Table Of ContentCharged particle’s pT spectra and elliptic flow in √sNN=200 GeV Au+Au collisions:
QGP vs. hadronic resonance gas
A. K. Chaudhuri∗ and Victor Roy†
Variable Energy Cyclotron Centre, 1-AF, Bidhan Nagar, Kolkata - 700 064, India
(Dated: January 6, 2011)
We show that if the hadronic resonance gas (HRG), with viscosity to entropy ratio η/s ≈0.24,
is physical at temperature T ≈220 MeV, charged particles pT spectra and elliptic flow in Au+Au
collisions at RHIC,overa widerange of collision centrality donot distinguish between initial QGP
fluidandinitialhadronicresonancegas. Unambiguousidentificationofbulkofthematterproduced
in Au+Au collisions require clear demonstration that HRG is unphysical at temperature T >200
MeV. It calls for precise lattice simulations with realistic boundary conditions.
1
ExperimentsatRelativisticHeavyIonCollider(RHIC) that hadrons retain their identity yet reproduce the ex-
1
0 produced convincing evidences that in central and mid perimentalmultiplicity. Inthe following,we showthatif
2 central Au+Au collisions, a hot, dense, strongly inter- the hadrons retain their identity at temperature T 220
≈
acting matter is created[1],[2],[3],[4]. Whether the mat- MeV, charged particles p spectra and elliptic flow in
n T
a ter canbe identified as the lattice QCD predictedQuark 0-40% Au+Au collisions, do not distinguishes between
J Gluon Plasma(QGP) or not is still a question of debate. initial HRG with viscosity to entropy ratio η/s 0.24
≈
5 The problem is closely related to quark confinement; and initial QGP with η/s=0.08. Unambiguous identifi-
quarks are unobservable and any information about the cation of the matter with QGP will require clear proof
] initial state has to be obtained from observed hadrons that HRG has a limiting temperature T 200 MeV or
h ≤
only. QGP, even if produced in Au+Au collisions, is a theconfinement-deconfinementtransitionoccuratatem-
t
- transient state, it expands, cools, hadronises, cools fur- perature 200 MeV.
cl ther till interactions between the hadrons become too ≤
u weak to continue the evolution. Hadronisation is a non- Most reliable information about the confinement-
n
perturbative process. Whether or not the hadronisation deconfinement transition temperature is obtained in lat-
[
process erases any memory of the constituent quarks is tice simulations of QCD. It is now established that the
2 uncertain. If the hadronisation process erases the mem- confinement-deconfinement transition is not a thermo-
v ory, fromthe observedhadrons one cannot comment on dynamic phase transition, rathera cross-over. Since it is
3 the initialQGPphase. PresentsearchforQGPatRHIC cross-over, there is no unambiguous temperature where
2
isonthepremisethatthehadronisationprocessdoesnot thetransitiontakeplace. InflectionpointofthePolyakov
2
5 erase the memory and from the observed hadrons, using loopis generallyquoted as the (pseudo)critical tempera-
. a dynamical model like hydrodynamics, one can back- ture for the confinement-deconfinement transition. Cur-
9
trace to the initial QGP phase. Hydrodynamic equa- rently, there is debate over the value of the pseudo crit-
0
tions are closed with an Equation of State (EOS) and ical temperature. HotQCD collaboration, with physical
0
1 one can incorporate the possibility of phase transition strange quark mass and somewhat larger than physical
: in the model. Indeed, a host of experimental data in u and d quark masses (ms/mu,d = 10) claimed that
v
Au+Au collisions at RHIC is well explained in a hydro- both the chiral and confinement-deconfinement tran-
i
X dynamical model with QGP in the initial state [5]. The sition takes place at a common temperature Tc =
r alternative,namelyhadronicresonancegas(HRG)inthe 192(4)(7). [12],[13],[14],[15],[16]. For more physical
a initialstate generallygive poorerdescriptionto the data u and d masses (ms/mu,d=20), the critical tempera-
[6], that too with HRG at very high temperature, T ture shift by 5 MeV to lower side [17]. Wuppertal-
270 MeV. At such high temperature hadrons densityi ≈is Budapest colla∼boration, with physical strange and light
verylargeρhad 4fm−3anditisdifficulttobelievethat quark masses (ms/mu,s = 28), on the other obtained
they retain thei∼r individual identity. However, situation a different result. [18],[19], [20], [21]. Chiral transi-
changes if viscous effects are included. Model calcula- tion temperature Tc=157 (3)(3) MeV is 20 MeV less
∼
tions[7],[8],[9],[10]indicatethatforahadronicresonance thanthe confinement-deconfinementtransitiontempera-
gas,viscositytoentropyratioisconsiderablylargerthan ture Tc=170(4)(4) MeV. Several other observables, e.g.
