Transverse Energy per Charged Particle in Heavy-Ion Collisions: Role of Collective Flow

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by  Swatantra Kumar Tiwari

Transverse Energy per Charged Particle in Heavy-Ion Collisions: Role of Collective
Flow
Swatantra Kumar Tiwari and Raghunath Sahoo∗
Discipline of Physics, School of Basic Sciences, Indian Institute of Technology Indore, Indore- 453552, INDIA
(Dated: February 3, 2017)
The ratio of (pseudo)rapidity density of transverse energy and the (pseudo)rapidity density of
charged particles, which is a measure of the mean transverse energy per particle, is an impor-
tant observable in high energy heavy-ion collisions, which reveals about the mechanism of particle
productionandthefreeze-outcriteria. Itscollisionenergyandcentralitydependenceisexactlylike
√
thechemicalfreeze-outtemperaturetilltopRHICenergy. TheLHCmeasurementat s =2.76
NN
TeV brings up new challenges to rule out the mechanisms of gluon saturation or non-equilibrium
phenomena being prevalent at high energies, which could contribute to the above observable. The
Statistical Hadron Gas Model (SHGM) with a static fireball approximation has been successful in
7 describing both the centrality and energy dependence till top RHIC energies. However, the SHGM
1 predictionsforhigherenergiesarehighlyunderestimatedbytheLHCdata. Inordertounderstand
0 this,wehaveincorporatedradialfloweffectinanexcludedvolumeSHGM.Thehard-coreradiusof
2 baryons at lower collision energies plays an important role in the description of a hadronic system.
Inviewof this, inordertomakeacompleteenergy dependencestudyfrom FAIR toLHCenergies,
b
we have considered an excluded volume SHGM. Our studies suggest that the collective flow plays
e
F an important role in describing ET/Nch and it could be one of the possible parameters to explain
the√jumpobservedinET/Nch fromRHICtoLHCenergies. PredictionsfortheLHCmeasurements
2 at s = 5.02 TeV are given.
NN

]
h PACSnumbers: 12.38.Mh,25.75.Nq,24.10.Pa
p
-
p I. INTRODUCTION loss patterns of heavy-quarks in the medium formed at
e theLHCarefewveryimportantaspectsofthenewstates
h
of matter formed at ultra-relativistic energies. The mat-
[
The transverse energy and charged particle multiplic- ter formed at LHC energies has seen to show properties

2 ityatmidrapidityareoneofthefirstmeasurementsdone very different from that is observed at RHIC in many
v in heavy-ion collision experiments at (ultra)relativistic aspects, i.e. suppression of J/ψ [7] and other quarkonia,
23 energies,asglobalobservablestocharacterizethesystem ddηE(yT)/ddηN(cyh) ≡ ET/Nch showing behaviour not expected
3 formed in these collisions. These two are very impor- by equilibrium Statistical Hadron Gas Model (SHGM)
3 tantobservables, astheyaredirectlyassociatedwiththe with a static fireball approximation [8–10], the collision
0 collision geometry and collision energy. Heavy-ion colli- energydependenceofdN /dη deviatingfromalogarith-
ch
1. sions at AGS, SPS, RHIC and LHC energies aim to pro- mic behaviour etc. There is a need to study the global
0 duce a partonic phase of matter and study the Quantum observableslikeE /N inmoredetailsincludingthecol-
T ch
7 Chromodynamics (QCD) deconfinement transition and lectivity in the system to rule out any possible effect of
1 explore/scan the QCD phase diagram for a possible lo- non-equilibrium phenomena or the effect of gluon satu-
v: cation of the critical point (CP), by controlling the tem- ration, which is expected at higher collision energies, to
i perature and baryochemical potential by changing the play a role at LHC energies.
