Viscous hydrodynamics description of $φ$ meson production in Au+Au and Cu+Cu collisions

0.35 MB·English

by  A. K. Chaudhuri

Viscous hydrodynamics description of φ meson production in Au+Au and Cu+Cu
collisions
∗
A. K. Chaudhuri
Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata 700 064, India
(Dated: January 27, 2009)
In the Israel-Stewart’s theory of 2nd order dissipative hydrodynamics, we have simulated φ pro-
duction in Au+Auand Cu+Cu collisions at √s =200 GeV. Evolution of QGP fluid with viscos-
NN
ity over the entropy ratio η/s=0.25, thermalised at τi=0.2 fm, with initial energy density εi=5.1
GeV/fm3 explainstheexperimentaldataonφmultiplicity,integratedv2,meanpT,pT spectraand
ellipticflowincentralandmid-centralAu+Aucollisions. η/s=0.25isalsoconsistentwithcentrality
dependenceofφpT spectrainCu+Cucollisions. ThecentralenergydensityinCu+Cucollisions is
εi=3.48 GeV/fm3.
9
0 PACSnumbers: 47.75.+f,25.75.-q,25.75.Ld
0
2
n I. INTRODUCTION are too weak to continue the evolution. Using suitable
a algorithm (e.g. Cooper-Frye) information at the freeze-
J out can be converted into particle spectra and can be
7 Experiments in Au+Au collisions at RHIC [1, 2, 3, directlycomparedwith experimentaldata. Thus, hydro-
2 4], produced convincing evidences that in non-central dynamics,in anindirect way,cancharacterizethe initial
Au+Aucollisions,ahot,dense,stronglyinteracting,col- conditionofthemediumproducedinheavyioncollisions.
h] lective QCD matter is created. Whether the matter Hydrodynamics equations are closed only with an equa-
t can be characterized as the lattice QCD [5, 6] predicted tionofstate(EOS)andonecaninvestigatethepossibility
l- Quark-Gluon-Plasma(QGP)ornot, is still a questionof ofphase transitioninthe medium. Ahostof experimen-
c debate. For long,strangenessenhancementis considered tal data produced in Au+Au collisions at RHIC, at c.m.
u
as a signature of QGP formation [7]. In QGP environ- energy√s=200GeV,havebeensuccessfullyanalysedus-
n
[ ment, gg → ss¯ is abundant. If not annihilated before ing ideal hydrodynamics [12], with an equation of state
hadronisation, early produced strange and anti-strange
with 1st order confinement-deconfinement phase transi-
1 quarks will coalescein to strange hadrons and compared tion. Multiplicity, mean p , p -spectra, elliptic flow etc.
v to elementary pp collisions, strange particle production T T
of identified particles, are well explained in the ideal hy-
1
will be enhanced. Recently, STAR collaboration pub-
8 drodynamic model with QGP as the initial state. Ideal
lished their measurements of φ(ss¯) mesons in Au+Au
1 hydrodynamics analysis of the RHIC data indicate that
4 [8, 9] and in Cu+Cu [10] collisions. Both in Au+Au and in central Au+Au collisions, at the equilibration time
. Cu+Cu collisions, compared to pp collisions, φ meson τ 0.6 fm, central energy density of the QGP fluid is
1 i
0 productionisenhanced. However,itisuncertainwhether εi ≈30 GeV/fm−3 [12]. It may be mentioned that ideal
ornottheenhancementisduetoincreasedproductionin ≈
9 hydrodynamics description of data are not unblemished.
0 QGPorduetocanonicalsuppressionofstrangenessinpp p spectra or the elliptic flow are explained only up to
T
: collisions. STAR measurements of φ mesons in Au+Au transversemomentap 1.5GeV. Athigherp descrip-
v T T
collisions found to be compatible with a model based on ≈
i tiondeteriorates. Alsoidealhydrodynamicdescriptionto
X recombinationofthermalsquarks[11],strengtheningthe data gets poorer in peripheral collisions.
r belief that in Au+Au collisions,a robustthermalparton However, estimate of initial condition of the fluid can
a
source is created.
