research article transverse momentum distributions of

Research ArticleTransverse Momentum Distributions of Hadrons Produced inPb-Pb Collisions at LHC Energy radicsNN = 276 TeV

Saeed Uddin Inam-ul Bashir and Riyaz Ahmed Bhat

Department of Physics Jamia Millia Islamia (Central University) New Delhi 110025 India

Correspondence should be addressed to Saeed Uddin saeed jmiyahoocoin

Received 12 August 2014 Revised 21 October 2014 Accepted 28 October 2014

Academic Editor Edward Sarkisyan-Grinbaum

Copyright copy 2015 Saeed Uddin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

The transversemomentum spectra of several types of hadrons 119901 119901119870+119870minus1198700119904ΛΩΩ Ξminus and Ξ produced inmost central Pb-Pb

collisions at LHC energyradicsNN = 276TeV have been studied at midrapidity (|119910| lt 05) using an earlier proposed unified statisticalthermal freeze-out model The calculated results are found to be in good agreement with the experimental data measured by theALICE experiment at LHC The model calculation fits provide the thermal freeze-out conditions in terms of the temperature andcollective flow effect parameters for different particle species Interestingly the model parameter fits to the experimental data revealstronger collective flow in the system and lesser freeze-out temperatures of the different particle species as compared to Au-Aucollisions at RHICThe strong increase of the collective flow appears to be a consequence of the increasing particle density at LHCThe model used incorporates a longitudinal as well as transverse hydrodynamic flow The chemical potential has been assumed tobe nearly equal to zero for the bulk of the matter owing to high degree of nuclear transparency effect at such collision energies Thecontributions from heavier decay resonances are also taken into account

1 Introduction

The study of identified particle spectra in heavy-ion collisionsat ultrarelativistic energies is an important tool to investigatethe properties of the strongly interacting system created insuch collisionsThe study also helps us to learn about the finalstate distribution of baryon numbers among various particlespecies at the thermochemical freeze-out after the collisionwhich is initially carried by the nucleons only [1]

Within the framework of the statistical model it isassumed that a hot and dense fireball is formed over anextended region for a brief period of time (sim a few fmc)after the initial collision and it undergoes collective expansionleading to a decrease in its temperature and finally to thehadronizationAfter the hot fireball formed in such collisionswhich initially has a very high density of partons (ie quarksand gluons) hadronizes the hadrons keep rescattering witheach other and continue to build up collective expansionConsequently the matter becomes dilute and the averagedistance between hadrons exceeds the range of the strong

interactions At this point of time all scattering processes stopand the hadrons decouple that is a freeze out occurs [2]

The hadronic abundance freeze-out (ie the chemicalfreeze-out) occurs earlier when the rates for inelastic pro-cesses in which secondary hadrons are produced or thehadrons change their identity (via strangeness exchange pro-cess etc) become too small to keep up with the expansionSince the corresponding inelastic cross sections are only asmall fraction of the total cross section at lower (thermal)energies the inelastic processes stop well before the elasticones leading to an earlier chemical freeze-out for the hadronabundances Finally at a later stage the hadrons completelydecouple from each other such that even the elastic processesalso come to a stop Consequently themomentum spectra getfrozen in time and a thermal (or hydrodynamical) freeze-outoccurs Thus chemicalfreeze-outprecedes thermal or kineticfreeze-out [3]

Transverse momentum (119901119879) distributions of identified

hadrons are themost common tools used to study the dynam-ics of high energy collisions

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 154853 7 pageshttpdxdoiorg1011552015154853

2 Advances in High Energy Physics

The identified particle spectra provide information aboutboth the temperature of the system and the collective flow atthe time of thermal freeze-out Collective flow depends onthe internal pressure gradients created in the collision andis addressed by hydrodynamic models [4ndash6] These effectsare species-dependent The produced hadrons are believedto carry information about the collision dynamics and thesubsequent space-time evolution of the system

Hence an accurate measurement of the transverse momen-tum distributions of identified hadrons along with the rapid-ity spectra is essential for the understanding of the dynamicsand the properties of the created matter up to the finalthermal or hydrodynamical freeze-out in case of collectiveflow [7]

It has been shown earlier [2] that this model successfullysimultaneously explains the rapidity and transverse momen-tum distributions of hadrons and their ratios in Au-Aucollisions at highest RHIC energy ofradicsNN = 200GeV In thispaper we briefly describe the model and use it to reproducethe transversemomentumdistributions of hadrons producedin Pb-Pb collisions atradicsNN = 276 TeV

2 Model

In order to obtain the particle spectra in the overall restframe of the hadronic fireball in our model we first define theinvariant cross-section for given hadronic specie in the localrest frame of a hadronic fluid element Since the invariantcross section will have the same value in all Lorentz frames[8] we can thus write

