Elliptic Flow and Shear Viscosity within a Transport Approach from RHIC to LHC Energy
ABSTRACT We have investigated the build up of anisotropic flows within a parton
cascade approach at fixed shear viscosity to entropy density \eta/s to study
the generation of collective flows in ultra-relativistic heavy ion collisions.
We present a study of the impact of a temperature dependent \eta/s(T) on the
generation of the elliptic flow at both RHIC and LHC. Finally we show that the
transport approach, thanks to its wide validity range, is able to describe
naturally the rise - fall and saturation of the v_2(p_T) observed at LHC.
arXiv:1110.2383v1 [hep-ph] 11 Oct 2011
Elliptic Flow and Shear Viscosity within a
Transport Approach from RHIC to LHC Energy
S. Plumari∗,†and V. Greco∗,†
∗Department of Physics and Astronomy, University of Catania, Via S. Sofia 64, I-95125 Catania
†Laboratorio Nazionale del Sud, INFN-LNS, Via S. Sofia 63, I-95125 Catania
Abstract. We have investigated the build up of anisotropic flows within a parton cascade approach
at fixed shear viscosity to entropy density η/s to study the generation of collective flows in ultra-
on the generation of the elliptic flow at both RHIC and LHC. Finally we show that the transport
approach, thanks to its wide validity range, is able to describe naturally the rise - fall and saturation
of the v2(pT) observed at LHC.
Keywords: Quark-Gluon plasma, Relativistic Heavy Ion Collisions, Viscosity, Collective flows.
PACS: 25.75.Nq, 25.75.-q, 25.75.Ld
The RHIC program at BNL has shown that the azimuthal asymmetry in momentum
space, namely the elliptic flow v2, is the largest ever seen in HIC suggesting that an
almost perfect fluid with a very small shear viscosity to entropy density ratio, η/s,
has been created . The first measurement at LHC in Pb+Pb at 2.76TeV  shows
that the integrated elliptic flow as a function of collision energy increase of about 30%
compared to the flow measured at RHIC energy of 200GeV, while the v2(pT) measured
at LHC comparted to that of RHIC does not change indicating an increase in the average
transverse momentum. It remains to be understood if this means an equal η/s of the
formed plasma or it is the result of different initial conditions and possible larger non-
The most common approach to study viscous correction is viscous hydrodynamics
at second order in gradient expansion according to the Israel-Stewart theory [3, 4, 5].
This approach has been implemented to simulate the RHIC collisions providing an
upper bound for η/s ≤ 0.4. Such an approach, apart from the limitation to 2+1D
simulations, has the more fundamental problem of a limited range of validity in η/s
and in the transverse momentum pT. In these proceedings we discuss results within
the relativistic transport approach that has the advantage to be a 3+1D approach not
based on a gradient expansion in viscosity that is valid also for large η/s and for
out of equilibrium momentum distribution allowing a reliable description also of the
intermediate pT range. In this pT region viscous hydrodynamics breaks its validity
because the relative deviation of the equilibrium distribution function δ f/feqincreases
the results obtained with a parton cascade approach where the EoS is fixed to the
one of a free massless gass ε −3P = 0 and the mean free path λ is finite. A more
Tbecoming large already at pT≥ 3T ∼ 1GeV. In the following we will show
quantitativecomparison with the experimental data would require the inclusion of mean
field dynamics associate to an equation of state P(ε) according to lQCD results .
A first step in this direction has been discussed in [7, 8] while the implementation of
a quasi-particle model  in the contest of transport theory is in progress along lines
similar to .
THE PARTON CASCADE AT FIXED η/s
Our approach is a 3 + 1 dimensional Montecarlo cascade  for on-shell partons
based on the stochastic interpretation of the transition rate discussed in Ref. . In
kinetic theory under ultra-relativistic conditions the shear viscosity can be expressed
as η = (4/15)ρ < p > λ with ρ the parton density, λ = [ρσtr]−1the mean free path
and < p > the average momentum. Therefore considering that the entropy density for a
massless gas is s = ρ(4−µ/T), µ being the fugacity, we get:
where σtris the transport cross section. In our approach we solve the relativistic Boltz-
mann equation with the constraint that η/s is fixed during the dynamics of the collisions
in a way similar to  but with an exact local implementation a more detailed discus-
sion of the method is in . In fact fixing η/s we can evaluate locally in space and
time the strength of the cross section σtr(ρ,T) needed to have η/s at the wanted value
by mean of the following formula:
< p >
< p >
This approach is equivalent to have a total cross section of the form σTot =
K(ρ,T)σpQCD> σpQCD where K takes into account the non perturbative effects
responsible for that value of viscosity. Note that this approach have been shown to
recover the viscous hydrodynamics evolution of the bulk system [4, 11], but implicitly
assume that also high pT particles collide with largely nonperturbative cross section.
We show here that both at RHIC and LHC there are signatures of the disappearence of
the large non perturbative physics with increasing pT.
Effect of temperature dependent η/s(T)
In our calculation the initial condition are longitudinal boost invariant with the initial
parton density dN/dη(b = 0) = 1250 at RHIC and dN/dη(b = 0) = 2250 at LHC.
