Relativistic Hydrodynamics at RHIC and LHC
Tetsufumi Hirano1,∗)
1 Department of Physics, The University of Tokyo, Tokyo 113–0033, Japan
Recent development of a hydrodynamic model is discussed by putting an emphasis on
realistic treatment of the early and late stages in relativistic heavy ion collisions. The model,
which incorporates a hydrodynamic description of the quark-gluon plasma with a kinetic
approach of hadron cascades, is applied to analysis of elliptic flow data at the Relativistic
Heavy Ion Collider energy. It is predicted that the elliptic flow parameter based on the
hybrid model increases with the collision energy up to the Large Hadron Collider energy.
§1. Introduction
One of the important discoveries made at the Relativistic Heavy Ion Collider
(RHIC) in Brookhaven National Laboratory is that the elliptic flow parameter,1)
namely, the second Fourier coefficient v2 = 〈cos(2φ)〉 of the azimuthal momentum
distribution dN/dφ,2) is quite large in non-central Au+Au collisions.3) Over the
past years, many studies have been devoted to understanding the elliptic flow from
dynamical models: (1) The observed v2 values near midrapidity at low transverse
momentum (pT ) in central and semi-central collisions are consistent with predictions
from ideal hydrodynamics.4) (2) The v2 data cannot be interpreted by hadronic cas-
cade models.5), 6) (3) A partonic cascade model7) can reproduce these data only with
significantly larger cross sections than the ones obtained from the perturbative cal-
culation of quantum chromodynamics. The produced dynamical system is beyond
the description of naive kinetic theories. Thus, a paradigm of the strongly cou-
pled/interacting/correlated matter is being established in the physics of relativistic
heavy ion collisions.8) The agreement between hydrodynamic predictions and the
data suggests that the heavy ion collision experiment indeed provides excellent op-
portunities for studying matter in local equilibrium at high temperature and for
drawing information of the bulk and transport properties of the quark-gluon plasma
(QGP). These kinds of phenomenological studies closely connected with experimen-
tal results, so to say, the “observational QGP physics”, will be one of the main
trends in modern nuclear physics in the eras of the RHIC and the upcoming Large
Hadron Collider (LHC). Then it is indispensable to sophisticate hydrodynamic mod-
eling of heavy ion collisions for making quantitative statements on properties of the
produced matter with estimation of uncertainties. In fact, the ideal fluid dynamical
description gradually breaks down as one studies peripheral collisions4) or moves
away from midrapidity.9), 10) This requires a more realistic treatment of the early
and late stages11)–14) in dynamical modeling of relativistic heavy ion collisions.
In this paper, recent studies of the state-of-the-art hydrodynamic simulations are
highlighted with emphases on the importance of the final decoupling stage (Sec. 3)
∗) e-mail address: hirano@phys.s.u-tokyo.ac.jp
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1699v1
2 T. Hirano
and of much better understanding of initial conditions (Sec. 4). A prediction of the
v2 parameter at the LHC energy will be made in Sec. 5. See also other reviews
15) to
complement other topics of hydrodynamics in heavy ion collisions at RHIC.
§2. A QGP fluid + hadronic cascade model
We have formulated a dynamical and unified model,13) based on fully three-
dimensional (3D) ideal hydrodynamics,9), 10) toward understanding the bulk and
transport properties of the QGP. During the fluid dynamical evolution one assumes
local thermal equilibrium. However, this assumption can be expected to hold only
during the intermediate stage of the collision. In order to extract properties of the
QGP from experimental data one must therefore supplement the hydrodynamic de-
scription by appropriate models for the beginning and end of the collision. In Sec. 3,
we employ the Glauber model for initial conditions in hydrodynamic simulations.
Initial entropy density is parametrized as a superposition of terms scaling with the
densities of participant nucleons and binary collisions, suitably generalized to ac-
count for the longitudinal structure of the initial fireball.13) Instead, in Secs. 4 and
5, we employ the Color Glass Condensate (CGC) picture17) for colliding nuclei and
calculate the produced gluon distributions18) as input for the initial conditions in
the hydrodynamical calculation.19) During the late stage, local thermal equilibrium
is no longer maintained due to expansion and dilution of the matter. We treat this
gradual transition from a locally thermalized system to free-streaming hadrons via
a dilute interacting hadronic gas by employing a hadronic cascade model20) below a
switching temperature of T sw=169 MeV. A massless ideal parton gas equation of
state (EOS) is employed in the QGP phase (T >Tc = 170 MeV) while a hadronic
resonance gas model is used at T <Tc. When we use the hydrodynamic code all the
way to final decoupling, we take into account10) chemical freezeout of the hadron
abundances at T ch = 170 MeV, separated from thermal freezeout of the momen-
tum spectra at a lower decoupling temperature T th, as required to reproduce the
experimentally measured yields.21)
§3. Success of a hybrid approach
Initial conditions in 3D hydrodynamic simulations are put so that centrality
and pseudorapidity dependences of charged particle yields are reproduced. A linear
combination of terms scaling with the number of participants and that of binary
collisions enables us to describe centrality dependence of particle yields at midra-
pidity. This agreement with the data still holds when the ideal fluid description is
replaced by a more realistic hadronic cascade below T sw. See also Fig. 3. When ideal
hydrodynamics is utilized all the way to kinetic freezeout, T th = 100 MeV is needed
to generate enough radial flow for reproduction of proton pT spectrum at midra-
pidity. One major advantage of the hybrid model over the ideal hydrodynamics is
that the hybrid model automatically describes freezeout processes without any free
parameters. The hybrid model works remarkably well in reproduction of pT spectra
for identified hadrons below pT ∼ 1.5 GeV/c.