not be creditable unless dissipative effects are accounted
Relativistic hydrodynamics provides a convenient tool for. Unlike in ideal fluid evolution, where initial and
to analyse Au+Au collision data. It is assumed that in final state entropy remains the same, entropy is gener-
the collision a fireball is produced. Constituents of the ated in viscous evolution. Consequently, to produce a
fireballcollide frequently to establish localthermal equi- fixed final state entropy, viscous fluid require less ini-
libriumsufficientlyfastandafteracertaintimeτ ,hydro- tial energy density than an ideal fluid. QGP viscosity is
i
dynamics become applicable. If the macroscopic proper- quiteuncertain. Theoreticalestimatecoverawiderange,
ties of the fluid e.g. energy density, pressure, velocity η/s 0-1. String theory based models (ADS/CFT) give
≈
etc. areknownatthe equilibrationtimeτ , the relativis- a lower bound on viscosity of any matter η/s 1/4π
i ≥
tic hydrodynamic equations can be solved to give the [13]. In a perturbative QCD, Arnold et al [14] esti-
space-time evolutionof the fireballtill a givenfreeze-out mated η/s 1. In a SU(3) gauge theory, Meyer [15]
∼
conditionsuchthatinteractionsbetweentheconstituents gave the upper bound η/s<1.0,and his best estimate is
η/s=0.134(33) at T = 1.165T . At RHIC region, Naka-
c
muraandSakai[16]estimatedtheviscosityofahotgluon
gas as η/s=0.1-0.4. Attempts have been made to es-
∗E-mail:[email protected] timate QGP viscosity directly from experimental data.
2
Gavin and Abdel-Aziz [17] proposed to measure viscos-
0.20
ity from transverse momentum fluctuations. From the Au+Au@b=7 fm
existing data on Au+Au collisions, they estimated that e 0=30 GeV/fm3, t 0=0.6 fm
QGP viscosity as η/s=0.08-0.30. Experimental data on
Song & Heinz[0805.1756]
0.15
elliptic flow has also been used to estimate QGP viscos- py AZHYDRO-KOLKATA
o
ity. Ellipticflowscaleswitheccentricity. Departureform otr
s
the scaling can be understood as due to off-equilibrium ni
a
effectandutilisedtoestimateviscosity[18]as,η/s=0.11- m 0.10
u
0.19. Experimental observation that elliptic flow scales nt
e
with transverse kinetic energy is also used to estimate m
o
QGP viscosity, η/s 0.09 0.015 [19], a value close m 0.05
∼ ±
to the ADS/CFT bound. From heavy quark energy
loss, PHENIX collaboration [20] estimated QGP viscos-
ity η/s 0.1-0.16. 0.00
Inrec≈entyears,considerableprogresshasbeenmadein 0 2 4 6 8 10
numerical implementation of dissipative hydrodynamics ( t - t i) (fm)
[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. From
FIG. 1: Viscous fluid (η/s=0.08) simulation for temporal
the viscous hydrodynamic simulation of elliptic flow in
evolution of momentum anisotropy in b=7 fm Au+Au col-
Au+Aucollisions,LuzumandRomatschke[34]obtained
lision at RHIC. The solid line is the simulation result from
an upper bound to the ratio η/s< 0.4-0.5. In [35] Song
VISH2+1 [33] and the dashed line is the simulation result
and Heinz also argued that η/s < 0.4 is a robust upper
from AZHYDRO-KOLKATA.Initial condition of thefluid is
bound of QGP viscosity. In a recent paper [36], we have
very similar in both thesimulations.
estimated QGP viscosity as η/s 0.25. It was shown
≈
that φ mean p is sensitive to QGP viscosity. STAR
T
data [8, 9] on centrality dependence of φ mean p in
T
Au+Aucollisionsdefinitelyrejectidealfluidorfluidwith
viscosity η/s 0.08-0.16. Data are explained only with
η/s=0.25. In≤itial central energy density of the fluid is ∂ Tµν = 0, (1)
µ
εi=350.1GGeVeV/f,mm3ucinh ildesesalthhayndrothdaytnatmheicse.stiEmvaotleudtiovnaluoef Dπµν = 1 (πµν 2η <µuν>)
−τ − ∇
∼ π
viscous fluid (η/s=0.25) also explains the centrality de-
[uµπνλ+uνπνλ]Du . (2)
pendence of φ multiplicity, integrated v2, pT spectra up − λ
to 3 GeV.
