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arXiv:1106.6350v1 [nucl-th] 30 Jun 2011

The QGP shear viscosity – elusive goal or just

around the corner?

Chun Shen1, Steffen A Bass2, Tetsufumi Hirano3,4, Pasi

Huovinen5, Zhi Qiu1, Huichao Song6and Ulrich Heinz1

1Department of Physics, The Ohio State University, Columbus, Ohio 43026, USA

2Department of Physics, Duke University, Durham, North Carolina 27708, USA

3Department of Physics, Sophia University, Tokyo 102-8554, Japan

4Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan

5ITP, J.W.Goethe-Universit¨ at, D-60438 Frankfurt a.M., Germany

6Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Abstract.

VISHNU,a rather precise (O(25%)) extraction of the QGP shear viscosity (η/s)QGPfrom

heavy-ion elliptic flow data is possible if the initial eccentricity of the collision fireball

is known with < 5% accuracy. At this point, eccentricities from initial state models

differ by up to 20%, leading to an O(100%) uncertainty for (η/s)QGP. It is shown

that a simultaneous comparison of elliptic and triangular flow, v2and v3, puts strong

constraints on initial state models and can largely eliminate the present uncertainty in

(η/s)QGP. The variation of the differential elliptic flow v2(pT) for identified hadrons

between RHIC and LHC energies provides additional tests of the evolution model.

With the new viscous hydrodynamic + hadron cascade hybrid code

Prologue – how to measure (η/s)QGP: Hydrodynamics converts the initial spatial

deformation of the fireball created in relativistic heavy-ion collisions into final state

momentum anisotropies. Viscosity degrades the conversion efficiency εx=??y2−x2??

εp=?Txx−Tyy?

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

charged hadron elliptic flow vch

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

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 of vch

To isolate the QGP viscosity (η/s)QGPwe therefore need a hybrid code that couples

viscous hydrodynamics of the QGP to a realistic model of the late hadronic stage, such

as UrQMD [4], that describes its dynamics microscopically. VISHNU [5] is such a code.

??y2+x2??→

?Txx+Tyy?of the fluid; for given initial fireball ellipticity εx, the viscous suppression

2 [1]. Its distribution in pT depends on the chemical

2from which we want to extract η/s.

Extraction of (η/s)QGP from 200AGeV Au+Au collisions at RHIC: The

left panel in Fig. 1 shows that such an approach yields a universal dependence of

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the ellipticity-scaled total charged hadron elliptic flow, vch

multiplicity density per overlap area, (1/S)(dNch/dy), that depends only on (η/s)QGP

but not on the details of the initial state model that provides εx and S [6].

equilibrium flow and bulk viscous effects on these curves are small [6].

2/εx, on the charged hadron

Pre-

0 10203040

(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)

0102030

(1/S) dNch/dy (fm-2)

0

0.05

0.1

0.15

0.2

0.25

v2/ε

0102030 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 [6].

The QGP viscosity can be extracted from experimental vch

with these universal curves. The right panels of Fig. 1 show this for MC-Glauber and

MC-KLN initial state models [6]. In both cases the slope of the data [7] is correctly

reproduced (not true for ideal nor viscous hydrodynamics with constant η/s). Due

to the ∼20% larger ellipticity of the MC-KLN fireballs, the magnitude of vch

differs between the two models. Consequently, the value of (η/s)QGPextracted 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 εx

more precisely, (η/s)QGPcannot be determined to better than a factor 2 from elliptic

flow data alone, irrespective of any other model improvements. Taking 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.

2data by comparing them

2,exp/εx

0246

b (fm)

8101214

0

0.05

0.1

0.15

0.2

0.25

0.3

π

p

K

v2/ε2

ideal hydro, MC−Glauber

single-shot,v2/¯ εpart

e-by-e, ?v2?/¯ εpart

0246

b (fm)

8 1012 14

0.1

0.15

0.2

0.25

0.3

0.35

π

p

K

v2/ε2

ideal hydro, MC−KLN

single-shot,v2/¯ εpart

e-by-e, ?v2?/¯ εpart

Figure 2.

parameter for pions, kaons and protons from single-shot and event-by-event ideal fluid

evolution of fluctuating initial conditions from the MC-Glauber (left) and MC-KLN

(right) models.

VISHNU with (η/s)QGP=

excellent description of all aspects of soft (pT<1.5GeV) hadron production (pT-

(Color online) Eccentricity-scaled elliptic flow as function of impact

1

4πfor MC-Glauber and

2

4πfor MC-KLN provides an

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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 [8]. Such a level of theoretical control is unprecedented.

