review article particle production in ultrarelativistic...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 805413 27 pageshttpdxdoiorg1011552013805413

Review ArticleParticle Production in Ultrarelativistic Heavy-Ion CollisionsA Statistical-Thermal Model Review

S K Tiwari and C P Singh

Department of Physics Banaras Hindu University Varanasi 221005 India

Correspondence should be addressed to S K Tiwari sktiwari4bhugmailcom

Received 13 June 2013 Accepted 23 August 2013

Academic Editor Bhartendu K Singh

Copyright copy 2013 S K Tiwari and C P Singh This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The current status of various thermal and statistical descriptions of particle production in the ultrarelativistic heavy-ion collisionsexperiments is presented in detail We discuss the formulation of various types of thermal models of a hot and dense hadron gas(HG) and themethods incorporated in the implementing of the interactions between hadrons It includes our new excluded-volumemodel which is thermodynamically consistent The results of the above models together with the experimental results for variousratios of the produced hadrons are compared We derive some new universal conditions emerging at the chemical freeze-out ofHG fireball showing independence with respect to the energy as well as the structure of the nuclei used in the collision Further wecalculate various transport properties of HG such as the ratio of shear viscosity-to-entropy using our thermal model and comparewith the results of other models We also show the rapidity as well as transverse mass spectra of various hadrons in the thermal HGmodel in order to outline the presence of flow in the fluid formed in the collision The purpose of this review article is to organizeand summarize the experimental data obtained in various experiments with heavy-ion collisions and then to examine and analyzethem using thermal models so that a firm conclusion regarding the formation of quark-gluon plasma (QGP) can be obtained

1 Introduction

One of the main purposes of various heavy-ion collisionprogrammes running at various places such as relativisticheavy-ion collider (RHIC) at Brookhaven National Labora-tory (BNL) and large hadron collider (LHC) at CERN is tounderstand the properties of strongly interacting matter andto study the possibility of a phase transition from a confinedhot dense hadron gas (HG) phase to a deconfined andorchiral symmetric phase of quark matter called quark-gluonplasma (QGP) [1ndash9] By colliding heavy-ions at ultrarelativis-tic energies such a phase transition is expected to materializeand QGP can be formed in the laboratory Unfortunatelythe detection of the QGP phase is still regarded as an uphilltask However the existence of a new form of a matter calledstrongly interacting QGP (sQGP) has been demonstratedexperimentally [10] There is much compelling evidencefor example elliptic flow high energy densities and verylow viscosity [11] However we do not have supportiveevidence that this fluid is associatedwith the properties quarkdeconfinement andor chiral symmetry restoration which

are considered as direct indications for QGP formation [11]Although various experimental probes have been devised buta clean unambiguous signal has not yet been outlined in thelaboratory So our prime need at present is to propose somesignals to be used for the detection of QGP However for thispurpose understanding the behaviour and the properties ofthe background HG is quite essential because if QGP is notformed matter will continue to exist in the hot and denseHG phase In the ultrarelativistic nucleus-nucleus collisionsa hot and dense matter is formed over an extended region fora very brief time and it is often called a ldquofireballrdquo The quarkmatter in the fireball after subsequent expansion and coolingwill be finally converted into HG phase Recently the studiesof the transverse momentum spectra of dileptons [12ndash21] andhadrons [22ndash27] are used to deduce valuable informationregarding temperature and energy density of the fireball Theschematic diagram for the conjectured space-time evolutionof the fireball formed in the heavy-ion collisions is shownin Figure 1 [28] The space-time evolution consists of fourdifferent stages as follows (i) In the initial stage of collisions

2 Advances in High Energy Physics

BeamBeam Space

Preequilibrium

Equilibrated QGP

Mixed phase

Hadron gas

Freeze-outTime

Figure 1 A schematic diagram of space-time evolution of ultrarel-ativistic heavy-ion collisions

labeled as ldquoPreequilibriumrdquo in Figure 1 processes of parton-parton hard scatterings may predominantly occur in theoverlap region of two colliding nuclei thus depositing a largeamount of energy in the medium The matter is still notin thermal equilibrium and perturbative QCD models candescribe the underlying dynamics as a cascade of freely collid-ing partons The time of the preequilibrium state is predictedto about 1 fmc or less (ii) After the short preequilibriumstage the QGP phase would be formed in which parton-parton andor string-string interactions quickly contributeto attain thermal equilibrium in the medium The energydensity of this state is expected to reach above 3ndash5GeVfm3equivalent to the temperature of 200ndash300MeV The volumethen rapidly expands and matter cools down (iii) If thefirst order phase transition is assumed the ldquomixed phaserdquo isexpected to exist between the QGP and hadron phases inwhich quarks and gluons are again confined into hadrons atthe critical temperature 119879

119888 In the mixed phase the entropy

density is transferred into lower degrees of freedom andtherefore the system is prevented from a fast expansionThis leads to a maximum value in the lifetime of themixed phase which is expected to last for a relatively longtime (120591 gt 10 fmc) (iv) In the hadronic phase the systemkeeps collective expansion via hadron-hadron interactionsdecreasing its temperature Then the hadronic interactionsfreeze after the system reaches a certain size and temperatureand hadrons that freely stream out from the medium are tobe detected There are two types of freeze-out stages Wheninelastic collisions between constituents of the fireball do notoccur any longer we call this a chemical freeze-out stageLater when the elastic collisions also cease to happen in thefireball this stage specifies the thermal freeze-out

Since many experiments running at various places mea-sure the multiplicity ratios and so forth of various hadronsit is necessary to know to which extent the measured hadronyields indicate equilibration The level of equilibration ofparticles produced in these experiments is tested by analyzingthe particle abundances [22 23 29ndash57] or their momentumspectra [22ndash27 37 38 46 47 58] using thermal modelsNow in the first case one establishes the chemical compo-sition of the system while in the second case additional

information on dynamical evolution and collective flow canbe extracted Furthermore study of the variations of multi-plicity of produced particles with respect to collision energythe momentum spectra of particles and ratios of various par-ticles have led to perhaps one of the most remarkable resultscorresponding to high energy strong interaction physics [6]

Recently various approaches have been proposed for theprecise formulation of a proper equation of state (EOS) forhot and dense HG Lacking lattice QCD results for the EOSat finite baryon density 119899

119861 a common approach is to con-

struct a phenomenological EOS for both phases of stronglyinteractingmatter Among those approaches thermalmodelsare widely used and indeed are very successful in describingvarious features of the HG These models are based on theassumption of thermal equilibrium reached in HG A simpleform of the thermal model of hadrons is the ideal hadron gas(IHG) model in which hadrons and resonances are treatedas pointlike and noninteracting particles The introductionof resonances in the system is expected to account for theexistence of attractive interactions among hadrons [59] Butin order to account for the realistic behaviour of HG a shortrange repulsion must also be introduced The importance ofsuch correction is more obvious when we calculate the phasetransition using IHG picture which shows the reappearanceof hadronic phase as a stable configuration in the simpleGibbs construction of the phase equilibrium between the HGand QGP phases at very high baryon densities or chemi-cal potentials This anomalous situation [60ndash62] cannot beexplained because we know that the stable phase at any given(119879 120583119861) is the phase which has a larger pressure Once the

system makes a transition to the QGP phase it is expectedto remain in that phase even at extremely large 119879 and 120583

119861due

to the property of asymptotic freedom of QCD Moreover itis expected that the hadronic interactions become significantwhen hadrons are densely packed in a hot and dense hadrongas One significant way to handle the repulsive interactionbetween hadrons is based on a macroscopic descriptionin which the hadrons are given a geometrical size andhence they will experience a hardcore repulsive interactionwhen they touch each other and consequently a van-derWaals excluded-volume effect is visible As a result theparticle yields are essentially reduced in comparison to thatof IHG model and also the anomalous situation in thephase transitionmentioned above disappears Recentlymanyphenomenological models incorporating excluded-volumeeffect have been widely used to account for the propertiesof hot and dense HG [63ndash76] However these descriptionsusually suffer from some serious shortcomings First mostlythe descriptions are thermodynamically inconsistent becauseone does not have a well-defined partition function orthermodynamical potential (Ω) fromwhich other thermody-namical quantities can be derived for example the baryondensity (119899

119861) = 120597Ω120597120583

119861 Secondly for the dense hadron gas

the excluded-volumemodel violates causality (ie velocity ofsound 119888

119904in the medium is greater than the velocity of light)

So although some of the models explain the data very wellsuch shortcomings make the models mostly unattractiveSun et al [76] have incorporated the effect of relativisticcorrection in the formulation of an EOS for HG However

Advances in High Energy Physics 3

such effect is expected to be very small because the massesof almost all the hadrons present in the hot and dense HGare larger than the temperature of the system so they areusually treated as nonrelativistic particles except pion whosemass is comparable to the temperature but most of the pionscome from resonances whosemasses are again larger than thetemperature ofHG [77] In [78] two-source thermalmodel ofan ideal hadron gas is used to analyze the experimental dataon hadron yields and ratios In this model the two sources acentral core and a surrounding halo are in local equilibriumat chemical freeze-out It has been claimed that the excluded-volume effect becomes less important in the formulation ofEOS for hadron gas in the two-source model

Another important approach used in the formulation ofan EOS for the HG phase is mean-field theoretical models[79ndash82] and their phenomenological generalizations [83ndash85] These models use the local renormalizable Lagrangiandensities with baryonic and mesonic degrees of freedom forthe description of HG phaseThesemodels rigorously respectcausality Most importantly they also reproduce the groundstate properties of the nuclear matter in the low-density limitThe short-range repulsive interaction in these models arisesmainly due to 120596-exchange between a pair of baryons It leadsto the Yukawa potential 119881(119903) = (11986624120587119903) exp(minus119898

120596119903) which

further gives mean potential energy as 119880119861= 1198662119899119861119898120596

It means that 119880119861is proportional to the net baryon density

119899119861 Thus 119880

119861vanishes in the 119899

119861rarr 0 limit In the bary-

onless limit hadrons (mesons) can still approach pointlikebehaviour due to the vanishing of the repulsive interactionsbetween them It means that in principle one can excite alarge number of hadronic resonances at large 119879 This willagain make the pressure in the HG phase larger than thepressure in the QGP phase and the hadronic phase wouldagain become stable at sufficiently high temperature andthe Gibbs construction can again yield HG phase at large119879 In some recent approaches this problem has been curedby considering another temperature-dependent mean-field119880VDW(119899 119879) where 119899 is the sum of particle and antiparticlenumber densities Here119880VDW(119899 119879) represents van-derWaalshardcore repulsive interaction between two particles anddepends on the total number density 119899which is nonzero evenwhen net baryon density 119899

119861is zero in the large temperature

limit [72 73 75] However in the high-density limit thepresence of a large number of hyperons and their unknowncouplings to the mesonic fields generates a large amount ofuncertainty in the EOS of HG in the mean-field approachMoreover the assumption of how many particles and theirresonances should be incorporated in the system is a crucialone in the formulation of EOS in this approach The mean-field models can usually handle very few resonances only inthe description of HG and hence are not as such reliable ones[75]

In this review we discuss the formulation of variousthermal models existing in the literature quite in detailand their applications to the analysis of particle productionin ultrarelativistic heavy-ion collisions We show that it isimportant to incorporate the interactions between hadronsin a consistent manner while formulating the EOS for hot

dense HG For repulsive interactions van-der Waals typeof excluded-volume effect is often used in thermal modelswhile resonances are included in the system to accountfor the attractive interactions We precisely demand thatsuch interactions must be incorporated in the models ina thermodynamically consistent way There are still somethermalmodels in the literaturewhich lack thermodynamicalself-consistency We have proposed a new excluded-volumemodel where an equal hardcore size is assigned to eachtype of baryons in the HG while the mesons are treatedas pointlike particles We have successfully used this modelin calculating various properties of HG such as numberdensity and energy density We have compared our resultswith those of the other models Further we have extractedchemical freeze-out parameters in various thermal modelsby analyzing the particle ratios over a broad energy rangeand parameterized them with the center-of-mass energy Weuse these parameterizations to calculate the particle ratiosat various center-of-mass energies and compare them withthe experimental data We further provide a proposal in theform of freeze-out conditions for a unified description ofchemical freeze-out of hadrons in various thermal modelsAn extension of the thermal model for the study of thevarious transport properties of hadronswill also be discussedWe also analyze the rapidity as well as transverse massspectra of hadrons using our thermal model and examinethe role of any flow existing in the medium by matchingthe theoretical results with the experimental data Thusthe thermal approach indeed provides a very satisfactorydescription of various features of HG by reproducing a largenumber of experimental results covering wide energy rangefrom alternating gradient synchrotron (AGS) to large hadroncollider (LHC) energy