The partons are initially distributed in coordinate space according to the Glauber model
while in the momentum space at RHIC (LHC) the partons with pT ≤ p0= 2GeV
(pT≤ p0=4GeV) are distributed according to a thermalized spectrum with a maximum
temperature in the center of the fireball of 2TC(3.5TC), while for pT> p0we take the
η/s(T) ∝ T
η/s(T) ∝ T2
account the quasi-particle model predictions for η/s .
Different temperature dependent parametrizations for η/s. The orange area take into
spectrum of non-quenched minijets according to standard NLO-pQCD calculations. We
also start our simulation at the time t0= 0.6fm/c at RHIC and t0= 0.3fm/c at LHC.
In order to study the effect of the kinetic freezeout on the generation of the elliptic
flow we have performed two calculations one with a constant 4πη/s = 1 during all
the evolution of the system (red dashed line of Fig.1) the other (shown by black solid
line in Fig.1) with 4πη/s = 1 in the QGP phase and an increasing η/s in the cross over
regiontowardstheestimatedvalueforhadronicmatter4πη/s=8 . Such an increase
allows for a smooth realistic realization of the kinetic freeze-out. In Fig. 2 it is shown the
elliptic flow v2(pT) at mid rapidity for 20%−30% centrality for both RHIC Au+Au at
√s = 200GeV (left panel) and LHC Pb+Pb at√s = 2.76TeV (rhigh panel). As we can
see at RHIC energies, left panel of Fig. 2, the v2is sensitive to the hadronic phase and
the effect of the freeze out is to reduce the v2of about of 25%, from red dashed line to
black solid line in left panel of Fig. 2. For the pTrange shown we get a good agreement
with the experimental data for a minimal viscosity η/s ≈ 1/(4π) once the f.o. condition
is included. At LHC energies, right panel of Fig. 2, the scenario is different, we have that
the v2is less sensitive to the increase of η/s at low temperature in the hadronic phase.
The effect of large η/s in the hadronic phase is to reduce the v2by less than 5% in the
low pTregion, from red dashed line to the black solid line in right panel of Fig. 2. This
different behaviour of v2between RHIC and LHC energies can be explained looking at
the life time of the fireball. In fact at RHIC energies the life time of the fireball is smaller
than that at LHC energies, 5fm/c at RHIC against the about 10fm/c at LHC. Therefore
at RHIC the elliptic flow has not enough time to fully develop in the QGP phase. While
at LHC we have that the v2can develop almost completely because the fireball spend
more time in the QGP phase.
Due to this large life time of the fireball at LHC and the larger initial temperature is
interesting to study the effect of a temperature dependence in η/s. In the QGP phase
η/s is expected to have a minimum of η/s ≈ (4π)−1close to TCas suggested by lQCD
calculation . While at high temperature quasi-particle models seems to suggest
a temperature dependence of the form η/s ∼ Tαwith α ≈ 1−1.5 . To analyze
these possible scenarios for η/s in the QGP phase we have considered two different
situation one with a linear dependence 4πη/s = T/T0= (ε/ε0)1/4(blue line) and the
NO f.o. η/s=(4π)-1
f.o. η/s(T) ∝ T
f.o. η/s(T) ∝ T2
RHIC: Au+Au@200 GeV
STAR (20-30) %
LHC: Pb+Pb@2.76 TeV
left panel, the orange band indicate RHIC results measured by STAR and the orange points on the right
panel are the LHC results measured by the ALICE collaboration, data taken by . The red dashed line is
thecalculationwith 4πη/s=1duringall the evolutionofthe fireball andwithoutthe freezeoutcondition,
while the black blue and green lines are calculations with the inclusion of the kinetic freeze out and with
4πη/s = 1, 4πη/s ∝ T and 4πη/s ∝ T2in the QGP phase respectively.
Differential elliptic flow v2(pT) at mid rapidity for 20%−30% collision centrality. On the
otherone with a quadratic dependence 4πη/s=(T/T0)2=(ε/ε0)1/2(green line) where
ε0= 1.7GeV/fm3is the energy density at the beginning of the cross over regions where
the η/s has its minimum, see Fig.1. At RHIC energies the v2is essentially not sensitive
to the dependence of η/s on temperature in the QGP phase, see the blu and green lines
in the left panel of Fig. 2. Howevertheeffect on average is to decrease thevalue ofv2but
at low pT< 1.5GeV the v2(pT) appears to be insensitive to η/s(T) while a quite mild
dependence appears at higher pTwhere however the transport approach tends always to
overpredicted the elliptic flow observed experimentally. At LHC energies the build-up
of v2is more affected by the η/s in the QGP phase and on average it is reduced of about
a 20%. In any case still a strong temperature dependence in η/s has a small effect on
the generation of v2we found that with a constant or at most linearly dependent on T
η/s the transport approach can describe the data at both RHIC and LHC at least up to
pT∼ 2GeV. However the transport approach should keeps its validity also at higher pT,
but as previously said, the agreement with data seem to weaken at pT> 2GeV both at
RHIC and LHC. We discuss the underlying reasons in the next section.
