kinetic description of the late
hadronic rescattering and freeze-out stage. A robust method for fixing
(eta/s)_QGP from the collision centrality dependence of the eccentricity-scaled
charged hadron elliptic flow is presented. The systematic uncertainties
associated with this extraction method are discussed, with specific attention
to our presently restricted knowledge of initial conditions. With the
(eta/s)_QGP extracted in this way, VISHNU yields an excellent description of
all soft-hadron data from Au+Au collisions at top RHIC energy. Extrapolations
to Pb+Pb collisions at the LHC, using both a purely hydrodynamic approach and
VISHNU, are presented and compared with recent experimental results from the
ALICE Collaboration. The LHC data are again well described by VISHNU, with the
same (eta/s)_QGP value as at RHIC energies.

Full text

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arXiv:1108.5323v1 [nucl-th] 26 Aug 2011
The Viscosity of Quark-Gluon Plasma at RHIC and the LHC
Ulrich Heinz∗, Chun Shen∗and Huichao Song†
∗Department of Physics, The Ohio State University, Columbus, Ohio 43026, USA
†Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Abstract. The specific shear viscosity (η/s)QGPof quark-gluon plasma (QGP) can be extracted from elliptic flow data in
heavy-ion collisions bycomparing themwiththedynamical model VISHNU whichcouples aviscous fluiddynamic description
of the QGP with a microscopic kinetic description of the late hadronic rescattering and freeze-out stage. A robust method for
fixing (η/s)QGPfrom the collision centrality dependence of the eccentricity-scaled charged hadron elliptic flow is presented.
The systematic uncertainties associated with this extraction method are discussed, with specific attention to our presently
restricted knowledge of initial conditions. With the (η/s)QGPextracted in this way, VISHNU yields an excellent description
of all soft-hadron data from Au+Au collisions at top RHIC energy. Extrapolations to Pb+Pb collisions at the LHC, using both
a purely hydrodynamic approach and VISHNU, are presented and compared with recent experimental results from the ALICE
Collaboration. The LHC data are again well described by VISHNU, with the same (η/s)QGPvalue as at RHIC energies.
Keywords: Hydrodynamics, quark-gluon plasma, heavy-ion collisions, eccentricity, shear viscosity, elliptic flow, hybrid approach
PACS: 25.75.-q, 12.38.Mh, 25.75.Ld, 24.10.Nz
HOW TO MEASURE ( ( (η η η/ / /s s s) ) )QGP
Relativistic heavy-ion collisions create spatially deformed fireballs of hot, dense matter – in both non-central and
(due to event-by-event shape fluctuations) central collisions. Hydrodynamics converts this initial spatial deformation
into final state momentum anisotropies. Viscosity degrades the conversion efficiency εx=??y2−x2??
of the fluid (x and y are the directions transverse to the beam direction z); for given initial fireball ellipticity εx,
the viscous suppression of the dynamically generated total momentum anisotropy εpis monotonically related to the
specific shear viscosity η/s. The observable most directly related to εpis the total chargedhadron elliptic flow vch
Its distribution in pT depends on the chemical composition and pT-spectra of the various hadron species; the latter
evolve in the hadronic stage due to continuously increasing radial flow (and so does v2(pT)), even if (as expected at
top LHC energy [2]) εpfully saturates in the QGP phase. When (as it happens at RHIC energies) εpdoes not reach
saturation before hadronization, dissipative hadronic dynamics [3] affects not only the distribution of εpover hadron
species and pT, but even the final value of εpitself, and thus vch
QGP viscosity (η/s)QGPwe thereforeneeda hybridcode that couplesviscoushydrodynamicsof theQGP to a realistic
modelofthelate hadronicstage, suchas UrQMD[4], thatdescribesits dynamicsmicroscopically.VISHNU[5], a hybrid
of VISH2+1(Viscous Israel-Stewart Hydrodynamics in 2+1 dimensions [6]) and UrQMD, is such a code.
??y2+x2??→ εp=?Txx−Tyy?
?Txx+Tyy?
2[1].
2from which we want to extract η/s. To isolate the
( ( (η η η/ / /s s s) ) )QGPAT RHIC
TheleftpanelinFig.1showsthat suchanapproachyieldsauniversaldependenceoftheellipticity-scaledtotal charged
hadronelliptic flow, vch
only on (η/s)QGPbut not on the details of the initial state model that provides εxand S [7]. Pre-equilibrium flow and
bulk viscous effects on these curves are small [7].
