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arXiv:1203.2452v1 [nucl-th] 12 Mar 2012

Influence of a temperature-dependent shear viscosity on the azimuthal asymmetries of

transverse momentum spectra in ultrarelativistic heavy-ion collisions

H. Niemia,b, G.S. Denicolc, P. Huovinenc, E. Moln´ arb,d, and D.H. Rischkeb,c

aDepartment of Physics, P.O. Box 35 (YFL) FI-40014 University of Jyv¨ askyl¨ a, Finland

bFrankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany

cInstitut f¨ ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨ at,

Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany and

dMTA Wigner Research Centre for Physics, H-1525 Budapest, P.O.Box 49, Hungary

We study the influence of a temperature-dependent shear viscosity over entropy density ratio η/s,

different shear relaxation times τπ, as well as different initial conditions on the transverse momentum

spectra of charged hadrons and identified particles. We investigate the azimuthal flow asymmetries

as a function of both collision energy and centrality. The elliptic flow coefficient turns out to be

dominated by the hadronic viscosity at RHIC energies. Only at higher collision energies the impact

of the viscosity in the QGP phase is visible in the flow asymmetries. Nevertheless, the shear viscosity

near the QCD transition region has the largest impact on the collective flow of the system. We also

find that the centrality dependence of the elliptic flow is sensitive to the temperature dependence

of η/s.

PACS numbers: 25.75.-q, 25.75.Ld, 12.38.Mh, 24.10.Nz

I.INTRODUCTION

Determining the properties of the quark-gluon plasma

(QGP) is nowadays one of the most important goals in

high-energy nuclear physics.

interacting particles reliable results can be obtained

from first-principle quantum field-theoretical calcula-

tions.Unfortunately, for strongly interacting matter

these tools provide only a limited amount of information.

It is, however, possible to calculate the thermodynamical

properties of such matter numerically from the theory of

strong interactions, quantum chromodynamics (QCD).

These lattice QCD calculations show that if the temper-

ature is sufficiently high, the matter undergoes a tran-

sition from a confined phase where the relevant degrees

of freedom are hadrons, to a deconfined phase where the

degrees of freedom are quarks and gluons, the so-called

QCD transition [1].

In recent years, experiments at the Relativistic Heavy-

Ion Collider (RHIC) at Brookhaven National Labora-

tory [2] and the Large Hadron Collider (LHC) at CERN

have provided a wealth of data from which one could in

principle obtain information about the QGP. However,

to compare these data with lattice QCD results is not

straightforward. So far, lattice calculations have pro-

vided reliable results for static thermodynamical prop-

erties of QCD matter, e.g. the equation of state (EoS).

The system created in heavy-ion collisions is, however,

not static but dynamical, because it expands and cools in

a very short time span of order 10−23seconds. Obviously,

in order to be able to properly interpret the experimental

results and infer the properties of QCD matter, we also

need a good understanding of the dynamics of heavy-ion

collisions.

Fluid dynamics is one of the most commonly used

frameworks to describe the space-time evolution of the

created fireball, because the complicated microscopic dy-

For a system of weakly

namics of the matter is encoded in only a few macroscopic

parameters like the EoS and the transport coefficients.

Currently, fluid-dynamical models give a reasonably

good quantitative description of transverse momentum

spectra of hadrons and their centrality dependence [3–7].

So far, most calculations assume that the shear viscosity

to entropy density ratio η/s is constant, and they show

that, in order to describe the azimuthal asymmetries of

the spectra, e.g. the elliptic flow coefficient v2, this con-

stant must be very small, of order 0.1. However, for real

physical systems, η/s depends (at least) on the temper-

ature [8]. A constant value of η/s can only be justified

as an average over the space-time evolution of the sys-

tem. It is not clear how this average is related to the

temperature dependence of η/s.

In previous work [9, 10], we have studied the conse-

quences of relaxing the assumption of a constant η/s.

We found that the relevant temperature region where the

shear viscosity affects the elliptic flow most varies with

the collision energy. At RHIC the most relevant region

is around and below the QCD transition temperature,

while for higher collision energies the temperature region

above the transition becomes more and more important.

In this work we shall extend our previous study and pro-

vide a more detailed picture of the temperature regions

that affect elliptic flow as well as higher harmonics at a

given collision energy.

This paper is organized in the following way. In Sec.

II, we describe our fluid-dynamical framework and its nu-

merical implementation. In Sec. III, we specify the EoS,

the transport coefficients, and the initialization. Sections

IV and V contain a detailed compilation of our results,

some of which were already shown in Refs. [9, 10]. We

present the transverse momentum spectra and the ellip-

tic flow of hadrons at various centralities with different

parameterizations of η/s as function of temperature. We

also study the impact of different initial conditions and

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of the choice of the relaxation time for the shear-stress

tensor. In Sec. VI, we investigate evolution of the elliptic

flow in more detail and, in Sec. VII, find the tempera-

ture regions where v2 and v4 are most sensitive to the

value of η/s. Finally, we summarize our results and give

some conclusions. We use natural units ? = c = k = 1

throughout the paper.

II.FLUID DYNAMICS

A.Formalism

In order to describe the evolution of a system on length

scales much larger than a typical microscopic scale, for

instance the mean-free path, it is sufficient to characterize

the state of matter by a few macroscopic fields, namely

the energy-momentum tensor Tµνand, possibly, some

charge currents Nµ

a. Fluid dynamics is equivalent to the

local conservation laws for these fields,

∂µTµν= 0,∂µNµ

a= 0.(1)

In the absence of conserved charges and bulk viscosity,

the energy-momentum tensor Tµνcan be decomposed as

Tµν= euµuν− P∆µν+ πµν,

where uµ= Tµνuν/e is the fluid four-velocity, e is the

energy density in the local rest frame of the fluid, i.e., in

the frame where uµ= (1,0,0,0), and P is the thermo-

dynamic pressure. The shear-stress tensor is defined as

πµν= T?µν?, where the angular brackets ?? denote the

symmetric and traceless part of the tensor orthogonal to

the fluid velocity. With the (+,−,−,−) convention for

the metric tensor gµν, the projector ∆µν= gµν− uµuν.