X
collision species/centrality and collision energy. In these
r effortsthefuturefacilitieslikeCBMexperimentatFAIR The pseudo(rapidity) density of transverse energy,
a
energies (Elab : 10 AGeV-40 AGeV) and the LHC and dET/dη(y) is an approximate Lorentz invariant measure
beyond (FCC, ILC) would play a pivotal role in explor- of the energy distribution and is a measure of the explo-
ing the QCD phase boundary, and establishing the na- sivenessofthecollision. Thisisanimportantobservable,
tureoftheQCDphasetransitionandthelocationofCP. as it is the energy of the produced particles in the trans-
The discovery of a strongly interacting partonic matter verse phase space, which was completely empty before
at RHIC, which behaves like a liquid with lowest η/s ra- the collision. The energy of the incoming nuclei in the
tio [1–4], and is comparable with the ADS/CFT calcula- longitudinal direction is converted to the energy of the
tions [5], small systems (p+p collisions) at LHC showing produced secondaries. The midrapidity measurement of
collectivity [6], which is a possible signature of Quark- dET/dy is related to the initial energy density of the
Gluon Plasma (QGP) (initially expected to be formed system. In longitudinal boost invariant Bjorken hydro-
only in top central heavy-ion collisions) and the energy dynamics, this helps in making a direct comparison with
the lattice QCD prediction of energy density for a de-
confinement transition and thereby giving a first hint of
a possible partonic medium. The study of the centrality
∗Correspondingauthor: [email protected] andcollisionenergydependenceofdET/dy(η)shedslight
2
on possible freeze-out criteria in heavy-ion collisions [8]. of E /N at top RHIC and LHC energies, which is re-
T ch
Recently using an excluded-volume statistical hadron lated to the chemical freeze-out of the system. We study
gas model (EV-SHGM) with the longitudinal flow, the energy dependence of the associated observables like
we have studied the limiting fragmentation (LF) phe- theparticipantpairnormalized-dET ,theBjorkenenergy
dy(η)
nomenon in the rapidity spectra of π+ in heavy-ion col- density ((cid:15) ) in order to study the created system at
Bj
lisions. Here, we found that the LF is violated at top different energies and the possible different behaviour at
RHIC and LHC energies [11], which goes inline with LHCenergies,whichcouldservethepurposeofrulingout
the expectations for higher energies, where the inelastic and/or establishing different production mechanisms. In
cross-section is shown to increase with collision energy. our model calculation, we assume that the chemical and
In this paper, we study the variation of ET/Nc√h with thermalfreeze-outsoccursimultaneouslywhichinfersthe
respect to centrality and centre-of-mass energy ( sNN) absence of the possible elastic scattering after chemical
over a broad energy range from 2.7 GeV to 5.02 TeV us- freeze-out [13, 14].
ingEV-SHGMwithradialflow. Experimentally,E /N
T ch Thepaperisorganizedasfollows: insection-II,wegive
increases at lower energies rapidly and then it saturates
the formulation of the SHGM with an excluded volume
around SPS energies upto top RHIC energy. Till lower
correction and the method of inclusion of collective flow.
SPS energies, the increase in collision energy increases
In section-III, we show the results and in section-IV, we
the mean energy or transverse mass (m ) of each par-
T discuss the relation of E /N with various freeze-out
T ch
ticle. From SPS to RHIC energies, the additional en-
criteria. InsectionV,wegivethesummarywithoutlook
ergy pumped into the system in terms of the increase
√ and open problems.
in s , goes for new particle production [8, 9]. Re-
NN
cently,theexperimentaldataatLHCenergyof2.76TeV
show a jump in this spectrum due to furthermore in-
crease in mean energy or m [9], and possible collective II. FORMULATION OF THE MODEL
T
effects which does not follow a static fireball expecta-
tions. In order to understand this at LHC energies, we The formula for the number density of the i-th baryon
use our recently proposed model, where we incorporate in the Maxwell-Boltzmann’s statistics can be written as
the attractive interactions by including the resonances [15] :
upto the mass of 2 GeV and repulsive interactions by
assigning the geometrical hard-core size to each baryon. ∂R
nex =(1−R)I λ −I λ2 , (1)
Mesons are treated as pointlike particles in our model. i i i i i∂λ
i
We also incorporate the collective flow in our model to
(cid:88)
explain the experimental data at various energies partic- where R = nexV0 is the fractional occupied volume
i i
ularlyatLHCenergies. Inref. [12],thestatisticalmodel
i
is also used to study the transverse energy per charged by the baryons [16]. V0 =(4π r(cid:48)3)/3 is the eigen-volume
i
particle at midrapidity with longitudinal and transverse of each baryon having a hard-core radius r(cid:48) and λ is the
i
flows for the wide range of energies from AGS to RHIC. fugacity of the i-th baryon. Here we take r(cid:48) = 0.8fm,
In our case, we do not take volume as a free parameter which is the only free parameter in the discussed model.
and extend our analysis upto LHC energies, where the I is the momentum integral for baryons in the Boltz-
i
role of collective flow is more pronounced than at lower mann’sstatistics. Eq. (1)canbereducedinthefollowing
energies. Inaddition,westudythecentralitydependence form [17, 18] :
dN g Vλ (cid:104)(cid:16) ∂R(cid:17) E (cid:105)
i = i i (1−R)−λ i . (2)
dy m dm dφ (2π)3 i∂λ (cid:104) (cid:18)E (cid:19)(cid:105)
T T i exp i
T
(cid:112)
Here y is the rapidity variable and m = m2+p 2 is Weassumethatthevolumeofthefireball, Visthesame
T T
the transverse mass. E is the energy of the i-th baryon, for all types of hadrons at the time of the homogeneous
i
V is the total volume of the fireball formed at chemical emissions.