2 Formulation of Thermal Models

Various types of thermal models for HG using excluded-volume correction based on van-der Waals type effect havebeen proposed Thermal models have often used the grandcanonical ensemble description to write the partition func-tion for the system because it suits well for systems withlarge number of produced hadrons [30] andor large volumeHowever for nonrelativistic statistical mechanics the useof a grand canonical ensemble is usually just a matter ofconvenience [86] Furthermore the canonical ensemble canbe used in case of small systems (eg peripheral nucleus-nucleus collisions) and for low energies in case of strangenessproduction [87 88] due to canonical suppression of thephase space Similarly some descriptions also employ isobaricpartition function in the derivation of their HG model Wesuccinctly summarize the features of somemodels as follows

21 Hagedorn Model In the Hagedorn model [67 68] it isassumed that the excluded-volume correction is proportionalto the pointlike energy density 1205980 It is also assumed that thedensity of states of the finite-size particles in total volume 119881can be taken as precisely the same as that of pointlike particlesin the available volume Δ where Δ = 119881 minus Σ

1198941198810

119894and 1198810

119894is the


eigen volume of the 119894th particle in the HG Thus the grandcanonical partition function satisfies the relation

ln119885 (119879 119881 120582) = ln1198850 (119879 Δ 120582) (1)

The sum of eigen volumes Σ1198941198810

119894is given by the ratio of the

invariant cluster mass to the total energy density and 120582 isthe fugacity that is 120582 = exp(120583119879) Hence Σ

1198941198810

119894= 1198644119861 =

1198811205984119861 and the energy density 120598 = Δ1205980119881 1205980 is the energydensity when particles are treated as pointlike Now using theexpression for Δ one finally gets

120598ex119894=

1205980

119894

1 + 12059804119861 (2)

When 12059804119861 ≫ 1 120598 = Σ119894120598ex119894= 4119861 which is obviously the

upper limit for 120598 since it gives the energy density existinginside a nucleon and is usually regarded as the latent heatdensity required for the phase transition Here 119861 representsthe bag constant The expressions of number density andpressure can similarly be written as follows

119899ex119894=

1198990

119894

1 + 12059804119861

119875ex119894=

1198750

119894

1 + 12059804119861

(3)

Here 1198990119894and 1198750

119894are the number density and pressure of

pointlike particles respectively

22 Cleymans-Suhonen Model In order to include van-derWaals type of repulsion between baryons Cleymans andSuhonen [63] assigned an equal hardcore radius to eachbaryon Consequently the available volume for baryons is119881 minus Σ

1198941198731198941198810

119894 here 1198810

119894is the eigen volume of 119894th baryon and

119873119894is the total number As a result the net excluded number

density pressure and the energy density of amulticomponentHG are given as follows

119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

(4)

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

(5)

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

(6)

where 1198990119894 1198750119894 and 1205980

119894are net baryon density pressure and

energy density of pointlike particles respectively and Σ1198941198990

1198941198810

119894

is the fraction of occupied volume Kuono and Takagi [66]modified these expressions by considering the existence ofa repulsive interaction either between a pair of baryons or

between a pair of antibaryons onlyTherefore the expressions(4) (5) and (6) take the folowing forms

119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198990

119894

1 + sum1198990

1198941198810

119894

+ 1198990

119872

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

+ 1198750

119872

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+ 1205980

119872

(7)

where 1198990119894and 1198990

119894are the number density of the pointlike

baryons and antibaryons respectively and 1205980119894(1205980

119894) and 1198750

119894(1198750

119894)

are the corresponding energy density and pressure Similarly1198990

119872 1198750119872 and 1205980

119872are the number density pressure and energy

density of pointlike mesons respectively

23 Rischke-Gorenstein-Stocker-Greiner (RGSG) Model Theabove discussed models possess a shortcoming that they arethermodynamically inconsistent because the thermodynam-ical variables like 119899

119861cannot be derived from a partition

function or thermodynamical potential (Ω) for example119899119861= 120597Ω120597120583

119861 Several proposals have come to correct such

types of inconsistencies Rischke et al [69] have attemptedto obtain a thermodynamically consistent formulation Inthis model the grand canonical partition function 119885

119866for

pointlike baryons can be written in terms of canonicalpartition function 119885

119862as follows

1198850

119866(119879 120583 119881) =

infin

sum

119873=0

exp(120583119873119879)119885119862 (119879119873119881) (8)

They furthermodified the canonical partition function119885119862by

introducing a step function in the volume so as to incorporateexcluded-volume correction into the formalism Thereforethe grand canonical partition function (8) finally takes thefollowing form

119885ex119866(119879 120583 119881 minus 119881

0119873)

=

infin

sum

119873=0

exp(120583119873119879)119885119862(119879119873119881 minus 119881

0119873)

times 120579 (119881 minus 1198810119873)

(9)

The above ansatz ismotivated by considering119873 particles witheigen-volume 1198810 in a volume 119881 as 119873 pointlike particles inthe available volume 119881 minus 1198810119873 [69] But the problem in thecalculation of (9) is the dependence of the available volumeon the varying number of particles119873 [69] To overcome thisdifficulty one should use the Laplace transformation of (9)Using the Laplace transform one gets the isobaric partitionfunction as follows

119885119875= int

infin

0

119889119881 exp (minus120585119881)119885ex119866(119879 120583 119881 minus 119881

0119873) (10)


or

119885119875= int

infin

0

119889119909 expminus119909[120585 minusln1198850119866(119879 120583)

119909] (11)

where 119909 = 119881 minus 1198810119873 and 120583 = 120583 minus 1198791198810120585 Finally we get atranscendental type of equation as follows

119875ex(119879 120583) = 119875

0(119879 120583) (12)

where

120583 = 120583 minus 1198810119875ex(119879 120583) (13)

The expressions for the number density entropy density andenergy density in thismodel can thus take a familiar form like

119899ex(119879 120583) =

1205971198750(119879 120583)

120597120583

120597120583

120597120583=

1198990(119879 120583)

1 + 11988101198990 (119879 120583)

119904ex1(119879 120583) =

1205971198750(119879 120583)

120597119879=

1199040

1(119879 120583)

1 + 11988101198990 (119879 120583)

120598ex(119879 120583) =

1205980(119879 120583)

1 + 11988101198990 (119879 120583)

(14)

These equations resemble (4) and (6) as given in Cleymans-Suhonen model [63] with 120583 being replaced by 120583 The abovemodel can be extended for a hadron gas involving severalbaryonic species as follows

119875ex(119879 1205831 120583

ℎ) =

ℎ

sum

119894=1

1198750

119894(119879 120583119894) (15)

where

120583119894= 120583119894minus 1198810

119894119875ex(119879 120583119894) (16)

with 119894 = 1 ℎ Particle number density for the 119894th speciescan be calculated from following equation

119899ex119894(119879 120583119894) =

1198990

119894(119879 120583119894)

1 + sumℎ

119895=11198810

1198951198990

119895(119879 120583119895)

(17)

Unfortunately the above model involves cumbersome tran-scendental expressions which are usually not easy to calcu-late Furthermore this model fails in the limit of 120583

119861= 0

because 120583119861becomes negative in this limit

24 New Excluded-Volume Model Singh et al [70] haveproposed a thermodynamically consistent excluded-volumemodel in which the grand canonical partition function usingBoltzmann approximation can be written as follows

ln119885ex119894=119892119894120582119894

61205872119879int

119881minussum1198951198731198951198810119895

1198810119894

119889119881

times int

infin

0

1198964119889119896

radic1198962 + 1198982

119894

exp(minusradic1198962 + 119898

2

119894

119879)

(18)

where 119892119894and 120582

119894= exp(120583

119894119879) are the degeneracy factor

and the fugacity of 119894th species of baryons respectively 119896is the magnitude of the momentum of baryons and 1198810

119894

is the eigenvolume assigned to each baryon of 119894th specieshencesum

1198951198731198951198810

119895becomes the total occupied volume where119873

119895

represents the total number of baryons of 119895th species We canrewrite (18) as follows

ln119885ex119894= 119881(1 minussum

119895

119899ex1198951198810

119895)119868119894120582119894 (19)

where integral 119868119894is

119868119894=119892119894

21205872(119898119894

119879)

2

11987931198702(119898119894

119879) (20)

Thuswe have obtained the grand canonical partition functionas given by (19) by incorporating the excluded-volume effectexplicitly in the partition function The number densityof baryons after excluded-volume correction (119899ex

119894) can be

obtained as

119899ex119894=120582119894

119881(120597 ln119885ex

119894

120597120582119894

)

119879119881

(21)

So our prescription is thermodynamically consistent andit leads to a transcendental equation

119899ex119894= (1 minus 119877) 120582119894119868119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

(22)

Here 119877 = sum119894119899ex1198941198810

119894is the fractional occupied volume It is

clear that if we put the factor 120597119877120597120582119894= 0 and consider only

one type of baryons in the system then (22) can be reduced tothe thermodynamically inconsistent expression (4)The pres-ence of 120597119877120597120582

119894in (22) thus removes the thermodynamical

inconsistency For single-componentHG the solution of (22)can be taken as follows

119899ex=1

119881

int120582

0119889120582 exp [minus11198681198810120582]120582 exp [minus11198681198810120582]

(23)

For a multicomponent hadron gas (22) takes the followingform

119877 = (1 minus 119877)sum

119894

1198681198941198810

119894120582119894minussum

119894

1198681198941198810

1198941205822

119894

120597119877

120597120582119894

(24)

Using the method of parametric space [89] we write

120582119894 (119905) =

1

(119886119894+ 1198681198941198810

119894119905) (25)

where 119886119894is the parameter and 119905 gives the space We finally get

the solution of (24) as follows

119877 = 1 minus

intinfin

119905[exp (minus1199051015840) 119866 (1199051015840)] 1198891199051015840

exp (minus119905) 119866 (119905) (26)


where 119905 is a parameter such that

119889120582119894(119905)

119889119905= minus1198681198941205822

1198941198810

119894

119866 (119905) = 119905

ℎ

prod

119894=2

(119886119894+ 1198681198941198810

119894119905)

(27)

If 120582119894rsquos and 119905 are known one can determine 119886

119894 The quantity 119905

is fixed by setting 1198861= 0 and one obtains 119905 = 1119868

111988111205821 here

the subscript 1 denotes the nucleon degree of freedom andℎ is the total number of baryonic species Hence by using 119877and 120597119877120597120582

119894one can calculate 119899

119894 It is obvious that the above

solution is not unique since it contains some parameterssuch as 119886

119894 one of which has been fixed to zero arbitrarily

Alternatively one can assume that [70]

120597119877

120597120582119894

=

120597sum119895119899ex1198951198810

119895

120597120582119894

= (120597119899

ex119894

120597120582119894

)1198810

119894 (28)

Here an assumption is made that the number density of 119894thbaryonwill only depend on the fugacity of same baryonThen(24) reduces to a simple form as

120597119899ex119894

120597120582119894

+ 119899ex119894(1

1198681198941198810

1198941205822

119894

+1

120582119894

) =1

1205821198941198810

119894

(1 minus sum

119894 = 119895

119899ex1198951198810

119895)

(29)

The solution of (29) can be obtained in a straight forwardmanner as follows [70]

119899ex119894=

119876119894(1 minus sum

119895 = 119894119899ex1198951198810

119895)

1205821198941198810

119894

exp( 1

1198681198941198810

119894120582119894

) (30)

where

119876119894= int

120582119894

0

exp(minus 1

1198681198941198810

119894120582119894

)119889120582119894 (31)

Now 119877 can be obtained by using the following relation

119877 = sum

119895

119899ex1198951198810

119895=119883

1 + 119883 (32)

where

119883 =sum119894119899ex1198941198810

119894

1 minus sum119894119899ex1198941198810

119894

(33)

Here 119883 is the ratio of the occupied volume to the availablevolume Finally 119899ex

119894can be written as

119899ex119894=(1 minus 119877)

1198810

119894

119876119894

120582119894exp (minus1119868

1198941198810

119894120582119894) minus 119876119894

(34)

The solution obtained in this model is very simple and easyThere is no arbitrary parameter in this formalism so itcan be regarded as a unique solution However this theorystill depends crucially on the assumption that the number

density of 119894th species is a function of the 120582119894alone and it is

independent of the fugacities of other kinds of baryons Asthe interactions between different species become significantin hot and dense HG this assumption is no longer validMoreover one serious problem crops up since we cannotdo calculation in this model for 119879 gt 185MeV (and 120583

119861gt

450MeV) This particular limiting value of temperatureand baryon chemical potential depends significantly on themasses and the degeneracy factors of the baryonic resonancesconsidered in the calculation