The QGP viscosity can be extracted from experimental vch
The right panels of Fig. 1 show this for MC-Glauber and MC-KLN initial state models (please see [7] and references
therein for a description of these models). In both cases the slope of the data [8] is correctly reproduced;this is not the
case for ideal nor for viscous hydrodynamics with constant η/s. Due to the ∼20% larger ellipticity of the MC-KLN
fireballs, the magnitude of vch
from this comparison changes by more than a factor 2 between them. Relative to the initial fireball ellipticity all other
model uncertainties are negligible. Without constraining εxmore precisely, (η/s)QGPcannot be determined to better
2/εx, on the chargedhadronmultiplicity density per overlaparea, (1/S)(dNch/dy), that depends
2data by comparing them with these universal curves.
2,exp/εxdiffers between the two models. Consequently, the value of (η/s)QGPextracted

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01020 30 40
(1/S) dNch/dy (fm-2)
0
0.05
0.1
0.15
0.2
0.25
v2/ε
0.0 0.4 810
0.08 0.6 810
0.16 0.9 810
0.24 0.9 810
0.24 1.2 810
.
.
.
.
hydro (η/s)+UrQMD
η/s τ0 dN/dy
0.0 0.4 810
Glauber / KLN
(fm/c) max.
0.16 0.9 810
0.08 0.6 810
η/s
0.0
0.08
0.16
0.24
(b)
0 102030
(1/S) dNch/dy (fm-2)
0
0.05
0.1
0.15
0.2
0.25
v2/ε
010 2030 40
(1/S) dNch/dy (fm-2)
hydro (η/s) + UrQMD
hydro (η/s) + UrQMD
MC-Glauber
MC-KLN
0.0
0.08
0.16
0.24
0.0
0.08
0.16
0.24
η/s
η/s
v2{2} / 〈ε2
〈v2〉 / 〈εpart〉Gl
part〉1/2
Gl
(a)
(b)
v2{2} / 〈ε2
〈v2〉 / 〈εpart〉KLN
part〉1/2
KLN
FIGURE 1.
(Color online) Centrality dependence of eccentricity-scaled elliptic flow [7].
than a factor 2 from elliptic flow data alone, irrespective of any other model improvements.1Taking the MC-Glauber
and MC-KLN models to represent a reasonable range of initial ellipticities, Fig. 1 gives 1<4π(η/s)QGP<2.5 for
temperatures Tc<T <2Tcprobed at RHIC.
All calculations in Fig. 1 and following below were done in "single-shot" mode, where the ensemble of fluctuating
MonteCarloinitialstates wasfirst averagedintheparticipantplane[7]toobtainasmoothaverageinitialdensityprofile
and then evolved just once through the hydrodynamic stage. Event-by-event evolution of each fluctuating initial state
separately and performing the ensemble average only at the end may produce somewhat less elliptic flow and thus
slightly reduce the (η/s)QGPvalues extracted from comparison with the data [13]. The magnitude of this reduction
depends on (η/s)QGP[10] but is not expected to exceed (0.2-0.3)/4π [13].
????
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FIGURE 2.
identified pions and protons from 200AGeV Au+Au collisions at RHIC for different centralities. Experimental data are from STAR
and PHENIX, theoretical lines from VISHNU (see [14] for details and references). After appropriate adjustment of initial conditions
[14] the pT-spectra are seen to be insensitive to the QGP viscosity whereas the elliptic flow depends strongly on it. For MC-Glauber
initial conditions (upper right panels) (η/s)QGP=0.08 works well, for MC-KLN (bottom right panels) (η/s)QGP=0.16 works well
for all collision centralities. In each case, changing (η/s)QGPby 0.08 destroys the agreement between theory and data.
(Color online) Transverse momentum spectra (left panel) and differential elliptic flow v2(pT) (right panel) for
VISHNUwith (η/s)QGP=1
of soft (pT<1.5GeV) hadron production (pT-spectra and differential v2(pT) for all charged hadrons together as
well as for individual identified species) in 200AGeV Au+Au collisions at all but the most peripheral collision
centralities [14]. As an example we show in Fig. 2 pT-spectra and differential elliptic flow for identified pions and
4πfor MC-Glauber and
2
4πfor MC-KLN provides an excellent description of all aspects
1It has been suggested [9, 10, 11, 12] that the ambiguity between the MC-Glauber and MC-KLN ellipticities which lies at the origin of this
uncertainty can be resolved by simultaneously analyzing elliptic and triangular flow, v2and v3.