1198641198893119873

1198893119901= 1198641015840 1198893119873

11988931199011015840 (1)

where 119864(1198641015840) is the energy of the particle and 119901(119901

1015840) is the

momentum The primed quantities on the RHS refer to theinvariant spectra of given hadronic specie in the rest frameof the local hadronic fluid element while the unprimedquantities on the LHS refer to the invariant spectra of thesame hadronic specie in the overall rest frame of the hadronicfireball The occupation number distribution of the hadronsin the momentum space follows the distribution function

1198641015840 1198893119873

1198893

1199011015840sim

1198641015840

119890((1198641015840minus120583)119879) plusmn 1

(2)

where (+) sign and (minus) sign are for fermions and bosonsrespectively and 120583 is the chemical potential of the givenhadronic specie For the temperatures under considerationand the large masses of hadrons it is safe to work withBoltzman distribution

In recent works [7 9] it has been clearly shown that thereis a strong evidence of increasing baryon chemical potential120583119861 along the collision axis in the RHIC experiments In view

of this fact we write the expression for chemical potentialas 120583119861

= a + b11991020[7 9 10] where y0 is the rapidity of

the expanding hadronic fluid element Here a and b arethe two model parameters which can be fixed by fitting theexperimental data In the model the value of a essentially

defines the baryon chemical potential in the central regionof the bulk hadronic matter formed while b determinesthe rate of increase of baryon chemical potential along the(longitudinal) rapidity axis with y0 In case of very highdegree of nuclear transparency the values of a and bwill tendto vanish for the bulk of the matter Further it is assumedthat [7] the rapidity of the expanding hadronic fluid element1199100120572119911 or 119910

0= 120585119911 where 119911 is the longitudinal coordinate

of the hadronic fluid element and 120585 is a proportionalityconstant The above conditions also ensure that under thetransformation 119911 rarr minus119911 we will have 119910

0rarr minus119910

0 thereby

preserving the symmetry of the hadronic fluid flow about119911 = 0 along the rapidity axis in the centre of massframe of the colliding nuclei This leads to an expression forthe longitudinal velocity component of the hadronic fluidelement

120573119911(119911) = 1 minus

2

exp (2120585119911) + 1= tanh (119910

0) (3)

The transverse velocity component of the hadronic fireball120573119879is assumed to vary with the transverse coordinate 119903 in

accordance with the blast wave model as 120573119879(119903) = 120573

119904

119879(119903119877)119899

[11] where 119899 is an index which fixes the profile of 120573119879(119903)

in the transverse direction and 120573119904

119879is the hadronic fluid

surface transverse expansion velocity and is fixed in themodel by using the parameterization 120573

119904

119879= 1205730

119879radic1 minus 1205732

119911[7]

This relation is also required to ensure that the net velocity120573 of any fluid element must satisfy 120573 = radic1205732

119879+ 1205732119911

lt 1We also parameterize 119877 that is the transverse radius offireball as 119877 = 119903

0exp(minus11991121205902) where 120590 fixes the width

of the matter distribution in the transverse direction [7 9]and 119911 as described above is the longitudinal coordinateof hadronic fluid element This is required as the collidingnuclei when passing through each other may still feel somedrag thus resulting only in a partial transparency Conse-quently the collision axis will be populated by an extendedhadronic matter rapidly moving away from each other withits transverse size decreasing rapidly following a Gaussiandistribution along the 119911-axis

In our analysis the contributions of various heavierhadronic resonances [10 12] which decay after the thermalfreeze-out of the hadronic matter has occurred are also takeninto accountThe invariant spectrumof a given decay productof a given parent hadron in the local rest frame of a hadronicfluid element is written as [7 10 12]

1198641015840 1198893119873

decay

11988931199011015840=

1

21199011015840119898ℎ

119901lowastint

119864+

119864minus

119889119864ℎ119864ℎ1198893119873ℎ

1198893119901ℎ

(4)

where the subscript ℎ stands for the decaying (parent)hadron The two body decay kinematics gives the producthadronrsquos momentum and energy in the ldquorest frame of thedecaying hadronrdquo as 119901lowast = (119864

lowast2minus 1198982)12 and 119864

lowast= (1198982

ℎminus

1198982

119895+ 1198982)2119898ℎ where 119898

119895indicates the mass of the other

decay hadron produced along with the first one The limitsof integration are 119864

plusmn= 119898ℎ11989821198641015840119864lowast

plusmn 1199011015840119901lowast

The 1198641015840(119864ℎ)

and 1199011015840(119901ℎ) are respectively the product (decaying parent)

Advances in High Energy Physics 3

0 1000 2000 3000 4000 5000

001

01

1

pT (MeV)

1E minus 3

proton

d2N(2120587pTdpTdy)

(1G

eV2)

ALICE (Pb-Pb) 276TeV |y| lt 05

(a)

0 1000 2000 3000 4000 5000

001

01

1

pT (MeV)

1E minus 3

antiproton

d2N(2120587pTdpTdy)

(1G

eV2)


(b)