Relativistic Hydrodynamics at RHIC and LHC 3
partN
0 50 100 150 200 250 300 350 400
QGP + hadron fluids
QGP fluid and hadron gas
Hadron gas
PHOBOS(hit)
PHOBOS(track)
-6 -4 -2 0 2 4 6
hydro+cascade
=100MeV, hydrothT
=169MeV, hydrothT
PHOBOS 25-50%
b=8.5fm
Fig. 1. (Left) Centrality dependence of v2. The solid (dashed) line results from a full ideal fluid
dynamic approach (a hybrid model). For reference, a result from a hadronic cascade model6)
is also shown (dash-dotted line). (Right) Pseudorapidity dependence of v2. The solid (dashed)
line is the result from a full ideal hydrodynamic approach with T th = 100 MeV (T th = 169
MeV). Filled circles are the result from the hybrid model. All data are from the PHOBOS
Collaboration.22)
The centrality dependences of v2 at midrapidity (| η |< 1) from (1) a hadronic
cascade model6) (dash-dotted), (2) a QGP fluid with hadronic rescatterings taken
through a hadronic cascade model (dashed), and (3) a QGP+hadron fluid with
T th = 100 MeV (solid) are compared with the PHOBOS data.22) A hadronic cascade
model cannot generate enough elliptic flow to reproduce the data. This is observed
also in other hadronic cascade calculations.5) Thus it is almost impossible to interpret
the v2 data from a hadronic picture only. Models based on a QGP fluid generate
large elliptic flow and gives v2 values which are comparable with the data. When
a hadronic matter is also treated by ideal hydrodynamics, v2 is overpredicted in
peripheral collisions. This is improved by dissipative effects in the hadronic matter.
Note that there could exist effects of eccentricity fluctuation,23) which is not taken
into account in the current approach. Deviation between the data and the QGP-
fluid-based results above Npart ∼ 200 could be attributed to these effects.
From the integrated elliptic flow data at midrapidity, initial push from QGP
pressure turns out to be important at midrapidity. In Fig. 1 (right), the pseu-
dorapidity dependence of v2 data in 25-50% centrality observed by PHOBOS
22) are
compared with QGP fluid models. Ideal hydrodynamics with T th = 169 MeV, which
is just below the transition temperature Tc = 170 MeV, underpredicts the data in
the whole pseudorapidity region. Hadronic rescatterings after QGP fluid evolution
generate the right amount of elliptic flow and, consequently, the triangle pattern of
the data is reproduced well. If the hadronic matter is also assumed to be described
by ideal hydrodynamics until T th = 100 MeV, v2 overshoots in forward/backward
rapidity regions (| η |∼ 4). This is simply due to the fact that, in ideal hydrody-
namics, v2 is approximately proportional to the initial eccentricity which is almost
independent of space-time rapidity. So the hadronic dissipation is quite important in
forward/backward rapidity regions as well as at midrapidity in peripheral collisions
(Npart < 100). From these studies, the perfect fluidity of the QGP is needed to
4 T. Hirano
obtain enough amount of the integrated v2, while the dissipation (or finite values
of the mean free path among hadrons) in the hadronic matter is also important to
obtain less elliptic flow coefficients when the multiplicity is small at midrapidity in
peripheral collisions and/or in forward/backward rapidity regions. This is exactly
the novel picture of dynamics in relativistic heavy ion collisions, namely, the nearly
perfect fluid QGP core and the highly dissipative hadronic corona, addressed in
Ref. 16)
 (GeV/c)Tp
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.05
πSTAR, 
STAR, K
STAR, p
Fig. 2. Transverse momentum dependence of v2 for pions, kaons and protons. Filled plots are the
results from the hybrid model. The impact parameter in the model simulation is 7.2 fm which
corresponds to 20-30% centrality. Data (open plots) for pions, kaons and protons are obtained
by the STAR Collaboration.24)
As a cross-check on the picture, we also study pT dependence of v2 for identi-
fied hadrons in semi-central collisions to see whether the hybrid model works. We
correctly reproduce mass ordering behavior of differential elliptic flow below pT ∼ 1
GeV/c as shown in Fig. 2. Here experimental data are from STAR.24) Although
we also reproduce the data in 10-20% and 30-40% well (not shown), it is hard to
reproduce data in very central collisions (0-5%) due to a lack of initial eccentricity
fluctuation in this model. It is worth mentioning that, recently, the hybrid model
succeeds to describes differential elliptic flow data for identified hadrons at forward
rapidity observed by BRAHMS.25)
§4. Challenge for a hydrodynamic approach
So far, an ideal hydrodynamic description of the QGP fluid with the Glauber
type initial conditions followed by an kinetic description of the hadron gas describes
the space-time evolution of bulk matter remarkably well. The CGC,17) whose cases
are growing both in deep inelastic scatterings and in d+Au collisions recently, is one
of the relevant pictures to describe initial colliding nuclei in high energy collisions. In
this section, novel hydrodynamic initial conditions19) based on the CGC are employed
for an analysis of elliptic flow.