≈
Purpose of the present paper is to show that viscos- Eq.1 is the conservation equation for the energy-
ity over entropy ratio η/s=0.25, is consistent with the momentum tensor, Tµν = (ε + p)uµuν pgµν + πµν,
−
STARmeasurementsofφellipticflowincentralandmid- ε, p and u being the energy density, pressure and fluid
central Au+Au collisions. It is also consistent with the velocity respectively. πµν is the shear stress tensor (we
recent STAR data on φ production in Cu+Cu collision haveneglectedbulkviscosityandheatconduction). Eq.2
at 200 GeV. The central energy density however, is less istherelaxationequationfortheshearstresstensorπµν.
in Cu+Cu collisions, εi=3.48 GeV/fm3. The paper is In Eq.2, D = uµ∂µ is the convective time derivative,
organisedasfollows: insectionII,webrieflydescribethe <µuν> = 1( µuν + νuµ) 1(∂.u)(gµν uµuν) is a
∇ 2 ∇ ∇ − 3 −
hydrodynamicalequationsusedtocomputetheevolution symmetric traceless tensor. η is the shear viscosity and
of ideal and viscous fluid. We have used a lattice moti- τ is the relaxation time. It may be mentioned that in
π
vated equation of state. Construction of the EOS is also a conformally symmetric fluid relaxation equation can
discussed in section II. Simulation results are discussed contain additional terms [33].
in section III. Summary and conclusions are given in
Assuming boost-invariance, Eqs.1 and 2 are solved in
section IV. (τ = √t2 z2,x,y,η = 1lnt+z) coordinates, with a
− s 2 t−z
code ”‘AZHYDRO-KOLKATA”’, developed at the Cy-
clotron Centre, Kolkata. Details of the code can be
II. HYDRODYNAMICAL EQUATIONS, foundin[27]. ToshowthatAZHYDRO-KOLKATAcom-
EQUATION OF STATE AND INITIAL putes the evolution correctly, in Fig.1, we have com-
CONDITIONS
pared the temporal evolution of momentum anisotropy
ε = <Txx−Tyy> of a QGP fluid with a calculation of
p <Txx+Tyy>
A. Hydrodynamical equations Song and Heinz [33]. Initial conditions are same for
both the simulations. Within 10% or less, AZHYDRO-
In the Israel-Stewart’s theory of 2nd order dissipative KOLKATA simulation reproduces Song and Heinz’s [33]
hydrodynamics, space-time evolution of the fluid is ob- result for temporal evolution of momentum anisotropy
tained by solving, ε .
p
3
8
sSB/T3 Au+Au
20 3) Cu+Cu
m
V/f 6 critical energy density
e
G
15 0.4 y (
3s/T 0.3 densit 4
10 gy
ep/ 0.2 er
n
e
0.1 al 2
5 ntr
0.0 Ce
0 1 2 3
e1/4[(GeV/fm3)1/4]
0 0
0.0 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 300 350 400
T (GeV) Npart
FIG. 2: Black circles are lattice simulation [6] for entropy
FIG.3: (coloronline)Theblackandredsolidlinesareinitial
density. The black line is the parametric representation to
centralenergydensityasafunctionofnumberofparticipants
the lattice simulations. In the inset, the solid and dashed
in Au+Au and Cu+Cu collisions. The dashed line shows
linesarethesquaredspeedofsoundinlatticebasedEOSand
thecriticalenergydensityfortheconfinement-deconfinement
in an EOS incorporating 1st order transition [12].
cross-over transition.
B. Equation of state
hadrongascomprisingalltheresonancesbelow2-3GeV.
Asthelatticesimulationcoverawidetemperaturerange
One of the most important inputs of a hydrodynamic
belowthe crossovertemperature,T =196(3)MeV, we
co
modelistheequationofstate(EOS).Throughthisinput
chooseto use the lattice basedEOS(Eq.3-5) bothin the
macroscopic hydrodynamic models make contact with
QGP and in the hadronic phase. The idea is to expose
the microscopic world. Most of the hydrodynamical cal-
the lattice simulation of EOS to experimental scrutiny.
culationsareperformedwithEOSwitha1storderphase
Indeed,lattice simulationsarevagueaboutthe natureof
transition. Huovinen [37] reported an ’ideal’ hydrody-
the confined(T <T ) phase. The confinedphase is cer-
co
namic simulation with 2nd order phase transition. He
tainlyunlikehadronicresonancegas. Thetraceanomaly
concluded that the experimental data (e.g. elliptic flow (ε 3p)/T4inthetemperaturerange140-200MeVisap-
of proton or antiproton) are better explained with EOS −
proximately 30% less than that of a hadronic resonance
with 1st order phase transition than with EOSwith 2nd
gas [6]. One also note that at low temperature, effective
order phase transition. However, lattice simulations [6]
degrees of freedom in the confined phase is g 2. In
h
indicate that confinement to deconfinement transition is ≈
contrast, in hadronic resonance gas, g 40. In the in-
h
a crossover,rather thana 1stor2ndorderphasetransi- setofFig.2,thesquaredspeedofsound(≥c2 p/ε)inthe
tion. It is then essential that hydrodynamic simulations lattice based EOS is compared with c2 insa≈n EOS with
s
aredonewithEOSwithcross-overtransitionratherthan
1st order phase transition [12], which model the quark
with EOS with 1st or 2nd order transition. In Fig.2, a
phase with bag model, and the hadronic phase by the
recent lattice simulation [6] for the entropy density is hadronicresonancegas. In1storderEOS,c2 fallsharply
s
shown. The solid line in Fig.2 is a parameterisation of
nearthecriticaltemperature. Thefallissmoothenedout
the entropy density.