Impact of high pTpartons on v2
In our approach we have that σTot= KσpQCDtherefore we have large cross section
independently of the pT of the colliding particles. But we know that particles with
high energies should collide with the pQCD cross section. In order to take into account
the proper scattering cross section for hard collisions we extend our previous approach
allowing for a K factor that depends on the invariant energy of the collision K(s) which
gives the connection between the non pertubative interacting bulk and the asymptotic
pQCD limit. We choose this function in such a way that at high energies K(s) → 1 and
we get the correct assumed pQCD limit. For the function K(s) we choose an exponential
K(s/Λ2) η/s(T) ∝T
K=const η/s(T) ∝T
RHIC: Au+Au@200 GeV
STAR (20-30) %
K(s/Λ2) η/s(T) ∝ T
K=const η/s(T) ∝ T
LHC: Pb+Pb @ 2.76 TeV
ALICE (20 - 30 )%
are the calculations with K = const and for 4πη/s = 1 and 4πη/s ∝ T with f.o. respectively for black
and blue curves while the solid lines are the same but with K(s/Λ2). Right: v2(pT) at mid rapidity and for
20%−30% collision centrality at LHC with the same legend, data taken from .
Left: v2(pT) at mid rapidity for 20%−30% collision centrality at RHIC. The dashed lines
form K(s/Λ2) = 1+γe−s/Λ2, where Λ is a scale parameter that fix the energy scale
at which the pQCD behaviour begins to be reached. While γ plays the same role of
K in the old calculations and it is determined again in order to keep fixed locally the
η/s. Therefore we can repeat the same procedure as described in the previous section
but now with σTot= K(s/Λ2)σpQCD. Due to its physical meaning we of course expect
Λ to be greater than 2GeV, in particular we have performed different calculation for
different value of Λ and we have obtained that for Λ > 4GeV the elliptic flow becomes
less sensitive to the value of the parameter Λ. Specifically in our calculation we have
considered the value Λ=4GeV. As we can see at RHIC energies, left panel of Fig. 3, we
have that K(s/Λ2) does not affect at all the v2(pT) for pT< 2GeV, in other words high
pTparton at RHIC energies does not affect the generation of the v2of the bulk. Instead
we have a reduction of the v2for pT> 3GeV and with the inclusion of K(s/Λ2) the v2
becomes a decreasing function of pTfor pT> 3GeV in perfect agreement with what is
observed experimentally (orange band). In Fig. 3 (right) we compare in a large range
the v2(pT) at LHC energy with (solid) and without (dashed) the inclusion of an energy
dependent K factor and for two T dependence of the η/s. We notice that the two sets of
experimental data refer to different method of v2measurements, namely v2 (circle)
and v2 (square) and our theorethical results should be compared to v2 because
event-by-event fluctuations are not considered. As we can see at LHC energies the v2
is sensitiveto K(s/Λ2) already at pT≈1.5GeV quite lower than the RHIC case, in other
words the many high pTpartons that we have at LHC energies affect the generation of
the v2of the bulk. Similar results we have when we include a η/s(T) in the QGP phase.
At low pTthe raise of the v2is an effect of a strong interacting fluid with a very small
viscosity. In this regime we have that particles with low pTinteract non perturbatively
with large cross sections and therefore we get a description in agreement with hydrody-
namics. With theincrease of the pTof the partons thepQCD limitbegins to be important
andfor pT>3−4GeV theellipticflowsstarts tobeadecreasing functionof pT. Thedis-
appearance of the non perturbative effect significantly affects the v2(pT) making faster
and stronger (∼ 20−25%) the fall in the elliptic flow in the range 3GeV < pT< 8GeV.
Finally for pT> 8GeV in our calculation seems to appear the saturation of the v2simi-
larly to the experimental data and typical of a path-lengh mechanism as in jet quenching
models . An analysis with better statistics is required. In this range of pTthe only
effect is that given by the pQCD limit.
We have investigatedwithin a transport approach at fixed η/s the effect of a temperature
dependent η/s at RHIC and LHC energies. At RHIC we have seen that the elliptic
flow is more sensitive to the kinetic freeze out (hadronic phase) and still of 25% of
v2depends on it. At LHC we get an opposite effect, nearly all the v2comes from the
QGP phase and the η/s of the hadronic phase is irrilevant. We get for both at RHIC
and LHC a good agreement with the data when the ratio η/s ≈ 1/(4π) or a linear T
dependence is considered, in general we observe not a large sensitivity of v2to the T
dependencein η/s. Furthermorewe haveseen that at LHCthelarge ammountof particle
with pT> 4GeV/c interacting nearly perturbatively cannot be neglected. The interplay
between perturbative and non-perturbative behaviour seems to have an important effect
on the generation of v2at intermediate pTand it could explain the rapid raise and fall of
v2(pT) in 0 < pT< 8GeV/c shown in the experiments.
This work was supported in part by funds provided by the Italian Ministry of Education,
Universities and Reserach under the Firb Research Grant RBFR0814TT.
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