Figure 1 Transverse momentum spectra of protons (a) and antiprotons (b) for the centrality class (0ndash5)

hadronrsquos energy and momentum in the local rest frame ofthe hadronic fluid element A Boltzmann type distributionfor the massive decaying hadron in the local rest frame of thehadronic fluid element leads to the following final expressionfor the invariant cross section of the product hadron

1198641015840 1198893119873

11988931199011015840

=1

21199011015840119898ℎ

119901lowast120582ℎ119892ℎ119890minus120572120579119864

1015840119864lowast

times 120572

120579[1198641015840

119864lowast sinh (1205721205791199011015840119901lowast) minus 119901

1015840

119901lowast cosh (1205721205791199011015840119901lowast)]

+1198792 sinh (1205721205791199011015840119901lowast)

(5)

where 120572 and 120579 are given by119898ℎ1198982 and 1119879 respectively

3 Results and Discussions

We employ the minimum 1199092DoF method to fit the exper-

imental data We find that the model calculations results(shown by solid curves in all the cases) fit the experimentaldata quite well (shown by filled circles in all the cases) Theexperimental data are taken from the ALICE Collaborationfor Pb-Pb collisions at radicsNN = 276TeV [13ndash15] We haveshown the (statistical + systematic) errors in all the cases

Over a fairly large 119901119879range the hydrodynamical calcu-

lations show an approximate exponential behavior whereasthe tails of measured spectra show a significant deviation inthe slope beyond 5GeV at LHC At RHIC this transition fromexponential behavior takes place at 119901

119879≳ 3GeV The fraction

of hadronswith very large119901119879(ge3GeV at RHIC andge5GeV at

LHC) is however small We have considered the (maximum)119901119879range up to 5GeV in the present analysis It is because that

the statistical hydrodynamic calculations cannot describethe hadron spectra at such large transverse momenta Thehadrons detected in this region are essentially formed by thepartonswhich are result of the hard processesThese originatefrom the direct fragmentation of high-energy partons of thecolliding beams and therefore are not able to thermalizethrough the process of multiple collisions [16] We thereforeturn to softer hadrons which are assumed to be reasonablythermalized and form the bulk of the secondary matterproduced

The applicability region of hydrodynamics at LHC istherefore predicted to be for 119901

119879le 4-5GeV depending on

the particlersquos massThis range is wider than at RHIC [17]Thetransverse momentum distributions are found to be sensitiveto the values of the thermalkinetic freeze-out temperature 119879and the transverse flow parameter 1205730

119879 whereas it is found to

be insensitive to the change in the values of 120590 in ourmodel Inour analysiswe have therefore fixed the value of the parameter120590 = 50 This value essentially determines the size of thehadronic matter distributed along the 119911-axis and has a strongeffect on the shape of the rapidity spectra of the particlesIn our earlier analysis [2] of the RHIC data the value of 120590turned out to be nearly 42 However it is expected to belarge at the LHC energy The insensitivity of the transversemomentum distribution to the parameter 120590 has been testedand it is found that the minimum 119909

2DoF for protons varies

only from 0611 to 0614 if 120590 is varied from 40 to 60 We havetaken the values of a and b both to be zero for all the hadronsunder the assumption of a baryon symmetricmatter expectedto be formed under the condition of a high degree of nucleartransparency in the nucleus-nucleus collisions at LHC energy


0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

K+

d2N(2120587pTdpTdy)

(1G

eV2)


(a)

0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

Kminus

d2N(2120587pTdpTdy)

(1G

eV2)


(b)

Figure 2 Transverse momentum spectra of119870+ (a) and 119870minus (b) for the centrality class (0ndash5)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1

10

100

K0s

pT (MeV)

1E minus 3

d2N(dpTdy)

(1G

eV)


Figure 3 Transverse momentum spectra of 1198700119904for the centrality

class (0ndash5)

that is an ideal Bjorken picture Unlike the previous workswe have in our present analysis treated the index parameter119899 as a free parameter The values of the parameters 119879 1205730

119879

and 119899 at freeze-out are determined by obtaining a best fit toa given hadronrsquos transverse momentum spectrum The valueof 120585 = 1 is fixed for all the hadrons studied in this paperThe theoretical fits for the transverse momentum spectra ofall the hadrons have been normalized at the first data point(ie at the lowest 119901

119879) to facilitate a proper comparison with

the experimental data setIn Figure 1 we have shown the transverse momen-

tum spectra of protons and antiprotons The values of

0 1000 2000 3000 4000 5000

001

01

1

10Λ

pT (MeV)

d2N(dpTdy)

(1G

eV)


Figure 4 Transverse momentum spectra of lambda Λ for the cen-trality class (0ndash5)

the thermalkinetic freeze-out temperature 119879 the transverseflow parameter 1205730

119879 and the index parameter 119899 for protons as

well as antiprotons are found to be same that is 102MeV088 and 140 respectively with a minimum 119909