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protons (resonance decay contributions are included). Such a level of theoretical control is unprecedented.2
( ( (η η η/ / /s s s) ) )QGPAT THE LHC
The successful comprehensivefit of soft hadron spectra and elliptic flow in Au+Au collisions at RHIC shown in Fig. 2
and elaborated on in more detail in Refs. [14, 15] allows for tightly constrained LHC predictions. Fig. 3 shows such
predictionsforbothpureviscoushydrodynamicsVISH2+1[15]and VISHNU[16].A straightforwardextrapolationwith
020406080
centrality (%)
0
0.02
0.04
0.06
0.08
0.1
v2
RHIC: η/s=0.16
LHC: η/s=0.16
LHC: η/s=0.20
LHC: η/s=0.24
STAR
ALICE
v2{4}
MC-KLN
Reaction Plane
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??
FIGURE 3.
flow for identified hadrons for 20-30% centrality (VISH2+1, right [15]) for 200AGeV Au+Au collisions at RHIC and 2.76ATeV
Pb+Pb collisions at the LHC. Experimental data are from [17].
fixed (η/s)QGPoverpredicts the LHC vch
KLN) gives better agreement with the ALICE data [17]. However, at LHC energies v2becomes sensitive to details of
the initial shear stress profile [15], and no firm conclusion can be drawn yet whether the QGP turns more viscous (i.e.
less strongly coupled) at higher temperatures. Furthermore, ALICE [18] has noted a discrepancy between the ¯ p/π−
ratio measured in Pb+Pb collisions at the LHC and the value observed by STAR in Au+Au collisions at RHIC. The
latter has a strong influence on the value of the chemical decoupling temperature implemented in the model. We use
Tchem=165MeV which nicely fits the normalization of the proton spectra from STAR but overpredicts those from
PHENIX by a factor ∼ 1.5−2 (left panel in Fig. 2). The ¯ p/π−ratio measured by ALICE at the LHC agrees with the
PHENIX value measured at RHIC (see Fig. 7 in [18]) but is smaller by a factor ∼ 1.5−2 than what is implemented
in the LHC predictions from Refs. [15, 16]. Correspondingly, our predictions of the ¯ p spectra for Pb+Pb@LHC [15]
overpredict the measured spectra by this factor [18]. Reducing the ¯ p/π−ratio to the measured value will reduce the
chargedhadronelliptic flow [19]. To to the larger radial flow, this reductionsis stronger at the LHC than at RHIC. This
may account for the ∼ 15% overprediction of vch
The right panel of this figure shows that, at fixed pT<1GeV, v2(pT) increases from RHIC to LHC for pions but
decreases for all heavier hadrons. The similarity at RHIC and LHC of vch
noted in Ref. [17] thus appears accidental. As a result of this shift of the elliptic flow to larger pTfor heavier particles,
which is caused by the strongerradial flow at the LHC, the mass-splitting betweenthe v2(pT) curves fordifferentmass
hadrons grows from RHIC to LHC. This predicted growth has been confirmed by ALICE (see Fig. 6 in [20]).
As mentioned in footnote 2, the purely hydrodynamic simulations based on VISH2+1with constant η/s=0.2 fail
to correctly reproduce the centrality dependence of the proton elliptic flow vp
vp
(Color online) Total charged hadron ellipticflow asfunction of centrality(VISHNU, left [16]) and differential elliptic
2values by 10-15%; a slight increase of (η/s)QGPfrom 0.16 to 0.20 (for MC-
2for (η/s)QGP=0.16 at the LHC seen in the left panel of Fig. 3.
2(pT) for the sum of all charged hadrons
2(pT). Especially in central collisions,
2(pT) is overpredicted at small pT(see Fig. 2 in [21]), i.e. the radial flow pushing the elliptic flow to higher pT(and
2We note that the purely hydrodynamic model VISH2+1 does almost equally well, with (η/s)QGP=0.2 for MC-KLNinitial conditions [15], except
for the centrality dependence of the differential elliptic flow v2(pT) for protons. We will see a similar failure of VISH2+1 for Pb+Pb collisions at
the LHC further below. The main difference to VISHNU is that, in order to generate enough radial flow at freeze-out, VISH2+1 must be started
earlier (τ0=0.6 instead of 1.05fm/c) because it lacks the highly dissipative hadronic phase that generates additional radial flow in VISHNU at late
times (in the VISH2+1 simulations η/s is held constant at 0.2 until hadronic freeze-out). The variation with collision centrality of the final balance
between radial and elliptic flow turns out to be correct in VISHNU (where more of the radial flow develops later) but incorrect in VISH2+1 (where
more of it is created early).