If the system is sufficiently close to local thermody-

namical equilibrium, the energy-momentum conservation

equations can be closed by providing the EoS, P(T), the

equations determining πµν, and the transport coefficients

entering these equations, e.g. the shear viscosity η(T).

The EoS P(T) and the shear viscosity η(T) can in prin-

ciple be computed by integrating out the dynamics on

microscopic length scales.

While the conservation laws are exact for any system,

the equations determining the shear-stress tensor require

certain approximations, so that the only variables en-

tering the equations of motion are those that appear in

the energy-momentum tensor, namely e, uµ, and πµν.

In the so-called relativistic Navier-Stokes approximation,

the shear-stress tensor is directly proportional to the gra-

dients of the four-velocity,

(2)

πµν= 2ησµν≡ 2η∂?µuν?.(3)

We note that in this approximation the shear-stress ten-

sor is not an independent dynamical variable.

Unfortunately, this approximation results in parabolic

equations of motion, and subsequently the signal speed is

not limited in this theory. In relativistic fluid dynamics

this violation of causality leads to the existence of linearly

unstable modes, which make relativistic Navier-Stokes

(NS) theory useless for practical applications [11, 12].

A commonly used approach that cures these instabil-

ity and acausality problems is Israel-Stewart (IS) theory

[13]. In this approach the shear-stress tensor, the heat

flow and bulk viscous pressure are introduced as inde-

pendent dynamical variables and fulfill coupled, so-called

relaxation-type differential equations of motion. Assum-

ing vanishing heat-flow and bulk viscosity, the relaxation

equation for the shear-stress tensor can be written as [15],

τπ˙ π?µν?+ πµν= 2ησµν+ λ1πµνθ + λ2σ?µ

+ λ3π?µ

απν?α

απν?α+ λ4ω?µ

απν?α,(4)

where ˙A = uµ∂µA denotes the comoving derivative of

A and θ = ∂µuµis the expansion scalar. The shear-

relaxation time τπ is the slowest time scale of the un-

derlying microscopic theory [14].

can be derived by neglecting all faster microscopic time

scales [15]. Like τπ, the coefficients λican in principle be

calculated from the underlying microscopic theory, i.e.,

in our case QCD. Unfortunately, for QCD the transport

coefficients appearing in Eq. (4) are still largely unknown.

For the sake of simplicity, in this work we use λ1= −4/3,

obtained from the Boltzmann equation for a massless

gas [13], and λ2 = λ3 = λ4 = 0. The shear-relaxation

time and the shear viscosity are left as free parameters.

Instead of the full (3+1)–dimensional treatment we

consider a simplified evolution where the expansion in

the z-direction is described by boost-invariant scaling

flow [16], i.e., the longitudinal velocity is given by vz=

z/t, and the scalar densities are independent of the space-

time rapidity ηs =

2log

measured in laboratory coordinates. In this approxima-

tion the full evolution depends only on the coordinates

(τ,x,y), where x and y are the transverse coordinates

and τ =√t2− z2is the longitudinal proper time.

Formally, IS theory

1

?

t+z

t−z

?

.Here, t is the time

B. Numerical implementation

Once the initial values of the components of the energy-

momentum tensor are specified at a given initial time τ0,

the space-time evolution of the system is obtained by

solving the conservation laws (1) together with the IS

equations (4).

The conservation laws are solved using the algorithm

developed in Refs. [17] and generalized to more than one

dimension in Ref. [18]. This method, known as SHASTA

for ”SHarp and Smooth Transport Algorithm”, solves

equations of the type

∂tU + ∂i(viU) = S(t,x), (5)

where U = U(t,x) is for example T00, T0i, ..., viis the

ith component of three-velocity, and S(t,x) is a source

term, for more details see Ref. [19].

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3

We can further stabilize SHASTA by letting the an-

tidiffusion coefficient Aad which controls the amount of

numerical diffusion to be proportional to

1

(k/e)2+ 1,(6)

where e is the energy density in the local rest frame, and

k is some constant of order 10−5GeV/fm3. In this way,

Aad goes smoothly to zero near the boundaries of the

grid, i.e., we increase the amount of numerical diffusion in

that region. We have checked that this neither affects the

solution nor produces more entropy inside the decoupling

surface.

The relaxation equation (4) could also be solved using

SHASTA. However, we noticed that solving it by replac-

ing the spatial gradient at grid point i on the left-hand

side of Eq. (4) by a centered second-order difference,

∂xUi=Ui+1− Ui−1

2∆x

, (7)

where U = πµν, yields a more stable algorithm. Time

derivatives in the source terms are simply taken as first-

order backward differences. Like in SHASTA, all spatial

gradients in the source terms are discretizised according

to Eq. (7).

C.Freeze-out

We assume that freeze-out, i.e., the transition from the

fluid-dynamical system to free-streaming particles hap-

pens on a hypersurface of constant temperature. Unless

otherwise stated, we assume that the freeze-out temper-

ature is Tdec= 100 MeV. We include all 2- and 3-particle

decays of hadronic resonances according to Ref. [20].

The transverse momentum distribution of hadrons

is calculated using the Cooper-Frye description [21].