freeze-out and N is the total number of the i-th baryon. ByusingE =m coshy,eq. (2)canbewrittenas[18]:
i i T
3
dN g Vλ (cid:104)(cid:16) ∂R(cid:17) m coshy (cid:105)
i = i i (1−R)−λ T . (3)
dy m dm dφ (2π)3 i∂λ (cid:104) (cid:18)m coshy(cid:19)(cid:105)
T T i exp T
T
A. Transverse Energy of Hadrons in a Thermal try measurement of E , one redefines the energy of the
T
Model individual secondaries as [19–21]
The transverse energy, E in an event is defined as:
T 
E −m for baryons
 total
ET =(cid:88)Eisinθi, (4) Ei = Etotal+m for anti-baryons (5)
E for all other particles.
i total
with θ as the polar angle made by the i-th particle in Considering the above experimental formulae, we pro-
i
an event with the detector. The sum is taken over all ceedwiththeformulationofthetransverseenergyinour
the particles emitted into a fixed solid angle within the excluded-volume model. Using eq. (3), we write the en-
detector acceptance. Taking into account the calorime- ergy of the i-th baryon per unit rapidity as :
dE g Vλ (cid:104)(cid:16) ∂R(cid:17)(cid:105)(cid:90) m3 cosh2y dm
i = i i (1−R)−λ T T . (6)
dy (2π2) i∂λ (cid:104) (cid:18)m coshy(cid:19)(cid:105)
i exp T
T
Similarly, the energy of the m-th meson per unit ra- transverseenergyoftheparticlesinastationarythermal
pidity can be calculated as : source.
dE g Vλ (cid:90) m3 cosh2y dm
m = m m T T . (7)
dy (2π2) (cid:104) (cid:18)m coshy(cid:19)(cid:105) B. Transverse Energy of Hadrons in a Thermal
exp T Model with Flow
T
Here, g and λ are the degeneracy factor and fugac- Theinvariantyieldwiththeinclusionsofcollectiveflow
m m
ity of the m-th meson. The above equations give the can be written as [22] :
dN g Vλ m (cid:104) ∂R(cid:105) (cid:90) (cid:16) m cosh(y−η)coshρ−p sinhρcosφ(cid:17)
i = i i T (1−R)−λ exp − T T r dr dζ. (8)
m dm dy dφ (2π)3 i∂λ T
T T i
Here,ρistheparametergivenbyρ=tanh−1β ,withthe For the sake of simplicity, we take n=1 in our model
r
(cid:16) (cid:17)n
calculation. After incorporating the collective flow in
profilechosenasβ =β ξ [22,23]. β ,themaximum
r s s
our thermal model, we get the expressions for energy of
surfacevelocityappearsasafreeparameterinourmodel
(cid:16) (cid:17) baryons per unit rapidity as follows :
calculation and ξ = r/R , R is the maximum radial
0 0
position of the fireball at freeze-out. The average of the
radialflowcanbecalculatedbythefollowingformula[24]
:
(cid:82) β ξnξ dξ (cid:16) 2 (cid:17)
s
<βr >= (cid:82) ξ dξ = 2+n βs. (9)
4
dE g Vλ (cid:104) ∂R(cid:105) (cid:90) (cid:16) m cosh(y−η)coshρ−p sinhρcosφ(cid:17)
i = i i (1−R)−λ exp − T T m3 cosh2y r dr dφdζdm , (10)
dy (2π)3 i∂λ T T T
i
where r lies between 0 and R i.e. the radius of the ζ varies between −η and η . In a similar fashion,
0 max max
fireballatfreeze-out,theazimuthalangleφliesbetween0 we can calculate the energy of the m-th meson by using
and2π,andthelongitudinalspace-timerapidityvariable the following formula :
dE g Vλ (cid:90) (cid:16) m cosh(y−η)coshρ−p sinhρcosφ(cid:17)
m = m m exp − T T m3 cosh2y r dr dφdζdm . (11)
dy (2π)3 T T T
Here, E , g , and λ represent the energy, degeneracy III. RESULTS AND DISCUSSIONS
m m m
factor and fugacity of the m-th meson. Now, eq. (4) can
be reduced in the following form :
In order to calculate the ratio E /N , we estimate
T ch
(cid:88) N inourmodelintermsofthenumberofprimarilypro-
(cid:104)E (cid:105)=(cid:104) E sinθ (cid:105). (12) ch
T i i
ducedparticles,N.Wefollowthesameprocedureasdone
i
in ref. [8]. We first estimate the ratio of the total num-
The average of sinθ can be calculated as follows : berofhadronsinfinalstate,N tothetotalnumber
decays √
of primordial hadrons, N with respect to s over a
1 (cid:90) 1 (cid:90) NN
(cid:104)sinθ(cid:105)= sinθdΩ = sin2θ dθdφ, broadenergyrangefromAGStoLHCenergiesusingEV-
4π 4π SHGM as shown in the upper panel of figure 1. Now, we
studytheratioofthenumberofchargedhadrons,N to
wheredΩ(=sinθdθdφ)isthesolidangle. Now,integrat- √ ch
N with s fromAGStoLHCenergiesasshown
decays NN
ing above equation we get :
in the lower panel of the figure 1. In order to calculate
these ratios, we use the chemical freeze-out temperature
1 (cid:90) π (cid:90) 2π π √
(cid:104)sinθ(cid:105)= sin2θ dθ dφ = . (T),andbaryonchemicalpotential(µB)ateach sNN as
4π 0 0 4 mentioned in ref. [26]. We find t√hat the ratio Ndecays/N
initially increases rapidly with s , because the pro-
Now,wecanwritetheexpressionofthetransverseenergy NN
duction of resonances increases with energy and become
of hadrons as follows [8] :
saturated at SPS energies around a value of 1.7, where
π(cid:104) (cid:105) chemicalfreeze-outtemperaturebecomesindependentof
(cid:104)ET(cid:105)= 4 (cid:104)E(cid:105)−mN(cid:104)NB −NB¯(cid:105) . (13) collision ener√gy. Similarly, the ratio Nch/Ndecays also in-
crease with s and gets saturated at SPS energies
NN
(cid:104)E(cid:105) is the total energy of hadrons. NB −NB¯ is the net- around0.6. Although, thesefindingsarethesameasob-
baryon which can be calculated by using eq. (1). After served in [8] but the difference occurs at lower energies
obtainingthetransverseenergy,wecalculatetheBjorken where the excluded-volume correction is more effective.