incrossovertransition. LatticebasedEOSisalsosofter.
s T T
c
=α+[β+γT][1+tanh − ], (3)
T3 ∆T
C. Initial conditions
Fromtheparametricformoftheentropydensity,pres-
sure and energy density can be obtained using the ther-
Solution of partial differential equations (Eqs.1,2) re-
modynamic relations,
quires initial conditions, e.g. transverse profile of the
energy density (ε(x,y)), fluid velocity (v (x,y),v (x,y))
x y
T and shear stress tensor (πµν(x,y)) at the initial time τ .
i
p(T) = s(T)ds (4)
Z One also need to specify the viscosity (η) and the relax-
0
ation time (τ ). A freeze-out prescription is also needed
ε(T) = Ts p. (5) π
− to convert the information about fluid energy density
Generally, in hydrodynamic simulations, hadronic and velocity to particle spectra and compare with ex-
phase is approximated by a (non-interacting) resonance periment.
4
surements on φ mesons in Cu+Cu collisions. It is inter-
Y (fm)105 0-10% Au+Au 105 50-60% Au+Au einsittiinagl etonenrgoytedtehnastityεaisuc/aεlcieus ≈witAh1an/u3u/cAle1cau/r3.raAdipupsa.reWntiltyh,
0.05 boost-invariance,theinitialsystemisacylinderofradius
0 0 1.5 R=r0A1/3,infinitely extendedintherapiditydirection.
-5 0.015.5 -5 Phenomenologicalrelationεaiu/εciu ≈A1a/u3/A1cu/3 indicate
that at the freeze-out also the system can be approxi-
-10-10 -5 0 5 10-10-10 -5 0 5 10 mated again by a cylinder with radius R′ R.
10 10 Beforewecomparehydrodynamicsimula∝tionforφpro-
0-10% Cu+Cu 50-60% Cu+Cu
ductionwithexperiment,itisinterestingtocompareini-
5 5 tial central energy density in different centrality rages of
0.05
1.5 Au+Au and Cu+Cu collisions. In Fig.3, we have com-
0.67
0 0
pared the initial central energy density in Au+Au and
0.05
-5 -5 in Cu+Cu collisions as a function of participant num-
ber. TheblackandredlinesareforAu+AuandCu+Cu
-10-10 -5 0 5 10-10-10 -5 0 5 10 collisions respectively. In the region where they overlap,
X (fm) initialcentralenergydensity inAu+AuandCu+Cucol-
lisions are similar. One then expects that φ spectra in
FIG. 4: The top two panels show the energy density con-
mid central Au+Au and central Cu+Cu collisions will
toursin0-10% and50-60% centralityAu+Aucollisions. The
contoursaredrawnatε=1.5and0.05GeV/fm3. Theycorre- be similar. The expectation is fulfilled in STAR experi-
ment (e.g. φ p spectra in 40-50% Au+Au and 10-20%
spondtoconfinement-deconfinementcross-overandfreeze-out T
respectively. Thebottomtwopanelsshowtheenergydensity Cu+Cu collisions are nearly identical). In Fig.3, the
contours in Cu+Cu collisions. In 50-60% centrality Cu+Cu dashed line is the energy density (εco 1.5 GeV/fm3)
≈
collisions, initially QGP is not produced. for the confinement-deconfinement cross-over. In most
of the collisions, in the central region, initially the fluid
is produced in the deconfined state. However, energy
In [36], we assumed that the fluid is thermalised at density has a distribution, fluid in the central region is
τ =0.2fmandthe initialfluidvelocityis zero,v (x,y)= at higher density than the fluid at periphery. Thus in
i x
v (x,y) = 0. Initial energy density was assumed to be mid-central collisions, only a small portion of the fluid
y
distributed as [12] will be in the deconfined phase. In Fig.4, in four pan-
els, we have shown the contours of initial energy density
in 0-10% and 50-60% centrality Au+Au and in Cu+Cu
ε(b,x,y)=ε0[0.75Npart(b,x,y)+0.25Ncoll(b,x,y)], collisions. Contoursaredrawnatεco=1.5GeV/fm3 and
(6) εfo=0.05 GeV/fm3, corresponding to cross-over energy
where b is the impact parameter of the collision. N density and freeze-out. In 0-10% centrality Au+Au and
part
and N are the average participant and collision num- Cu+Cu collisions, initially, fluid in the central region is
coll
ber respectively. The shear stress tensor was initialised inQGPstate. ButfractionoffluidinQGPstateislarger
with boost-invariant value. For the relaxation time, we in Au+Au than in Cu+Cu collisions. One can immedi-
usedtheBoltzmannestimateτ =3η/4p. Thefreeze-out ately say that hard probe signature of QGP formation
π
was fixed at TF=150 MeV. In Eq.6, ε0 is a parameter will be less prominent in 0-10% centrality Cu+Cu col-
which does not depend on the impact parameter of the lisions than in 0-10% centrality Au+Au collisions. For
collision. Assuming thatη/s remainaconstantthrough- example,one canconjecture that in 0-10%Cu+Cu colli-
out the evolution, for a set of values η/s=0 (ideal fluid), sions J/ψ’s will be less suppressed than in a 0-10% cen-
0.08, 0.16 and 0.25, we fit ε0 to reproduce STAR mea- trality Au+Au collision. Experiments do vindicate the
surements of φ multiplicity in 0-5% centrality Au+Au conjecture [38, 39, 40] .