2DoF of

061 for protons and 055 for antiprotons The same valuesof the freeze-out parameters for protons and antiprotonsindicate a simultaneous freeze-out of these particles inthe dense hadronic medium

The transverse momentum spectra for119870+ and119870minus shownin Figure 2 gives the value of (119879 1205730

119879 and 119899) as (103MeV

089 and 180) for Kaons and (105MeV 088 and 180) foranti-Kaons The minimum 119909

2DoF for both the two cases


0 1000 2000 3000 4000 5000 6000

001

01

1Ξminus

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

0 1000 2000 3000 4000 5000 6000

001

01

1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ξminus


(b)

Figure 5 Transverse momentum spectra of Ξminus (a) and Ξminus (b) for the centrality class (0ndash10)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01 Ω

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ω

(ALICE (Pb-Pb) 276TeV |y| lt 05)

(b)

Figure 6 Transverse momentum spectra ofΩ (a) and Ω (b) for the centrality class (0ndash10)

turns out to be 034The almost similar freeze-out parametersobtained for protons antiprotons Kaons and anti-Kaonsindicate a near simultaneous freeze-out of these particles

The transverse momentum spectrum of neutral Kaonthat is 1198700

119904 is shown in Figure 3 The values of 119879 1205730

119879 and 119899

obtained from the spectra of 1198700119904are respectively 125MeV

084 and 161 with the minimum 1199092DoF = 170 The

1198700

119904shows a larger thermal freeze-out temperature than the

charged Kaons indicating its earlier freeze-out than119870plusmn

The transverse momentum spectra of hyperons (ie ΛΞminus Ξminus Ω and Ω) are shown in Figures 4 5 and 6 The

spectrum of Λ gives the values of 119879 1205730119879 and 119899 as 127MeV

084 and 106 respectively with a minimum 1199092DoF =

052 These values for Ξminus are found to be 133MeV 081

and 090 while for Ξminus these are 149MeV 080 and 125respectively The parameters for Ω and Ω are (155MeV

077 and 122) and (154MeV 077 and 123) The minimum1199092DoF for Ξminus and Ξminus are 038 and 050 whereas for Ω and

Ω the minimum 1199092DoF are 010 and 020 respectively The

relatively smaller values of minimum 1199092DoF for Ω119904 are due

to larger experimental error barsThe values of the thermalkinetic freeze-out temperature

119879 the transverse flow parameter 1205730119879 and the index parameter

119899 for all the hadrons studied in this paper are again presentedin Table 1 to facilitate a proper comparison

It is evident from Table 1 that the lighter particles thatis (anti)protons and Kaons exhibit a lower thermal freeze-out temperature and a higher surface transverse expansionvelocity compared to the heavy (multi)strange hyperonsThe reason for this can be attributed to an early freeze-out for the massive particles (hyperons) when the thermaltemperature is high and the collective flow is in the early


0 1000 2000 3000 4000 5000

001

01

1

Our modelEPOS Krakow

VISH2 + 1

pT (MeV)

d2N(dpTdy)

(1G

eV)

(Ξminus + Ξminus)2 (0ndash10)ALICE (Pb-Pb) 276TeV |y| lt 05

(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01


VISH2 + 1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

(Ω + Ω)2 (0ndash10)ALICE (Pb-Pb) 276TeV |y| lt 05

(b)

Figure 7 Transverse momentum spectra for (Ξminus +Ξminus)2 (a) and (Ω+Ω)2 (b) are compared to hydrodynamic models EPOS VISH2+1 andKRAKOW for the (0ndash10) most central Pb-Pb collisions atradicsNN = 276TeV

Table 1 Freeze-out parameters of various hadrons obtained fromtheir transverse momentum spectra

Particle 119879 (MeV) 1205730

119879119899 119909

2

DoF119901 102 088 140 061119901 102 088 140 055119870

+ 103 089 180 034119870minus 105 088 180 034

1198700

119904125 084 161 170

Λ 127 084 106 052Ξminus 133 081 090 038

Ξminus 149 080 125 048Ω 155 077 122 010Ω 154 077 123 020

stage of development and consequently 1205730119879is small The early

freeze-out of these particles is due to their smaller cross-sectionwith the hadronicmatterThis can also be understoodin terms of the mean free path 120582 of a particle in a thermalenvironment which is given by 120582 = 1]120588 where ] is the meanthermal cross section of the particle with the surroundingmatter having density 120588 The transverse flow velocity profileindex 119899 using the best fit method for the hadrons consideredhere lies in the range 161ndash180 for the lightest particlesconsidered here (ie Kaons) for (anti)protons it is 140 andfor the heavier multistrange hyperon it lies in the range 090ndash125 Hence in general 119899 appears to be smaller for the heavierparticles Another observation is that for the particles withlower 119899 the 119901

