Multiplicity, mean $p_T$, $p_T$-spectra and elliptic flow of identified particles in Pb+Pb collisions at LHC

0.27 MB·English

by  A. K. Chaudhuri

Multiplicity, mean p , p -spectra and elliptic flow of identified particles in Pb+Pb
T T
collisions at LHC
∗
A. K. Chaudhuri
Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata 700 064, India
(Dated: January 1, 2009)
Israel-Stewart’s causal theory of dissipative hydrodynamics, with the ADS/CFT lower limit of
shearviscositytoentropyratio(η/s=0.08), giveconsistent descriptionofanumberofexperimental
observablesinAu+AucollisionsatRHIC(c.m. energy√s=200GeV)[1]. AssumingthatinPb+Pb
collisionsatLHC(c.m. energy√s=5.5TeV),exceptfortheinitialtemperature,otherparametersof
thefluidremainunchanged,wehavepredictedforthecentralitydependenceofmultiplicity,meanpT,
pT-spectra,ellipticflow. ThecentraltemperatureofthefluidisadjustedtoTi=421MeVsuchthat
inaPb+Pbcollision, withparticipantnumberNpart=350,averagechargeparticlemultiplicityis
∼
900 and is consistent with the experimental trend observed at lower energies. Compare to Au+Au
9
0 collisions at RHIC, in Pb+Pb collisions at LHC, on the average, particle multiplicity increases by
0 a factor of 1.6 , the mean pT is increased by 10% only. The elliptic flow on the other hand
∼ ∼
2 decreases by 15%.
∼

n
PACSnumbers: 47.75.+f,25.75.-q,25.75.Ld
a
J

1 I. INTRODUCTION hydrodynamics analysis of the RHIC data indicate that

in central Au+Au collisions, at the equilibration time
]
-th 5],EpxrpoedriumceedntscoinnviAnuci+ngAuevcidolelniscieosnsthaattRinHInCon[2-c,e3n,tr4a,l τεii ≈≈300.6GfemV,/fcemnt−r3al[7e]n.erItgymdaeynbsietymoefntthioeneQdGtPhafltuiiddeaisl
l hydrodynamics description of data are not unblemished.
c Au+Aucollisions,ahot,dense,stronglyinteracting,col-
p spectra or the elliptic flow are explained only up to
u lective QCD matter is created. Whether the matter can T
n be characterizedasthelattice QCD[6]predictedQuark- transversemomentapT ≈1.5GeV. AthigherpT descrip-
[ tiondeteriorates. Alsoidealhydrodynamicdescriptionto
Gluon-Plasma(QGP)ornot,isstillaquestionofdebate.
data gets poorer in peripheral collisions.
3 Relativistichydrodynamicsprovidesaconvenienttoolto
Hydrodynamics implicitly assume local thermal equi-
v analyse Au+Au collision data. It is assumed that in the
3 collision a fireball is produced. Constituents of the fire- libration. In the rest frame of the fluid, particle mo-
4 ball collide frequently to establish local thermal equilib- menta are isotropic. But at the early stage of the evo-
6 riumsufficientlyfastandafteracertaintimeτ ,hydrody- lution, assumption of isotropic momentum distribution
0 namics become applicable. Ifthe macroscopiciproperties cannotbeveryaccurate. Systemcouldonlybeinpartial
.
3 ofthefluide.g. energydensity,pressure,velocityetc. are equilibration and for a creditable analysis of the exper-
0 knownattheequilibrationtimeτ ,therelativistichydro- imental data, dissipative effects must be accounted for.
8 dynamic equations can be solvedito give the space-time Shear viscosity is the most important dissipative effect
0
in heavy ion collisions. Shear viscosity of QGP matter
evolution of the fireball till a given freeze-out condition
:
v such that interactions between the constituents are too is quite uncertain. In recent years, string theory moti-
i vated calculations [8] indicated that shear viscosity over
X weaktocontinuetheevolution. Usingsuitablealgorithm
entropy ratio of any matter is bounded from the lower
(e.g. Cooper-Frye) information at the freeze-out can be
r
a converted into particle spectra and can be directly com- side, η/s 1/4π. It is then expected that at the mini-
≥
mum,shearviscosityoverentropyofQGPmattershould
pared with experimental data. Thus, hydrodynamics,
be 1/4π.