energy density by using the following formula:
In this paper, we want to study the variations of
√
(cid:15) = dET 1 , (14) ET/Nch withrespecttocentralityforvarious sNN. For
Bj dy τ π R2 thispurpose,weconnectcentralityintermsofnumberof
participants(N )withTandµ bystudyingthecen-
part B
where τ is the formation time and π R2 is the transverse trality dependence of particle ratios in our model. Fig-
overlap area of the colliding nuclei. There are various ure 2 represents the variations of various hadron ratios
ways to quantify the overlap area. Here, R is the radius suchasK+/π+, K−/π−, K−/K+, andp¯/pwithrespect
of the colliding nuclei given by R = R0 A1/3. Replac- t√o number of participants, Npart for Au-Au collisions at
ing A by Npart/2, where Npart is the number of nucleon sNN = 200 GeV. In order to calculate Npart depen-
participants [25], (cid:15) becomes dence of ratios, we relate the freeze-out parameters T
Bj
andµ withN atthisenergy. Weselectthreecentral-
B part
dE 1 ity bins with participant numbers 328±6 (most-central),
(cid:15) = T . (15)
Bj dy (cid:16) (cid:17)2/3 140±11 (mid-central), and 62±10 (peripheral) while cal-
τ π R2 N /2
0 part culating the particle ratios. We find that the freeze-out
5
1.8
0.25 K+/p + 0.25 K-/p -
1.6 Experimental Data Experimental Data
0.2 Our Model 0.2 Our Model
N 1.4
/s
cay 1.2 0.15 0.15
e
d
N 1 0.1 0.1
0.8 0.05 0.05
1 10 102 103 50 100 150 200 250 300 350 50 100 150 200 250 300 350
sNN (GeV) K-/K+Npart p/p Npart
0.8 1.5 Experimental Data1.5 Experimental Data
s Our Model Our Model
y
a
c
Nde 0.6 1 1
/h
c
N
0.5 0.5
0.4
1 10 sNN (1G02eV) 103 100 N2p0a0rt 300 100 N2p0a0rt 300
FIG. 1: The variation of N /N (upper panel) and FIG. 2: The variations of particle ratios for Au-Au collisions
d√ecays √
N /N (lowerpanel)with s atchemicalfreeze-out. withrespecttonumberofparticipantsat s =200GeV.
ch decays NN NN
Symbols show the experimental data [27] and lines are our
model calculations.
parametersTandµ donotvarymuchwithN . Hav-
B part
ing done these, we compare our model predictions with
the experimental data [27]. We find that our model ex-
plainsthedataverywelloverallthecentralities. Forthe 2
sake of convenience, we take the strangeness saturation
s = 200 GeV
factor (γ ) equals 1 over all the centrality at this energy. NN
s
Experimental Data
In ref. [28], the detailed analysis of variations of particle
ratios with centrality is done using the variation of γ Our Model
s 1.5
with Npart. Our Model+Flow
)
In figure 3, we show the variations of E /N with V
√ T ch
N forAu-Aucollisionsat s =200GeV.Wetake e
part NN G
the same value of T and µ as used in the calculation
of the hadron ratios with reBspect to Npart at this energy (ch 1
whilestudyingtheN dependenceofE /N . Wealso N
part T ch
compare our results with the experimental data [19] and /T
E
find a very good agreement between these two. In our
calculation, we take the same centrality bins as used in 0.5
thecalculationofparticleratiosatthisenergy. E /N is
T ch
almost independent of centrality except at lower central-
ity bins with the participant number less than 100. The
present model explains the data well except at a lower
0
N . We also show the obtained results with collective 100 200 300 400
part
flow which explains the data within the experimental er-
N
rors. part
Figure 4 represents the centrality dependence of vari-
√
ous hadrons ratios for Pb-Pb collisions at s = 2.76 FIG. 3: Transverse energy per charged particle vs number
NN √
TeV. To calculate the particle ratios with respect to of participants for Au-Au collisions at sNN = 200 GeV.