We first calculate the centrality dependence of the multiplicity to see that the
CGC indeed correctly describes the initial entropy production and gives proper initial
conditions for the fluid dynamical calculations. Both CGC and Glauber model initial
Relativistic Hydrodynamics at RHIC and LHC 5
partN
0 50 100 150 200 250 300 350 400
=100MeVthGlauber+hydro, T
=100MeVthCGC+hydro, T
CGC+hydro+cascade
PHOBOS
Fig. 3. Centrality dependence of charged particle multiplicity per number of participant nucleons.
The solid (dashed) line results from Glauber-type (CGC) initial conditions. The dash-dotted
line results from our hybrid model. Experimental data are from PHOBOS.27)
conditions, propagated with ideal fluid dynamics, reproduce the observed centrality
dependence of the multiplicity,27) see Fig. 3. In the hydrodynamic simulations, the
numbers of stable hadrons below T ch are designed to be fixed by introducing chemical
potential for each hadron.10) On the other hand, the number of charged hadrons is
approximately conserved during hadronic cascades. So the centrality dependence of
charged particle yields is also reasonably reproduced by the hybrid approach.
In the left panel of Fig. 4 we show the impact parameter dependence of the
eccentricity of the initial energy density distributions at τ0 = 0.6 fm/c. We neglect
event-by-event eccentricity fluctuations although these might be important for very
central and peripheral events.23) Even though both models correctly describe the
centrality dependence of the multiplicity as shown in Fig. 3, they exhibit a signifi-
cant difference: The eccentricity from the CGC is 20-30% larger than that from the
Glauber model.13), 28) The situation does not change even when we employ the “uni-
versal” saturation scale29) in calculation of gluon production. The initial eccentricity
is thus quite sensitive to model assumptions about the initial energy deposition which
can be discriminated by the observation of elliptic flow. The centrality dependence
of v2 from the CGC initial conditions followed by the QGP fluid plus the hadron gas
is shown in Fig. 4 (right). With Glauber model initial conditions,4) the predicted
v2 from ideal fluid dynamics overshoots the peripheral collision data.
22) Hadronic
dissipative effects within hadron cascade model reduce v2 and, in the Glauber model
case, are seen to be sufficient to explain the data (Fig. 1 (left)).13) Initial conditions
based on the CGC model, however, lead to larger elliptic flows which overshoot the
data even after hadronic dissipation is accounted for,13) unless one additionally as-
sumes significant shear viscosity also during the early QGP stage. Therefore precise
understanding of the bulk and transport properties of QGP from the elliptic flow
data requires a better understanding of the initial stages in heavy ion collisions.
6 T. Hirano
b (fm)
0 2 4 6 8 10 12 14
no diffuseness
CGC (IC-e)
=0.85)αGlauber (
partN
0 50 100 150 200 250 300 350 400
hydro+cascade, CGC
hydro+cascade, Glauber
PHOBOS(hit)
PHOBOS(track)
Fig. 4. (Left) Impact parameter dependence of the eccentricity of the initial energy density dis-
tributions. The solid (dashed) line results from Glauber-type (CGC) initial conditions. The
dotted line assumes a box profile for the initial energy density distribution. (Right) Centrality
dependence of v2. The solid (dashed) line results from CGC (Glauber model) initial conditions
followed by ideal fluid QGP dynamics and a dissipative hadronic cascade. The data are from
PHOBOS.22)
§5. Elliptic flow at LHC
The elliptic flow parameter plays a very important role in understanding global
aspects of dynamics in heavy ion collisions at RHIC. It must be also important to
measure elliptic flow parameter at the LHC energy toward comprehensive under-
standing of the degree and mechanism of thermalization and the bulk and transport
properties of the QGP.