In order to remove above discrepanciesMishra and Singh[32 33] have proposed a thermodynamically consistent EOSfor a hot and dense HG using Boltzmannrsquos statistics Theyhave proposed an alternative way to solve the transcendentalequation (22) We have extended this model by using quan-tum statistics into the grand canonical partition functionso that our model works even for the extreme values oftemperature and baryon chemical potentialThus (20) can berewritten as follows [31]

119868119894=119892119894

61205872119879int

infin

0

1198964119889119896

radic1198962 + 1198982

119894

1

[exp (119864119894119879) + 120582

119894] (35)

and (22) takes the following form after using the quantumstatistics in the partition function

119899ex119894= (1 minus 119877) 119868119894120582119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

+ 1205822

119894(1 minus 119877) 119868

1015840

119894 (36)

where 1198681015840119894is the partial derivative of 119868

119894with respect to 120582

119894 We

can write 119877 in an operator equation form as follows [32 3390 91]

119877 = 1198771+ Ω119877 (37)

where 1198771= 1198770(1 + 119877

0) with 1198770 = sum1198990

1198941198810

119894+ sum 1198681015840

1198941198810

1198941205822

119894 1198990119894is

the density of pointlike baryons of 119894th species and theoperator Ω has the following form

Ω = minus1

1 + 1198770sum

119894

1198990

1198941198810

119894120582119894

120597

120597120582119894

(38)

Using theNeumann iterationmethod and retaining the seriesupto Ω2 term we get

119877 = 1198771+ Ω1198771+ Ω21198771 (39)

After solving (39) we finally get the expression for totalpressure [70] of the hadron gas as follows

119875ex= 119879 (1 minus 119877)sum

119894

119868119894120582119894+sum

119895

119875meson119895 (40)

Here 119875meson119895

is the pressure due to 119895th type of mesonHere we emphasize that we consider the repulsion arising

only between a pair of baryons andor antibaryons becausewe assign each of them exclusively a hardcore volume Inorder to make the calculation simple we have taken anequal volume 1198810 = 412058711990333 for each type of baryons with


a hardcore radius 119903 = 08 fm We have considered in ourcalculation all baryons and mesons and their resonanceshaving masses up to a cutoff value of 2GeVc2 and lying inthe HG spectrum Here only those resonances which possesswell-defined masses and widths have been incorporated inthe calculations Branching ratios for sequential decays havebeen suitably accounted and in the presence of several decaychannels only dominant mode is included We have alsoimposed the condition of strangeness neutrality strictly byputtingsum

119894119878119894(119899119904

119894minus119899119904

119894) = 0 where 119878

119894is the strangeness quantum

number of the 119894th hadron and 119899119904119894(119899119904

119894) is the strange (anti-

strange) hadron density respectively Using this constraintequation we get the value of strange chemical potential interms of 120583

119861 Having done all these things we proceed to

calculate the energy density of each baryon species 119894 by usingthe following formula

120598ex119894=1198792

119881

120597 ln119885ex119894

120597119879+ 120583119894119899ex119894 (41)

Similarly entropy density is

119904 =120598ex119894+ 119875

exminus 120583119861119899119861minus 120583119878119899119878

119879 (42)

It is evident that this approach is more simple in compari-son to other thermodynamically consistent excluded-volumeapproaches which often possess transcendental final expres-sions [69 86] Our approach does not involve any arbitraryparameter in the calculation Moreover this approach can beused for extremely low as well as extremely large values of119879 and 120583

119861 where all other approaches fail to give a satisfying

result since we do not use Boltzmannrsquos approximation in ourcalculation

3 Statistical and ThermodynamicalConsistency

Recently question of statistical and thermodynamical consis-tency in excluded-volume models for HG has widely beendiscussed [77] In this section we reexamine the issue ofthermodynamical and statistical consistency of the excluded-volume models In RGSG model [69] the single particlegrand canonical partition function (9) can be rewritten asfollows

119885ex119866(119881 119879 120583) =

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(43)

where

119911 (119879) =119892

21205872int

infin

0

1198962119889119896 exp[

[

minus

(1198962+ 1198982)12

119879

]

]

(44)

Here in this model119873 in the available volume (119881 minus 1198810119873) isindependent of 120583 Differentiating (43) with respect to 120583 weget the following equation

120597119885ex119866

120597120583=

infin

sum

119873=0

119873

119879exp(

120583119873

119879)

(119881 minus 1198810119873)119873

119873120579 (119881 minus 119881

0119873)119911119873

(45)

Multiplying both sides of (45) by 119879119885ex119866 we get

119879

119885ex119866

120597119885ex119866

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(46)

We know that the expressions for statistical and thermody-namical averages of number of baryons are as follows

⟨119873⟩ =1

119885ex119866

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

119873 = 119879120597 ln119885ex

119866

120597120583=119879

119885ex119866

120597119885ex119866

120597120583

(47)

respectively Using (47) in (46) we get [77]

119873 = ⟨119873⟩ (48)

Thus we see that in RGSG model thermodynamical averageof number of baryons is exactly equal to the statistical averageof number of baryons Similarly in this model we can showthat

119864 = ⟨119864⟩ (49)

Now we calculate the statistical and thermodynamicalaverages of number of baryons in our excluded-volumemodel The grand canonical partition function in our model(ie (18)) can take the following form

119885ex(119881 119879 120583) =

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873 (50)

where 119911 is given by (44) We use Boltzmannrsquos statistics for thesake of convenience and consider only one species of baryonsIn our model119873 present in the available volume (119881 minus 1198810119873)is 120583 dependent However for multicomponent system onecannot use ldquofixed 119873rdquo because in this case van-der Waalsapproximation is not uniquely defined [92 93] So we useaverage 119873 in our multicomponent grand partition functionHowever at high temperatures it is not possible to use onecomponent van-derWaals description for a system of variousspecies with different masses [92 93] Now differentiating(50) with respect to 120583 we get

120597119885ex

120597120583=

infin

sum

119873=0

119873

119879exp (

120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(51)


Multiplying both sides of (51) by 119879119885ex we get

119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)

Using the definitions (47) (52) can take the following form

119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)

Here 119899 is the thermal average of number density of baryons1198990 is the number density of pointlike baryons and

⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)

The second term in (54) is the redundant one and arisesbecause 119873 present in the available volume (119881 minus 1198810119873)is a function of 120583 We call this term ldquocorrection termrdquo InFigure 2 we have shown the variation of thermodynam-ical average of the number density of baryons and theldquocorrection termrdquo with respect to 119879 at 120583

119861= 400MeV We

see that there is an almost negligible contribution of thisldquocorrection termrdquo to thermodynamical average of numberdensity of baryons Although due to this ldquocorrection termrdquothe statistical average of the number density of baryonsis not exactly equal to its thermodynamical average thedifference is so small that it can be neglected Similarly wecan show that such redundant terms appear while calculatingstatistical average of energy density of baryons and arise dueto the temperature dependence of 119873 Such terms again givenegligible contribution to thermodynamical average of theenergy density of baryons Here we see that the statisticaland thermodynamical averages of physical quantities such asnumber density and energy density are approximately equalto each other in our model also Thus our excluded-volumemodel is not exactly thermodynamically consistent but it cansafely be taken as consistent because the correction term inthe averaging procedure appears as negligibly small

60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV

n (1fm3)Correction term (1fm3)

Figure 2 Variation of thermodynamical average of the numberdensity of baryons and the ldquocorrection termrdquo with respect to 119879 atconstant 120583

119861

4 Comparisons between Model Results andExperimental Data

In this section we review various features of hadron gas andpresent our comparisons between the results of various HGmodels and the experimental data

41 Chemical Freeze-Out Criteria Thermal models providea systematic study of many important properties of hot anddense HG at chemical freezeout (where inelastic interactionscease) To establish a relation between chemical freezeoutparameters (119879 120583

119861) and radic119904119873119873 a common method is used to

fit the experimental hadron ratios Many papers [30 49 5094ndash96] have appeared in which 119879 and 120583

119861are extracted in

terms of radic119904119873119873 In [30] various hadron ratios are analyzedfrom radic119904119873119873 = 27GeV to 200GeV and chemical freeze-outparameters are parameterized in terms of radic119904119873119873 by usingfollowing expressions

119879 (MeV) = 119879lim (1 minus1

119886 + (exp (radic119904119873119873 (GeV)) minus 119887) 119888)

120583119861=

119889

1 + 119890radic119904119873119873

(56)

Here 119886 119887 119888 119889 119890 and 119879lim (the limiting temperature) arefitting parameters Various authors [94 95] have includedthe strangeness suppression factor (120574

119904) in their model while

extracting the chemical freeze-out parameters In the thermalmodel 120574

119904is used to account for the partial equilibration of the

strange particles Such situation may arise in the elementary


120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210

Andronic et al (dNdy)

Andronic et al (4120587)Becattini et al (4120587)

Letessier et al (4120587)Dumitru et al (4120587)

radicsNN (GeV)

radicsNN (GeV)

Figure 3The energy dependence of chemical freeze-out temperature and baryon chemical potential in various studies Figure is taken from[30]

p-p collisions andor peripheral A-A collisions and mostlythe use of 120574

119904is warranted in such cases [94 97] Moreover

120574119904asymp 1 has been found in the central collisions at RHIC

[98 99] We do not require 120574119904in our model as an additional

fitting parameter because we assume that strangeness is alsofully equilibrated in the HG Also it has been pointed outthat inclusion of 120574

119904in thermal model analysis does not

affect the values of fitting parameters 119879 and 120583119861much [30]

Dumitru et al [96] have used inhomogeneous freeze-outscenario for the extraction of 119879 and 120583

119861at various radic119904119873119873

In a recent paper [100] condition of vanishing value of1205811205902 or equivalently 119898

4= 3120594

2 is used to describe thechemical freezeout line where 120581 120590119898

4 and 120594 are kurtosis the

standard deviation fourth order moment and susceptibilityrespectively In [101] first time experimental data on 1205811205902and 119878120590 here 119878 is skewness has been compared with thelattice QCD calculations and hadron resonance gas model todetermine the critical temperature (119879

119888) for the QCD phase

transition Recently it is shown that the freeze-out parametersin heavy-ion collisions can be determined by comparingthe lattice QCD results for the first three cumulants of netelectric charge fluctuations with the experimental data [102]In Figure 3 we have shown the energy dependence of thermalparameters 119879 and 120583

119861extracted by various authors In all the

studies similar behaviour is found except in the Letessierand Rafelski [95] which may be due to usage of manyadditional free parameters such as light quark occupancyfactor (120574

119902) and an isospin fugacity We have also extracted

freeze-out parameters by fitting the experimental particle-ratios from the lowest SIS energy to the highest RHIC energyusing our model [31] For comparison we have shown thevalues obtained in other models for example IHG modelCleymans-Suhonen model and RGSG model in Table 1 Wethen parameterize the variables 119879 and 120583

119861in terms ofradic119904119873119873 as

follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)

where the parameters 119886 119887 119888 119889 and 119890 have been determinedfrom the best fits 119886 = 1482 plusmn 00037GeV 119887 = 03517 plusmn0009GeVminus1 119888 = 0163 plusmn 00021GeV 119889 = 0170 plusmn

002GeVminus1 and 119890 = 0015plusmn001GeVminus3 The systematic errorof the fits can be estimated via quadratic deviation 1205752 [30]defined as follows

1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894

are the experimental data and thermalmodel result of either the hadron yield or the ratio of hadronyields respectively

In this analysis we have used full phase space (4120587)data at all center-of-mass energies except at RHIC energieswhere only midrapidity data are available for all the ratiosMoreover the midrapidity and full phase space data at these


Table 1 Thermal parameters (119879 120583119861) values obtained by fitting the experimental particle ratios in different model calculations

radic119904NN (GeV) IHG model RGSG model Cleymans-Suhonen model Our model119879 120583

1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)

1 10210radicsNN (GeV)

Figure 4 Variation of baryon chemical potential with respect tocentre-of-mass energy in our model

energies differ only slightly as pointed out by Alt and NA49Collaboration for119870+120587+ and119870minus120587minus ratios [104] In Figure 4we have shown the parametrization of the freeze-out valuesof baryon chemical potential with respect to radic119904119873119873 andsimilarly in Figure 5 we have shown the chemical freeze-outcurve between temperature and baryon chemical potential[31]

42 Hadron Ratios In an experimental measurement ofvarious particle ratios at various centre-of-mass energies[105ndash111] it is found that there is an unusual sharp variationin the Λ120587minus ratio increasing up to the peak value Thisstrong variation of the ratio with energy indicates the criticaltemperature of QCD phase transition [112] between HG andQGP [113 114] and a nontrivial information about the criticaltemperature 119879

119862asymp 176MeV has been extracted [114] Figure 6

shows the variation of Λ120587minus with radic119904119873119873 We compare theexperimental data with various thermal models [31 63 69]