N , we relate T and µ with N at this energy. Symbolsaretheexperimentaldata[19]. Solidlineistheresult
part B part
obtainedinourmodelwithflowwhiledottedlineistheresult
Onceweget the variations offreeze-outparameters with
of our model calculations.
N , we calculate various particle ratios. While cal-
part
culating ratios with respect to N , we select three
part
6
8000
0.25 K+/p + 0.25 K-/p -
Experimental Data Experimental Data
HBT Data
0.2 Our Model 0.2 Our Model
Our Model
0.15 0.15 6000
0.1 0.1
)
3
m
0.05 0.05 f
(

y 4000
d
50 100150200 250300 350400450 50 100 150 200 250 300 350 400 450 /
K-/K+Npart p/p Npart V-p
1.5 Experimental Data1.5 Experimental Data d
Our Model Our Model
2000
1 1
0.5 0.5
1 10 102 103
100 200 300 400 100 200 300 400 s (GeV)
N N NN
part part
FIG. 4: The variations of hadron ratios for Pb-Pb collisions FIG. 5: The variations of freeze-out volume for π−. Solid
√
with respect to N at s = 2.76 TeV. Symbols show symbols are HBT data points [32, 33] and open symbols are
part NN
the experimental data [29] and lines are our model results. those obtained in our model calculations.
centrality bins in our model with participant numbers 102
2.7 GeV
382±17 (most-central), 128±16 (mid-central), and 7±4
(peripheral). Again, we take γ equal to one over all Experimental Data
s
the centrality for the sake of simplicity. We compare our -2)V 10 Our Model + Flow
results with the experimental data [29] and again, find e
G
that our model explains the data very well over all the y (
d
centralities. T 1
√ m
Now, we need to know s dependence of chemi- d
NN mT
cal freeze-out volume in unit slice of rapidity to calcu- N/
late transverse momentum spectra, transverse energy at 2d 10-1
midrapidity etc. Figure 5 describes the chemical freeze-
out volume (dV/dy) for π− for the most central colli-
√
sionsatvarious sNN startingfrom2.7GeVto5.02TeV 10-2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
calculated in our model with Boltzmann approximation.
m -m (GeV)
To calculate the dV/dy for π−, we use the experimental T 0
midrapidity yield of π− [30] and divide it by the corre-
√
sponding number density calculated in our model. At FIG. 6: The transverse mass spectra of π+ at s = 2.7
√ NN
s =5.02 TeV, we use the AMPT data for the midra- GeV. Symbols are experimental data [34] while line is our
NN
pidityyieldforπ+ [31]intheabsenceoftheexperimental result.
data. We compare our model results with the HBT data
[32, 33], which represents the kinetic freeze-out volume,
and find that our model results follow the same trend as model calculations. After comparison, we get the value
observed in the HBT data. While HBT data correspond of the surface velocity, βs which gives the value of the
to the kinetic freeze-out volume, our calculations refer radial flow at this energy.
to the volume of the fireball at chemical freeze-out. A In figure 7, we show the p spectra of π− for the most
T√
lower value from our estimation is thus expected. Figure central collisions at various s from 7.7 GeV to 62.4
NN
6demonstratesthetransversemassspectraofpositivepi- GeV. We use our model with the effect of flow to cal-
√
ons for the most central collisions at s = 2.7 GeV. culate p -spectra at all energies where we take only the
NN T
We compare our calculations with flow, represented by surfacevelocity, β asafreeparameter. Wecompareour
s
the solid line, with the experimental data [34]. We no- results with the experimental data [35–38] and find that
tice a good agreement between experimental data and our model with flow explains the data satisfactorily. Af-
7
102 7.7 GeV 102 11.5 GeV 2
11001--1021 EOxupr eMriomdeenl t+a lF Dlo-22p)dy (GeVdppN/2dawTTta1100-1-2110 EOxupr eMriomdeenl t+a lF Dloawta sNN =EO x2up.r7 eM6ri omTdeeeVnltal Data
-22pN/2d)dy (GeVdppTT111110001100---10102322100 00..55 11 31pp9TT9 11((GGG...556eeEOVVe EO))GxVuxpurep erVM erMi22romiomdedeeneln tl+a t+a l22F ..l55DF lDola-22-2po)dy (GeVdpp)dy (GeVdppN/2dwatwTTTTata11111000100--1-0132221103300 00..55 11 6pp2TT2 11((.GG..7455eeEO VV GG))xEOuperexu eVpVMrr eiMom22rdiomeednele t+nal t l+aF 22D l..l55F oaDlwotaawta 33 (GeV)/NETch1.51 Our Model+Flow
10-1 p2N/2d10-1 0.5
10-20 0.5 1 1.5 2 2.5 10-230 0.5 1 1.5 2 2.5 3
p (GeV) p (GeV)
T T
FIG. 7: The transverse momentum spectra of π− at various
energies. Symbols are experimental data while lines are our
0
model results with flow. 100 200 300 400 500
N
part
FIG. 9: Transverse energy per charged particle vs N for
√ part
0.6 Our Model Pb-Pb collisions at sNN = 2.76 TeV. Symbols are the ex-
perimentaldata[20]. Thesolidlineistheresultofourmodel
Fitting Function calculationswithflowanddottedlineisthatobtainedinour
0.5 model calculation without flow.