the ADS/CFT limit, η/s 1/4π [11]. Since entropy is quarksusceptibility,pressureetcinWuppertal-Budapest
generatedduringevolution≥,unlikean’ideal’HRG,a’vis- simulations is also shifted by 20-30 MeV compared to
cous’HRGcanbeinitializedatalowertemperaturesuch HotQCD. The 20 MeV shift in the confinement-
∼
deconfinementtransitiontemperatureisnotproperlyun-
derstood. Possiblereasonscouldbelargerphysicalquark
masses in HotQCD simulations. Also, one notes that
∗Electronicaddress: [email protected] though both the collaborations use staggered fermions,
†Electronicaddress: [email protected] the fermion actions are different. It is still unproven
thatinthecontinuumlimit,staggeredfermionsrepresent
2
QCD. The difference in the pseudo critical temperature changes to that of the hadronic resonance gas.
may also be due to different ’staggered’ fermion action Solution of Eq.1 and 2 require initial energy density,
in the two simulations. Additionally, lattice results for velocitydistributioninthe transverseplaneattheinitial
the transition temperature can be up by 30 MeV. All time. A freeze-out prescription is also needed. In the
∼
the lattice simulation uses periodic boundary condition, PT scenario, we assume that QGP fluid is thermalised
while in a realistic situation e.g. heavy ion collisions, in the time scale τ =0.6 fm [5]. Compared to QGP, a
i
the deconfined region is bordered by a confined phase. hadronic resonance gas is expected to thermalise late.
Exploratory quenched study suggests that with realistic We assume the canonical value, τ =1 fm as the initial
i
boundaryconditiontransitiontemperaturecanbe upby time in the NPT scenario. Both in the PT and NPT
30 MeV [22]. Considering the uncertainties associated scenario,initialfluidvelocityisassumedtobezero. Inan
with lattice simulations, the pseudo critical temperature impact parameter b collision, the initial energy density
for the confinement-deconfinement transition tempera- is assumed to be distributed as [5],
ture could be as high as T 220 MeV. Qualitatively,
c
≈
for hadron size 0.5 fm, limiting hadron density (such
that hadrons ar≈e not overlapped) is ρhad = 1/V 2 ε(b,x,y)=ε0[(1 f)Npart(b,x,y)+fNcoll(b,x,y)] (3)
fm−3. For HRG, ρ 2fm−3 corlrimesiptonds to l≈im- −
iting temperature T had=≈220 MeV, value close to the where Npart(b,x,y) and Ncoll(b,x,y) are the transverse
limit
profile of participant numbers and binary collision num-
highest possible pseudo critical temperature, as argued
bers. f inEq.3isthefractionofhardscattering. Mostof
above. T 220 MeV is possibly the highest temperature
≈ the hydrodynamic simulations are performed with hard
atwhich hadronscanretaintheir individual identity. At
scattering fraction f=0.25 or 0.13 [5],[29]. We assumed
higher temperature hadrons will overlapextensively and
f = 0.25 in PT scenario. Hard scattering fraction is as-
lose their identity.
In the following we simulate Au+Au collisions in two sumed to be zero in the NPT scenario. ε0 in Eq.3 is the
central energy density of the fluid in impact parameter
scenarios,(i)no phasetransition(NPT) scenariowhena
b=0 collision. In the PT scenario, for a fixed freeze-
HRGwithlimitingtemperatureT=220MeVisproduced
in the initial collisions and (ii) a phase transition (PT) out temperature TF=150 MeV, ε0 is varied to best re-
produce experimental charged particles p spectra in 0-
scenario when QGP is produced. Space-time evolution T
10%Au+Au collision. Inthe NPT scenario,we usedthe
of HRG/QGP fluid is obtained by solving,
limiting value, ε0=5.1 GeV/fm3, corresponding to cen-
tral temperature T =220 MeV. HRG initialised at cen-
i
∂ Tµν = 0, (1) tralenergydensityε0=5.1GeV/fm3 donotproducead-
µ
equate number of hadrons from the freeze-out surface
1
Dπµν = (πµν 2η <µuν>) at TF=150 MeV. We lower the freeze-out temperature
−τ − ∇
π toT =110MeV.Dissipativehydrodynamicsalsorequire
F
[uµπνλ+uνπνλ]Duλ. (2) initialisation of the shear stress tensors πµν. In both
−
the scenarios, we initialise πµν at the boost-invariant
Eq.1 is the conservation equation for the energy- values, πxx = πyy = 2η(x,y)/3τ , πxy = 0 [27]. For
i
momentum tensor, Tµν = (ε + p)uµuν pgµν + πµν, the relaxation time τ , we use the Boltzmann estimate,
− π
ε, p and u being the energy density, pressure and fluid τ =6η/4p.