119879spectra show a shoulder-like shape at low

transverse momentaA comparison with a similar fit to the RHIC data [2 6]

shows that for the most central collisions the flow velocityincreases significantly at LHC reaching almost 09 and that

the kinetic freeze-out temperature drops below the one atRHIC For RHIC [2] the values of 119879 and 120573

0

119879were found to

be in the range 163ndash188MeV and 058ndash067 respectivelyWe have also compared the results of some other model

calculations like VISH2+1 HKM and Krakow models [1318] with our model results in Figure 7 for the transversemomentum spectra of (Ξminus+Ξminus)2 and (Ω+Ω)2TheKrakowmodel results are available up to nearly 3GeV for (Ξminus +Ξminus)2and nearly 25 GeV for (Ω + Ω)2

In the first case the KRAKOW model results are seento fit the experimental data reasonably well but provides abad fit for the second case that is (Ω + Ω)2 The VISH2+1and HKM describe the shape of these spectra somewhatbetter however the VISH2+1 overestimates the yield of (Ω +

Ω)2 at larger 119901119879 Also it is found [18] that Krakow model

overestimates the yield of protons up to 175GeV In contrastourmodel successfully reproduces the shape of the transversemomentum distributions of these hadrons in a wider 119901

119879

range up to 5GeV

4 Conclusion

The transverse momentum spectra of the hadrons 119901 119901 119870+119870minus 1198700119904 Λ Ω Ω Ξminus and Ξminus are fitted quite well by using

our model The assumption of vanishing chemical potentialat midrapidity shows the effects of almost complete trans-parency in Pb-Pb collisions at LHC energy of 276 TeV Wealso observe an earlier freeze-out of hyperons as compared tolightermass particles that is Kaons andprotonsTheprotonsantiprotons Kaons and anti-Kaons have similar freeze-outconditions which indicate their near simultaneous freeze-out from the dense hadronic medium The larger valuesof 1205730119879at the LHC energy as compared to those at RHIC

indicates a stronger flow effect present in the system at LHC


The spectra are compared with the predictions from someother hydrodynamic models also and it is found that a betterfit is obtained by using our model covering a wider 119901

119879range

up to 5GeV

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Inam-ul Bashir is thankful to University Grants Commissionfor awarding him the Basic Scientific Research (BSR) Fellow-ship Riyaz Ahmed Bhat is grateful to Council of Scientificand Industrial Research NewDelhi for awarding him SeniorResearch Fellowship Saeed Uddin is grateful to UniversityGrants Commission for financial assistance

References

[1] E Kornas ldquoEnergy dependence of proton and antiproton pro-duction in central Pb+Pb collisions from NA49rdquoThe EuropeanPhysical Journal C vol 49 pp 293ndash296 2007

[2] S Uddin R A Bhat I ul Bashir W Bashir and J S AhmadldquoSystematic of particle thermal freeze-out in a hadronic fireballat RHICrdquo In press httpxxxtauacilabs14010324

[3] U W Heinz ldquoConcepts of heavy-ion physicsrdquo httparxivorgabshep-ph0407360

[4] W Broniowski and W Florkowski ldquoDescription of the RHIC119901perpspectra in a thermal model with expansionrdquo Physical Review

Letters vol 87 Article ID 272302 2001[5] W Broniowski and W Florkowski ldquoDescription of strange

particle production in Au+Au collisions of radicsNN = 130GeVin a single-freeze-out modelrdquo Physical Review C vol 65 ArticleID 064905 2002

[6] D Teaney J Lauret and E V Shuryak ldquoFlow at the SPS andRHIC as a quark-gluon plasma signaturerdquo Physical Review Let-ters vol 86 no 21 pp 4783ndash4786 2001

[7] S Uddin J S AhmadW Bashir and R Ahmad Bhat ldquoA unifiedapproach towards describing rapidity and transverse momen-tum distributions in a thermal freeze-out modelrdquo Journal ofPhysics G vol 39 no 1 Article ID 015012 2012

[8] S Sarkar H Satz and B SinhaThe Physics of the Quark-GluonPlasma vol 785 of Lecture Notes in Physics Spinger BerlinGermany 2010

[9] F Becattini J Cleymans and J Strumpfer ldquoRapidity varia-tion of thermal parameters at SPS and RHICrdquo httparxivorgabs07092599

[10] S Uddin J S Ahmad M Ali W Bashir R A Bhat andM F Mir ldquoLongitudinal hadronic flow at RHIC in extendedstatistical thermal model and resonance decay effectsrdquo ActaPhysica Polonica B vol 41 no 11 pp 2433ndash2448 2010

[11] C Ristea A Jipa I Lazanu et al ldquoHubble flow in relativisticheavy ion collisionsrdquo Journal of Physics Conference Series vol420 no 1 Article ID 012040 2013

[12] S Uddin N Akhttar and M Ali ldquoPion production and collec-tive flow effects in intermediate energy nucleus-nucleus colli-sionsrdquo International Journal of Modern Physics A vol 21 no 7p 1471 2006