collisions. Centrality dependence of φ multiplicity and
φ mean p are simultaneously explained only with vis-
T
cosity over entropy ratio η/s=0.25. The corresponding III. RESULTS
centralenergydensity,inb=0Au+Aucollisionisε =5.1
i
GeV/fm3. A. centrality dependence of φ multiplicity, mean pT
For Cu+Cu collisions, we only change the central en- and integrated v2 in Au+Au and Cu+Cu collisions
ergy density, other parameters remain unchanged. The
initial energy density in Cu+Cu collisions is obtained In [36], we have shown the viscous (η/s=0.25) hydro-
by fitting φ multiplicity in central 0-10% Cu+Cu col- dynamics fit to the STAR data on φ multiplicity, mean
lisions. The fitted value corresponds to central energy p and integrated flow in Au+Au collisions. For com-
T
density ε =3.48 GeV/fm3 in b=0 Cu+Cu collisions. pleteness purpose, here also, we show the fits along with
i
As it will be shown below, QGP fluid, initialised with the fit obtained to Cu+Cu data. In Figs.5a and 5b, the
ε =3.48 GeV/fm3 reproduces most of the STAR mea- STAR measurements for the centrality dependence of φ
i
5
10 3 1.5
Au+Au@RHIC (b) Cu+Cu@RHIC (a) Au+Au (b) Cu+Cu
1.4
8 1.3
2 1.2
6 V) 1.1
dN/dy > (Ge 1.0
4 pT0.9
1 <
0.8
2
0.7
0.6
0 0
0 100 200 300 4000 20 40 60 80 100120 0.5
0 100 200 300 4000 50 100
Npart
Npart
FIG. 5: (a) Filled circles are STAR data on the centrality
FIG. 6: (a) Filled circles are STARdata [8] on centrality de-
dependenceofφmesonmultiplicity. Theblacklineisviscous
(η/s=0.25) hydrodynamic fit to the data. The initial time pendence of mean pT of φ mesons in Au+Au collisions. The
τi=0.2 fm, initial central energy density εi=5.1 GeV/fm3, black line is the fit obtained to the data in viscous hydrody-
namics. (b) same as in (a) but for Cu+Cu collisions.
freeze-outtemperatureisTF=150MeV.(b)sameasin(a)but
for Cu+Cu collisions. The initial energy density is εi=3.48
GeV/fm3.
0.14
(a) Au+Au (b) Cu+Cu
0.12
multiplicity (dN/dy) in Au+Au [8, 9] and Cu+Cu [10]
collisions are shown. In the region where N over- 0.10
part
lap, φ multiplicity is nearly identical in Au+Au and in 2
v
Cu+Cu collisions. In Fig.5a and ,b the solid lines are ed 0.08
at
the hydrodynamic predictions for φ multiplicity. Evo- egr 0.06
lution of viscous (η/s=0.25) QGP fluid thermalised at int
τ =0.2 fm and initialised with central energy density 5.1 0.04
i
GeV/fm3 in Au+Au collisions and 3.48 GeV/fm3 in
0.02
Cu+Cu collisions, reproduces the data in all the cen-
trality ranges of collisions. As indicated above, we have 0.00
used only the most central collision data (0-5% in case 0 100 200 300 4000 50 100
of Au+Au collisions and 0-10% in case of Cu+Cu col- Npart
lisions) to fix the initial energy density. Glauber model
FIG.7: (a)filledcirclesareSTARdataonintegratedelliptic
initialcondition(Eq.6)correctlyincorporatethe central-
flowinAu+Aucollisions. Theblacklineshowsthecentrality
itydependenceandφmultiplicityisreproducedinallthe
dependence of integrated v2. (b) centrality dependence of
centrality ranges of collisions.
integrated v2 in Cu+Cu collisions.