For the final spectra we need to know the lo-

cal single-particle momentum distribution functions of

hadrons on the freeze-out surface.

ploy the widely used 14-moment ansatz where the

correction to the local-equilibrium distribution f0i =

{exp[(uµpµ

four-momentum pµ

iis given by [22]

Here, we em-

i− µi)/T] ± 1}−1of a hadron of species i with

δfi= f0i

pµ

T2(e + P).

ipν

iπµν

(8)

We note that this functional form for δf is merely an

ansatz. If dissipative fluid dynamics is derived from the

Boltzmann equation without assuming the 14-moment

approximation, the full expansion of δf contains an infi-

nite number of terms, for details see Ref. [15]. The effect

of this will be studied in a future work.

III.PARAMETERS

A.Equation of State

As EoS we use the recent s95p-PCE-v1 parameteriza-

tion of lattice QCD results [23]. In this parameteriza-

tion, the high-temperature part is matched to recent re-

sults of the hotQCD collaboration [24, 25] and smoothly

connected to the low-temperature part described as a

hadron resonance gas. All hadrons listed in Ref. [26] up

to a mass of 2 GeV are included in the hadronic part of

the EoS. The system is assumed to chemically freeze-out

at Tchem = 150 MeV. Below this temperature the EoS

is constructed according to Refs. [27–29]. This construc-

tion assumes that the evolution below Tchemis isentropic.

Strictly speaking this is not the case in viscous hydrody-

namics since dissipation causes an increase in entropy.

However, we have checked that in our calculations the

viscous entropy production from all fluid cells with tem-

peratures below Tchem= 150MeV is less than 1% of the

initial entropy, whereas the entropy production during

the entire evolution ranges from 3 −14 %, depending on

the collision energy and the η/s parameterization.

B.Transport coefficients

Thetemperature-dependent

parametrized as follows.

minimum of η/s to be at Ttr= 180 MeV. Unless other-

wise stated, the value of η/s at the minimum is assumed

to equal the lower bound η/s = 0.08 conjectured in the

framework of the AdS/CFT correspondence [31].

shearviscosityis

In all cases, we take the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.100.20 0.300.400.500.60

η/s

T [GeV]

LH-LQ

LH-HQ

HH-LQ

HH-HQ

FIG. 1. (Color online) Different parameterizations of η/s as a

function of temperature. The (LH-LQ) line is shifted down-

wards and the (HH-HQ) line upwards for better visibility.

The parameterization of the hadronic viscosity is based

on Ref. [32] where the authors consider a hadron reso-

nance gas with additional Hagedorn states. In practice,

we use a temperature dependence of η/s of the following

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functional form [9, 33],

η

s

???

HRG= 0.681 − 0.0594T

Ttr

− 0.544

?T

Ttr

?2

. (9)

At T = 100 MeV this coincides with the η/s value given

in Ref. [32], and decreases smoothly to the minimum

value η/s = 0.08 at Ttr. We note that many authors ob-

tain considerably larger values for the shear viscosity of

hadronic matter, see e.g. Refs. [34]. Our motivation here

is to illustrate the effects of hadronic viscosity rather than

to use a parameterization that is as realistic as possible.

We shall see that even this low η/s leads to considerable

effects for hadronic observables in Au + Au collisions at

RHIC. We further note that, since we are considering

a chemically frozen hadron resonance gas below Tchem,

while in Ref. [32] chemical equilibrium is assumed at all

temperatures, the entropy densities, and therefore the

values of η, differ between the two calculations at a given

value of T < Tchem.

The high-temperature QGP viscosity is parametrized

according to lattice QCD results [35] in such a way that

it connects to the minimum of η/s at Ttr. The functional

form used is

η

s

???

QGP= −0.289+ 0.288T

Ttr

+ 0.0818

?T

Ttr

?2

.(10)

We take the following four parameterizations of the

shear viscosity:

• (LH-LQ) η/s = 0.08 for all temperatures,

• (LH-HQ) η/s = 0.08 in the hadron gas, and above

T = 180 MeV η/s increases according to Eq. (10),

• (HH-LQ) below T = 180 MeV, η/s is given by Eq.

(9), and above we set η/s = 0.08,

• (HH-HQ) we use Eqs. (9) and (10) for the hadron

gas and the QGP, respectively.

These parameterizations are shown in Fig. 1. Besides

these four cases we also study the effect of varying the

value of the minimum of η/s, see Secs. V and VII.

In order to complete the description, we also need to

specify the relaxation time. In this work we use a func-

tional form suggested by kinetic theory,

τπ= cτ

η

e + p,

(11)

where cτ is a constant. Causality requires that cτ ≥

2 [12]. Unless otherwise stated, we shall use the value

cτ= 5 which coincides with the value obtained from the

Boltzmann equation in the 14-moment approximation for

a massless gas of classical particles [36]. The relaxation

times corresponding to the parameterizations above are

shown in Fig. 2. The effect of varying the relaxation time

separately from η is also studied in Sec. V.

0.0

1.0

2.0

3.0

4.0

5.0

0.100.200.30

T [GeV]

0.400.500.60

τπ [fm]

LH-LQ

LH-HQ

HH-LQ

HH-HQ

FIG. 2. (Color online) Relaxation times corresponding to the

different parameterizations of η/s, for cτ = 5. The (LH-LQ)

line is shifted downwards and the (HH-HQ) line upwards for

better visibility.

C.Initial state

We still need to specify the initial state at some proper

time τ0. For a boost-invariant system it is sufficient to

provide the components of the energy-momentum tensor

in the transverse plane at z = 0, i.e., ηs = 0. Within

our approximations these are the local energy density,

the initial transverse velocity, and the three independent

components of the shear-stress tensor. Here, we will as-

sume that the initial transverse velocity is zero and, un-

less otherwise stated, the initial shear-stress tensor is also

assumed to be zero.