π
velocity respectively. πµν is the shear stress tensor (we Results of our simulations for charged particles p
T
haveneglectedbulkviscosityandheatconduction). Eq.2 spectra and elliptic flow in 0-40% Au+Au collisions are
istherelaxationequationfortheshearstresstensorπµν. shown in Fig.1 and 2. We have assumed that through
In Eq.2, D = uµ∂µ is the convective time derivative, out the evolution, viscosity to entropy ratio remains a
∇<µuν> = 12(∇µuν + ∇νuµ) − 13(∂.u)(gµν − uµuν) is constant. InPTscenario,wehavesimulatedAu+Aucol-
a symmetric traceless tensor. η is the shear viscosity lisions for four values of viscosity, (i)η/s=0 (ideal fluid)
and τπ is the relaxation time. It may be mentioned (ii)η/s = 1/4π 0.08 (ADS/CFT limit), (iii)η/s=0.12
that in a conformally symmetric fluid relaxation equa- and (iv)η/s=0.1≈6. Corresponding central energy densi-
tion can contain additional terms [23]. Assuming longi- ties are ε0=35.5,29.1, 25.6 and 20.8 (GeV/fm3) respec-
tudinal boost-invariance, the equations are solved with tively. In the NPT scenario, we simulate Au+Au colli-
the code ’AZHYDRO-KOLKATA’ [24],[25],[26],[27] in sions for five values of viscosity, η/s=0, 0.08, 0.12, 0.24
(τ =√t2 z2,x,y,η = 1lnt+z) coordinates. and 0.30.
− 2 t−z
Hydrodynamic equations (Eq.1 and 2) are closed only In Fig.1, we have compared simulated charged par-
with an equation of state (EOS) p = p(ε). In the NPT ticles p spectra in 0-40% Au+Au collisions with the
T
scenario, we use EOS for the non-interacting hadronic PHENIX data [30]. The left panels (a-d) show the fit
resonance gas, comprising all the resonances with mass to the data in the NPT scenario. The solid lines from
m 2.5 GeV. In the PT scenario, we use a lattice bottom to top corresponds to η/s=0, 0.08, 0.16, 0.24
res
≤
based EOS [28], where the high temperature phase is and 0.30 respectively. Simulation results in the PT sce-
modeled by the recent lattice simulation [13]. At the nario are shown in the right panel (e-h). The lines from
cross-overtemperatureT =196MeV,theEOSsmoothly bottom to top are for viscosity to entropy ratio η/s=0,
co
3
-2V)102 (a) 0-10% (e) 0-10% 0.4 (a) 0-10% (e) 0-10%
e v(BBC}
G101 0.3 2
y (100 )T v2{ZDC-SDM}
2N/dpdT1100--21 v (p200..12 v2{2}
d
10-3 0.0
102 (b) 10-20% (f) 10-20% 0.4 (b) 10-20% (f) 10-20%
101 0.3
100
0.2
10-1
10-2 0.1
10-3 0.0
102 (c) 20-30% (g) 20-30% 0.4 (c) 20-30% (g) 20-30%
101
0.3
100
0.2
10-1
10-2 0.1
10-3
0.0
(d) 30-40% (h) 30-40%
102 0.4 (d) 30-40% (h) 30-40%
101
0.3
100
10-1 0.2
10-2 0.1
10-3
0.0
0 1 2 30 1 2 3
p (GeV) 0 1 2 30 1 2 3
T p (GeV)
T
FIG. 1: PHENIX data [30] for charged particles pT spectra FIG. 2: (color online) same as in Fig.1 but for elliptic flow.
in0-40%Au+Aucollisions arecomparedwithhydrodynamic
simulationsinNPT(theleftpanel)andPTscenario(theright
panel). Seethetext for details.