Figure 5 shows the excitation function of the charged particle elliptic flow v2,
scaled by the initial eccentricity ε, for Au+Au collisions at b = 6.3 fm impact pa-
rameter, using three different models: (i) a pure 3D ideal fluid approach with a
typical kinetic freezeout temperature T th = 100 MeV where both QGP and hadron
gas are treated as ideal fluids (dash-dotted line); (ii) 3D ideal fluid evolution for the
QGP, with kinetic freezeout at T th = 169 MeV and no hadronic rescattering (dashed
line); and (iii) 3D ideal fluid QGP evolution followed by hadronic rescattering below
T sw = 169MeV (solid line). Although applicability of the CGC model for SPS ener-
gies might be questioned, we use it here as a systematic tool for obtaining the energy
dependence of the hydrodynamic initial conditions. By dividing out the initial eccen-
tricity ε, we obtain an excitation function for the scaled elliptic flow v2/ε whose shape
should be insensitive to the facts that CGC initial conditions produce larger eccen-
tricities and the resulting integrated v2 overshoots the data at RHIC and also that
experiments with different collision system (Pb+Pb) will be performed at the LHC.
Figure 5 shows the well-known bump in v2/ε at SPS energies (
sNN ∼ 10GeV)
predicted by the purely hydrodynamic approach, as a consequence of the softening
of the equation of state (EOS) near the quark-hadron phase transition region,30) and
that this structure is completely washed out by hadronic dissipation,12) consistent
with the experimental data.31), 32) Even at RHIC energies, hadronic dissipation still
reduces v2 by ∼ 20%. The hybrid model predicts a monotonically increasing excita-
Relativistic Hydrodynamics at RHIC and LHC 7
tion function for v2/ε which keeps growing from RHIC to LHC energies,
12) contrary
to the ideal fluid approach whose excitation function almost saturates above RHIC
energies.30)
 (GeV)NNs
10 210 310 410
=100MeV
CGC+hydro, T
=169MeV
CGC+hydro, T
CGC+hydro+cascade
Au+Au Charged, b=6.3fm
LHCRHIC
Fig. 5. Excitation function of v2/ε in Au+Au collisions at b = 6.3 fm. The solid line results from
CGC initial conditions followed an ideal QGP fluid and a dissipative hadronic cascade. The
dashed (dash-dotted) line results from purely ideal fluid dynamics with thermal freezeout at
T th = 169MeV (100MeV).
§6. Conclusions
We have studied the recent elliptic flow data at RHIC by using a hybrid model
in which an ideal hydrodynamic treatment of the QGP is combined with a hadronic
cascade model. With the Glauber-type initial conditions, the space-time evolution of
the bulk matter created at RHIC is well described by the hybrid model. The agree-
ment between the model results and the data includes v2(Npart), v2(η), pT spectra
for identified hadron below pT ∼ 1.5 GeV/c and v2(pT ) for identified hadrons at
midrapidity and in the forward rapidity region. If the Glauber type initial condi-
tions are realized, we can establish a picture of the nearly perfect fluid of the QGP
core and the highly dissipative hadronic corona. However, in the case of the CGC
initial conditions, the energy density profile in the transverse plane is more “eccen-
tric” than that from the conventional Glauber model. This in turn generates large
elliptic flow, which is not consistent with the experimental data. Without viscous
effects even in the QGP phase, we cannot interpret the integrated elliptic flow at
RHIC. If one wants to extract informations on the properties of the QGP, a better
understanding of the initial stages is required. We have also calculated an excitation
function of elliptic flow scaled by the initial eccentricity and found that the function
continuously increases with collision energy up to the LHC energy when hadronic
dissipation is taken into account.
Acknowledgments
The author would like to thank M. Gyulassy, U. Heinz, D. Kharzeev, R. Lacey
and Y. Nara for collaboration and fruitful discussions. He is also much indebted to
8 T. Hirano
T. Hatsuda and T. Matsui for continuous encouragement to the present work. He
also thanks M. Isse for providing him with a result from a hadronic cascade model
shown in Fig. 1. This work was partially supported by JSPS grant No.18-10104.