In Fig.6a and b, we have shown the STAR measure-
mentsofφmeanp inAu+Au[8,9]andCu+Cu[10]col-
T
lisions. Withintheerrorbars,centralitydependenceofφ
v2 is not measured yet. However, simulation results in-
mean p in Au+Au and in Cu+Cu collisions are nearly
T dicate that compared to Au+Au collisions, φ meson in-
identical (though central value is consistently higher in
tegrated flow is less in Cu+Cu collisions. The reason is
Au+Au collisions). The black lines in Fig.6a,b are fit to
understood. Initial eccentricity is small in Cu+Cu than
the data in viscous hydrodynamics. In viscous hydrody-
in Au+Au collisions. Elliptic flow has size dependence,
namics also,φ mean p do not showany appreciable de-
T smaller the system, less is the flow.
pendence on system size. STAR measurements of mean
p in Au+Au collisions are nicely reproduced. φ mean
T
p in Cu+Cu are reproduced within 10% or less.
T
Black lines in Fig.7a and b, are the viscous hydrody- B. φ pT-spectra in Au+Au and Cu+Cu collisions
namics predictions for the centrality dependence of inte-
grated v2 in Au+Au and Cu+Cu collisions. In Au+Au STAR measurements [8] for φ meson pT spectra in 0-
collisions,STARmeasuredintegratedv2in0-5%,10-40% 5%,5-10%,10-20%,20-30%,30-40%,40-50%,50-60%and
and 40-80% centrality collisions [8, 9]. Filled circles in 60-70%centralityAu+Au collisions areshowninFig.8a.
Fig.7a arethe STAR measurements. Exceptfor the very As noted by the STAR collaboration[8] φ p -spectra up
T
peripheral collision, hydrodynamic prediction is agree- to 30-40% centrality collisions are well fitted by an ex-
ment with STAR data. In Cu+Cu collisions, integrated ponential, indicating thermal production of φ in central
6
C. N(Ω)/N(φ) vs. pT
101
100 (a) Au+Au@RHIC (b) Cu+Cu@RHIC
10-1 STAR collaboration measured the transverse momen-
10-2 tumdependenceoftheratioN(Ω)/N(φ)inAu+Aucolli-
-2V)10-3 sions [8]. STAR measurementsof the ratioin 0-12%,20-
Ge10-4 40% and 40-60% centrality Au+Au collisions are shown
2p (T10-5 in the three panels, a,b,c of Fig.9. The ratio increases
dyd10-6 with p till p 3-4 GeV then drops. In peripheralcol-
dN/10-7 lisions,Tthe raTtio≈drops at lower p than in more central
10-8 T
10-9 collisions. Both Ω(sss) and φ(ss¯) are strange particles,
10-10 devoidofanynon-strangequarks. TheratioN(Ω)/N(φ)
10-11 can shed light on the production mechanism of strange
0 1 2 3 0 1 2 3 particles, specifically, strange baryon and mesons. The
pT (GeV)
ratio can also test a model. Correct reproduction of the
ratiowillindicatethatthestrangenesssectoriscorrectly
FIG.8: (color online) (a) STARdata[8]φmeson pT spectra
in 0-5%, 0-10%, 10-20%, 20-30%, 30-40%, 40-50%, 50-60% modeled. Models based on recombination of thermal
and 60-70% centrality Au+Aucollisions. The black lines are strange quarks [11] can reproduce the φ meson pT spec-
φ spectra from evolution of viscous fluid. (b) STAR data tra up to p = 5 GeV. The model [11] also reproduces
T
on pT spectra of φ in Cu+Cu collisions in 0-10%, 10-20%, the ratio N(Ω)/N(φ) up to pT 4 GeV, reproduces the
20-30%, 30-40%, 40-50% and 50-60% Cu+Cu collisions. The decreasing trend at p > 4 Ge≈V. But at p > 4 GeV,
T T
black lines are viscous hydrodynamicpredictions. the model largely over predict the ratio, indicating that
at large p , in recombination models Ω’s are more pro-
T
duced than in experiment. In Fig.9, the black lines are
hydrodynamic predictions for the ratio in Au+Au colli-
sions. Hydrodynamics predictions are shown up to p =
T
5 GeV. The ratio increases with p , and continue to in-
T
collisions. Exponentialfit worsenin moreperipheralcol- creaseevenatlargep . EvolutionofviscousQGPdonot
T
lisions. ALevyfunction(whichhasanexponentialshape reproducetheexperimentaltrendthattheratiodecreases
at low pT and power law shape at large pT) fits the pe- beyond a certain pT. It is not expected also. As noted
ripheral data. Apparently, in peripheral collisions non- earlier, φ spectra at p > 3 GeV are not reproduced.