0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)

Figure 5 Variation of chemical freeze-out temperature with respectto baryon chemical potential in our model

and find that our model calculation gives much better fit tothe experimental data in comparison to other models Weget a sharp peak around centre-of-mass energy of 5GeVand our results thus almost reproduce all the features of theexperimental data

In Figure 7 we have shown the variations of 120601120587 ratiowith radic119904119873119873 The 120601 yields in the thermal models are oftenmuch higher in comparison to data We notice that nothermal model can suitably account for the multiplicity ratioof multistrange particle since 120601 is 119904119904 hidden-strange quarkcombination However quark coalescence model assuminga QGP formation has been claimed to explain the results[115 116] successfully In the thermal models the results forthe multistrange particles raise doubt over the degree ofchemical equilibration for strangeness reached in the HGfireball We can use an arbitrary parameter 120574

119904as used in

several models The failures of thermal models in these casesmay indicate the presence of QGP formation but it is stillnot clear In Figure 8 we have shown the energy dependenceof antiparticle to particle ratios for example 119870minus119870+ 119901119901


1 100

005

01

015

02

025

AGS (Au-Au)SPS (Pb-Pb)RHIC (Au-Au)

Our modelCleymans-Suhonen modelRGSG model

102 103 104

Λ120587

minus

radicsNN (GeV)

Figure 6 The energy dependence of the Λ120587minus ratio Lines arethe results of various thermal models [31 63 69] Points are theexperimental data [104 117ndash121] RHIC data are at midrapidity

ΛΛ and Ξ+Ξminus These ratios increase sharply with respecttoradic119904119873119873 and then almost saturate at higher energies reachingthe value equal to 10 at LHC energy On comparison withthe experimental data we find that almost all the thermalmodels describe these data successfully at all the center-of-mass energies However RGSG model [69] fails to describethe data at SPS and RHIC energies in comparison to othermodels [31]

43 Thermodynamical Properties We present the thermalmodel calculations of various thermodynamical properties ofHG such as entropy per baryon (119904119899

119861) and energy density

and compare the results with the predictions of amicroscopicmodel URASiMA event generator developed by Sasaki [122]URASiMA (ultrarelativistic AA collision simulator basedon multiple scattering algorithm) is a microscopic modelwhich includes the realistic interactions between hadronsIn URASiMA event generator molecular-dynamical simu-lations for a system of a HG are performed URASiMAincludes the multibody absorptions which are the reverseprocesses of multiparticle production and are not included inany other model Although URASiMA gives a realistic EOSfor hot and dense HG it does not include antibaryons andstrange particles in their simulation which is very crucial InFigure 9 we have plotted the variation of 119904119899

119861with respect

to temperature (119879) at fixed net baryon density (119899119861) 119904119899

119861

calculated in our model shows a good agreement with theresults of Sasaki [122] in comparison to other excluded-volume models It is found that thermal model approachwhich incorporates macroscopic geometrical features gives

0

5

10

15


Our modelRGSG modelCleymans-Suhonen model

1 10 102 103 104

1000

120601120587

radicsNN (GeV)

Figure 7 The energy dependence of 120601120587 ratio Lines are the resultsof various thermal models [31 63 69] Points are the experimentaldata [104 117ndash121] RHIC data are at midrapidity

a close results with the simulation involving microscopicinteractions between hadrons There are various parameterssuch as coupling constants of hadrons appear in URASiMAmodel due to interactions between hadrons It is certainlyencouraging to find an excellent agreement between theresults obtained with two widely different approaches

Figure 10 represents the variation of the energy density ofHGwith respect to119879 at constant 119899

119861 Again ourmodel calcula-

tion is more closer to the result of URASiMA in comparisonto other excluded-volume models Energy density increasesvery slowly with the temperature initially and then rapidlyincreases at higher temperatures

44 Causality One of the deficiencies of excluded-volumemodels is the violation of causality in the hot and densehadron gas that is the sound velocity 119888

119904is larger than the

velocity of light 119888 in themedium In other words 119888119904gt 1 in the

unit of 119888 = 1 means that the medium transmits informationat a speed faster than 119888 [74] Since in this paper we arediscussing the results of various excluded-volume modelsit would be interesting to see whether these models respectcausality or not In Figure 11 we have plotted the variationsof the total hadronic pressure 119875 as a function of the energydensity 120598 of the HG at a fixed entropy per particle usingour model calculation [31] We find for a fixed 119904119899 that thepressure varies linearly with respect to energy density InFigure 12 we have shown the variation of 119888

119904(1198882119904= 120597119875120597120598 at

fixed 119904119899) with respect to 119904119899 We find that 119888119904le 058 in our

model with interacting particles We get 119888119904= 058 (ie 1radic3)

for an ideal gas consisting of ultrarelativistic particles This


0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104

SPS (Pb-Pb)RHIC (Au-Au)Our model

RGSG modelCleymans-Suhonen model

AGS (Au-Au)SPS (Pb-Pb)RHIC (Au-Au)LHC (Pb-Pb)


Ξ+Ξ

minusAGS (Au-Au)SPS (Pb-Pb)RHIC (Au-Au)LHC (Pb-Pb)




radicsNN (GeV)radicsNN (GeV)

radicsNN (GeV) radicsNN (GeV)

Figure 8 The energy dependence of antihadron to hadron ratios Points are the experimental data [104 117ndash121] and lines are results ofvarious models RHIC and LHC data are at midrapidity

feature endorses our viewpoint that our model is not onlythermodynamically consistent but it does not also involveany violation of causality even at large density Similarly inRGSG model [69] we do not notice that the value of 119888

119904

exceeds 1 as shown in Figure 12 It should be mentioned thatwe are using full quantum statistics in all the models takenfor comparisons here However we find that the values in

the RGSG model cannot be extracted when temperature ofthe HG exceeds 250MeV No such restriction applies for ourmodel

45 Universal Freeze-Out Criteria One of the most remark-able successes of thermal models is in explaining the multi-plicities and the particle ratios of various particles produced


0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB

Our model (nB = 0157 fmminus3)RGSG model (nB = 0157 fmminus3)Cleymans-Suhonen model (nB = 0157 fmminus3)Data points from URASiMA (nB = 0157 fmminus3)

Figure 9 Variation of 119904119899119861with respect to temperature at constant

net baryondensity Lines show the results of various thermalmodelsand points are the data calculated by Sasaki using URASiMA eventgenerator

in heavy-ion experiments from the lowest SIS energy tomaximum LHC energy Some properties of thermal fireballare found to be common to all collision energies which give auniversal freeze-out conditions in heavy-ion collisions Nowwe review the applicability of thermal models in derivingsome useful chemical freeze-out criteria for the fireballRecent studies [39 48 103 123ndash125] predict that the followingempirical conditions are to be valid on the entire freeze-outhypersurface of the fireball (i) energy per hadron always hasa fixed value at 108GeV (ii) sum of baryon and anti-baryondensities is 119899

119861+ 119899119861= 012fm3 (iii) normalized entropy

density is 1199041198793 asymp 7 Further Cleymans et al [103] have foundthat all the above conditions separately give a satisfactorydescription of the chemical freeze-out parameters 119879 and 120583

119861

in an IHG picture only Moreover it was also found thatthese conditions are independent of collision energy and thegeometry of colliding nuclei Furthermore Cleymans et al[103] have hinted that incorporation of excluded-volumecorrection leads to wild as well as disastrous effects on theseconditions The purpose in this section is to reinvestigatethe validity of these freeze-out criteria in excluded-volumemodels Along with these conditions a condition formulatedby using percolation theory is also proposed as a chemicalfreeze-out condition [124] An assumption is made that inthe baryonless region the hadronic matter freezes out dueto hadron resonances and vacuum percolation while inthe baryon rich region the freeze-out takes place due tobaryon percolation Thus the condition which describes the

0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)

Figure 10 Variation of energy densitywith respect to temperature atconstant net baryon density Lines are the results of various thermalmodels and points are the data calculated by Sasaki usingURASiMAevent generator

chemical freeze-out line is formulated by following equation[124]

119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)

where 119881ℎis the volume of a hadron The numbers 124 and

034 are obtained within percolation theory [126]In Figure 13 we have shown the variation of 119864119873 with

respect to radic119904119873119873 at the chemical freeze-out point of thefireball The ratio 119864119873 shows a constant value of 10 in ourmodel and it shows also a remarkable energy independenceSimilarly the curve in IHG model shows that the value for119864119873 is slightly larger than the one as reported in [103]However results support the finding that 119864119873 is almostindependent of energy and also of the geometry of thenuclei Most importantly we notice that the inclusion ofthe excluded-volume correction does not change the resultmuch which is contrary to the claim of Cleymans et al [103]The condition 119864119873 asymp 10GeV was successfully used in theliterature to make predictions [45] of freeze-out parametersat SPS energies of 40 and 80A GeV for Pb-Pb collisionslong before the data were taken [31] Moreover we have alsoshown in Figure 13 the curves in the Cleymans-Suhonenmodel [63] and the RGSG model [69] and we notice asmall variation with radic119904119873119873 particularly at lower energiesIn Figure 14 we study a possible new freeze-out criterionwhich was not proposed earlier We show that the quantityentropy per particle that is 119878119873 yields a remarkable energy


05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20

Energy density of HG (GeVfm3)

Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112

Figure 11 Variations of total hadronic pressure with respect toenergy density of HG at fixed entropy per particle 119904119899 Ourcalculations show linear relationship and slope of the lines givessquare of the velocity of sound 1198882

119904[31]

independence in our model calculation The quantity 119878119873 asymp70 describes the chemical freeze-out criteria and is almostindependent of the centre-of-mass energy in our modelcalculation However the results below radic119904119873119873 = 6GeV donot give a promising support to our criterion and reveal someenergy dependence also This criterion thus indicates thatthe possible use of excluded-volume models and the thermaldescriptions at very low energies is not valid for the HG Sim-ilar results were obtained in the RGSG Cleymans-Suhonenand IHG models also [31] The conditions that is 119864119873 asymp10 GeV and 119878119873 asymp 70 at the chemical freeze-out form aconstant hypersurface from where all the particles freeze outand all kinds of inelastic collisions cease simultaneously andfly towards the detectors Thus all particles attain thermalequilibrium at the line of chemical freeze-out and when theycome out from the fireball they have an almost constantenergy per particle (asymp10) and entropy per particle (asymp70)Moreover these values are independent of the initial collisionenergy as well as the geometry of the colliding nuclei

Our finding lends support to the crucial assumption ofHG fireball achieving chemical equilibrium in the heavy-ion collisions from the lowest SIS to RHIC energy andthe EOS of the HG developed by us indeed gives a properdescription of the hot and dense fireball and its subsequentexpansion However we still do not get any informationregarding QGP formation from these studies The chemicalequilibrium once attained by the hot and dense HG removesany memory regarding QGP existing in the fireball beforeHG phase Furthermore in a heavy-ion collision a largeamount of kinetic energy becomes available and part of it is

01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)

Figure 12 Variation of velocity of sound in the hot dense HGmedium with respect to entropy per particle (119904119899)

06

08

1

12

14

Our modelCleymans-Suhonen model

RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)

Figure 13 Variation of 119864119873 with radic119904119873119873 IHG model calculation isshown by dash-dotted line Cleymans-Suhonen and RGSG modelscalculations are shownbydashed anddotted lines respectively Solidline shows the result of our model

always lost during the collision due to dissipative processes Inthermal description of the fireball we ignore the effect of suchprocesses and we assume that all available kinetic energy (ormomentum) is globally thermalized at the freeze-out densityExperimental configuration of the collective flow developed


4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)

Figure 14 Variation of 119878119873 with radic119904119873119873 IHG model calculation isshown by dash-dotted line Cleymans-Suhonen and RGSG modelscalculations are shownbydashed anddotted lines respectively Solidline shows the calculation by our model

in the hot dense matter reveals the unsatisfactory nature ofthe above assumption

46 Transport Properties of HG Transport coefficients arevery important tools in quantifying the properties of stronglyinteracting relativistic fluid and its critical phenomena thatis phase transition and critical point [127ndash129] The fluc-tuations cause the system to depart from equilibrium anda nonequilibrated system that is created for a brief timeThe response of the system to such fluctuations is essentiallydescribed by the transport coefficients for example shear vis-cosity bulk viscosity speed of sound and so forth Recentlythe data for the collective flow obtained from RHIC andLHC experiments indicate that the system created in theseexperiments behaves as strongly interacting perfect fluid[130 131] whereas we expected that QGP created in theseexperiments should behave like a perfect gas The perfectfluid created after the phase transition indicates a very lowvalue of shear viscosity to entropy ratio so that the dissipativeeffects are negligible and the collective flow is large as it wasobtained by heavy ion collision experiments [10 132 133]There were several analytic calculations for 120578 and 120578119904 ofsimple hadronic systems [134ndash140] along with some sophis-ticated microscopic transport model calculations [141ndash143]in the literature Furthermore some calculations predict thatthe minimum of shear viscosity to entropy density is relatedwith the QCD phase transition [144ndash149] Similarly soundvelocity is an important property of the matter created inheavy-ion collision experiments because the hydrodynamicevolution of this matter strongly depends on it A minimum