Event-by-event hydrodynamics of fluctuating fireballs: In Fig. 1 we evolved

a smooth averaged initial profile (“single-shot hydrodynamics”). This overestimates

the conversion efficiency v2/ε [9, 10].Fig. 2 shows that event-by-event ideal fluid

dynamical evolution of fluctuating fireballs reduces v2/ε by a few percent [10]. The

effect is only ∼ 5% for pions but larger for heavier hadrons. We expect it to be less in

viscous hydrodynamics which dynamically dampens large initial fluctuations. A reduced

conversion efficiency v2/ε from event-by-event evolution will reduce the value of (η/s)QGP

extracted from vch

for (η/s)QGPwill at most be of order 0.02-0.03.

2; based on what we see in ideal fluid dynamics, the downward shift

Predictions for spectra and flow at the LHC: The successful comprehensive fit

of spectra and elliptic flow at RHIC [8] allows for tightly constrained LHC predictions.

Fig. 3 shows such predictions for both pure viscous hydrodynamics VISH2+1 [11] and

VISHNU [12]. A straightforward extrapolation with fixed (η/s)QGPoverpredicts the LHC

02040 6080

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

?

?

?

?

?

?

?

?

?

??

Figure 3. (Color online) Total charged hadron elliptic flow as function of centrality

(VISHNU, left [12]) and differential elliptic flow for identified hadrons for 20-30%

centrality (VISH2+1, right [11]) for 200AGeV Au+Au collisions at RHIC and

2.76ATeV Pb+Pb collisions at the LHC. Experimental data are from [13].

vch

gives better agreement with the ALICE data [13]. However, at LHC energies v2becomes

sensitive to details of the initial shear stress profile [11], and no firm conclusion can be

drawn yet whether the QGP turns more viscous (i.e. less strongly coupled) at higher

temperatures. The right panel 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

2 values by 10-15%; a slight increase of (η/s)QGPfrom 0.16 to 0.20 (for MC-KLN)

2(pT) for the sum of all charged hadrons thus appears accidental.

Constraining initial state models by simultaneous measurement of v2and v3:

While the ellipticities ε2differ by about 20% between MC-KLN and MC-Glauber models,

their triangularities ε3(which are entirely due to event-by-event fluctuations) are almost

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0.2 0.40.60.8 1.0

0

2

4

6

8

10

12

14

pT (GeV)

v2(%)

Au+Au @ RHIC, 20-30%

MC-KLN-like, η/s = 0.217

MC-Glb.-like, η/s = 0.111

MC-Glb.-like, η/s = 0.224

0.20.40.60.81.0

0

1

2

3

4

5

pT(GeV)

v3(%)

Au+Au @ RHIC, 20-30%

MC-KLN-like, η/s = 0.217

MC-Glb.-like, η/s = 0.224

MC-Glb.-like, η/s = 0.111

Figure 4.

hydrodynamics for initial eccentricities from the MC-KLN and MC-Glauber models.

(Color online) pT-differential elliptic and triangular flow from viscous

identical [10]. This suggests to use triangular flow v3(which is almost entirely [10] driven

by ε3) to obtain a model-independent measurement of (η/s)QGP. Fig. 4 shows vπ

and vπ

random relative angle) taken from the fluctuating Glauber (“MC-Glauber-like”) and

KLN (“MC-KLN-like”) models. It demonstrates that a given set of flow data requires

shear viscosities that differ by a factor 2 to reproduce v2(pT) and but the same shear

viscosities in both models to reproduce v3(pT). A good fit by both models to v2(pT)

produces dramatically different curves for v3(pT), and vice versa. The figure illustrates

the strong discriminating power for such simultaneous studies and gives hope for a much

more precise extraction of (η/s)QGPin the near future.

2(pT)

3(pT) for deformed Gaussian fireballs with average eccentricities ε2and ε3(with

Acknowledgments: This work was supported by the U.S. Department of Energy under grants No.

DE-AC02-05CH11231, DE-FG02-05ER41367, DE-SC0004286, and (within the framework of the JET

Collaboration) DE-SC0004104; by the Japan Society for the Promotion of Science through Grant-in-

Aid for Scientific Research No. 22740151; by the ExtreMe Matter Institute (EMMI); and by BMBF

under project No. 06FY9092. We gratefully acknowledge extensive computing resources provided to

us by the Ohio Supercomputer Center. C. Shen thanks the Quark Matter 2011 organizers for support.

References

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[7] Ollitrault J Y, Poskanzer A M and Voloshin S A 2009 Phys. Rev. C 80 014904

[8] Song H, Bass S A, Heinz U, Hirano T and Shen C 2011 Phys. Rev. C 83 054910

[9] Andrade R et al. 2006 Phys. Rev. Lett. 97 202302; Andrade R et al. 2008 ibid. 101 112301

[10] Qiu Z and Heinz U 2011 Preprint 1104.0650

[11] Shen C, Heinz U, Huovinen P and Song H 2011 Preprint 1105.3226

[12] Song H, Bass S A and Heinz U 2011 Phys. Rev. C 83 054912

[13] Aamodt K et al. [ALICE Collaboration] 2010 Phys. Rev. Lett. 105 252302