T
thermal source contribute to φ meson production. The In p range p 3 GeV, viscous hydrodynamics appear
T T
blacklinesinFig.8aarepT spectrafromevolutionofvis- to under predict≤the ratio in pT range pT 3 GeV. For
cous QGP fluid. Except for very peripheral collisions, example in 0-12%centrality collisions, the≤ratio is under
data up to pT=3 GeV are well explained in viscous hy- predicted by 40%. In more peripheral collisions, the
drodynamics. At larger pT (not shownin Fig.8), viscous ratio is less u∼nder predicted. The result is interesting.
hydrodynamics under predict the pT-spectra. At large φ pT-spectra, up to pT=3 GeV, are well reproduced in
pT, other sources e.g. pQCD processes can contribute viscoushydrodynamics. Apparently,inviscoushydrody-
and hydrodynamic models may not be reliable.
namics, Ω’s are not produced in sufficient number. The
ratio N(Ω)/N(φ) is not measured in Cu+Cu collisions.
The blue lines in Fig.9 are the predictions for the ratio
STAR collaborationrecently published their measure-
inCu+Cucollisions. Theratioisnearlyidenticaltothat
ments for φ p -spectra in Cu+Cu collisions [10]. STAR
T
in Au+Au collisions. The ratioN(Ω)/N(φ) do not show
measurements [10] for φ p spectra in 0-10%, 10-20%,
T
any system size dependence.
10-20%, 20-30%, 30-40%, 40-50% and 50-60% centrality
Cu+Cu collisions are shown in Fig.8b. A Levy function
alsofitsthep spectrainCu+Cucollisions[10]. Forsim-
T
ilarN ,φspectrainAu+AuandCu+Cucollisionsare D. φ elliptic flow in Au+Au and Cu+Cu collisions
part
similar. The parameters of the Levy function in Au+Au
collision and Cu+Cu collision are also similar for nearly STAR collaborationmeasuredφ meson elliptic flow in
identical for participant numbers. Black lines in Fig.8b 0-5%, 10-40% and 40-80% and 0-80% (minimum bias)
are the predictions from viscous hydrodynamics. Here centrality Au+Au collisions [8]. STAR measurements
again,except for the very peripheral(50-60%)collisions, for the elliptic flow are shown in Fig.10. In Fig.10, the
data, up to p =3 GeV are well explained. The results black lines are the elliptic flow from evolution of viscous
T
indicate that φ pT spectra, up to pT= 3GeV, both in fluidwithcentralenergydensityε0=5.1GeV/fm3. Even
Au+Au and in Cu+Cu collisions are consistent with hy- though as mentioned earlier, viscous hydrodynamics is
drodynamic evolution of QGP fluid with viscosity over not reliable beyond p =3 GeV (φ p -spectra are un-
T T
entropy ratio η/s=0.25. The central energy density of der predicted), we have shown predictions for flow up
thefluidinb=0Au+Au(Cu+Cu)collisionsis 5.1(3.48) to p =5 GeV. In 0-5% collisions, elliptic flow is very
T
GeV/fm3 . ≈ small, and viscous fluid evolution reproduces the flow.
7
0.3 Blue lines in Fig.10, are the predictions for flow in 0-
(a)0-12% (b)20-40% (c) 40-60%
5%, 10-40% and 40-80%and minimum bias Cu+Cu col-
lisions. Comparedto Au+Aucollisions,elliptic flowis
∼
10% less in Cu+Cu collisions. It is consistent with our
0.2
predictions for integrated v2. Integrated v2 is also less
F)N( in Cu+Cu than in Au+Au collisions (see Fig.7). φ me-
/
W)N( sonellipticflowinCu+Cucollisionsarenotmeasuredyet
0.1 andthepredictionscannotbetestedagainstexperiment.
Future experiments can verify the predictions.