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http://arxiv.org/abs/nucl-th/0110037
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	Introduction
	A QGP fluid + hadronic cascade model
	Success of a hybrid approach
	Challenge for a hydrodynamic approach
	Elliptic flow at LHC
	Conclusions

  Recent development of a hydrodynamic model is discussed by putting an
emphasis on realistic treatment of the early and late stages in relativistic
heavy ion collisions. The model, which incorporates a hydrodynamic description
of the quark-gluon plasma with a kinetic approach of hadron cascades, is
applied to analysis of elliptic flow data at the Relativistic Heavy Ion
Collider energy. It is predicted that the elliptic flow parameter based on the
hybrid model increases with the collision energy up to the Large Hadron
Collider energy.

Introduction
One of the important discoveries made at the Relativistic Heavy Ion Collider
(RHIC) in Brookhaven National Laboratory is that the elliptic flow parameter,1)
namely, the second Fourier coefficient v2 = 〈cos(2φ)〉 of the azimuthal momentum
distribution dN/dφ,2) is quite large in non-central Au+Au collisions.3) Over the
past years, many studies have been devoted to understanding the elliptic flow from
dynamical models: (1) The observed v2 values near midrapidity at low transverse
momentum (pT ) in central and semi-central collisions are consistent with predictions
from ideal hydrodynamics.4) (2) The v2 data cannot be interpreted by hadronic cas-
cade models.5), 6) (3) A partonic cascade model7) can reproduce these data only with
significantly larger cross sections than the ones obtained from the perturbative cal-
culation of quantum chromodynamics. The produced dynamical system is beyond
the description of naive kinetic theories. Thus, a paradigm of the strongly cou-
pled/interacting/correlated matter is being established in the physics of relativistic
heavy ion collisions.8) The agreement between hydrodynamic predictions and the
data suggests that the heavy ion collision experiment indeed provides excellent op-
portunities for studying matter in local equilibrium at high temperature and for
drawing information of the bulk and transport properties of the quark-gluon plasma
(QGP). These kinds of phenomenological studies closely connected with experimen-
tal results, so to say, the “observational QGP physics”, will be one of the main
trends in modern nuclear physics in the eras of the RHIC and the upcoming Large
Hadron Collider (LHC). Then it is indispensable to sophisticate hydrodynamic mod-
eling of heavy ion collisions for making quantitative statements on properties of the
produced matter with estimation of uncertainties. In fact, the ideal fluid dynamical
description gradually breaks down as one studies peripheral collisions4) or moves
away from midrapidity.9), 10) This requires a more realistic treatment of the early
and late stages11)–14) in dynamical modeling of relativistic heavy ion collisions.
In this paper, recent studies of the state-of-the-art hydrodynamic simulations are
highlighted with emphases on the importance of the final decoupling stage (Sec. 3)
∗) e-mail address: hirano@phys.s.u-tokyo.ac.jp
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1699v1
2 T. Hirano
and of much better understanding of initial conditions (Sec. 4). A prediction of the
v2 parameter at the LHC energy will be made in Sec. 5. See also other reviews
15) to
complement other topics of hydrodynamics in heavy ion collisions at RHIC.
§2. A QGP fluid + hadronic cascade model
We have formulated a dynamical and unified model,13) based on fully three-
dimensional (3D) ideal hydrodynamics,9), 10) toward understanding the bulk and
transport properties of the QGP. During the fluid dynamical evolution one assumes
local thermal equilibrium. However, this assumption can be expected to hold only
during the intermediate stage of the collision. In order to extract properties of the
QGP from experimental data one must therefore supplement the hydrodynamic de-
scription by appropriate models for the beginning and end of the collision. In Sec. 3,
we employ the Glauber model for initial conditions in hydrodynamic simulations.
Initial entropy density is parametrized as a superposition of terms scaling with the
densities of participant nucleons and binary collisions, suitably generalized to ac-
count for the longitudinal structure of the initial fireball.13) Instead, in Secs. 4 and
5, we employ the Color Glass Condensate (CGC) picture17) for colliding nuclei and
calculate the produced gluon distributions18) as input for the initial conditions in
the hydrodynamical calculation.19) During the late stage, local thermal equilibrium
is no longer maintained due to expansion and dilution of the matter. We treat this
gradual transition from a locally thermalized system to free-streaming hadrons via
a dilute interacting hadronic gas by employing a hadronic cascade model20) below a
switching temperature of T sw=169 MeV. A massless ideal parton gas equation of
state (EOS) is employed in the QGP phase (T >Tc = 170 MeV) while a hadronic
resonance gas model is used at T <Tc. When we use the hydrodynamic code all the
way to final decoupling, we take into account10) chemical freezeout of the hadron
abundances at T ch = 170 MeV, separated from thermal freezeout of the momen-
tum spectra at a lower decoupling temperature T th, as required to reproduce the
experimentally measured yields.21)
§3. Success of a hybrid approach
Initial conditions in 3D hydrodynamic simulations are put so that centrality
and pseudorapidity dependences of charged particle yields are reproduced. A linear
combination of terms scaling with the number of participants and that of binary
collisions enables us to describe centrality dependence of particle yields at midra-
pidity. This agreement with the data still holds when the ideal fluid description is
replaced by a more realistic hadronic cascade below T sw. See also Fig. 3. When ideal
hydrodynamics is utilized all the way to kinetic freezeout, T th = 100 MeV is needed
to generate enough radial flow for reproduction of proton pT spectrum at midra-
pidity. One major advantage of the hybrid model over the ideal hydrodynamics is
that the hybrid model automatically describes freezeout processes without any free
parameters. The hybrid model works remarkably well in reproduction of pT spectra
for identified hadrons below pT ∼ 1.5 GeV/c.