For the initial time we choose τ0= 1 fm. The energy

density e(τ0,x,y) is based on the optical Glauber model

by assuming that the energy density is a function of the

density of binary nucleon-nucleon collisions nBC, or the

density of wounded nucleons nWN, or both,

e(τ0,x,y) = Cef(nBC,nWN).(12)

The overall normalization, Ce, is fixed in order to re-

produce the observed multiplicities in the most central

√sNN = 200 GeV Au+Au collisions at RHIC, and in

√sNN= 2.76 GeV Pb+Pb collisions at LHC.

The centrality dependence of the multiplicity is repro-

duced in this work in two different ways:

• BCfit: choosing f to be a polynomial in nBC,

f(nBC) = nBC+ c1n2

BC+ c2n3

BC. (13)

• GLmix: using a superposition of nBCand nWN,

f(nBC,nWN) = d1nBC+ (1 − d1)nWN. (14)

Here, the coefficient c2is introduced in order to guarantee

that the parameterizations are monotonically increasing

with increasing binary-collision or wounded-nucleon den-

sity. This ensures that the highest energy density is in

the center of the system, i.e., at x = y = 0.

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5

√sNN [GeV] c1 [fm−2] c2 [fm−4]

200

−0.032

2760

−0.01

5500

d1

0.1

0.7

1.0

Tmax [MeV]

313

430

504

0.00035

0.0001

00

TABLE I. Initialization parameters for different collision en-

ergies. The maximum temperature Tmaxis given for the BCfit

initialization with the (LH-LQ) parameterization of η/s. For

the other initializations Tmax differs less than 5%.

For a given impact parameter, the optical Glauber

model yields a different number of participants and differ-

ent centrality classes than the Monte Carlo Glauber mod-

els commonly used by the experimental collaborations.

Using the optical Glauber model, we can either choose

to reproduce the multiplicity as a function of the num-

ber of participants or as a function of centrality classes.

In general, this leads to different coefficients ci and d1.

Here, we choose to determine the initial conditions by

requiring that the centrality dependence of the charged

particle multiplicity as a function of the number of partic-

ipants [37, 38] is reproduced. We have checked that, if we

determine the centrality dependence by matching to the

centrality classes given by the optical Glauber model, the

elliptic flow is more suppressed in central and enhanced

in peripheral collisions at RHIC energies, while at LHC

energies it remains practically unchanged. In order to be

fully consistent with the experimental determination of

the centrality classes, one would need to generate fluctu-

ating initial conditions via a Monte Carlo Glauber model,

see e.g. Refs. [39, 40].

For√sNN= 5.5 TeV Pb+Pb collisions we use the mul-

tiplicity in the most central collisions as predicted by the

EKRT model [41]. In this case the centrality dependence

is assumed to follow binary scaling, i.e., c1 = c2 = 0

in Eq. (13). All initialization parameters are shown in

Table I.

Different parameterizations of η/s lead to different en-

tropy production and therefore different final multiplic-

ity, even if the initial state is kept the same. This is es-

pecially true for different parameterizations of the high-

temperature shear viscosity, since most of the entropy

is produced during the early stages of the collision [42].

We compensate this using different overallnormalizations

e.g. between the (HH-LQ) and (HH-HQ) parameteriza-

tions. Entropy production during the hadronic evolution

is small and not compensated. The centrality dependence

of the entropy production is also different for different η/s

parameterizations. Since it leads to at most a 5% differ-

ence in the final multiplicities and is hardly visible in the

results, it is not corrected here.

IV.RESULTS AND COMPARISON WITH

EXPERIMENTAL DATA

In this section we use the initializations and parame-

terizations of η/s given above, and compare the results

with experimental data from RHIC and LHC.

A. Transverse momentum spectra and elliptic flow

at RHIC

In Fig. 3 we show the pT-spectra of pions for different

centrality classes for RHIC√sNN = 200 GeV Au+Au

collisions and compare them with PHENIX data [37]. We

only show results using the BCfit initialization; those for

the GLmix initialization are very similar. The freeze-out

temperature is chosen as Tdec = 100 MeV. This choice

reproduces the slopes of the pT-spectra quite well.

Once we correct the normalization of the initial energy

density profile for different entropy production, the slopes

of the pT-spectra are practically unaffected by the η/s pa-

rameterizations. We note that in our earlier work [9] this

correction was not made, and the different η/s param-

eterizations lead not only to different multiplicities but

also to different slopes for the pT-spectra. This effect was

even more pronounced at LHC than at RHIC, due to an

increase in entropy production caused by larger gradients

appearing with an earlier initialization time τ0= 0.6 fm.

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

00.51 1.52 2.53

dN/dydpT

2 [1/GeV2]

pT [GeV]

π+

RHIC 200 AGeV

0-5 %

5-10 %

20-30 %

30-40 %

40-50 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

PHENIX

FIG. 3.

initialization.

(Color online) Pion spectra at RHIC, with BCfit

The kaon spectra are shown in Fig. 4 and the proton

spectra in Fig. 5 with the BCfit initialization. Both are

compared with PHENIX data [37]. Because we do not

consider net-baryon number in our calculations, the pro-

ton and anti-proton spectra are identical. For this rea-

son we show both the proton and the anti-proton data in

Fig. 5.

For both kaons and protons the calculated spectra are

slightly more curved than the data and they also lie above

the data. As for the pions, the slopes of the spectra are

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6

practically independent of the η/s parameterization.

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

00.51 1.522.5

dN/dydpT

2 [1/GeV2]

pT [GeV]

K+

RHIC 200 AGeV

0-5 %

5-10 %

20-30 %

30-40 %

40-50 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

PHENIX

FIG. 4. (Color online) Kaon spectra at RHIC, with BCfit

initialization.

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

00.511.5

pT [GeV]

22.53

dN/dydpT

2 [1/GeV2]

p

RHIC 200 AGeV

0-5 %

5-10 %

20-30 %

30-40 %

40-50 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

PHENIX

FIG. 5. (Color online) Proton spectra at RHIC, with BCfit

initialization.