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thereby reducing v2(pT) at low pT) as generated by the model is not strong enough in central collisions. The same
problem is seen in Fig. 4 for the extrapolation of VISH2+1to LHC energies: whereas the pion and kaon elliptic flows
at pT<1.5GeV/c are well described at all collision centralities, there appears to be a lack of radial flow in central and
semi-central collisions such that the proton v2is not pushed towards larger pT as strongly as seen in the data. This
problem disappears in the more peripheral bins, indicating an incorrect centrality dependence of the balance between
radial and elliptic flow in the VISH2+1model.
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FIGURE 4.
collisions at different centralities, as measured by ALICE (preliminary data reported at Quark Matter 2011 [20, 21]), compared
with VISH2+1 calculations using MC-KLN initial conditions with η/s=0.2 [15].
(Color online) Differential elliptic flow, v2{2}(pT), for pions, kaons and anti-protons from 2.76ATeV Pb+Pb
As was the case at RHIC energies,this problemis removedwhen replacingthe hydrodynamicdescriptionof the late
hadronic phase with low η/s=0.2 by a microscopic kinetic description using VISHNU(see Fig. 5). The calculations
shown in Fig. 5 were done primarily to understand systematic differences between the predictions for spectra and
elliptic flow from VISH2+1and VISHNU. For this reason they were performed with the same value η/s=0.2 in the
QGP phase, even thoughthe VISHNUcalculations for identified hadronsfromAu+Au at RHIC [14] had showna slight
preference for the smaller value (η/s)QGP=0.16. Fig. 5 shows that the problem with the lack of radial flow in central
collisions that was seen in Fig. 4 has been resolved: for the 5%-10% and 10%-20% centrality bins, VISHNUdescribes
the differential v2up to pT=2GeV/c almost perfectly, for all three particle species. Looking closely, one observes
a slight (6%) underprediction of v2(pT) for all three particle species. This underprediction gets stronger in more
peripheralcollisions (reaching9% at 50%-60%centrality)– a clearsign that we have slightlyoverestimated(η/s)QGP.
(Since, for fixed η/s, viscous effects increase in inverse proportion to the fireball size [6], an overestimate of η/s will
lead to an underprediction of v2(pT) that grows with the impact parameter of the collisions.) We are confident that,
after reducing(η/s)QGPto 0.16,the data will be well describedat all collision centralities. Correspondingsimulations
are in progress.
CONCLUSIONS
The hybrid model VISHNU, which describes the evolution of the dense and strongly coupled quark-gluon plasma
phase macroscopically using viscous fluid dynamics and that of the dilute late hadronic rescattering and freeze-out
stage microscopically using a kinetic approach, provides a comprehensive quantitative description of the bulk matter
created in relativistic heavy-ion collisions at RHIC and LHC. Transverse momentum spectra and elliptic flow of
soft charged hadrons, pions, kaons, and protons are well reproduced at all collision centralities, with a QGP shear

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viscosity (η/s)QGP=2
of (η/s)QGPbetween RHIC and LHC that would reflect the different temperature ranges probed. Overall, the QGP
liquid created in heavy-ion collisions at the LHC appears to be as strongly coupled as at RHIC energies.
4π=0.16 if MC-KLN initial conditions are used. So far the data yield no evidence for a change
0
0.1
0.2
0.3
0.4
0.5
0.6
v2/ε
Pb+Pb @ 2.76 A TeV
5%-10%
pions
kaons
protons
VISHNU
012
Pb+Pb @ 2.76 A TeV
10%-20%
pions
kaons
protons
VISHNU
012
0
0.1
0.2
0.3
0.4
0.5
0.6
Pb+Pb @ 2.76 A TeV
20%-30%
pions
kaons
protons
VISHNU
012
pT (GeV)
0
0.1
0.2
0.3
0.4
0.5
0.6
v2/ε
Pb+Pb @ 2.76 A TeV
30%-40%
pions
kaons
protons
VISHNU
012
pT (GeV)
Pb+Pb @ 2.76 A TeV
40%-50%
pions
kaons
protons
VISHNU
012
pT (GeV)
Pb+Pb @ 2.76 A TeV
50%-60%
pions
kaons
protons
VISHNU
FIGURE 5.
calculations with (η/s)QGP=0.2, using the same MC-KLN initial conditions as in Fig. 3. Shown is the eccentricity-scaled elliptic
flow, i.e. v2{2}/εx{2} for the experimental data and ?v2?/?εx? for the theoretical curves.
(Color online) Same preliminary data from ALICE [20, 21] as in Fig. 4, but now compared with VISHNU
Acknowledgments: This work was supported by the U.S. Department of Energy under grants No. DE-AC02-
05CH11231, DE-SC0004286 and (within the framework of the JET Collaboration) No. DE-SC0004104. Extensive
computing resources provided by the Ohio SupercomputingCenter are gratefully acknowledged.
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