0.4 √
> with respect to s from lower AGS energies to LHC
r NN
energies. We notice that it increases monotonically with
b<
0.3 the collision energy with it’s lowest value at AGS to
the highest value at LHC. This shows significant col-
lectivity in high energy heavy-ion collisions. We fit a
0.2
phenomenologically motivated power-law function, i. e.
√
a+b( s )c totheenergydependenceof(cid:104)β (cid:105). Here, a,
NN r
0.1 b and c are the fit parameters. For the best fit we get,
a = 0.072 ± 0.057, b = 0.141 ± 0.049, and c = 0.148 ±
√
0.029. The predicted value of (cid:104)β (cid:105) at s = 5.02 TeV
r NN
1 10 102 103 104 is 0.569 for the top central Pb+Pb collisions.
s (GeV) Figure 9 demonstrates the centrality dependence of
√
NN
E /N forPb-Pbcollisionsat s =2.76TeV.Wecal-
T ch NN
culateE /N vsN usingthesamevalueofTandµ
T ch part B
FIG. 8: The variations of the radial√flow velocity ((cid:104)βr(cid:105)) ex- as used in fitting the hadron ratios with respect to Npart
tractedinourmodelwithrespectto sNN. Thesolidlineis at this energy. We compare EV-SHGM predictions with
a power-law fit as mentioned in the text.
the experimental data [20] and find that it could not ex-
plainthedataoverallthenumberofparticipants. These
findings suggest that we need to incorporate collective
ter comparison with the experimental data, we get the flow in EV-SHGM to explain the centrality dependence
value of the βs, from which we calculate the average ra- ofET/Nch atLHCenergies. Aftercomparisonwithdata,
dialflowvelocity. Here,ourmodelwithflowexplainsthe we find that our model with flow explains the data very
experimentaldatasuccessfullyexceptatlowerpT,where well over all the centrality bins within the experimental
the contribution of resonance decay plays an important uncertainties. In ref. [39], it is argued that in a kinetic
role [30]. In this paper, we do not take the contributions freeze-out scenario, the temperature should depend on
of resonance decays while calculating pT spectra and we centrality because during the kinetic freeze-out process
reserve this for a future work. thereisacompetitionbetweenlocalscatteringandglobal
Figure 8 shows the variations of average radial flow expansion. Thus, the kinetic temperature is sensitive to
velocity, (cid:104)β (cid:105) extracted in the framework of EV-SHGM the freeze-out process and hence centrality dependent.
r
8
20 20
Experimental Data
Experimental Data Fitting function
V) 15 EKRT Model 15 Our Model
e
G Our Model 2) Our Model + Flow
m
(
2) Our Model + Flow V/f
/
part 10 Ge 10
N
(
(
y/ tBj
d
/ ˛
T
E 5 5
d
0 0
1 10 102 103 1 10 102 103
s (GeV) s (GeV)
NN NN
√
FIG. 10: The variation of dE /dy per number of participant FIG. 11: The variation of (cid:15) τ with respect to s for
√ T Bj NN
with respect to s for the most central collisions. Solid the most central collisions. Solid line represents our model
NN
line represents our model calculation with flow and dashed calculationwithflowanddashedlineistheresultobtainedin
lineisthe resultobtainedinourmodelwithoutflow. Dotted ourstationarythermalmodel. Thedottedlinerepresentsthe
line presents the results of EKRT model. Symbols are the result of the logarithmic fitting function as described in the
experimental data points [20]. text. Symbols are the experimental data points [10, 20, 21].