Figure 6 shows the pT-differential elliptic flow v2(pT) of

charged hadrons for different centrality classes using the

BCfit initialization. Similarly, Fig. 7 shows the elliptic

flow for the GLmix initialization. The calculations are

compared with the four-particle cumulant data from the

STAR collaboration [43].

As was observed in Ref. [9], the differential elliptic flow

is largely independent of the high-temperature η/s pa-

rameterization, but highly sensitive on the hadronic η/s

at RHIC. This holds for all centrality classes. The sup-

pression of the elliptic flow due to the hadronic viscos-

ity is even more enhanced in more peripheral collisions.

Note that with the BCfit initialization, the elliptic flow

in the most central collision class is reproduced by the

parameterizations with a large hadronic viscosity, while

with the GLmix initialization the elliptic flow in the same

centrality class is better described by taking a constant

η/s = 0.08. However, with the latter choice the elliptic

flow tends to be overestimated in more peripheral col-

lisions. On the other hand, the temperature-dependent

hadronic η/s gives the centrality dependence correctly

down to the 30 − 40 % centrality class. In even more

peripheral collisions a large hadronic viscosity tends to

suppress the elliptic flow too much.

Figure 8 shows v2(pT) for protons with the BCfit ini-

tialization compared to the two-particle cumulant data

from the STAR collaboration [44].

qualitatively the same response to the different η/s pa-

rameterizations as all charged hadrons, i.e., v2(pT) de-

pends strongly on the hadronic viscosity, but is almost

independent of the high-temperature η/s. Since we use a

smooth initialization, with no initial-state fluctuations

included, quantitative comparisons with two- or four-

particle cumulant data are not straightforward.

The protons show

0.00

0.05

0.10

0.15

0.20

0.25

v2(pT)

charged hadrons

RHIC 200 AGeV

10-20 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

STAR v2{4}

20-30 %

0.00

0.05

0.10

0.15

0.20

0.25

0123

v2(pT)

pT [GeV]

30-40 %

0123

pT [GeV]

40-50 %

FIG. 6. (Color online) Charged hadron v2(pT) at RHIC, with

BCfit initialization.

B.Transverse momentum spectra and elliptic flow

at LHC

Transverse momentum spectra of charged hadrons in

most central Pb+Pb collisions with√sNN = 2.76 TeV

at LHC are shown in Fig. 9. At LHC, both initializa-

tions BCfit and GLmix give very similar results for both

elliptic flow and the spectra, because the contribution

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7

0.00

0.05

0.10

0.15

0.20

0.25

v2(pT)

charged hadrons

RHIC 200 AGeV

10-20 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

STAR v2{4}

20-30 %

0.00

0.05

0.10

0.15

0.20

0.25

0123

v2(pT)

pT [GeV]

30-40 %

0123

pT [GeV]

40-50 %

FIG. 7. (Color online) Charged hadron v2(pT) at RHIC, with

GLmix initialization.

0.00

0.05

0.10

0.15

v2(pT)

anti-protons

RHIC 200 AGeV

10-20 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

STAR v2{2}

20-30 %

0.00

0.05

0.10

0.15

012

v2(pT)

pT [GeV]

30-40 %

012

pT [GeV]

40-50 %

FIG. 8. (Color online) Proton v2(pT) at RHIC, with BCfit

initialization.

from binary collisions is large, of order ∼ 70 %, see Ta-

ble I. Therefore, we show only results with the BCfit ini-

tialization; these are compared to data from the ALICE

collaboration [45]. The calculated spectra are somewhat

flatter than the data. Here, we have used the same decou-

pling temperature as at RHIC, i.e., Tdec= 100 MeV. We

could improve the agreement with the data by decoupling

at even lower temperature than at RHIC. Another way

to improve the agreement is choosing a larger chemical

freeze-out temperature. This would give steeper spectra,

but the proton multiplicity at RHIC would then be over-

estimated. However, we have tested that the dependence

of the spectra and the elliptic flow on η/s is unchanged

by these details.

As was the case at RHIC, at LHC the slopes of the

spectra are practically independent of the η/s parame-

terization. We note that here we have used the initial-

ization time τ0= 1.0 fm, i.e., the same as at RHIC. In

Ref. [9] we observed a quite visible correlation between

the shear viscosity and the spectral slopes. Here, the later

initialization time and the fact that we now compensate

for the entropy production between different η/s param-

eterizations almost completely removes this correlation.

However, the earlier the evolution starts, the more the

viscosity will affect the slopes.

10-1

100

101

102

103

0.511.52

pT [GeV]

2.533.54

dN/dηdpT

2 [1/GeV2]

charged hadrons

LHC 2760 AGeV

0-5 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

ALICE

FIG. 9. (Color online) Charged hadron spectra at LHC, with

BCfit initialization.

The pT-differential elliptic flow for all charged hadrons

is shown in Fig. 10 and for protons in Fig. 11.

charged hadron elliptic flow is compared with ALICE

four-particle cumulant data [46]. We can see that in the

10 − 20 % centrality class, changing the hadronic η/s or

changing the high-temperature η/s has quite a similar

impact on the elliptic flow, e.g. the difference between

the LH-LQ and LH-HQ and between the LH-LQ and

HH-LQ curves is nearly the same. However, the more pe-

ripheral the collision is, the more the viscous suppression

is dominated by the hadronic η/s. This is confirmed by

comparing the grouping of the flow curves in the 40−50

% centrality class at LHC with that at RHIC, cf. Figs. 6

and 10. As was the case in Au+Au collisions at RHIC,

also here the grouping of the curves for the protons is

similar to that of all charged hadrons, cf. Fig. 11.