Present analysis indicate that QGP fluid, with viscos-
ity over entropy ratio η/s=0.25, is consistent with most
0.0 of the published STAR data on φ production in central
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
and mid-centralAu+Au and Cu+Cu collisions. The ini-
pT (GeV) tial central energy density of the fluid is 5.1 GeV/fm3
FIG.9: (coloronline)Filled circlesareSTARdataonthepT in Au+Au collisions and ≈ 3.48 GeV/f≈m3 in Cu+Cu
dependence of the ratio N(Ω)/N(φ) in Au+Au collisions in collisions. Present estimate of viscosity and initial en-
0-12%,20-40% and40-80% centralityAu+Aucollisions. The ergydensityareobtainedbyfitting experimentaldatain
black lines are the viscous hydrodynamic predictions for the a hydrodynamic model. Limitations of the model must
ratio in Au+Aucollisions. The blue lines are the predictions be discussed. We have neglected bulk viscosity. In gen-
for the ratio in Cu+Cu collisions. eral bulk viscosity is much smaller than shear viscosity.
However,recent lattice simulation [6] indicate that trace
anomalyisnon-zeronearthecross-overtemperature. Us-
0.3 (a) 0-5% (b) 1100--4400%% Au+Au ing the lattice data, Kharzeev et al [41] computed bulk
viscosity of QGP. Near the cross over temperature, bulk
w 0.2
o viscosity can be significantly large. Experimental data
ptic fl 0.1 include effect of both the shear and bulk viscosity. Ne-
elli glectingbulk viscositywillresultinto overestimatingthe
0.0
shear viscosity. Then η/s=0.25 is an upper bound of
-0.1 QGP viscosity. However, since bulk viscosity is appre-
ciable only near the cross over temperature, we do not
0.3 (c) 40-8400%-8 0% Au+Au (d) 0-80% expect substantial entropy production due to bulk vis-
0.2 al v2 cosity and estimate of shear viscosity will largely remain
nti unaltered.
0.1 e
er
0.0 diff
-0.1 IV. SUMMARY AND CONCLUSIONS
0 1 2 3 4 50 1 2 3 4 5
pT (GeV)
To summarise, in the Israel-Stewart’s theory of dissi-
pative hydrodynamics, we have simulated φ production
FIG. 10: (color online) Filled circles in panels a,b,c and d
from Au+Au and Cu+Cu collisions at √s=200 GeV. In
are STAR measurements for φ meson elliptic flow in 0-10%,
10-40% and 40-80% and 0-80% centrality Au+Au collisions. anearlierpublication[36],wehaveshownthattheSTAR
The black lines in the figure are the hydrodynamic model data on φ mean pT in Au+Au collisions is sensitive to
predictions for elliptic flow in Au+Au collisions. Blue lines viscosity and estimated QGP viscosity as η/s=0.25. For
are thepredicted flow in Cu+Cu collisions. η/s=0.25, QGP fluid, thermalised at τ =0.2 fm and ini-
i
tialised with central energy density 5.1 GeV/fm3, ex-
plain centrality dependence of φ mean p , multiplicity,
T
Elliptic flow in 10-40% collisions is also reproduced in integrated v2 and pT spectra (up to pT 3 GeV). It is
≈
the model. However, flow is largely over predicted in nowshownthatthe STAR dataonφ elliptic flowin cen-
40-80% centrality collisions. 40-80% centrality collisions tralandmid-centralAu+Aucollisionsarealsoexplained
approximatelycorrespondstob=11fmAu+Aucollision. in evolution of QGP fluid with viscosity to entropy ratio
Hydrodynamicmodelsarenotreliableatsuchperipheral η/s=0.25. η/s=0.25 is also consistent with STAR data
collisions. Interestingly, elliptic flow in 0-80% centrality on φ meson p spectra in Cu+Cu collisions. In Cu+Cu
T
collisionsisalsowellreproducedinviscoushydrodynam- collisions, central energy density is 3.48 GeV/fm3, 1.5
∼
ics. Considering that only the central energy density times less that in Au+Au collisions. We have given pre-
is fixed to reproduce φ multiplicity in 0-5% centrality dictions for elliptic flow in Cu+Cu collisions. Predicted
Au+Au collisions, reproduction of φ p spectra and el- elliptic flow in Cu+Cu collisions is 10% less than that
T
∼
liptic flow in central and mid central collisions can be in Au+Au collisions. In conclusion, STAR data on in-
considered as a great success of viscous hydrodynamics. termediate range p 3 GeV, φ meson production is
T
≤
8
Au+Au and Cu+Cu collisions is consistent with hydro- ratio η/s 0.25.
≈
dynamicevolutionofQGPfluidwithviscositytoentropy
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