[13] B Abelev J Adam and D Adamova ldquoMulti-strange baryonproduction at mid-rapidity in Pb-Pb collisions at radicsNN =

276TeVrdquo Physics Letters B vol 728 pp 216ndash227 2013[14] Iouri Belikov (for the ALICE Collaboration) ldquo1198700

119904and Λ

production in Pb-Pb collisions with the ALICE experimentrdquoJournal of Physics G Nuclear and Particle Physics vol 3 no 12Article ID 124078 2011

[15] B Abelev J Adam andD Adamova ldquoCentrality dependence of120587119870 and p production in Pb-Pb collisions atradicsNN = 276TeVrdquoPhysical Review C vol 88 Article ID 044910 2013

[16] P Huovinen and P V Ruuskanen ldquoHydrodynamic models forheavy ion collisionsrdquo Annual Review of Nuclear and ParticleScience vol 56 pp 163ndash206 2006

[17] K J Eskola H Honkanen H Niemi P V Ruuskanen andS S Rasanen ldquoHadron multiplicities pT-spectra and net-baryon number in central Pb+Pb collisions at the LHCrdquo httparxivorgabs07051770

[18] M Rybczynski W Florkowski and W Broniowski ldquoSingle-freeze-out model for ultrarelativistic heavy-ion collisions atradicsNN = 276TeVrdquo Physical Review C vol 85 no 5 Article ID054907 9 pages 2012

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The identified particle spectra provide information aboutboth the temperature of the system and the collective flow atthe time of thermal freeze-out Collective flow depends onthe internal pressure gradients created in the collision andis addressed by hydrodynamic models [4ndash6] These effectsare species-dependent The produced hadrons are believedto carry information about the collision dynamics and thesubsequent space-time evolution of the system

Hence an accurate measurement of the transverse momen-tum distributions of identified hadrons along with the rapid-ity spectra is essential for the understanding of the dynamicsand the properties of the created matter up to the finalthermal or hydrodynamical freeze-out in case of collectiveflow [7]

It has been shown earlier [2] that this model successfullysimultaneously explains the rapidity and transverse momen-tum distributions of hadrons and their ratios in Au-Aucollisions at highest RHIC energy ofradicsNN = 200GeV In thispaper we briefly describe the model and use it to reproducethe transversemomentumdistributions of hadrons producedin Pb-Pb collisions atradicsNN = 276 TeV

2 Model

In order to obtain the particle spectra in the overall restframe of the hadronic fireball in our model we first define theinvariant cross-section for given hadronic specie in the localrest frame of a hadronic fluid element Since the invariantcross section will have the same value in all Lorentz frames[8] we can thus write

1198641198893119873

1198893119901= 1198641015840 1198893119873

11988931199011015840 (1)

where 119864(1198641015840) is the energy of the particle and 119901(119901

1015840) is the

momentum The primed quantities on the RHS refer to theinvariant spectra of given hadronic specie in the rest frameof the local hadronic fluid element while the unprimedquantities on the LHS refer to the invariant spectra of thesame hadronic specie in the overall rest frame of the hadronicfireball The occupation number distribution of the hadronsin the momentum space follows the distribution function

1198641015840 1198893119873

1198893

1199011015840sim

1198641015840

119890((1198641015840minus120583)119879) plusmn 1

(2)

where (+) sign and (minus) sign are for fermions and bosonsrespectively and 120583 is the chemical potential of the givenhadronic specie For the temperatures under considerationand the large masses of hadrons it is safe to work withBoltzman distribution

In recent works [7 9] it has been clearly shown that thereis a strong evidence of increasing baryon chemical potential120583119861 along the collision axis in the RHIC experiments In view

of this fact we write the expression for chemical potentialas 120583119861

= a + b11991020[7 9 10] where y0 is the rapidity of

the expanding hadronic fluid element Here a and b arethe two model parameters which can be fixed by fitting theexperimental data In the model the value of a essentially

defines the baryon chemical potential in the central regionof the bulk hadronic matter formed while b determinesthe rate of increase of baryon chemical potential along the(longitudinal) rapidity axis with y0 In case of very highdegree of nuclear transparency the values of a and bwill tendto vanish for the bulk of the matter Further it is assumedthat [7] the rapidity of the expanding hadronic fluid element1199100120572119911 or 119910

0= 120585119911 where 119911 is the longitudinal coordinate

of the hadronic fluid element and 120585 is a proportionalityconstant The above conditions also ensure that under thetransformation 119911 rarr minus119911 we will have 119910

0rarr minus119910

0 thereby

preserving the symmetry of the hadronic fluid flow about119911 = 0 along the rapidity axis in the centre of massframe of the colliding nuclei This leads to an expression forthe longitudinal velocity component of the hadronic fluidelement

120573119911(119911) = 1 minus

2

exp (2120585119911) + 1= tanh (119910

0) (3)

