Influence of a temperature-dependent shear viscosity on the azimuthal asymmetries of transverse momentum spectra in ultrarelativistic heavy-ion collisions
ABSTRACT We study the influence of a temperature-dependent shear viscosity over
entropy density ratio $\eta/s$, different shear relaxation times $\tau_\pi$, 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 $\eta/s$.
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
PACS numbers: 25.75.-q, 25.75.Ld, 12.38.Mh, 24.10.Nz
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 .
In recent years, experiments at the Relativistic Heavy-
Ion Collider (RHIC) at Brookhaven National Labora-
tory  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
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 . 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
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
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,
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,
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
. 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 ,
τπ˙ π?µν?+ πµν= 2ησµν+ λ1πµνθ + λ2σ?µ
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 .
can be derived by neglecting all faster microscopic time
scales . 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 , 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 , i.e., the longitudinal velocity is given by vz=
z/t, and the scalar densities are independent of the space-
time rapidity ηs =
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
. Here, t is the time
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
The conservation laws are solved using the algorithm
developed in Refs.  and generalized to more than one
dimension in Ref. . 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. .
We can further stabilize SHASTA by letting the an-
tidiffusion coefficient Aad which controls the amount of
numerical diffusion to be proportional to
(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
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,
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).
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. .
The transverse momentum distribution of hadrons
is calculated using the Cooper-Frye description .
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 =
iis given by 
Here, we em-
i− µi)/T] ± 1}−1of a hadron of species i with
T2(e + P).
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. . The effect
of this will be studied in a future work.
A. Equation of State
As EoS we use the recent s95p-PCE-v1 parameteriza-
tion of lattice QCD results . 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.  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
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 .
In all cases, we take the
0.10 0.200.300.40 0.500.60
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.  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
functional form [9, 33],
HRG= 0.681 − 0.0594T
At T = 100 MeV this coincides with the η/s value given
in Ref. , 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. . 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.  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  in such a way that
it connects to the minimum of η/s at Ttr. The functional
form used is
QGP= −0.289+ 0.288T
We take the following four parameterizations of the
• (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,
e + p,
where cτ is a constant. Causality requires that cτ ≥
2 . 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 . 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.
0.10 0.20 0.30
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
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
• 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.
√sNN [GeV] c1 [fm−2] c2 [fm−4]
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 . 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
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 .
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
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
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 . 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  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.
0 0.51 1.522.53
RHIC 200 AGeV
(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 . 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
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