Note that, within our set-up, the best agreement with

the ALICE data is obtained with the HH-HQ parameter-

ization, i.e., with a temperature-dependent η/s in both

hadronic and high-temperature phases. However, in the

low-pT region our calculations systematically underesti-

mate the elliptic flow in all centrality classes. As was the

case with the pT-spectrum, decoupling at a lower tem-

The

Page 8

8

perature and choosing a higher chemical freeze-out tem-

perature would improve the agreement, without changing

the grouping of the elliptic flow curves with the η/s pa-

rameterizations.

0.00

0.05

0.10

0.15

0.20

0.25

v2(pT)

charged hadrons

LHC 2760 AGeV

10-20 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

ALICE v2{4}

20-30 %

0.00

0.05

0.10

0.15

0.20

0.25

0123

v2(pT)

pT [GeV]

30-40 %

0123

pT [GeV]

40-50 %

FIG. 10. (Color online) Charged hadron v2(pT) at LHC, with

BCfit initialization.

0.00

0.05

0.10

0.15

0.20

0.25

v2(pT)

protons

LHC 2760 AGeV

10-20 %

LH-LQ

LH-HQ

HH-LQ

HH-HQ

20-30 %

0.00

0.05

0.10

0.15

0.20

0.25

0123

v2(pT)

pT [GeV]

30-40 %

0123

pT [GeV]

40-50 %

FIG. 11. (Color online) Proton v2(pT) at LHC, with BCfit

initialization.

In Fig. 12 we show the pT-differential elliptic flow for

√sNN = 5.5 TeV Pb+Pb collisions. In this case the

viscous suppression of v2(pT) is dominated by the high-

temperature η/s in central collisions, while peripheral

collisions resemble more the lower-energy central colli-

sions at LHC, i.e., both hadronic and high-temperature

viscosity contribute similarly to the suppression. Fur-

thermore, the higher the pT, the more the hadronic

viscosity contributes to the suppression. This happens

mainly because δf increases with both viscosity and pT.

0.00

0.35

0.05

0.10

0.15

0.20

0.25

0.30

0.35

v2(pT)

charged hadrons

10-20 %

LHC 5500 AGeV

LH-LQ

LH-HQ

HH-LQ

HH-HQ

20-30 %

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0123

v2(pT)

pT [GeV]

30-40 %

0123

pT [GeV]

40-50 %

FIG. 12. (Color online) Charged hadron v2(pT) at LHC 5.5

A TeV, with BC initialization.

V.

MINIMUM OF η/s AND RELAXATION TIME

EFFECTS OF SHEAR INITIALIZATION,

One of the main results of Ref. [9] is that, at RHIC,

the high-temperature shear viscosity has very little effect

on the elliptic flow. In this section we elaborate more

on this analysis, and explicitly show that this statement

holds for an out-of-equilibrium initialization of the shear-

stress tensor as well. We also study the effect of varying

the relaxation time.

0.00

0.05

0.10

0.15

0.20

0.25

0123

v2(pT)

pT [GeV]

NS - initialization

charged hadrons

20-30 %

RHIC 200 AGeV

LH-LQ

LH-HQ

HH-LQ

HH-HQ

STAR v2{4}

FIG. 13. (Color online) Charged hadron v2(pT) at RHIC,

with BCfit and NS initialization.

Figure 13 shows the elliptic flow of charged hadrons in

the 20 − 30 % centrality class at RHIC. Instead of set-

ting πµνto zero initially, here the so-called Navier-Stokes

Page 9

9

(NS) initialization where the initial values of the shear-

stress tensor are given by the first-order, asymptotic solu-

tion of IS theory, Eq. (3). For all η/s parameterizations,

the NS initialization increases the entropy production (up

to 30 %), especially for the parameterizationswith a large

high-temperature viscosity. This is corrected by adjust-

ing the initial energy density to produce approximately

the same final multiplicity. Although for the parame-

terizations with a large hadronic η/s the different shear

initializations give slightly different v2(pT) curves, the

grouping of these curves remains intact. We emphasize

that the NS initialization gives very different initial con-

ditions for each viscosity parameterization. If we use the

same non-zero initial shear stress, e.g. πµν= const.×σµν,

for each parameterization, the resulting v2(pT) curves in

each group in Fig. 13 would be even closer to each other.

The NS initialization with a constant η/s = 0.08 has

a relatively short relaxation time, see Fig. 2. Hence for

τπ≪ τ0the NS initialization is not a completely unrealis-

tic assumption for the initial values of πµν. However, for

larger values of η/s the relaxation times are considerably

larger, τπ ? τ0, and there is no reason to assume that

the asymptotic solution could have been reached already

at very early times.

0.0

0.2

0.4

0.6

0.8

0.100.200.300.40

η/s

T [GeV]

HH-HQ

HH-HQ, η/smin = 0.16

FIG. 14. (Color online) Parameterizations of η/s as a function

of temperature. The (HH-HQ) line the same as in Fig. 1.

So far we have changed the shear-viscosity parameteri-

zation by keeping the minimum fixed. In Fig. 14 we show

the original HH-HQ parameterization and one where η/s

around the minimum is twice as large. Figure 15 shows

three v2(pT) curves for Au+Au collisions at RHIC: one

with the original HH-HQ parameterization, one with the

larger minimum value of η/s, and the last one with the

same large minimum value of η/s, but with a larger re-

laxation time, i.e., the constant in the relaxation time

formula (11) is cτ = 10 instead of cτ= 5. We note that

even a relatively small change in the η/s parameteriza-

tion near the minimum produces quite a visible change

in v2(pT). At RHIC, this change can be almost com-

pletely compensated by adjusting the relaxation time.

This shows that in small, rapidly expanding systems like

the one formed in heavy-ion collisions, transient effects

have considerable influence on the evolution. In other

words, the relaxation time cannot be merely considered

as a way to regularize the unstable Navier-Stokes theory:

it has real physical effects that cannot be completely dis-

tinguished from the effects of η/s. In√sNN= 2.76 TeV

Pb+Pb collisions at LHC, the effect of changing the min-

imum or the relaxation time is practically the same.