in an indirect way, can characterize the initial condition
of the medium produced in heavy ion collisions. Hy- In recent years there has been significant progress in
drodynamics equations are closed only with an equation numerical implementation of viscous dynamics [1, 9, 10,
of state and one can investigate the possibility of phase 11, 12, 13, 14, 15, 16, 17, 18, 19] . At the Cyclotron
transition in the medium. Centre,Kolkata,wehavedevelopedacode”AZHYDRO-
KOLKATA”, to solve Israel-Stewart’s 2nd order theory
A host of experimental data produced in Au+Au col-
for dissipative hydrodynamics in 2+1 dimensions. In a
lisions at RHIC, at c.m. energy √s=200 GeV, have
recent publication [1], it was shown that minimally vis-
beensuccessfullyanalysedusingidealhydrodynamics[7].
cous hydrodynamics (η/s=0.08), with QGP as the ini-
Multiplicity, mean p , p -spectra, elliptic flow etc. of
T T
tialstate consistentlyexplaina largepartofRHIC data,
identified particles, are well explained in the ideal hy-
e.g. p -spectra of identified particles, elliptic flow in
drodynamic model with QGP as the initial state. Ideal T
minimum-bias/mid-centralcollisionsetc. Indeed,thede-
scription is better than that obtained in ideal dynamics,
particularly at large p .
T
∗E-mail:[email protected] Innottoodistantfuture,LargeHadronCollider(LHC)
2
is expected to be operational. It is being planned to col-
lide Lead beams at c.m. energy √s=5.5 TeV, which is
27timeslargerthantheRHICenergycollisions. Oneex- ∂ Tµν = 0, (1)
µ
pectsthatconclusiveevidenceforQGPformationcanbe 1
obtainedinPb+PbcollisionsatLHC. Inthe presentpa- Dπµν = (πµν 2η <µuν>). (2)
−τ − ∇
π
per,intheframeworkofminimallyviscoushydrodynam-
ics, we have given predictions for several experimental Eq.1 is the conservation equation for the energy-
observables,whichwill be measuredin Pb+Pbcollisions momentum tensor,Tµν =(ε+p)uµuν pgµν+πµν, ε, p
atLHCenergy. Itwillbeshownthatevenif,fromRHIC and u being the energy density, pressur−e and fluid veloc-
to LHC collisions, energy is increased by a factor of 27, ity respectively. Eq.2 is the relaxation equation for the
the experimental observables e.g. multiplicity, pT spec- shear stress tensor πµν. In Eq.2, D = uµ∂µ is the con-
tra, mean pT, elliptic flow etc. do show rather modest vective time derivative, <µuν> = 1( µuν + νuµ)
variation from RHIC energy collisions. The reason can 1(∂.u)(gµν uµuν) is a∇symmetric 2tra∇celess te∇nsor. −η
beunderstoodalso. Multiplicityorentropyofthesystem 3 −
is the shear viscosity and τ is the relaxation time. In
π
increases logarithmically with energy. Thus, from RHIC [1] Eqs.1 and 2 are solved in (τ = √t2 z2,x,y,η =
s
to LHC, even though collision energy increases by a fac- 1lnt+z) coordinates, assuming boost-inv−ariance.
tor 27,the entropyormultiplicity increase by a factorof 2 t−z
We note that presently there is disagreement about
1.6 only. Experimental observables do not show very
∼ the form of the relaxation equation to be used in heavy
large variation from RHIC energy collisions.