The transverse velocity component of the hadronic fireball120573119879is assumed to vary with the transverse coordinate 119903 in

accordance with the blast wave model as 120573119879(119903) = 120573

119904

119879(119903119877)119899

[11] where 119899 is an index which fixes the profile of 120573119879(119903)

in the transverse direction and 120573119904

119879is the hadronic fluid

surface transverse expansion velocity and is fixed in themodel by using the parameterization 120573

119904

119879= 1205730

119879radic1 minus 1205732

119911[7]

This relation is also required to ensure that the net velocity120573 of any fluid element must satisfy 120573 = radic1205732

119879+ 1205732119911

lt 1We also parameterize 119877 that is the transverse radius offireball as 119877 = 119903

0exp(minus11991121205902) where 120590 fixes the width

of the matter distribution in the transverse direction [7 9]and 119911 as described above is the longitudinal coordinateof hadronic fluid element This is required as the collidingnuclei when passing through each other may still feel somedrag thus resulting only in a partial transparency Conse-quently the collision axis will be populated by an extendedhadronic matter rapidly moving away from each other withits transverse size decreasing rapidly following a Gaussiandistribution along the 119911-axis

In our analysis the contributions of various heavierhadronic resonances [10 12] which decay after the thermalfreeze-out of the hadronic matter has occurred are also takeninto accountThe invariant spectrumof a given decay productof a given parent hadron in the local rest frame of a hadronicfluid element is written as [7 10 12]

1198641015840 1198893119873

decay

11988931199011015840=

1

21199011015840119898ℎ

119901lowastint

119864+

119864minus

119889119864ℎ119864ℎ1198893119873ℎ

1198893119901ℎ

(4)

where the subscript ℎ stands for the decaying (parent)hadron The two body decay kinematics gives the producthadronrsquos momentum and energy in the ldquorest frame of thedecaying hadronrdquo as 119901lowast = (119864

lowast2minus 1198982)12 and 119864

lowast= (1198982

ℎminus

1198982

119895+ 1198982)2119898ℎ where 119898

119895indicates the mass of the other

decay hadron produced along with the first one The limitsof integration are 119864

plusmn= 119898ℎ11989821198641015840119864lowast

plusmn 1199011015840119901lowast

The 1198641015840(119864ℎ)

and 1199011015840(119901ℎ) are respectively the product (decaying parent)


0 1000 2000 3000 4000 5000

001

01

1

pT (MeV)

1E minus 3

proton

d2N(2120587pTdpTdy)

(1G

eV2)


(a)

0 1000 2000 3000 4000 5000

001

01

1

pT (MeV)

1E minus 3

antiproton

d2N(2120587pTdpTdy)

(1G

eV2)


(b)



1198641015840 1198893119873

11988931199011015840

=1

21199011015840119898ℎ

119901lowast120582ℎ119892ℎ119890minus120572120579119864

1015840119864lowast

times 120572

120579[1198641015840


1015840


+1198792 sinh (1205721205791199011015840119901lowast)

(5)



















0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

K+

d2N(2120587pTdpTdy)

(1G

eV2)


(a)

0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

Kminus

d2N(2120587pTdpTdy)

(1G

eV2)


(b)


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1

10

100

K0s

pT (MeV)

1E minus 3

d2N(dpTdy)

(1G

eV)



class (0ndash5)


119879





0 1000 2000 3000 4000 5000

001

01

1

10Λ

pT (MeV)

d2N(dpTdy)

(1G

eV)






2DoF of



119879 and 119899) as (103MeV




0 1000 2000 3000 4000 5000 6000

001

01

1Ξminus

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

0 1000 2000 3000 4000 5000 6000

001

01

1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ξminus


(b)


1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01 Ω

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ω


(b)





119879 and 119899



1198700
















0 1000 2000 3000 4000 5000

001

01

1


VISH2 + 1

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01


VISH2 + 1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(b)



Particle 119879 (MeV) 1205730

119879119899 119909

2

DoF119901 102 088 140 061119901 102 088 140 055119870

+ 103 089 180 034119870minus 105 088 180 034

1198700

119904125 084 161 170

Λ 127 084 106 052Ξminus 133 081 090 038

Ξminus 149 080 125 048Ω 155 077 122 010Ω 154 077 123 020







0

119879were found to






119879

range up to 5GeV

4 Conclusion






119879range

up to 5GeV



Acknowledgments


References
















119904and Λ











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 1000 2000 3000 4000 5000

001

01

1

pT (MeV)

1E minus 3

proton

d2N(2120587pTdpTdy)

(1G

eV2)


(a)

0 1000 2000 3000 4000 5000

001

01

1

pT (MeV)

1E minus 3

antiproton

d2N(2120587pTdpTdy)

(1G

eV2)


(b)



1198641015840 1198893119873

11988931199011015840

=1

21199011015840119898ℎ

119901lowast120582ℎ119892ℎ119890minus120572120579119864

1015840119864lowast

times 120572

120579[1198641015840


1015840


+1198792 sinh (1205721205791199011015840119901lowast)