0.00

0.05

0.10

0.15

0.20

0.25

0123

v2(pT)

pT [GeV]

charged hadrons

20-30 %

RHIC 200 AGeV

HH-HQ

η/smin = 0.16, cτ=5

η/smin = 0.16, cτ=10

STAR v2{4}

FIG. 15. (Color online) Charged hadron v2(pT) at RHIC,

with BCfit initialization and for different minima of η/s and

relaxation times.

VI.TIME EVOLUTION OF THE ELLIPTIC

FLOW

One way to probe the effects of shear viscosity on the

elliptic flow is to calculate the time evolution of the lat-

ter. Typically this is done by calculating the so-called

momentum-space anisotropy from the energy-momentum

tensor,

εp=?Txx− Tyy?

?Txx+ Tyy?,(15)

where the ?···? denotes the average over the transverse

plane. The problem is, however, that one cannot make a

direct connection of εpto the actual value of v2obtained

from the decoupling procedure. Also, this way of study-

ing the time evolution does not take into account that,

at fixed time, part of the matter is already decoupled,

i.e., the average over the transverse plane includes also

matter that is outside the decoupling surface.

To overcome these two shortcomings of εp, we in-

stead calculate the v2 of pions from a constant-time

hypersurface that is connected smoothly to a constant-

temperature hypersurface at the edge of the fireball, see

Fig. 16 for examples of such hypersurfaces. Although, the

pions do not exist as real particles before hadronization,

the advantage is that the final v2we obtain matches the

one of thermal pions from the full decoupling calculation.

Figure 17 shows the time evolution of v2 in Au+Au

collisions at RHIC, in√sNN = 2.76 TeV Pb+Pb colli-

Page 10

10

0

2

4

6

8

10

12

14

16

18

τ [fm]

20-30 %

RHIC 200 AGeV

Tdec = 100 MeV

τ = 8 fm / T = 100 MeV

Tchem = 150 MeV

Ttr = 180 MeV

0

2

4

6

8

10

12

14

16

18

τ [fm]

LHC 2.76 ATeV

0

2

4

6

8

10

12

14

16

18

02468 1012

τ [fm]

x [fm]

LHC 5.5 ATeV

FIG. 16. (Color online) Constant-temperature hypersurfaces

at decoupling (Tdec= 100 MeV), chemical freeze-out (Tchem=

150 MeV), and at the minimum of η/s (Ttr = 180 MeV) at

different collision energies. Also, examples of surfaces that are

used in the calculation of the time evolution of v2 are shown

(dotted lines).

sions at LHC, and in√sNN= 5.5 TeV Pb+Pb collisions

at LHC. In all cases, the evolution is calculated in the

20 − 30 % centrality class. These results confirm our

earlier conjecture: at RHIC, the different η/s parameter-

izations create very little difference in the elliptic flow in

the early stages of the collision, while at later stages the

suppression due to the hadronic viscosity takes over and

groups the v2curves according to the hadronic viscosity.

At the intermediate LHC energy the impact of the QGP

viscosity is larger, and the final v2 still has a memory

0

0.02

0.04

0.06

0.08

v2 ("π")

20-30 %

RHIC 200 AGeV

LH-LQ

LH-HQ

HH-LQ

HH-HQ

0

0.02

0.04

0.06

0.08

v2 ("π")

LHC 2.76 ATeV

0

0.02

0.04

0.06

0.08

0246810 1214

v2 ("π")

τ [fm]

LHC 5.5 ATeV

FIG. 17.

collision energies.

(Color online) Time evolution of v2 at different

of this difference. The hadronic viscosity has a similar

impact on v2as the QGP viscosity. At the highest LHC

energy the hadronic suppression is small and the effect

of the QGP viscosity clearly dominates the grouping of

the v2 curves. Interestingly, both LHC evolutions show

an increase of v2around τ = 10 fm/c. This is when the

system is going through the chemical decoupling stage.

In the chemically frozen system v2tends to increase more

rapidly than in chemical equilibrium [29, 30]. At RHIC,

the chemical decoupling happens earlier, and also the

hadronic suppression is stronger, and the increase in v2

is washed out.

Page 11

11

VII.PROBING THE EFFECTS OF A

TEMPERATURE-DEPENDENT η/s ON THE vn’S

In this section, we try to probe the effects of a

temperature-dependent η/s on the azimuthal asymme-

tries in a more detailed way. To this end, we introduce

a modified η/s. Our baseline is a constant η/s|c= 0.08

that we then modify near some temperature Tiaccording

to

η

s(T) =η

s

???

c

?

1 + 2

?

exp

?|T − Ti| − δT

∆

?

+ 1

?−1?

,

(16)

where the parameters are taken to be δT = 10 MeV and

∆ = 1.5 MeV. One example of this η/s parameteriza-

tion is shown in Fig. 18.We note that, although we

use smooth initial conditions from the optical Glauber

model, we still get non-zero vnfor all even n. Although

these are much smaller than the ones obtained with the

fluctuations included, we can still probe the effects of vis-

cosity on these coefficients. By changing the temperature

Tiand comparing the simulations with a constant η/s we

can find the temperature regions where v2or v4are most

sensitive to changes of η/s at different collision energies.

0.0

0.1

0.2

0.3

0.10 0.20

T [GeV]

0.30

η/s

x

Ti = 170 MeV

η/s = 0.08

η/s + mod.

FIG. 18. (Color online) Shear viscosity with a modified tem-

perature dependence.