ion collisions. In [15, 18] an extra term R = [uµπνλ +
uνπνλ]Du is included in the relaxationequation. Shear
λ
stress tensor is traceless (πµ = 0) and transverse to 4-
µ
PB+Pb@LHC Au+Au@RHIC velocity (u πµν = 0). The term is needed to maintain
µ
centrality <Npart > <b> <Npart > <b> the transversality and tracelessness condition. Israel-
0-10 346.8 3.51 318.9 3.30 Stewart [20] developed the theory on gradient expan-
10-20 245.1 6.13 223.1 5.79 sion of entropy, gradients of equilibrium thermodynami-
calvariablesareassumedtobesmall. Theterm[uµπνλ+
20-30 169.9 7.93 154.1 7.49
uνπνλ]Du does not contribute to entropy and is missed
30-40 113.4 9.38 102.8 8.87 λ
in Israel-Stewart’s theory. Moreover, in Israel-Stewart’s
40-50 71.6 10.64 65.0 10.06 theory,thetermcanbeneglected(bothπµν andDu are
µ
50-60 41.5 11.76 37.8 11.11 small and their combination is neglected). We have also
60-70 21.5 12.79 19.8 12.09 checked that for minimally viscous fluid, contribution of
70-80 9.6 13.76 9.08 13.0 thetermisnegligible. Forminimallyviscousfluid,energy
80-90 3.1 14.80 2.97 14.03 density evolution is hardly affected whether the term is
present or not in the relaxation equation (see Fig.8 and
9 of [1]).
TABLE I: Glauber model calculation for the average par-
Details of the analysis of RHIC data in Au+Au col-
ticipant number < Npart > and average impact parame- lisions can be found in [1]. In brief, minimally viscous
ter < b > (in fm) for different ranges of centrality cuts in
QGP fluid was initialised with a Glauber model initial
Pb+Pb/Au+Aucollisions at LHC/RHICenergy.
condition, with 75% soft collisions and 25% hard colli-
sions. In a b=0 collision, this corresponds to central en-
tropy density S =110 fm−3, at the initial time τ =0.6
ini i
fm. The transverse fluid velocity at the initial time was
assumed to be zero, v = v = 0. The shear stress ten-
x y
sor πµν was assumed to attain boost-invariant value at
II. INITIAL CONDITIONS FOR PB+PB
the initial time τ . For the relaxation time τ , Boltz-
COLLISIONS AT LHC i π
mann approximation τ =3η/2p is used. The freeze-out
π
temperaturewasvariedto fitelliptic flowin16-23%cen-
DetailsofoursolutionofIsrael-Stewart’s[20]2ndorder trality Au+Au collisions. It was seen that for TF=130
theory of dissipative hydrodynamics can be found in [1]. MeV,elliptic flowin16-23%centralityAu+Aucollisions
Briefly, assuming longitudinal boost-invariance, we have as well as a host of other data are explained. For the
solvedthe Israel-Stewart’s2ndordertheoryfor abaryon equation of state we have used EOS-Q developed in [7],
free fluid with dissipation due to shear viscosity only. In with bag model EOS for the QGP phase and hadronic
Israel-Stewart’s theory, dissipative flows are treated as resonance gas for the hadronic phase. EOS-Q has a 1st
extended thermodynamic variables. For a baryon free order phase transition at Tc=164 MeV.
fluid with only shear viscosity as the dissipative effect, We assume that in Pb+Pb collisions at LHC, except
energy-momentum conservation equation is required to for the central entropy density, other parameters of the
besolvedsimultaneouslywiththerelaxationequationfor model remain unchanged. In Pb+Pb collisions also, the
the shear stress tensor, QGP fluid is thermalised at the same time as in Au+Au
3
7 1000
PHENIX data for Npart=350 Au+Au@RHIC
fit to PHENIX data Pb+Pb@LHC
6 extrapolated value at 5.5 TeV p-
N) dN/dypartch345 dN/dy 100 prKot+on
1/(.5 2 10
1
0 1
1 10 100 1000 10000 0 100 200 300 400
c.m. energy (GeV) Npart
FIG. 1: Filled circles are the PHENIX data for the charged FIG. 2: The black, red and green solid and dashed lines are
particle multiplicity per participant 1 dN as a function minimallyviscoushydrodynamicspredictionsforthecentral-
of c.m. energy for participant numbe.r5NNpapratrtd=y350. The solid itydependenceofπ−,K+ andprotonmultiplicityin Pb+Pb
lineisafittothePHENIXdatabyEq.3. Theunfilledcircleis collisions at LHC and in Au+Au collisions at RHIC. The
the extrapolated value of 1 dN at LHC energy √s=5.5 yileds are normalised by a factor of N=1.4 to account for
.5Npart dy theneglect of resonance production.