Relativistic Hydrodynamics at RHIC and LHC 3
partN
0 50 100 150 200 250 300 350 400
QGP + hadron fluids
QGP fluid and hadron gas
Hadron gas
PHOBOS(hit)
PHOBOS(track)
-6 -4 -2 0 2 4 6
hydro+cascade
=100MeV, hydrothT
=169MeV, hydrothT
PHOBOS 25-50%
b=8.5fm
Fig. 1. (Left) Centrality dependence of v2. The solid (dashed) line results from a full ideal fluid
dynamic approach (a hybrid model). For reference, a result from a hadronic cascade model6)
is also shown (dash-dotted line). (Right) Pseudorapidity dependence of v2. The solid (dashed)
line is the result from a full ideal hydrodynamic approach with T th = 100 MeV (T th = 169
MeV). Filled circles are the result from the hybrid model. All data are from the PHOBOS
Collaboration.22)
The centrality dependences of v2 at midrapidity (| η |< 1) from (1) a hadronic
cascade model6) (dash-dotted), (2) a QGP fluid with hadronic rescatterings taken
through a hadronic cascade model (dashed), and (3) a QGP+hadron fluid with
T th = 100 MeV (solid) are compared with the PHOBOS data.22) A hadronic cascade
model cannot generate enough elliptic flow to reproduce the data. This is observed
also in other hadronic cascade calculations.5) Thus it is almost impossible to interpret
the v2 data from a hadronic picture only. Models based on a QGP fluid generate
large elliptic flow and gives v2 values which are comparable with the data. When
a hadronic matter is also treated by ideal hydrodynamics, v2 is overpredicted in
peripheral collisions. This is improved by dissipative effects in the hadronic matter.
Note that there could exist effects of eccentricity fluctuation,23) which is not taken
into account in the current approach. Deviation between the data and the QGP-
fluid-based results above Npart ∼ 200 could be attributed to these effects.
From the integrated elliptic flow data at midrapidity, initial push from QGP
pressure turns out to be important at midrapidity. In Fig. 1 (right), the pseu-
dorapidity dependence of v2 data in 25-50% centrality observed by PHOBOS
22) are
compared with QGP fluid models. Ideal hydrodynamics with T th = 169 MeV, which
is just below the transition temperature Tc = 170 MeV, underpredicts the data in
the whole pseudorapidity region. Hadronic rescatterings after QGP fluid evolution
generate the right amount of elliptic flow and, consequently, the triangle pattern of
the data is reproduced well. If the hadronic matter is also assumed to be described
by ideal hydrodynamics until T th = 100 MeV, v2 overshoots in forward/backward
rapidity regions (| η |∼ 4). This is simply due to the fact that, in ideal hydrody-
namics, v2 is approximately proportional to the initial eccentricity which is almost
independent of space-time rapidity. So the hadronic dissipation is quite important in
forward/backward rapidity regions as well as at midrapidity in peripheral collisions
(Npart < 100). From these studies, the perfect fluidity of the QGP is needed to
4 T. Hirano
obtain enough amount of the integrated v2, while the dissipation (or finite values
of the mean free path among hadrons) in the hadronic matter is also important to
obtain less elliptic flow coefficients when the multiplicity is small at midrapidity in
peripheral collisions and/or in forward/backward rapidity regions. This is exactly
the novel picture of dynamics in relativistic heavy ion collisions, namely, the nearly
perfect fluid QGP core and the highly dissipative hadronic corona, addressed in
Ref. 16)
 (GeV/c)Tp
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-0.05
πSTAR, 
STAR, K
STAR, p
Fig. 2. Transverse momentum dependence of v2 for pions, kaons and protons. Filled plots are the
results from the hybrid model. The impact parameter in the model simulation is 7.2 fm which
corresponds to 20-30% centrality. Data (open plots) for pions, kaons and protons are obtained
by the STAR Collaboration.24)
As a cross-check on the picture, we also study pT dependence of v2 for identi-
fied hadrons in semi-central collisions to see whether the hybrid model works. We
correctly reproduce mass ordering behavior of differential elliptic flow below pT ∼ 1
GeV/c as shown in Fig. 2. Here experimental data are from STAR.24) Although
we also reproduce the data in 10-20% and 30-40% well (not shown), it is hard to
reproduce data in very central collisions (0-5%) due to a lack of initial eccentricity
fluctuation in this model. It is worth mentioning that, recently, the hybrid model
succeeds to describes differential elliptic flow data for identified hadrons at forward
rapidity observed by BRAHMS.25)
§4. Challenge for a hydrodynamic approach
So far, an ideal hydrodynamic description of the QGP fluid with the Glauber
type initial conditions followed by an kinetic description of the hadron gas describes
the space-time evolution of bulk matter remarkably well. The CGC,17) whose cases
are growing both in deep inelastic scatterings and in d+Au collisions recently, is one
of the relevant pictures to describe initial colliding nuclei in high energy collisions. In
this section, novel hydrodynamic initial conditions19) based on the CGC are employed
for an analysis of elliptic flow.