0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)

Our model (r = 05 fm)Gorenstein (r = 05 fm)

120578s

Figure 15 Variation of 120578119904with temperature for120583119861= 0 in ourmodel

and a comparisonwith the results obtained byGorenstein et al [157]

occurred in the sound-velocity has also been interpreted interms of a phase transition [29 145 150ndash155] and furtherthe presence of a shallow minimum corresponds to a cross-over transition [156] In view of the above it is worthwhileto study in detail the transport properties of the HG in orderto fully comprehend the nature of the matter created in theheavy-ion collisions as well as the involved phase transitionphenomenon In this section we have used thermal modelsto calculate the transport properties of HG such as shearviscosity to entropy ratio [31]

We calculate the shear viscosity in our thermal model asit was done previously by Gorenstein et al [157] using RGSGmodel According to molecular kinetic theory we can writethe dependence of the shear viscosity as follows [158]

120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)

where 119899 is the particle density 119897 is the mean free pathand 119901 is the average thermal momentum of the baryons orantibaryons For the mixture of particle species with differentmasses and with the same hardcore radius 119903 the shearviscosity can be calculated by the following equation [157]

120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)

where 119899119894is the number density of the ith species of baryons

(antibaryons) and 119899 is the total baryon density In Figure 15we have shown the variation of 120578119904 with respect to temper-ature as obtained in our model for HG having a baryonichardcore size 119903 = 05 fm and compared our results with thoseof Gorenstein et al [157]We find that near the expectedQCDphase transition temperature (119879

119888= 170ndash180MeV) 120578119904 shows a

lower value in our HGmodel than the value in other modelsIn fact 120578119904 in our model looks close to the lower bound(14120587) suggested by AdSQCD theories [159 160] Recentlymeasurements in Pb-Pb collisions at the large hadron collider(LHC) support the value 120578119904 asymp 14120587when comparedwith theviscous fluid hydrodynamic flow [161]


0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s

Our model (T = 10MeV)K Itakura (T = 10MeV)

Figure 16Variation of 120578119904with respect to baryon chemical potential(120583119861) at very low temperature 10MeV Solid line represents our

calculation [31] and dotted curve is for the calculations done byItakura et al [139]

In Figure 16 we have shown the variation of 120578119904 withrespect to 120583

119861at a very low temperature (asymp10MeV) [31]

Here we find that the 120578119904 is constant as 120583119861increases upto

700MeV and then sharply decreases This kind of valley-likestructure at low temperature and at 120583

119861around 950MeV was

also obtained by Chen et al [137] and Itakura et al [139]Theyhave related this structure to the liquid-gas phase transitionof the nuclear matter As we increase the temperature above20MeV this valley-like structure disappears They furthersuspect that the observation of a discontinuity in the bottomof 120578119904 valley may correspond to the location of the criticalpoint Our HG model yields a curve in complete agreementwith these results Figure 17 represents the variation of 120578 and120578119904 with respect to temperature at a fixed 120583

119861(=300MeV)

for HG having a baryonic hardcore size 119903 = 08 fm Wehave compared our result with the result obtained in [139]Here we find that 120578 increases with temperature in our HGmodel as well as in the simple phenomenological calculation[139] but it decreases with increasing temperature in low-temperature effective field theory (EFT) calculations [137139] However 120578119904 decreases with increasing temperaturein all three calculations and 120578119904 in our model gives thelowest value at all the temperatures in comparison to othermodels In Figure 18 we have shown a comparison between120578 calculated in our HG model with the results obtained in amicroscopic pion gas model used in [141] Our model resultsshow a fair agreement with the microscopic model results forthe temperature higher than 160MeV while at lower tem-peratures the microscopic calculation predicts lower valuesof 120578 in comparison to our results The most probable reasonmay be that the calculations have been done only for piongas in the microscopic model while at low temperaturesthe inclusion of baryons in the HG is very important inorder to extract a correct value for the shear viscosity [31]Figure 19 shows the variation of 120578119904 with respect to radic119904119873119873 inour model calculation We have compared our results with

10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)

120578 (GeV3) our model120578 (GeV3) phenomenological120578 (GeV3) EFT

120578s our model120578s phenomenological120578s EFT

Figure 17 Variation of 120578 in unit of (GeV)3 and 120578119904 with respect totemperature at 120583

119861= 300MeV in our model [31] and a comparison

with the results obtained by Itakura et al [139]

120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)

120578 (microscopic pion gas model)120578 (GeV3) our model

Figure 18 Variation of 120578 with respect to temperature at 120583119861=

300MeV in our model and a comparison with the results obtainedby Muronga [141]

that calculated in [157] There is similarity in our resultsat lower energies while our results significantly differ athigher energies However both the calculations show that 120578119904decreases with increasingradic119904119873119873

The study of the transport properties of nonequilibriumsystems which are not far from an equilibrium state hasyielded valuable results in the recent past Large values ofthe elliptic flow observed at RHIC indicate that the matterin the fireball behaves as a nearly perfect liquid with a smallvalue of the 120578119904 ratio After evaluating 120578119904 in strongly coupledtheories using AdSCFT duality conjecture a lower boundwas reported as 120578119904 = 14120587 We surprisingly notice that thefireball with hot dense HG as described in our model givestransport coefficients which agree with those given in differ-ent approaches Temperature and baryon chemical potential


0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)

Figure 19 Variation of 120578119904 with respect to radic119904119873119873 in our model [31]and other model calculation [157]

dependence of the 120578119904 are analyzed and compared with theresults obtained in other models Our results lend supportto the claim that knowledge of the EOS and the transportcoefficients of HG is essential for a better understandingof the dynamics of the medium formed in the heavy-ioncollisions [31]

47 Rapidity and TransverseMass Spectra In order to suggestany unambiguous signal for QGP the dynamics of the col-lisions should be understood properly Such information canbe obtained by analyzing the properties of various particleswhich are emitted from various stages of the collisionsHadrons are produced at the end of the hot and dense QGPphase but they subsequently scatter in the confined hadronicphase prior to decoupling (or ldquofreeze-outrdquo) from the collisionsystem and finally a collective evolution of the hot and densematter occurs in the form of transverse radial or elliptic flowwhich is instrumental in shaping the important features ofthe particle spectra The global properties and dynamics offreeze-out can be at best studied via hadronic observablessuch as rapidity distributions and transverse mass spectra[162] There are various approaches for the study of rapidityas well as transverse mass spectra of HG [28 37 38 163ndash185]Hadronic spectra from purely thermal models usually revealan isotropic distribution of particles [186] and hence therapidity spectra obtained with the purely thermal models donot reproduce the features of the experimental data satisfac-torily Similarly the transverse mass spectra from the thermalmodels reveal a more steeper curve than that observedexperimentally The comparisons illustrate that the fireballformed in heavy-ion collisions does not expand isotropicallyin nature and there is a prominent input of collective flow

in the longitudinal and transverse directions which finallycauses anisotropy in the rapidity and transverse mass distri-butions of the hadrons after the freeze-out Here we mentionsome kinds of models of thermal and collective flow used inthe literature Hydrodynamical properties of the expandingfireball have been initially discussed by Bjorken and Landaufor the central-rapidity and stopping regimes respectively[28 163] However collisions even at RHIC energies revealthat they are neither fully stopped nor fully transparent Asthe collision energy increases the longitudinal flow growsstronger and leads to a cylindrical geometry as postulated in[37 38 164 165]They assume that the fireballs are distributeduniformally in the longitudinal direction and demonstratethat the available data can consistently be described in athermal model with inputs of chemical equilibrium and flowalthough they have also used the experimental data for smallsystems only They use two simple parameters transverseflow velocity (120573

119903) and temperature (119879) in their models In

[166 167] nonuniform flow model is used to analyze thespectra specially to reproduce the dip at midrapidity in therapidity spectra of baryons by assuming that the fireballs aredistributed nonuniformly in the longitudinal phase space In[175ndash179] the rapidity-dependent baryon chemical potentialhas been invoked to study the rapidity spectra of hadrons Incertain hydrodynamical models [180] measured transversemomentum (119901

119879) distributions in Au-Au collisions atradic119904119873119873 =

130GeV [181ndash183] have been described successfully by incor-porating a radial flow In [184] rapidity spectra of mesonshave been studied using viscous relativistic hydrodynamicsin a 1+1 dimension assuming a nonboost invariant Bjorkenrsquosflow in the longitudinal direction They have also analyzedthe effect of the shear viscosity on the longitudinal expansionof the matter Shear viscosity counteracts the gradients of thevelocity field as a consequence it slows down the longitudinalexpansion Ivanov [185] has employed 3 FD model [187] forthe study of rapidity distributions of hadrons in the energyrange from 27GeV to 624GeV In 3 FD model three differ-ent EOS (i) a purely hadronic EOS (ii) the EOS involvingfirst order phase transition from hot dense HG to QGPand (iii) the EOS with smooth crossover transition are usedWithin all three scenarios they reproduced the data at thealmost same extent In [188] rapidity distributions of varioushadrons in the central nucleus-nucleus collisions have beenstudied in the Landaursquos and Bjorkenrsquos hydrodynamicalmodelThe effect of speed of sound (119888

119904) on the hadronic spectra and

the correlation of 119888119904with freeze-out parameters are indicated

In this section we study the rapidity and transverse massspectra of hadrons using thermal approach We can rewrite(36) in the following manner [58]

119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)


It means that the invariant distributions are [37 38 164]

119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)

If we use Boltzmannrsquos approximation (63) differs fromthe one used in the paper of Schnedermann et al [164] bythe presence of a prefactor [(1 minus 119877) minus 120582

119894120597119877120597120582

119894] However

we measure all these quantities precisely at the chemicalfreeze-out using our model and hence quantitatively wedo not require any normalizing factor as required in [164]We use the following expression to calculate the rapiditydistributions of baryons in the thermal model [58]

(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)

Here 119910 is the rapidity variable and 119898119879is the transverse

mass (119898119879= radic1198982 + 119901

1198792) Also 119881 is the total volume of the

fireball formed at chemical freeze-out and 119873119894is the total

number of 119894th baryonsWe assume that the freeze-out volumeof the fireball for all types of hadrons at the time of thehomogeneous emissions of hadrons remains the same It canbe mentioned here that in the above equation there occursno free parameter because all the quantities 119892119881 120582 119877 and soforth are determined in the model However (64) describesthe experimental data only at midrapidity while it fails atforward and backward rapidities so we need to modify it byincorporating a flow factor in the longitudinal directionThusthe resulting rapidity spectra of ith hadron is [37 38 58 164]

119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)

where (119889119873119894119889119910)th can be calculated by using (64)The expres-

sion for average longitudinal velocity is [166 167 189]

⟨120573119871⟩ = tanh(

120578max2) (66)

Here 120578max is a free parameter which provides the upperrapidity limit for the longitudinal flow velocity at a particularradic119904119873119873 and its value is determined by the best experimentalfit The value of 120578max increases with the increasing radic119904119873119873and hence 120573

119871also increases Cleymans et al [179] have

extended the thermal model [175 176] in which the chemical

freeze-out parameters are rapidity-dependent to calculatethe rapidity spectra of hadrons They use the followingexpression for rapidity spectra

119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)

where 1198891198731198941119889119910 is the thermal rapidity distribution of particles

calculated by using (64) and 120588(119910FB) is a Gaussian distributionof fireballs centered at zero and given by

120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)

Similarly we calculate the transverse mass spectra of hadronsby using following expression [58]

119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)

where1198701(119898119879119879) is the modified Besselrsquos function

1198701(119898119879

119879) = int

infin

0

cosh119910 [exp(minus119898119879cosh119910119879

)]119889119910 (70)

The above expression for transverse mass spectra arises froma stationary thermal source alone which is not capable ofdescribing the experimental data successfully So we incor-porate flow velocity in both the directions in (69) longitudi-nal as well as transverse in order to describe the experimentaldata satisfactorily After defining the flow velocity field wecan calculate the invariantmomentum spectrum by using thefollowing formula [164 190]

119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int exp(minus119896120583119906120583

119879)119896120582119889120590120582

(71)

While deriving (71) we assume that the local fluid velocity119906120583 gives a boost to an isotropic thermal distribution of

hadrons Now the final expression of transverse mass spectraof hadrons after incorporation of flow velocity in our modelis [58]

