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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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2
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].
shearviscosity is
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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4
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
00.511.522.5
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
0 0.51 1.5
pT [GeV]
2 2.53
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
0.51 1.52
pT [GeV]
2.53 3.54
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
0.100.20 0.300.40
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
0246810 12
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
02468 1012 14
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
100150 200
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]
250 300
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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