We first calculate the centrality dependence of the multiplicity to see that the
CGC indeed correctly describes the initial entropy production and gives proper initial
conditions for the fluid dynamical calculations. Both CGC and Glauber model initial
Relativistic Hydrodynamics at RHIC and LHC 5
partN
0 50 100 150 200 250 300 350 400
=100MeVthGlauber+hydro, T
=100MeVthCGC+hydro, T
CGC+hydro+cascade
PHOBOS
Fig. 3. Centrality dependence of charged particle multiplicity per number of participant nucleons.
The solid (dashed) line results from Glauber-type (CGC) initial conditions. The dash-dotted
line results from our hybrid model. Experimental data are from PHOBOS.27)
conditions, propagated with ideal fluid dynamics, reproduce the observed centrality
dependence of the multiplicity,27) see Fig. 3. In the hydrodynamic simulations, the
numbers of stable hadrons below T ch are designed to be fixed by introducing chemical
potential for each hadron.10) On the other hand, the number of charged hadrons is
approximately conserved during hadronic cascades. So the centrality dependence of
charged particle yields is also reasonably reproduced by the hybrid approach.
In the left panel of Fig. 4 we show the impact parameter dependence of the
eccentricity of the initial energy density distributions at τ0 = 0.6 fm/c. We neglect
event-by-event eccentricity fluctuations although these might be important for very
central and peripheral events.23) Even though both models correctly describe the
centrality dependence of the multiplicity as shown in Fig. 3, they exhibit a signifi-
cant difference: The eccentricity from the CGC is 20-30% larger than that from the
Glauber model.13), 28) The situation does not change even when we employ the “uni-
versal” saturation scale29) in calculation of gluon production. The initial eccentricity
is thus quite sensitive to model assumptions about the initial energy deposition which
can be discriminated by the observation of elliptic flow. The centrality dependence
of v2 from the CGC initial conditions followed by the QGP fluid plus the hadron gas
is shown in Fig. 4 (right). With Glauber model initial conditions,4) the predicted
v2 from ideal fluid dynamics overshoots the peripheral collision data.
22) Hadronic
dissipative effects within hadron cascade model reduce v2 and, in the Glauber model
case, are seen to be sufficient to explain the data (Fig. 1 (left)).13) Initial conditions
based on the CGC model, however, lead to larger elliptic flows which overshoot the
data even after hadronic dissipation is accounted for,13) unless one additionally as-
sumes significant shear viscosity also during the early QGP stage. Therefore precise
understanding of the bulk and transport properties of QGP from the elliptic flow
data requires a better understanding of the initial stages in heavy ion collisions.
6 T. Hirano
b (fm)
0 2 4 6 8 10 12 14
no diffuseness
CGC (IC-e)
=0.85)αGlauber (
partN
0 50 100 150 200 250 300 350 400
hydro+cascade, CGC
hydro+cascade, Glauber
PHOBOS(hit)
PHOBOS(track)
Fig. 4. (Left) Impact parameter dependence of the eccentricity of the initial energy density dis-
tributions. The solid (dashed) line results from Glauber-type (CGC) initial conditions. The
dotted line assumes a box profile for the initial energy density distribution. (Right) Centrality
dependence of v2. The solid (dashed) line results from CGC (Glauber model) initial conditions
followed by ideal fluid QGP dynamics and a dissipative hadronic cascade. The data are from
PHOBOS.22)
§5. Elliptic flow at LHC
The elliptic flow parameter plays a very important role in understanding global
aspects of dynamics in heavy ion collisions at RHIC. It must be also important to
measure elliptic flow parameter at the LHC energy toward comprehensive under-
standing of the degree and mechanism of thermalization and the bulk and transport
properties of the QGP.