In the case of chemical freeze-out, the temperature is EKRT model [42] which is based on the calculation of
observed to be centrality independent [40]. This is be- perturbative QCD with gluon saturation mechanism
causeduringthisprocess,thechemicalreactionsdecrease and hydrodynamics. We observe that the EKRT model
abruptlyleavingbehindthechemicallyfrozenstateatthe underestimates the experimental data upto top RHIC
freeze-out and thus the chemical freeze-out temperature energy but seems to explain the data at LHC energies.
is insensitive to the collective dynamics but depends on Figure 11 represents the variations of the product of
thermodynamical variables. The observation of a cen- Bjorken energy density ((cid:15) ) and formation time (τ)
√ Bj
trality independence of E /N at RHIC and LHC thus with respect to s from AGS to LHC energies. Fur-
T ch NN
indicates to the chemical freeze-out scenario. This ar- thermore, our model explains the experimental data
gument could be strengthened further in the subsequent [10, 20, 21] very well upto RHIC energies while it fails
section, when we make a direct comparison of E /N to explain the data at LHC energies. We also show
T ch
values with the universal freeze-out criteria. the results obtained after incorporating collective flow
In figure 10, we present the variations of in our model and found a very good agreement with
√
((dE /dy)/0.5N ) with respect to s over a experimental data including that at LHC. We also fit
T part NN
broad energy range from AGS to LHC energies. We the experimental data using the logarithmic function
√
compare our model predictions with the expe√rimental A+B ln( sNN),whereA=-2.32±0.51andB=1.46±
data [20]. The open symbol in the figure at s = 0.12 are fit parameters. We notice that this function fits
NN
2.76 TeV represents the ALICE data while solid symbol the data only upto RHIC energies and fails at LHC en-
is the measurement by the CMS experiment. Here, we ergies which suggests that logarithmic behaviour is not
take care of conversion of the dE /dη to dE /dy by valid at LHC energies in this case. This could be an
T T
using the Jacobian factor J(η,y), which is 1.09 at LHC indication of a different particle production mechanism
[41]. We notice that our model results agree well with playing a role at LHC energies, which needs further in-
the data upto RHIC energies while it is unsuccessful vestigations.
in explaining the LHC data. Now, we show the results In figure 12, we demonstrate the ratio E /N for
√T ch
obtained after incorporating the collective flow in our the most central collisions with respect to s start-
NN
model and find that it explains the ALICE data [20] ing from lower AGS energies to LHC energies. We con-
within experimental error but lies below to CMS data frontEV-SHGMcalculationswiththeexperimentaldata
[41] at LHC. We also show the results obtained in the [10, 20, 21, 41]. Here, the solid symbol at LHC is the
9
1.4 s = 2.76 TeV
NNExperimental Data
Experimental Data 1.2 Our Model
V) Our Model + Flow
1.5 Our Model Ge 1
(
Our Model + Flow > 0.8
T
p
<
) 0.6
V
e 0.4
G 1
( 1 50 100 150 200 250 300 350 400 450
h s = 20N0p aGrteV
c NN
N Experimental Data
/ET eV) 0.8 OOuurr MMooddeell + Flow
G
(
0.5 > 0.6
T
p
<
0.4
50 100 150 200 250 300 350 400 450
0 N
part
1 10 102 103
s (GeV) FIG. 13: Variations of average transverse momentum ((cid:104)p (cid:105))
NN √ T
with respect to N at s = 200 GeV and 2.76 TeV.
part NN
Symbols are the experimental data and lines are model re-
FIG.12: Transverseenergype√rchargedparticle(ET/Nch)as sults.
a function of collision energy s . Experimental data are
NN
compared with the expectations from EV-SHGM with and
without the effect of collective flow.
CMS data while open symbol is the ALICE data. The 1.5 S/N » 7.0
model seems to explain the data at SPS and RHIC ener- (n + n) » 0.08 (fm-3)
B B
gies but fails at LHC energies. These findings may hint s/T3 » 6.0
for possible non-equilibrium phenomena playing an im- (E/N) » 1.0 GeV
portant role at LHC. We also show the results obtained Experimental Data
in our thermal model with the effect of flow. We notice
)
V 1
that our model with flow explains the ALICE data [20]
e
within the experimental errors while it could not explain G
(
the CMS data [41] at LHC energies, which indicates a h
c
more precise estimation of E /N at the LHC. N
T ch
/
We have studied the average transverse momentum ET
((cid:104)p (cid:105)) of hadrons in EV-SHGM with and without the 0.5
T
effect of flow to understand the increase in E /N from
T ch
RHIC to LHC. Figure 13 demonstrates the N depen-
part
denceofaveragetransversemomentum((cid:104)p (cid:105))ofcharged
√ T
hadrons at top RHIC and LHC s = 2.76 TeV ener-
NN
gies. The lines are model calculations and symbols are
the experimental data [19, 43]. We find that, there is 1 10 102 103
an almost 63% increase in (cid:104)p (cid:105) from top RHIC to LHC
T s (GeV)
at 2.76 TeV energy in EV-SHGM model with collective NN
flow. Wealsogetanalmost57%increasein(cid:104)β (cid:105)fromtop
r
RHICtoLHCat2.76TeV,whichcouldbethereasonfor FIG. 14: The comparison between the experimental data on
increase in (cid:104)pT(cid:105). Further, no change in (cid:104)pT(cid:105) is observed ET/Nch and the expectations from various universal freeze-
in a static SHGM model from RHIC to LHC. This is be- out criteria in EV-SHGM.
cause a static SHGM uses T and µ as the inputs for
B
the estimation of various thermodynamical observables.