TeV, for participant number Npart=350.
collisionsatRHIC i.e. τ =0.6fm. The initialfluidveloc- Glaubermodelcalculationsforaverageparticipantnum-
i
ity is zero: vx = vy = 0, and the shear stress tensor has ber < Npart > and average impact parameter < b > for
attained the boost-invariant value. And as in Au+Au different centrality cuts are given in table I. For Pb+Pb
collisions, in Pb+Pb collisions also, the hadronic fluid collisionsatLHCenergy,wehaveusedσinel=70mb. For
freezes-out at T =130 MeV. The central energy density comparison, in table I, we have also shown the same re-
F
or entropy density in Pb+Pb collisions at LHC energy sults for Au+Au collisions at RHIC, when σinel=44 mb.
(√s=5.5 TeV) cannot be same as in Au+Au collisions
at RHIC (√s=200 GeV). One expects larger energy de-
position in Pb+Pb collisions. To obtain the initial en- III. MULTIPLICITY AND MEAN pT IN PB+PB
ergy/entropy density of the fireball in LHC energy colli- COLLISIONS AT LHC
sions we proceed as follows:
PHENIX collaboration [22] has tabulated the average
Particle multiplicity is one of the important observ-
charged particle multiplicity as a function of collision
ables in heavy ion collisions. It is a measure of the en-
energy for a range of collision centrality. In Fig.1, for
tropy of the system. In Fig.2, the black, red and green
participant number N =350, the average multiplicity
part lines are minimally viscous hydrodynamic model predic-
.5N1partdNdηch is shown as a function of collision energy . tionsforthecentralitydependenceofπ−,K+andproton
The multiplicity increases logarithmically with energy, multiplicityinPb+PbcollisionsatLHC.Forcomparison,
minimally viscoushydrodynamicpredictionsforπ−, K+
andprotonmultiplicityinAu+AucollisionsatRHICare
dN
ch =A+Bln√s, (3) alsoshown(thedashedlines)inFig.2. Wehaveneglected
dη
resonance production. To account for the neglect of res-
with A = 0.33 and B = 0.75. We use the relation onance production, yields are normalised by a factor of
−
to extrapolate to LHC energy √s=5.5 TeV. The extrap- N =1.4. Fromthepredictions,itappearsthatcompared
olated value of average charged particle multiplicity in toAu+Aucollisions,incentral/mid-centralPb+Pbcolli-
LHCenergyis 927 70. Weadjustthecentralentropy sions,particleyieldsareenhancedbyafactorof 1.6-1.8
densitytoS =∼180f±m−3suchthataN =350Pb+Pb . It is expected. Multiplicity increases logarit∼hmically
ini part
collision produce 900 charged particles. Entropy den- with energy.
sity S =180 fm∼−3 corresponds to central temperature Centrality dependence of mean p is another impor-
ini T
T =421 MeV. Compared to Au+Au collisions at RHIC tant observable. In Au+Au collisions at RHIC, in min-
i
(central temperature T =357 MeV), in Pb+Pb collisions imally viscous hydrodyanmics, centrality dependence of
i
at LHC, central temperature is 20% higher. < p > of identified particles are reasonably well expa-
T
∼
With the initial condition as described above we have lined. In Fig.3 minimally hydrodynamics predictions for
solved the hydrodynamic equations and calculate invari- mean p for π−, K+ and protons, in Pb+Pb colliisons
T
ant particle yield from the freeze-out surface at T =130 atLHC areshown. In collisionsbeyondN =50,mean
F part
MeV. It may be noted that we are assuming boost- < p > is approximately constant; < p > 0.5, 0.75
T T
invariance. Consequently, our predictions are valid only and 1 for π−, K+ and protons respectively.≈For com-
in the mid-rapidity range. In the following, we will parison, predictions for mean p in Au+Au collisions at
T
show our predictions as a function of collision central- RHICareshowninFig.3asthedashedlines. FromRHIC
ity or rather as a function of number of participants. to LHC, even though collision energy is increased by a
4
1.4 104
Pb+Pb@LHC 103
1.2 Au+Au@RHIC 102 vviissccoouuss hhyyddrroo APub++APub@@RLHHCIC
101 0-10%, 10-20%,20-30%,
<p> (GeV)T001...680 prKop+to-n 2-2N/dydp (GeV)T111111000000-----054321 30-40%,40(t-o5p0 %to ,b 5o0tt-o6m0%)
0.4 d 10-6
10-7
0.2 10-8
10-9
0.0 10-10
0 50 100150200250300350400450 0 1 2 3 4 5
Npart pT (GeV)
FIG.3: ThesolidlinesarepredictedmeanpT forπ−,K+and FIG. 4: The solid−lines are minimally viscous hydrodynamic
proton in Pb+Pb collisions at LHC energy. For comparison, predictions for π pT spectra in 0-10%, 10-20%, 20-30%, 30-
minimally viscous hydrodynamicspredictionsfor mean pT in 40%,40-50%and50-60%centralityPb+PbcollisionsatLHC.