119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)

Here 1198680(119901119879sinh 120588119879) is the modified Besselrsquos function

1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)

where 120588 is given by 120588 = tanhminus1120573119903 with the velocity profile

chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573

119904is themaximum surface


1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)

Figure 20 Energy dependence of the freeze-out volume for thecentral nucleus-nucleus collisions The symbols are the HBT datafor freeze-out volume 119881HBT for the 120587+ [191] 1198601015840 1198611015840 and 1198621015840 are thetotal freeze-out volume and 119860 119861 and 119862 depict the 119889119881119889119910 as foundin our model for 120587+119870+ and119870minus respectively119863 represents the totalfreeze-out volume for 120587+ calculated in the Ideal HGmodel1198631015840 is thethe total freeze-out volume for 120587+ in our model calculation usingBoltzmannrsquos statistics [58]

velocity and is treated as a free parameter and 120585 = (1199031198770)The

average of the transverse velocity can be evaluated as [182]

⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)

In our calculation we use a linear velocity profile (119899 =1) and 119877

0is the maximum radius of the expanding source at

freeze-out (0 lt 120585 lt 1) [182]In Figure 20 we have shown the variations of 119881 and

119889119881119889119910 with the radic119904119873119873 calculated in our excluded-volumemodel and compared with the results of various thermalmodels We show the total freeze-out volume for 120587+ cal-culated in our model using Boltzmannrsquos statistics We seethat there is a significant difference between the resultsarising from quantum statistics and Boltzmannrsquos statistics[58] We also show the total freeze-out volume for 120587+ inIHG model calculation by dash-dotted line 119863 We clearlynotice a remarkable difference between the results of ourexcluded-volume model and those of IHG model also Wehave also compared predictions fromourmodel with the dataobtained from the pion interferometry (HBT) [191] which infact reveals thermal (kinetic) freeze-out volumes The resultsof thermal models support the finding that the decoupling ofstrange mesons from the fireball takes place earlier than the120587-mesons Moreover a flat minimum occurs in the curvesaround the center-of-mass energy asymp8GeV and this featureis well supported by HBT data In Figure 21 we present therapidity distribution of 120587+ for central Au+Au collisions atradic119904119873119873 = 200GeV over full rapidity range Dotted line shows

Rapidity (y)

0

50

100

150

200

250

300

350

400

minus5 minus4 minus3 minus2 minus1 0 1 3 4 52

dN120587dy

120587+

Our model + flow (120573L = 092)Our model + flow (120573L = 088)Our model

Figure 21 Rapidity distribution of 120587+ at radic119904119873119873 = 200GeV Dottedline shows the rapidity distribution calculated in our thermal model[58] Solid line and dashed line show the results obtained afterincorporating longitudinal flow in our thermal model Symbols arethe experimental data [192]

the distribution of 120587+ due to purely thermal model Solid lineshows the rapidity distributions of 120587+ after the incorporationof longitudinal flow in our thermal model and results give agood agreement with the experimental data [192] In fittingthe experimental data we use the value of 120578max = 32 andhence the longitudinal flow velocity 120573

119871= 092 at radic119904119873119873 =

200GeV For comparison and testing the appropriateness ofthis parameter we also show the rapidity distributions at adifferent value that is 120578max = 28 (or 120573119871 = 088) by a dashedline in the figure We find that the results slightly differ andhence it shows a small dependence on 120578max [58] Figure 22represents the rapidity distributions of pion at various radic119904119873119873calculated by using (67) [179] There is a good agreementbetween themodel results and experimental data at allradic119904119873119873

In Figure 23 we show the transverse mass spectra for120587+ and proton for the most central collisions of Au+Au atradic119904119873119873 = 200GeV We have neglected the contributionsfrom the resonance decays in our calculations since thesecontributions affect the transverse mass spectra only towardsthe lower transverse mass side that is 119898

119879lt 03GeV We

find a good agreement between our calculations and theexperimental results for all 119898

119879except 119898

119879lt 03GeV after

incorporating the flow velocity in purely thermal modelThis again shows the importance of collective flow in thedescription of the experimental data [193] At this energythe value of 120573

119904is taken as 050 and transverse flow velocity

120573119903= 033 This set of transverse flow velocity is able

to reproduce the transverse mass spectra of almost all the


400

300

200

100

0minus4 minus2 0 2 4

Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088

Figure 22 Rapidity distribution of pion at various radic119904119873119873 Lines aremodel results and points are experimental data Figure 22 is takenfrom [179]

hadrons at radic119904119873119873 = 200GeV We notice that the transverseflow velocity slowly increases with the increasing radic119904119873119873 Ifwe take 120573

119904= 060 we find that the results differ with data

as shown in Figure 23 In Figure 24 we show the transversemomentum (119901

119879) spectra for 120587+ 119870+ and 119901 in the most

central collisions of Au-Au at radic119904119873119873 = 200GeV Our modelcalculations reveal a close agreement with the experimentaldata [193] In Figure 25 we show the119901

119879spectra of120587minus119870minus and

119901 for the Pb-Pb collisions at radic119904119873119873 = 276TeV at the LHCOur calculations again give a good fit to the experimentalresults [194] We also compare our results for 119901 spectrumwith the hydrodynamical model of Shen et al [195] whichsuccessfully explains 120587minus and 119870minus spectra but strongly failsin the case of 119901 spectrum [58] In comparison our modelresults show closer agreement with the experimental dataShen et al [195] have employed (2+1)-dimensional viscoushydrodynamics with the lattice QCD-based EOS They useCooper-Frye prescription to implement kinetic freeze-out inconverting the hydrodynamic output into the particle spectraDue to lack of a proper theoretical and phenomenologicalknowledge they use the same parameters for Pb-Pb collisionsat LHC energy which was used for Au-Au collisions atradic119904119873119873 = 200GeV Furthermore they use the temperature-independent 120578119904 ratio in their calculation After fitting theexperimental data we get 120573

119904= 080 (120573

119903= 053) at this energy

which indicates the collective flow as becoming stronger atLHC energy than that observed at RHIC energies In this plotwe also attempt to show how the spectra for 120587minus will change ata slightly different value of the parameter that is 120573

119904= 088

[58]

Our model

Our model + flowOur model

05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+

Our model + flow (120573s = 06)Our model + flow (120573s = 05)

Figure 23 Transverse mass spectra for 120587+ and proton for the mostcentral collisions at radic119904119873119873 = 200GeV Dashed and dotted lines arethe transverse mass spectra due to purely thermal source for 120587+ andproton respectively Solid and dash-dotted lines are the results for120587+ and proton respectively obtained after incorporation of flow in

thermal model [58] Symbols are the experimental data [193]

5 Summary and Conclusions

Themain aim in this paper is to emphasize the use of the ther-mal approach in describing various yields of different particlespecies that have been measured in various experimentsrunning at various places We have discussed various typesof thermal approaches for the formulation of EOS for HGWe have argued that incorporation of interactions betweenhadrons in a thermodynamically consistent way is importantfor the realistic formulation of HG from both qualitativelyand quantitatively point of view We have presented thesystematic study of the particle production in heavy-ion col-lisions fromAGS to LHC energyWe have observed from thisanalysis that the production of the particles seems to occurraccording to principle of equilibrium Yields of hadrons andtheir ratios measured in heavy-ion collisions match with thepredictions of thermal models assured the thermalization ofthe collision fireball formed in heavy-ion collisions Further-more various experimental observables such as transversemomentum spectra and elliptic flow indicate the presenceof the thermodynamical pressure developed in the earlystage and correlations which are expected in a thermalizedmedium

We have discussed a detailed formulation of variousexcluded-volume models and their shortcomings Someexcluded-volume models are not thermodynamically con-sistent because they do not possess a well defined partitionfunction from which various thermodynamical quantities


103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+

Our model + flowOur modelK+

p

Our model + flow

Our model


pT (GeV)

Figure 24 Transverse momentum spectra for 120587+ 119901 and119870+ for themost central Au-Au collision atradic119904119873119873 = 200GeV [58] Lines are theresults of our model calculation and symbols are the experimentalresults [193]

such as number density can be calculated However some ofthem are the thermodynamically consistent but suffer fromsome unphysical situations cropping up in the calculationsWe have proposed a new approximately thermodynamicallyconsistent excluded-volume model for a hot and dense HGWe have used quantum statistics in the grand canonicalpartition function of our model so that it works even atextreme values of119879 and 120583

119861where all other models fail More-

over our model respects causality We have presented thecalculations of various thermodynamical quantities such asentropy per baryon and energy density in various excluded-volume models and compare the results with those of amicroscopic approach URASiMA We find that our modelresults are in close agreement with that of the entirelydifferent approach URASiMA model We have calculatedvarious particle ratios at various radic119904119873119873 and we confrontedthe results of various thermal models with the experimentaldata and find that they are indeed successful in describingthe particle ratios However we find that our model resultsare closer to the experimental data in comparison to thoseof other excluded-volume models We have calculated someconditions such as 119864119873 and 119878119873 at chemical freeze-outpoints and attempted to test whether these conditions involveenergy independence as well as independence of structureof the nuclei involved in the collisions We find that 119864119873 asymp10 GeV and 119878119873 asymp 70 are the two robust freeze-out criteriawhich show independence of the energy and structure ofnuclei Moreover the calculations of transport properties in

103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus

Our model + flow (120573s = 088)Our model + flow (120573s = 08)Kminus

p

Viscoushydrodynamics

Figure 25 Transverse momentum spectra of various hadrons forthe most central collisions of Pb-Pb at radic119904119873119873 = 276TeV from LHC[58] Lines are the results of model calculations and symbols arethe experimental results [194] Thick-dashed line is the predictionof viscous-hydrodynamical model [195] for 119901

our model match well with the results obtained in otherwidely different approaches Further we present an analysis ofrapidity distributions and transverse mass spectra of hadronsin central nucleus-nucleus collision at various radic119904119873119873 usingour EOS for HG We see that the stationary thermal sourcealone cannot describe the experimental data fully unless weincorporate flow velocities in the longitudinal as well as inthe transverse direction and as a result our modified modelpredictions show a good agreement with the experimentaldata Our analysis shows that a collective flow develops ateachradic119904119873119873which increases furtherwith the increasingradic119904119873119873The description of the rapidity distributions and transversemass spectra of hadrons at each radic119904119873119873 matches very wellwith the experimental dataThus we emphasize that thermalmodels are indeed an important tool to describe the variousfeatures of hot and dense HG Although these models arenot capable of telling whether QGP was formed before HGphase they can give an indirect indication of it by showingany anomalous feature as observed in the experimental data

In conclusion the net outcome of this review is indeed asurprising one The excluded-volume HG models are reallysuccessful in describing all kinds of features of the HGformed in ultrarelativistic heavy-ion collisions The mostimportant property indicated by such description is thechemical equilibrium reached in such collisions Howeverthe description is still a geometrical one and does notinvolve anymicroscopic picture of interactions Moreover itsrelativistic and field theoretic generalizations are still needed


in order to make the picture a more realistic descriptionBut it is amazing to find that these models still work muchbetter than expected Notwithstanding these remarks weshould add that Lattice QCD results are now available forthe pressure entropy density and energy density and soforth for the entire temperature range from 119879 = 0 tohigher values at 120583 = 0 Here low temperature phase ofQCD is the HG and recently our excluded-volume modelreproduces these properties quite in agreement with Latticeresults [196] We have also used these calculations in theprecise determination of the QCD critical end point [197198] Thus we conclude that the excluded-volume modelsare successful in reproducing numerical results obtainedin various experiments and therefore further research isrequired to show how these descriptions are connected withthe microscopic interactions

Acknowledgment

S K Tiwari is grateful to the Council of Scientific andIndustrial Research (CSIR) New Delhi for providing aresearch grant

References

[1] C P Singh ldquoSignals of quark-gluon plasmardquo Physics Report vol236 no 3 pp 147ndash224 1993

[2] C P Singh ldquocurrent status of properties and signals of thequark-gluon plasmardquo International Journal of Modern PhysicsA vol 7 p 7185 1992

[3] M J Tannenbaum ldquoRecent results in relativistic heavy ioncollisions from ldquoa new state of matterrdquo to ldquothe perfect fluidrdquordquoReports on Progress in Physics vol 69 p 2005 2006

[4] B Muller ldquoPhysics and signatures of the quark-gluon plasmardquoReports on Progress in Physics vol 58 p 611 1995

[5] H Satz ldquoColour deconfinement in nuclear collisionsrdquo Reportson Progress in Physics vol 63 no 9 pp 1511ndash1574 2000

[6] H Satz ldquoquark matter and nuclear collisions a brief historyof strong interaction thermodynamicsrdquo International Journal ofModern Physics E vol 21 Article ID 1230006 23 pages 2012