Figure 5 shows the excitation function of the charged particle elliptic flow v2,
scaled by the initial eccentricity ε, for Au+Au collisions at b = 6.3 fm impact pa-
rameter, using three different models: (i) a pure 3D ideal fluid approach with a
typical kinetic freezeout temperature T th = 100 MeV where both QGP and hadron
gas are treated as ideal fluids (dash-dotted line); (ii) 3D ideal fluid evolution for the
QGP, with kinetic freezeout at T th = 169 MeV and no hadronic rescattering (dashed
line); and (iii) 3D ideal fluid QGP evolution followed by hadronic rescattering below
T sw = 169MeV (solid line). Although applicability of the CGC model for SPS ener-
gies might be questioned, we use it here as a systematic tool for obtaining the energy
dependence of the hydrodynamic initial conditions. By dividing out the initial eccen-
tricity ε, we obtain an excitation function for the scaled elliptic flow v2/ε whose shape
should be insensitive to the facts that CGC initial conditions produce larger eccen-
tricities and the resulting integrated v2 overshoots the data at RHIC and also that
experiments with different collision system (Pb+Pb) will be performed at the LHC.
Figure 5 shows the well-known bump in v2/ε at SPS energies (
sNN ∼ 10GeV)
predicted by the purely hydrodynamic approach, as a consequence of the softening
of the equation of state (EOS) near the quark-hadron phase transition region,30) and
that this structure is completely washed out by hadronic dissipation,12) consistent
with the experimental data.31), 32) Even at RHIC energies, hadronic dissipation still
reduces v2 by ∼ 20%. The hybrid model predicts a monotonically increasing excita-
Relativistic Hydrodynamics at RHIC and LHC 7
tion function for v2/ε which keeps growing from RHIC to LHC energies,
12) contrary
to the ideal fluid approach whose excitation function almost saturates above RHIC
energies.30)
 (GeV)NNs
10 210 310 410
=100MeV
CGC+hydro, T
=169MeV
CGC+hydro, T
CGC+hydro+cascade
Au+Au Charged, b=6.3fm
LHCRHIC
Fig. 5. Excitation function of v2/ε in Au+Au collisions at b = 6.3 fm. The solid line results from
CGC initial conditions followed an ideal QGP fluid and a dissipative hadronic cascade. The
dashed (dash-dotted) line results from purely ideal fluid dynamics with thermal freezeout at
T th = 169MeV (100MeV).
§6. Conclusions
We have studied the recent elliptic flow data at RHIC by using a hybrid model
in which an ideal hydrodynamic treatment of the QGP is combined with a hadronic
cascade model. With the Glauber-type initial conditions, the space-time evolution of
the bulk matter created at RHIC is well described by the hybrid model. The agree-
ment between the model results and the data includes v2(Npart), v2(η), pT spectra
for identified hadron below pT ∼ 1.5 GeV/c and v2(pT ) for identified hadrons at
midrapidity and in the forward rapidity region. If the Glauber type initial condi-
tions are realized, we can establish a picture of the nearly perfect fluid of the QGP
core and the highly dissipative hadronic corona. However, in the case of the CGC
initial conditions, the energy density profile in the transverse plane is more “eccen-
tric” than that from the conventional Glauber model. This in turn generates large
elliptic flow, which is not consistent with the experimental data. Without viscous
effects even in the QGP phase, we cannot interpret the integrated elliptic flow at
RHIC. If one wants to extract informations on the properties of the QGP, a better
understanding of the initial stages is required. We have also calculated an excitation
function of elliptic flow scaled by the initial eccentricity and found that the function
continuously increases with collision energy up to the LHC energy when hadronic
dissipation is taken into account.
Acknowledgments
The author would like to thank M. Gyulassy, U. Heinz, D. Kharzeev, R. Lacey
and Y. Nara for collaboration and fruitful discussions. He is also much indebted to
8 T. Hirano
T. Hatsuda and T. Matsui for continuous encouragement to the present work. He
also thanks M. Isse for providing him with a result from a hadronic cascade model
shown in Fig. 1. This work was partially supported by JSPS grant No.18-10104.
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http://arxiv.org/abs/nucl-th/0110037
http://arxiv.org/abs/nucl-th/0305084
http://arxiv.org/abs/nucl-th/0605008
http://arxiv.org/abs/nucl-th/0702082
http://arxiv.org/abs/hep-ph/0303204
http://arxiv.org/abs/nucl-ex/0312008
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http://arxiv.org/abs/nucl-ex/0701078
	Introduction
	A QGP fluid + hadronic cascade model
	Success of a hybrid approach
	Challenge for a hydrodynamic approach
	Elliptic flow at LHC
	Conclusions