Au+Aucollisions are also shown by thedashed lines. ThedashedlinesarethesameforAu+AucollisionsatRHIC.
To account for neglect of resonance production,theyield are
normalised bya factor of 1.4.
factorof27,themeanp isincreasedmarginally. Forex-
T 103
ample, in a central collision with Npart=350, for all the 110012 0-K1+0 %in, P1b0+-2P0b%@,2L0H-C30%,
species, mean pT is increased by 10% from RHIC to 100 30-40%(,t4o0p- 5to0 %bo, t5to0m-6)0%
LHC energy. ∼ -2V) 1100--21
Ge 10-3
2dN/dydp (T11110000----7654
10-8
IV. pT-SPECTRA IN PB+PB COLLISIONS AT 10-9
LHC 10-10
10-11
0 1 2 3 4 5
pT (GeV)
Minimallyviscoushydrodynamicmodelpredictionsfor
−
π pT-spectra, in 0-10%, 10-20%, 20-30%, 30-40%, 40- FIG.5: Thesolidlinesareminimallyviscoushydrodynamics
50%and50-60%centralityPb+PbcollisionsatLHCare predictionsfortheK+ invariantyieldin Pb+Pbcollisions at
shown in Fig.4. We have neglected resonance produc- LHCenergy.
tion. Resonance decay contribute to particle yield, more
at low p than at large p . For example, at freeze-out
T T
temperature T =150 MeV, nearly 50% of total pions V. ELLIPTIC FLOW IN PB+PB COLLISIONS
F
are from resonance decay at M =0∼.5 GeV, contribution AT LHC
T
of decay pions decreases to 20% at higher M =2 GeV
T
∼
[23]. Minimally viscous hydrodynamics predictions, nor- OneoftheimportantobservationsinAu+Aucollisions
−
malised by an factor of N=1.4, well explainedthe π pT atRHIC isthe significantelliptic flowinnon-centralcol-
spectra in Au+Au collisions. However, we must men- lisions. Qualitatively, elliptic flow is explained in a hy-
tion that overall normalisation do not account correctly drodynamicmodel,re-scatteringofsecondariesgenerates
forthe resonancecontributionto particlepT-spectra. pT pressure and drives the subsequent collective motion. In
spectra will be uncertain by 10-15%. For comparison,in non-central collisions, the reaction zone is asymmetric
Fig.4,predictedspectrainAu+AucollisionsatRHICare (almond shaped), pressure gradient is large in one direc-
shown as the dashed lines. pT spectra are slightly flat- tion and small in the other. The asymmetric pressure
tened at LHC. It is consistent with the predicted small gradients generate the elliptic flow. As the fluid evolve
increase in mean pT (see Fig.3) in LHC energy. and expands, asymmetry in the reaction zone decreases
InFig.5and6,wehaveshownthepredictedp spectra and comes a stage when the reaction zone become sym-
T
forK+ andprotons. Wehavenotshown,buthereagain, metricandsystemnolongergenerateelliptic flow. Ellip-
compared to RHIC energy p spectra is flattened. Be- ticflowisanearlytimephenomenaandasensitiveprobe
T
forewedigress,wewouldliketonotethateventhoughwe to the early stage of the fluid.
have shown predictions right up to p =5 GeV, at large InFig.7,solidlineisthepredictedellipticflowinmini-
T
p > 3 GeV, there may be other sources (e.g. jets) for mumbias Pb+PbcollisionsatLHC.For comparison,we
T
particle production. It is unlikely that viscous dynamics haveshowntheminimumbiasellipticflowinAu+Aucol-
will predict correctly particle production at high p > lisions at RHIC (the dahsed line). It is not shown here,
T
3GeV. Present predictions are expected to be in reason- but experimental minimum bias v2 in Au+Au collisions
able agreement with future experiments up to p 3 is well reproduced in minimally viscous hydrodynamics.