Figure 19 shows the results for v2 and v4 in the

20 − 30 % centrality class for RHIC and for both LHC

energies considered earlier. We plot the relative differ-

ence δvn/vn, where δvn= vn(η/s(T)) − vn(η/s|c). Each

point in the figure corresponds to a different calculation,

with a different value of Tiin Eq. (16). Similarly, Fig. 20

shows the same result, but without the δf contribution

to the freeze-out.

The viscosity can affect vn in two ways: by chang-

ing the space-time evolution of the integrated quantities

like the energy density, or by changing the local particle-

distribution function at freeze-out. With our small base-

-0.10

0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

δv2/v2

Tdec = 100 MeV

RHIC 200 AGeV

LHC 2.76 ATeV

LHC 5.5 ATeV

-0.15

-0.10

-0.05

0.00

0.05

100150200

Ti [MeV]

250300

δv4/v4

FIG. 19. (Color online) Effects of modified η/s on v2 and v4.

line viscosity the effect on the local distribution func-

tion is quickly washed out during the evolution below

the temperature Ti. Therefore, in these simulations, in

most of the temperature points, the change in η/s af-

fects vnthrough the space-time evolution, except at the

lowest-temperature point Ti= 110 MeV, where the peak

in η/s is close to the freeze-out temperature Tdec= 100

MeV. If we exclude the lowest temperature point in v4at

RHIC, we can read off from the figures that the temper-

ature region where viscosity affects both v2and v4most

is around the transition region T ∼ 150...200 MeV. For

v2this temperature region shifts slightly towards higher

temperatures with increasing collision energy, while for v4

the temperature where the effect is maximal is practically

unchanged. Other than this, the overall behavior of v2

and v4is quite similar. At high temperatures, the effect

of η/s increases with increasing collision energy, while at

low temperatures the viscous suppression decreases with

increasing collision energy, which is most notable for the

Ti = 110 MeV point where the viscosity effects on the

freeze-out distribution are strongest.

For v2we observed earlier that the suppression due to

the hadronic viscosity practically vanishes at the highest-

energy LHC collisions. This is again confirmed in Fig. 19.

This is, however, not true for higher harmonics. For v4

there is still a significant contribution from hadronic vis-

cosity at the full LHC energy. In this sense, higher har-

monics do not give direct access to the high-temperature

viscosity, but can rather help in constraining the hadronic

dynamics and viscosity as well as the correct form of δf.

Page 12

12

-0.10

0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

δv2/v2

Tdec = 100 MeV

RHIC 200 AGeV

LHC 2.76 ATeV

LHC 5.5 ATeV

-0.15

-0.10

-0.05

0.00

0.05

100 150200

Ti [MeV]

250300

δv4/v4

thermal π (no δf + no decays)

FIG. 20. (Color online) Same as Fig. 19, but without the δf

contribution.

This is also important since the hadronic evolution al-

ways tends to shadow the effects of the properties of the

high-temperature matter.

VIII.CONCLUSIONS

We have studied the effects of a temperature-

dependent η/s on the azimuthal asymmetries of hadron

transverse momentum spectra. We found earlier [9] that

the viscous suppression of the elliptic flow is dominated

by the hadronic viscosity in√sNN = 200 GeV Au+Au

collisions at RHIC, while in Pb+Pb collisions at the full

LHC energy√sNN= 5.5 TeV the suppression is mostly

due to the high-temperature shear viscosity. In this work

we have supplemented these earlier studies with more de-

tails.

First, we found that the suppression of the elliptic flow

due to the shear viscosity becomes more important in

more peripheral collisions. At least in our set-up, for

RHIC energies a temperature-dependent shear viscosity

improves the centrality dependence of the elliptic flow

compared to the data, similarly to what was found in the

hybrid approach of Ref. [5]. With a constant η/s = 0.08

and with the GLmix initialization, the measured v2(pT)

is reproduced in the most central collisions, but the cal-

culations give a too large elliptic flow for peripheral colli-

sions. However, with the BCfit initialization the elliptic

flow in the most central collisions is reproduced with a

temperature-dependent viscosity, and also the centrality

dependence is reproduced down to the 30 − 40 % cen-

trality class. Similarly, in Pb+Pb collisions at LHC both

a temperature-dependent hadronic η/s as well as an in-

creasing η/s in the high-temperature phase help in re-

producing the centrality dependence. Although there are

lots of uncertainties associated with the decoupling and

the initial state, at RHIC the centrality dependence of

v2(pT) may give access to the temperature dependence

of η/s in hadronic matter.

Furthermore, we have studied the effects of a

temperature-dependent η/s in a more detailed way. We

found that for a given collision energy both v2 and v4

are most sensitive to the shear viscosity near the transi-

tion temperature, i.e., T ∼ 150 − 200 MeV. For v2, this

region moves slightly to higher temperature and widens

with increasing collision energy, while for v4 it remains

practically unchanged. Other than that, the dependence

of v2 and v4 on η/s is similar with increasing collision

energy: the effect of the hadronic viscosity decreases and

the effect of the high-temperature viscosity increases.

For v2the effect of δf almost vanishes at the highest

collision energies, but for v4 it always remains signifi-

cant. At RHIC the δf corrections clearly dominate v4,

and even at the highest collision energies this effect is

comparable to the effects due to the modified space-time

evolution. In this sense, higher harmonics give access to

the δf corrections and the hadronic viscosity rather than

the high-temperature viscosity.

ACKNOWLEDGEMENT

This work was supported by the Helmholtz Inter-

national Center for FAIR within the framework of

the LOEWE program launched by the State of Hesse.

G.S.D., P.H., E.M., and D.H.R. acknowledge the hospi-

tality of the Department of Physics of Jyv¨ askyl¨ a Univer-

sity where part of this work was done. The work of H.N.

was supported by the Extreme Matter Institute (EMMI),

that of P.H. by BMBF under contract no. 06FY9092, and

that of E.M. by the Hungarian National Development

Agency OTKA/NF¨U 81655.

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