[7] P Braun-Munzinger K Redlich and J Stachel Invited Reviewfor Quark Gluon Plasma vol 3 Edited by R C Hwa and X-NWang World Scientific Publishing

[8] P Braun-Munzinger and J Wambach ldquoColloquiumml phasediagram of strongly interacting matterrdquo Reviews of ModernPhysics vol 81 no 3 pp 1031ndash1050 2009

[9] P Braun-Munzinger and J Stachel ldquoThe quest for the quark-gluon plasmardquo Nature vol 448 no 7151 pp 302ndash309 2007

[10] M Gyulassy and L McLerran ldquoNew forms of QCD matterdiscovered at RHICrdquo Nuclear Physics A vol 750 no 1 pp 30ndash63 2005

[11] K Adcoxbd SS Adlere S Afanasiev et al ldquoFormation of densepartonic matter in relativistic nucleus-nucleus collisions atRHIC experimental evaluation by the PHENIXCollaborationrdquoNuclear Physics A vol 757 pp 184ndash283 2005

[12] R Rapp ldquoSignatures of thermal dilepton radiation at ultrarela-tivistic energiesrdquo Physical Review C vol 63 Article ID 05490713 pages 2001

[13] V Ruuskanen ldquotransverse hydrodynamics with a first orderphase transition in very high-energy nuclear collisionsrdquo ActaPhysica Polonica B vol 18 p 551 1987

[14] I Krasnikova C Gale and D K Srivastava ldquoProduction ofintermediate mass dileptons in relativistic heavy ion collisionsrdquoPhysical Review C vol 65 Article ID 064903 2002

[15] W Cassing E L Bratkovskaya R Rapp and J WambachldquoProbing the120588 spectral function in hot anddense nuclearmatterby dileptonsrdquo Physical Review C vol 57 no 2 pp 916ndash921 1998

[16] J Wambach and R Rapp ldquoTheoretical interpretations of low-mass dileptonsrdquo Nuclear Physics A vol 638 no 1-2 p 171 1998

[17] K Gallmeister B Kampfer and O P Pavlenko ldquoIs there aunique thermal source of dileptons in Pb(158 A sdotGeV) + Au Pbreactionsrdquo Physics Letters B vol 473 no 1-2 pp 20ndash24 2000

[18] F Karsch K Redlich and L Turko ldquoChiral symmetry anddileptons in heavy ion collisionsrdquo Zeitschrift fur Physik C vol60 no 3 pp 519ndash525 1993

[19] K Kajantie J Kapusta L McLerran and A Mekjian ldquoDileptonemission and the QCD phase transition in ultrarelativisticnuclear collisionsrdquo Physical Review D vol 34 no 9 pp 2746ndash2754 1986

[20] T Hatsuda ldquoSpectral change of hadrons and chiral symmetryrdquoNuclear Physics A vol 698 p 243 2002

[21] K Yokokawa T Hatsuda A Hayashigaki and T KunihiroldquoSimultaneous softening of 120590 and 120588 mesons associated withchiral restorationrdquo Physical Review C vol 66 Article ID 0222012002

[22] U Heinz ldquoThe Little Bang searching for quark-gluon matter inrelativistic heavy-ion collisionsrdquoNuclear Physics A vol 685 pp414ndash431 2001

[23] P F Kolb J Sollfrank P V Ruuskanen and U Heinz ldquoHydro-dynamic simulation of elliptic flowrdquo Nuclear Physics A vol 661p 349 1999

[24] D Teaney J Lauret and E V Shuryak ldquoFlow at the SPSand RHIC as a quark-gluon plasma signaturerdquo Physical ReviewLetters vol 86 no 21 pp 4783ndash4786 2001

[25] U A Wiedemann and U Heinz ldquoParticle interferometry forrelativistic heavy-ion collisionsrdquo Physics Report vol 319 no 4-5pp 145ndash230 1999

[26] B Tomasik U A Wiedemann and U Heinz ldquoDynamics andsizes of the fireball at freeze-outrdquo Nuclear Physics A vol 663pp 753ndash756 2000

[27] J R Nix ldquoLow freeze-out temperature and high collectivevelocities in relativistic heavy-ion collisionsrdquo Physical Review Cvol 58 no 4 pp 2303ndash2310 1998

[28] J D Bjorken ldquoHighly relativistic nucleus-nucleus collisions thecentral rapidity regionrdquoPhysical ReviewD vol 27 no 1 pp 140ndash151 1983

[29] P Braun-Munzinger and J Stachel ldquoProbing the phase bound-ary between hadronic matter and the quark-gluon plasma inrelativistic heavy-ion collisionsrdquoNuclear Physics A vol 606 no1-2 pp 320ndash328 1996

[30] A Andronic P Braun-Munzinger and J Stachel ldquoHadronproduction in central nucleus-nucleus collisions at chemicalfreeze-outrdquo Nuclear Physics A vol 772 no 3-4 pp 167ndash1992006

[31] S K Tiwari P K Srivastava and C P Singh ldquoDescription ofhot and dense hadron-gas properties in a new excluded-volumemodelrdquo Physical Review C vol 85 no 1 Article ID 014908 2012

[32] M Mishra and C P Singh ldquoEffect of geometrical size of theparticles in a hot and dense hadron gasrdquo Physical Review C vol76 Article ID 024908 9 pages 2007


[33] M Mishra and C P Singh ldquoParticle multiplicities and ratios inexcluded volumemodelsrdquo Physical Review C vol 78 Article ID024910 9 pages 2008

[34] J Cleymans and K Redlich ldquoUnified description of freeze-outparameters in relativistic heavy ion collisionsrdquo Physical ReviewLetters vol 81 no 24 pp 5284ndash5286 1998

[35] P Braun-Munzinger D Magestro K Redlich and J StachelldquoHadron production in Au-Au collisions at RHICrdquo PhysicsLetters B vol 518 no 1-2 pp 41ndash46 2001

[36] D Magestro ldquoEvidence for chemical equilibration at RHICrdquoJournal of Physics G vol 28 no 7 pp 1745ndash1752 2002

[37] P Braun-Munzinger J Stachel J P Wessels and N Xu ldquoTher-mal equilibration and expansion in nucleus-nucleus collisionsat the AGSrdquo Physics Letters B vol 344 no 1ndash4 pp 43ndash48 1995

[38] P Braun-Munzinger J Stachel JP Wessels and N Xu ldquoTher-mal and hadrochemical equilibration in nucleus-nucleus colli-sions at the SPSrdquo Physics Letters B vol 365 pp 1ndash6 1996

[39] P Braun-Munzinger I Heppe and J Stachel ldquoChemical equi-libration in Pb+Pb collisions at the SPSrdquo Physics Letters B vol465 no 1ndash4 pp 15ndash20 1999

[40] S V Akkelin P Braun-Munzinger and YM Sinyukov ldquoRecon-struction of hadronization stage in Pb+Pb collisions at 158 AGeVcrdquo Nuclear Physics A vol 710 no 3-4 pp 439ndash465 2002

[41] F Becattini J Cleymans A Keranen E Suhonen and KRedlich ldquoFeatures of particle multiplicities and strangenessproduction in central heavy ion collisions between 17A and158AGeVcrdquoPhysical ReviewC vol 64 no 2 Article ID 0249012001

[42] A Keranen and F Becattini ldquoThe canonical effect in statisticalmodels for relativistic heavy ion collisionsrdquo Journal of Physics Gvol 28 p 2041 2002

[43] A Keranen and F Becattini ldquoChemical factors in canonicalstatistical models for relativistic heavy ion collisionsrdquo PhysicalReview C vol 65 Article ID 044901 7 pages 2002

[44] J Rafelski J Letessier and A Tounsi ldquoStrange particles fromdense hadronic matterrdquo Acta Physica Polonica B vol 27 pp1037ndash1140 1996

[45] K Redlich and A Tounsi ldquoStrangeness enhancement andenergy dependence in heavy ion collisionsrdquo European PhysicalJournal C vol 24 no 4 pp 589ndash594 2002

[46] J Sollfrank P Huovinen M Kataja P V Ruuskanen MPrakash and R Venugopalan ldquoHydrodynamical descriptionof 200A GeVc S+Au collisions hadron and electromagneticspectrardquo Physical Review C vol 55 no 1 pp 392ndash410 1997

[47] P Huovinena P F Kolbb U Heinzb P V Ruuskanend andS A Voloshine ldquoRadial and elliptic flow at RHIC furtherpredictionsrdquo Physics Letters B vol 503 pp 58ndash64 2001

[48] J Cleymans and K Redlich ldquoChemical and thermal freeze-outparameters from 1A to 200A GeVrdquo Physical Review C vol 60Article ID 054908 9 pages 1999

[49] J Cleymans H Oeschler and K Redlich ldquoInfluence of impactparameter on thermal description of relativistic heavy ioncollisions at (1-2)A GeVrdquo Physical Review C vol 59 no 3 pp1663ndash1673 1999

[50] J Cleymans D Elliott A Keranen and E Suhonen ldquoThermalmodel analysis of particle ratios in Ni+Ni experiments usingexact strangeness conservationrdquo Physical Review C vol 57 no6 pp 3319ndash3323 1998

[51] J Letessier and J Rafelski ldquoObserving quark-gluon plasmawithstrange hadronsrdquo International Journal ofModern Physics E vol9 no 2 pp 107ndash147 2000

[52] P Braun-Munzinger J Cleymans H Oeschler and K RedlichldquoMaximum relative strangeness content in heavy-ion collisionsaround 30A GeVrdquo Nuclear Physics A vol 697 no 3-4 pp 902ndash912 2002

[53] W Broniowski and W Florkowski ldquoDescription of the RHICpperp spectra in a thermal model with expansionrdquo Physical ReviewLetters vol 87 no 27 Article ID 272302 2001

[54] K Redlich ldquoStrangeness production in heavy ion collisionsrdquoNuclear Physics A vol 698 no 1ndash4 p 94 2002

[55] J Cleymans and H Satz ldquoThermal hadron production in highenergy heavy ion collisionsrdquo Zeitschrift fur Physik C vol 57 no1 pp 135ndash147 1993

[56] P Braun-Munzinger and J Stachel ldquo(Non) thermal aspects ofcharmonium production and a new look at J120595 suppressionrdquoPhysics Letters B vol 490 no 3-4 pp 196ndash202 2000

[57] P Braun-Munzinge and J Stachel ldquoOn charm production nearthe phase boundaryrdquo Nuclear Physics A vol 690 no 1ndash3 p 1192001

[58] S K Tiwari P K Srivastava and C P Singh ldquoThe effect of flowon hadronic spectra in an excluded-volume modelrdquo Journal ofPhysics G vol 40 Article ID 045102 2013

[59] G M Welke R Venugopalan and M Prakash ldquoThe speed ofsound in an interacting pion gasrdquo Physics Letters B vol 245 no2 pp 137ndash141 1990

[60] V V Dixit and E Suhonen ldquoBare-bones model for phasetransition in hadronic matterrdquo Zeitschrift fur Physik C vol 18no 4 pp 355ndash360 1983

[61] J Cleymans K Redlich H Satz and E Suhonen ldquoOnthe phenomenology of deconfinement and chiral symmetryrestorationrdquo Zeitschrift fur Physik C vol 33 no 1 pp 151ndash1561986

[62] J Cleymans R V Gavai and E Suhonen ldquoQuarks and gluonsat high temperatures and densitiesrdquo Physics Reports vol 130 no4 pp 217ndash292 1986

[63] J Cleymans and E Suhonen ldquoInfluence of hadronic hardcore radius on detonations and deflagrations in quark matterrdquoZeitschrift fur Physik C vol 37 no 1 pp 51ndash56 1987

[64] J Cleymans and D W Von Oertzen ldquoBose-Einstein condensa-tion of kaons in dense nuclearmatterrdquo Physics Letters B vol 249no 3-4 pp 511ndash513 1990

[65] N J Davidson H G Miller R M Quick and J CleymansldquoChemical equilibration in heavy-ion collisionsrdquo Physics LettersB vol 255 no 1 pp 105ndash109 1991

[66] H Kuono and F Takagi ldquoExcluded volume bag constant andhadron-quark phase transitionrdquo Zeitschrift fur Physik C vol 42pp 209ndash213 1989

[67] R Hagedorn and J Rafelski ldquoHot hadronic matter and nuclearcollisionsrdquo Physics Letters B vol 97 p 136 1980

[68] R Hagedorn ldquoThe pressure ensemble as a tool for describingthe hadron-quark phase transitionrdquo Zeitschrift fur Physik C vol17 pp 265ndash281 1983

[69] D H Rischke M I Gorenstein H Stocker and W GreinerldquoExcluded volume effect for the nuclear matter equation ofstaterdquo Zeitschrift fur Physik C vol 51 no 3 pp 485ndash489 1991