T
∼
GeV. It is interesting to note that in Pb+Pbcollisions,elliptic
5
110023 proton in Pb+Pb@LHC bias collisions, elliptic flow in different centrality ranges
101 03-01-04%0%, 1,400-2-500%%,2, 05-03-06%0%, of collisions is reduced at LHC.
100 (top to bottom)
-2V) 1100--21
Ge 10-3
2dN/dydp (T11110000----7654 VI. SUMMARY
10-8
10-9 To summarise, in a minimally viscous (η/s=0.08) hy-
10-10
10-11 drodynamics , we have given predictions for several ex-
0 1 2 3 4 5 perimental observables in Pb+Pb collisions at LHC en-
pT (GeV) 0.3
(a) 0-10% (b)10-20%
FIG. 6: same as in Fig.5 but for protons. elliptic flow00..12
20
0.0
0.3
15 (c)20-30% (d) 30-40%
0.2
w (v)2 0.1
elliptic flo 10 0.0
5 Au+Au@RHIC 0 1 2 3 4 5 0 1 2 3 4 5
pT (GeV)
Pb+Pb@LHC
FIG. 8: (color online) Predicted elliptic flow in 0-10%, 10-
0
0 1 2 3 4 5 6 20%,20-30%and30-40%Pb+PbcollisionsatLHCareshown.
pT (GeV) Theblack,redandgreenlinesareellipticflowforπ−,K+and
protons respectively.
FIG.7: Thesolidlineistheminimallyviscoushydrodynam-
ics prediction for the minimum bias elliptic flow in Pb+Pb
collisionsatLHC.Forcomparison,minimumbiasellipticflow
inAu+AucollisionsatRHICisalsoshown(thedashedline). ergy (√s=5.5 TeV). Assuming that particle multiplic-
ity increase logarithmically with c.m. energy, we have
extrapolated the lower energy data (tabulated by the
flow is reduced by 15%. Reduction of elliptic flow in PHENIXcollaboration)toobtainanestimateofparticle
LHC has also been∼predicted by Krieg and Bleicher [25]. multiplicity at LHC energy collisions. Our estimate in-
Inapartonrecombinationmodel,theystudiedtheenergy dicate that a Pb+Pb collision with participant number
dependence of elliptic flow in heavy ion collisions from Npart=350,produces 927chargedparticles. Theinitial
AGS to LHC energy. It was observed that from RHIC central entropy densit∼y of the fluid (Sini=180 fm−3) in
to LHC energy elliptic flow decreases. Parton transport Pb+Pbcollisionswasfixedtoreproducetheextrapolated
models also predict less elliptic flow in LHC energy than particle multiplicity. The initial time (τi) and freeze-out
in Au+Au collisions at RHIC [26]. temperature(TF)werekeptfixedatthevalue,τi=0.6fm
In Fig.8, in four panels, we have shown the minimally and TF=130 MeV, as it was obtained in the analysis of
viscous dynamics predictions for elliptic flow in 0-10%, Au+Au data in RHIC energy collisions.
10-20%,20-30%and,30-40%centralityPb+Pbcollisions OurpredictionsindicatethatcomparedtoAu+Aucol-
at LHC. The black, red and green lines corresponds to lisionsatRHIC,inPb+PbcollisionsatLHC,(i)particle
ellipticflowforπ−,K+ andprotons. Speciesdependence multiplicity will increase by a factor of 1.6, (ii) the
∼
is similar to that at RHIC energy. At large p , elliptic mean p will be enhanced by 10%, (iii) p spectra of
T T T
∼
flow between different species are marginally different. identifiedparticleswillbe(slightly)flattenedand(iv)el-
Species dependence is seen only at low p , lighter the liptic flow will decrease by 15%. We hope the results
T
∼
particle,moreistheellipticflow. Wehavenotshownany will be helpful in planning future experiments in LHC
comparison with v2 at RHIC. But as with the minimum energy.
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