gas, η/s=0.24-0.30 [7]. In the PT scenario also, best fit
to the 0-40%data is obtained in viscous QGP evolution,
0.08, 0.12 and 0.16 respectively. One observes that ideal with η/s=0.08, when (χ2/N) =5.6. Comparable fit
min
HRG,initialisedtocentraltemperatureT =220MeV,do
i is obtained with QGP viscosity η/s=0.12. Quantatively,
not explain the p spectra, data are largely under pre-
T with the initial conditions as used here, charged parti-
dicted. Discrepancy between simulated spectra and ex-
cles p spectra in 0-40% Au+Au collisions are better
T
periment diminishes with increasing viscosity and data
explained with HRG than QGP in the initial state.
are best explained with η/s=0.24. Description to the
In fig.2, we have compared simulated elliptic flow in
data deteriorates if viscosity is further increased. In the
0-40% Au+Au collisions with PHENIX measurements
PTscenarioalso,bestfittothe0-40%dataisobtainedin
viscous QGP evolution, with η/s=0.08-0.12. Ideal QGP [31]. PHENIX collaboration obtained v2 from two inde-
pendent analysis, (i) event plane method from two inde-
or QGP fluid with η/s>0.12 give comparatively poorer
description. To be quantative about the fit to the data pendent subdetectors, v2{BBC} and v2{ZDC−SMD}
in two different scenarios, we have computed χ2 values and (ii) two particle cumulant v2{2}. All the three mea-
for the fits obtained to the 0-40% data, surements of v2 are shown. They agree within the sys-
tematic errors. Simulated flows in the NPT scenario are
shown in the left panels, the lines (top to bottom) cor-
χ2/N = 1 X(ex(i)−th(i))2 (4) respond to HRG with η/s=0,0.08, 0.16, 0.24 and 0.3 re-
N err(i)2 spectively. Flowsin the PTscenariowith QGPviscosity
i
η/s=0, 0.08, 0.12 and 0.16 are shown in the right pan-
We include both the statistical and systematic errors els. For elliptic flow also, we have computed χ2 values
in the analysis. The χ2/N values are shown in Fig.3a. for the fits. The results are shown in Fig.3b. In the
In the NPT scenario, χ2/N as a function of η/s shows PT scenario, best fit to the 0-40% charged particles el-
a minima at η/s=0.24, when (χ2/N) =3.3. It is very lipticflowdataisobtainedwithQGPviscosityη/s=0.08
min
interesting to note that the η/s=0.24,obtainedfromthe ((χ2/N) =6.7). In the NPT scenario, the best fit is
min
analysis is in close agreement with theoretical estimates obtained with HRG with viscosity η/s=0.24, minimum
of viscosity to entropy ratio of a hot hadronic resonance (χ2/N) =11.1. Note that exactly, at these values of
min
4
150 (pa)- sc p2/eNc tfroar (b) c 2/N for (c) c 2/N for -2V)102 (a) 0-10% (e) 0-10% 0.4
T iinnii.. HQRGGP elliptic flow +peTl-lispptiecc ftlroaw y (Ge110001 0.3v2
d 0.2
pT10-1
100 2N/d10-2 0.1
N d10-3 0.0
c2/ 102 (b) 10-20% (f) 10-20% 0.4
101 0.3
50 100
0.2
10-1
0.1
10-2
10-3 0.0
0 102 (c) 20-30% (g) 20-30% 0.4
0.0 0.1 0.2 0.30.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 101 0.3
h /s 100
0.2
10-1
FIG. 3: The filled circles and squares are χ2/N for (a)pT 10-2 0.1
spectra,(b)ellipticflowand(c)pT spectra+ellipticflow,in0- 10-3 0.0
40%Au+Aucollisions, inPT(initialQGP)andNPT(initial 102 (d) 30-40% (h) 30-40% 0.4
HRG) scenarios. 101 0.3
100
0.2
10-1
0.1
viscosity, p spectra are also best explained. However, 10-2
T
as opposed to the p spectra, elliptic flow is better ex- 10-3 0.0
T 0 1 2 30 1 2 3
plained with QGP in the initial state rather than with p (GeV)
T
initial HRG. If both the spectra and elliptic flows are
analysed together, minimum χ2 is similar in both the FIG. 4: (color online) In left panels (a)-(d) charged particles
scenarios, χ2min ≈6.1 in the PT scenario and ≈7.2 in pT spectrain0-40%Au+Aucollisionsarecomparedwithhy-
the NPT scenario (see Fig.3c). It can not be claimed drodynamical simulations with initial HRGand initial QGP.