(5)



















0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

K+

d2N(2120587pTdpTdy)

(1G

eV2)


(a)

0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

Kminus

d2N(2120587pTdpTdy)

(1G

eV2)


(b)


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1

10

100

K0s

pT (MeV)

1E minus 3

d2N(dpTdy)

(1G

eV)



class (0ndash5)


119879





0 1000 2000 3000 4000 5000

001

01

1

10Λ

pT (MeV)

d2N(dpTdy)

(1G

eV)






2DoF of



119879 and 119899) as (103MeV




0 1000 2000 3000 4000 5000 6000

001

01

1Ξminus

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

0 1000 2000 3000 4000 5000 6000

001

01

1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ξminus


(b)


1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01 Ω

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ω


(b)





119879 and 119899



1198700
















0 1000 2000 3000 4000 5000

001

01

1


VISH2 + 1

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01


VISH2 + 1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(b)



Particle 119879 (MeV) 1205730

119879119899 119909

2

DoF119901 102 088 140 061119901 102 088 140 055119870

+ 103 089 180 034119870minus 105 088 180 034

1198700

119904125 084 161 170

Λ 127 084 106 052Ξminus 133 081 090 038

Ξminus 149 080 125 048Ω 155 077 122 010Ω 154 077 123 020







0

119879were found to






119879

range up to 5GeV

4 Conclusion






119879range

up to 5GeV



Acknowledgments


References
















119904and Λ











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

K+

d2N(2120587pTdpTdy)

(1G

eV2)


(a)

0 500 1000 1500 2000 2500 3000

001

01

1

10

pT (MeV)

Kminus

d2N(2120587pTdpTdy)

(1G

eV2)


(b)


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1

10

100

K0s

pT (MeV)

1E minus 3

d2N(dpTdy)

(1G

eV)



class (0ndash5)


119879





0 1000 2000 3000 4000 5000

001

01

1

10Λ

pT (MeV)

d2N(dpTdy)

(1G

eV)






2DoF of



119879 and 119899) as (103MeV




0 1000 2000 3000 4000 5000 6000

001

01

1Ξminus

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

0 1000 2000 3000 4000 5000 6000

001

01

1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ξminus


(b)


1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01 Ω

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ω


(b)





119879 and 119899



1198700
















0 1000 2000 3000 4000 5000

001

01

1


VISH2 + 1

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01


VISH2 + 1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(b)



Particle 119879 (MeV) 1205730

119879119899 119909

2

DoF119901 102 088 140 061119901 102 088 140 055119870

+ 103 089 180 034119870minus 105 088 180 034

1198700

119904125 084 161 170

Λ 127 084 106 052Ξminus 133 081 090 038

Ξminus 149 080 125 048Ω 155 077 122 010Ω 154 077 123 020







0

119879were found to






119879

range up to 5GeV

4 Conclusion






119879range

up to 5GeV



Acknowledgments


References
















119904and Λ











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 1000 2000 3000 4000 5000 6000

001

01

1Ξminus

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

0 1000 2000 3000 4000 5000 6000

001

01

1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ξminus


(b)


1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01 Ω

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)

Ω


(b)





119879 and 119899



1198700
















0 1000 2000 3000 4000 5000

001

01

1


VISH2 + 1

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01


VISH2 + 1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(b)



Particle 119879 (MeV) 1205730

119879119899 119909

2

DoF119901 102 088 140 061119901 102 088 140 055119870

+ 103 089 180 034119870minus 105 088 180 034

1198700

119904125 084 161 170

Λ 127 084 106 052Ξminus 133 081 090 038

Ξminus 149 080 125 048Ω 155 077 122 010Ω 154 077 123 020







0

119879were found to






119879

range up to 5GeV

4 Conclusion






119879range

up to 5GeV



Acknowledgments


References
















119904and Λ











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 1000 2000 3000 4000 5000

001

01

1


VISH2 + 1

pT (MeV)

d2N(dpTdy)

(1G

eV)


(a)

1000 1500 2000 2500 3000 3500 4000 4500 5000

001

01


VISH2 + 1

1E minus 3

pT (MeV)

d2N(dpTdy)

(1G

eV)


(b)



Particle 119879 (MeV) 1205730

119879119899 119909

2

DoF119901 102 088 140 061119901 102 088 140 055119870

+ 103 089 180 034119870minus 105 088 180 034

1198700

119904125 084 161 170

Λ 127 084 106 052Ξminus 133 081 090 038

Ξminus 149 080 125 048Ω 155 077 122 010Ω 154 077 123 020







0

119879were found to






119879

range up to 5GeV

4 Conclusion






119879range

up to 5GeV



Acknowledgments


References
















119904and Λ











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119879range

up to 5GeV



Acknowledgments


References
















119904and Λ











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article transverse momentum distributions of

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