From top RHIC to LHC energies T and µ are found
B
to be constant within systematics [44]. This reveals that
theincreaseinE /N atLHCcouldbeexplainedbythe the hadrons in our model while calculating the (cid:104)p (cid:105) and
T ch T
increasein(cid:104)p (cid:105)andthecollectiveflow. Here,wetakeall take assumption of centrality independent (cid:104)β (cid:105).
T r
10
IV. E /N AND FREEZE-OUT westudythetransverseenergyperunitrapidity. Wealso
T ch
calculate the Bjorken energy density using our model at
√
In this section, we discuss the comparison of the ex- various sNN. Finally,wecalculatetheratioET/Nch at
perimental data on the ratio E /N with that calcu- various energies using our excluded-volume SHGM with
T ch
lated in EV-SHGM using the various universal freeze- and without flow. Further we study various freeze-out
out criteria. Various observables such as the energy per criteria in the framework of EV-SHGM using ET/Nch as
hadrons (E/N)≈1.0 [45, 46], the sum of baryons and an- the observable.
tibaryons (nB + nB¯)≈ 0.12/fm3 [47], the normalized In conclusion, we successfully describe the Npart de-
entropy density, s/T3 ≈ 7.0 [48], and the entropy per pendence of various hadron ratios using our excluded-
hadron (S/N)≈7.0 [26] are proposed as universal freeze- volume model. We observe that although the model de-
outcriteriainheavy-ioncollisions. Theseobservablesare scribes the RHIC centrality data for the ratio, E /N ,
√ √ T ch
almostindependentof s exceptatlowerenergies. In it fails to explain the LHC data at s = 2.76 TeV.
NN √ NN
figure 14, we show the variations of E /N with s The inclusion of the collective flow in our model, how-
T ch NN
fromAGStoLHCenergies. Here,thesymbolsaretheex- everqualitativelyexplainsthecentralitydataattheLHC
perimentaldatawhilethelinesarethosecalculatedusing successfully, with some degree of deviations for higher
variousfreeze-outcriteriainourexcluded-volumemodel. centrality. While studying the energy dependence of
In our model, the values of these criteria are different E /N , we observe that the excluded-volume hadron
T ch
than that mentioned above in the text, which are shown gas model with radial flow does not explain the CMS
in the figure. We find a similar behaviour between the data at the LHC, whereas the ALICE data at the same
experimental data and those calculated using freeze-out energy is well-explained. This necessitates for precision
criteria[49]inourmodeluptotopRHICenergy,whileat measurement of E /N at LHC energies to see if mech-
T ch
LHC energies our calculations could not explain the ex- anisms other than the collective flow play a role. The
perimental data. This points to further investigation(s) energy dependence of Bjorken energy density, pseudora-
to understand the possible reason(s) for the deviation of piditydensitiesofchargedparticlesandtransverseenergy
LHC data from the universal freeze-out criteria. andtotalchargedparticlemultiplicityshowingdeviations
from a logarithmic behaviour [9, 50–56] at LHC may in-
dicate different multiparticle production mechanism at
V. SUMMARY AND CONCLUSION LHC, compared to that at RHIC and lower collision en-
ergies. The observed increase in E /N from RHIC to
T ch
In summary, we have performed the calculation of LHCisattributedduetoanincreasein(cid:104)pT(cid:105)andtheon-
global observables like transverse energy of hadrons, set of higher collective flow at the LHC. Our comparison
charged particle multiplicity and their ratios at midra- of energy dependence of ET/Nch with various universal
pidityusingourstatistical-thermalmodel. Westudythe freeze-out criteria reveals that further investigations are
centrality dependence of various hadron ratios using our necessary in order to have a proper understanding of the
model and extract T and µ at RHIC and LHC ener- LHC data and its connection with freeze-out.
B
gies. We use the same value of T and µ to study the
B
centrality dependence of the ratio E /N at those en-
T ch
ergies. Then we calculate the transverse mass spectra of
π− at various √s , except at √s = 2.7 GeV where VI. ACKNOWLEDGEMENT
NN NN
the transverse mass spectra of π+ is calculated by us-
ingEVSHGM.Wegettheaverageradialflowvelocityat The authors would like to acknowledge Dr. Aditya
various energies by comparing our calculations with the NathMishraandDr. DeepikaJenaforcarefulreadingof
experimental data. Once, we get the radial flow velocity the manuscript and providing useful comments.
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