thatexperimentaldataarebetterexplainedwithQGPin The solid and dashed lines corresponds to evolution of HRG
the initial state. Quantitatively, both the scenarios give with viscosity to entropy ratio η/s=0.5, thermalised at τi=3
fm, and QGP with viscosity to entropyratio η/s=0.08, ther-
nearlyidenticaldescriptiontothedata. Hydrodynamical
evolution of initial QGP fluid with viscosity to entropy malisedasτi=0.6fm. Inpanels(e)-(h)resultsforellipticflow
are shown.
ratio η/s=0.08, thermalised in the time scale τ =0.6 fm
i
to central energy density ε0=29.1 GeV/fm3 and that of
HRG with η/s=0.24, thermalised in the time scale τ =1
i
fmtocentralenergydensityε0=5.1GeV/fm3givenearly toRHICdataonpT spectraandellipticflowwillhowever
equivalent description to the PHENIX data for charged be changed if hadronic resonance gas is physical only at
particles pT spectra and elliptic flow in 0-40% Au+Au a lower temperature T 200 MeV. Even with large vis-
≤
collisions. We have not explored all possible initial con- cosity, η/s=0.5, hydrodynamical evolution of HRG, ini-
ditions in PT/NPT scenario. However, as the minimum tialisedatcentraltemperatureT =200MeV,do notpro-
i
χ2/N value in NPT/PT scenarios is reasonably small, duce comparable number of hadrons as in experiment,
it is unlikely that with a different initial condition, data unless initial or the thermalisation time is large τ 3
i
could be much better explained in the PT scenario, as fm. In Fig.4, the solid lines are simulation results ≥for
longasHRGatcentraltemperatureT =220MeVisphys- charged particles p spectra and elliptic flow with vis-
i T
ical. cous (η/s=0.5) HRG with central temperature T =200
i
One may argue that whether or not QGP is produced MeV, thermalised at the time scale τ = 3 fm. For com-
i
do not depend only on the p spectra and elliptic flow. parison, we have also shown in the simulations results
T
Jet quenching, high p suppression etc. do give addi- with minimally viscous (η/s=0.08) QGP at the initial
T
tional support to the claim of QGP formation at RHIC state(the dashedlines). Onenote thatin0-10%and10-
energy collisions. However, high p suppression or jet 20% collisions, HRG with central temperature T =200
T i
quenching phenomena samples only a restricted phase MeV,thermalisedinthetimescaleτ =3fm,givecompa-
i
space, as they are associated with high p trigger. Bulk rable description to the experimental charged particles
T
of the matter are at low p . To claim that QGP is pro- p spectra, as obtained with QGP in the initial state.
T T
duced in bulk, it is essential that charged particles p In 20-30% and 30-40% collisions however, data are bet-
T
spectraandellipticflowareexplainedinahydrodynamic ter explained with QGP in the initial state. Similarly
model. for the elliptic flow also. χ2 analysis also indicate that
The present result that if HRG is physical at T=220 charged particles p spectra and elliptic flow in 0-40%
T
MeV, NPT and PT scenario give equivalent description Au+Au collisions are better expalined with initial QGP
5
than with initial HRG with central temperature T =200 state, though possibly can not be neglected, can not be
i
MeV. χ2/N 22 with HRG at the initial state, factor of estimatedeither. The highlynon-equilibriumstate,with
≈
3 larger than that the value obtained with minimally unknown contribution to particle production make the
∼
viscous QGP in the initial state. Apparently, if HRG HRGmodelwithlimitingtemperatureT =200MeV,un-
i
is physical only at temperature T 200 MeV, PHENIX tenable. One can possibly exclude initial HRG if HRG
≤
data are better explained with initial QGP state than is physical only at temperature T 200 MeV. Though,
≤
initial HRG state. Large thermalisation time τ =3 fm, presentlattice simulationsindicate thatHRGis physical
i
also raises the issue of non-equilibrium contribution to only at 200 MeV, as discussed earlier, simulations are
≤
the particle production. Note that in dissipative hy- notpreciseenoughtoexclude HRGatT 220MeV.For
≈
drodynamics, τ is the time required by the system to confirmatory identification of matter produced in RHIC
i
achievenearequilibration. Largeτ implythatforasub- Au+AucollisionswithQGP,itisessentialtoexcludethe
i
stantialtime, system remainin a highly non-equilibrium HRG scenario. Lattice simulations with realistic bound-
state,whichcannotbemodeledtheoretically,andcontri- ary conditions, preferably with Wilson fermion actions,
bution to particle production from this non-equilibrium are urgently needed to exclude possible HRG scenario.
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