[70] C P Singh B K Patra and K K Singh ldquoThermodynamicallyconsistent EOS for hot dense hadron gasrdquo Physics Letters B vol387 no 4 pp 680ndash684 1996

[71] P K Panda M E Bracco M Chiapparini E Conte and GKrein ldquoExcluded volume effects in the quark meson couplingmodelrdquo Physical Review C vol 65 no 6 Article ID 0652062002


[72] D Anchishkin and E Suhonen ldquoGeneralization of mean-fieldmodels to account for effects of excluded volumerdquo NuclearPhysics A vol 586 no 4 pp 734ndash754 1995

[73] D Anchishkin ldquoParticle finite size effects as mean field approx-imationrdquo Soviet Physics JETP vol 75 p 195 1992

[74] N Prasad K K Singh and C P Singh ldquoCausality in anexcluded volume modelrdquo Physical Review C vol 62 no 3Article ID 037903 2000

[75] V K Tiwari K K Singh N Prasad and C P Singh ldquoMean fieldmodel with excluded volume correction for a multicomponenthadron gasrdquoNuclear Physics A vol 637 no 1 pp 159ndash172 1998

[76] B X Sun X F Lu and E G Zhao ldquoPerturbative calculationof the excluded volume effect for nuclear matter in a relativisticmean-field approximationrdquo Physical Review C vol 65 ArticleID 054301 2002

[77] M I Gorenstein ldquoExamination of the thermodynamical consis-tency of excluded-volume hadron gas modelsrdquo Physical ReviewC vol 86 Article ID 044907 3 pages 2012

[78] Z-D Lu A Faessler C Fuchs and E E Zabrodin ldquoAnalysisof particle production in ultrarelativistic heavy-ion collisionswithin a two-source statistical modelrdquo Physical Review C vol66 Article ID 044905 7 pages 2002

[79] J I Kapusta A P Vischer and R Venugopalan ldquoNucleation ofquark-gluon plasma from hadronic matterrdquo Physical Review Cvol 51 pp 901ndash910 1995

[80] J I Kapusta and K A Olive ldquoThermodynamics of hadronsdelimiting the temperaturerdquo Nuclear Physics A vol 408 no 3pp 478ndash494 1983

[81] J D Walecka ldquoA theory of highly condensed matterrdquo Annals ofPhysics vol 83 no 2 pp 491ndash529 1974

[82] B D Serot and J D Walecka ldquoProperties of finite nuclei in arelativistic quantum field theoryrdquo Physics Letters B vol 87 no3 pp 172ndash176 1979

[83] D H Rischke B L Friman H Stocker andW Greiner ldquoPhasetransition from hadron gas to quark-gluon plasma influence ofthe stiffness of the nuclear equation of staterdquo Journal of PhysicsG vol 14 no 2 pp 191ndash203 1988

[84] J Zimanyi B Lukacs P Levai J P Bondorf and N L BalazsldquoAn interpretable family of equations of state for dense hadronicmatterrdquo Nuclear Physics A vol 484 no 3-4 pp 647ndash660 1988

[85] K A Bugaev and M I Gorenstein ldquoThermodynamically self-consistent class of nuclear matter EOS and compression shocksin relativistic nuclear collisionsrdquo Zeitschrift fur Physik C vol 43no 2 pp 261ndash265 1989

[86] G D Yen M I Gorenstein W Greiner and S N YangldquoExcluded volume hadron gas model for particle number ratiosin a+a collisionsrdquo Physical Review C vol 56 no 4 pp 2210ndash2218 1997

[87] F Becattini ldquoAThermodynamical approach to hadron produc-tion in e+ e- collisionsrdquo Zeitschrift fur Physik C vol 69 pp 485ndash492 1996

[88] F Becattini and U Heinz ldquoThermal hadron production in ppand ppcollisionsrdquoZeitschrift fur Physik C vol 76 no 2 pp 269ndash286 1997

[89] S Uddin and C P Singh ldquoEquation of state of finite-sizehadrons thermodynamical consistencyrdquo Zeitschrift fur PhysikC vol 63 no 1 pp 147ndash150 1994

[90] C P Singh P K Srivastava and S K Tiwari ldquoQCD phaseboundary and critical point in a bagmodel calculationrdquoPhysicalReview D vol 80 Article ID 114508 2011

[91] C P Singh P K Srivastava and S K Tiwari ldquoErratum in QCDphase boundary and critical point in a bag model calculationrdquoPhysical Review D vol 83 Article ID 039904 1 page 2011

[92] G Zeeb K A Bugaev P T Reuter and H Stocker ldquoEquation ofstate for the two-component van der waals gas with relativisticexcluded volumesrdquo Ukrainian Journal of Physics vol 53 no 3pp 279ndash295 2008

[93] K A Bugaev PhD thesis ch 3[94] F Becattini M Gazdzicki A Keranen J Manninen and R

Stock ldquoChemical equilibrium study in nucleus-nucleus colli-sions at relativistic energiesrdquo Physical Review C vol 69 ArticleID 024905 19 pages 2004

[95] J Letessier and J Rafelski ldquoHadron production and phase chan-ges in relativistic heavy-ion collisionsrdquo The European PhysicalJournal A vol 35 pp 221ndash242 2008

[96] A Dumitru L Portugal and D Zschiesche ldquoInhomogeneousfreeze-out in relativistic heavy-ion collisionsrdquo Physical ReviewC vol 73 no 2 Article ID 024902 2006

[97] F Becattini and G Passaleva ldquoStatistical hadronization modeland transverse momentum spectra of hadrons in high energycollisionsrdquo European Physical Journal C vol 23 no 3 pp 551ndash583 2002

[98] M Kaneta and N Xu ldquoCentrality dependence of chemicalfreeze-out in Au+Au collisions at RHICrdquo httparxivorgabsnucl-th0405068

[99] J Adams and STAR Collaboration ldquoExperimental and theoret-ical challenges in the search for the quark gluon plasma TheSTAR Collaborationrsquos Critical Assessment of the Evidence fromRHIC Collisionsrdquo Nuclear Physics A vol 757 p 102 2005

[100] A Tawfik ldquoChemical freeze-out and higher order multiplicitymomentsrdquo httparxivorgabs13061025

[101] S Gupta X Luo B Mohanty H G Ritter and N Xu ldquoScale forthe phase diagram of quantum chromodynamicsrdquo Science vol332 no 6037 pp 1525ndash1528 2011

[102] A Bazavov H-T Ding and P Hegde ldquoFreeze-out conditionsin heavy ion collisions from QCD thermodynamicsrdquo PhysicalReview Letters vol 109 Article ID 192302 5 pages 2012

[103] J Cleymans H Oeschler K Redlich and S Wheaton ldquoCom-parison of chemical freeze-out criteria in heavy-ion collisionsrdquoPhysical Review C vol 73 no 3 Article ID 034905 2006

[104] C Alt and NA49 Collaboration ldquoPion and kaon production incentral Pb+Pb collisions at 20A and 30A GeV evidence for theonset of deconfinementrdquo Physical Review C vol 77 Article ID024903 10 pages 2008

[105] M Gazdzicki and the NA49 Collaboration ldquoReport fromNA49rdquo Journal of Physics G vol 30 p 701 2004

[106] S V Afanasiev T Anticic andD Barna ldquoEnergy dependence ofpion and kaon production in central Pb+Pb collisionsrdquo PhysicalReview C vol 66 Article ID 054902 2002

[107] T Anticic B Baatar D Barna et al ldquoΛ and Λ productionin central Pb-Pb collisions at 40 80 and 158A GeVrdquo PhysicalReview Letters vol 93 Article ID 022302 2004

[108] L Ahle L Ahle Y Akiba and K Ashktorab ldquoParticle produc-tion at high baryon density in central Au+Au reactions at 116AGeVcrdquo Physical Review C vol 57 pp R466ndashR470 1998

[109] L Ahle Y Akiba K Ashktorab et al ldquoCentrality dependence ofkaon yields in Si+A and Au+Au collisions at the AGSrdquo PhysicalReview C vol 60 Article ID 044904 1999

[110] S Albergo R Bellwied and M Bennett ldquoΛ spectra in 116AGeVc Au-Au collisionsrdquo Physical Review Lett vol 88 ArticleID 062301 2002


[111] J L Klay N N Ajitanand J M Alexander et al ldquoCharged pionproduction in 2A to 8AGeV central Au+Au collisionsrdquo PhysicalReview C vol 68 Article ID 054905 2003

[112] S Chatterjee R M Godbole and S Gupta ldquoStabilizing hadronresonance gas modelsrdquo Physical Review C vol 81 no 4 ArticleID 044907 2010

[113] R Stock ldquoRelativistic nucleus-nucleus collisions from theBEVALAC to RHICrdquo Journal of Physics G vol 30 no 8 ppS633ndashS648 2004

[114] J Noronha-Hostler H Ahmad J Noronha and C GreinerldquoParticle ratios as a probe of the QCD critical temperaturerdquoPhysical Review C vol 82 no 2 Article ID 024913 2010

[115] D Molnar and S A Voloshin ldquoElliptic flow at large transversemomenta from quark coalescencerdquo Physical Review Letters vol91 Article ID 092301 2003

[116] D Molnar ldquoParticle correlations at RHIC from parton coales-cence dynamicsmdashfirst resultsrdquo Journal of Physics G vol 30 pS1239 2004

[117] C Alt T Anticic and B Baatar ldquoEnergy dependence of Λand equiv production in central Pb+Pb collisions at 20A 30A40A 80A and 158A GeV measured at the CERN Super ProtonSynchrotronrdquo Physical Review C vol 78 Article ID 0349182008

[118] C Blume andThe NA49 collaboration ldquoReview of results fromthe NA49 Collaborationrdquo Journal of Physics G vol 31 no 6 ppS685ndashS691 2005

[119] M M Aggarwal and The STAR Collaboration ldquoStrange andmulti-strange particle production in Au+Au collisions atradicsNN= 624 GeVrdquo Physical Review C vol 83 Article ID 024901 2011

[120] A Andronic D Blaschke P Braun-Munzinger et al ldquoHadronproduction in ultra-relativistic nuclear collisions quarkyonicmatter and a triple point in the phase diagram of QCDrdquoNuclearPhysics A vol 837 no 1-2 pp 65ndash86 2010

[121] B Abelev N Arbor G Conesa Balbastre et al ldquoPion Kaon andProton production in central PbndashPb Collisions at radic119904119873119873 = 276TeVrdquo Physical Review Letters vol 109 Article ID 252301 2012

[122] N Sasaki ldquoA study of the thermodynamic properties of a hotdense hadron gas using an event generator hadro-molecular-dynamical calculationrdquo Progress of Theoretical Physics vol 10no 4 pp 783ndash805 2001

[123] P Braun-Munzinger and J Stachel ldquoParticle ratios equilibra-tion and the QCD phase boundaryrdquo Journal of Physics G vol28 no 7 pp 1971ndash1976 2002

[124] V Magas and H Satz ldquoConditions for confinement and freeze-outrdquoEuropean Physical Journal C vol 32 no 1 pp 115ndash119 2003

[125] A Tawfik ldquoInfluence of strange quarks on the QCD phasediagram and chemical freeze-outrdquo Journal of Physics G vol 31no 6 pp S1105ndashS1110 2005

[126] M B Isichenko ldquoPercolation statistical topography and trans-port in random mediardquo Reviews of Modern Physics vol 64 no4 pp 961ndash1043 1992

[127] J Kapusta ldquoViscous heating of expanding fireballsrdquo PhysicalReview C vol 24 no 6 pp 2545ndash2551 1981

[128] AM Polyakov Journal of Experimental andTheoretical Physicsvol 57 p 2144 1969

[129] F Karsch D Kharzeev and K Tuchin ldquoUniversal properties ofbulk viscosity near the QCD phase transitionrdquo Physics Letters Bvol 663 no 3 pp 217ndash221 2008

[130] U Heinz and P Kolb ldquoEarly thermalization at RHICrdquo NuclearPhysics A vol 702 no 1ndash4 p 269 2002

[131] A Adare and PHENIX Collaboration ldquoEnergy loss and flow ofheavy quarks in Au+Au collisions atradicsNN= 200GeVrdquo PhysicalReview Letters vol 98 Article ID 172301 2007

[132] E Shuryak ldquoWhat RHIC experiments and theory tell us aboutproperties of quark-gluon plasmardquo Nuclear Physics A vol 750no 1 pp 64ndash83 2005

[133] P Romatschke andURomatschke ldquoViscosity information fromrelativistic nuclear collisions how perfect is the fluid observedat RHICrdquo Physical Review Letters vol 99 no 17 Article ID172301 2007

[134] S Gavin ldquoTransport coefficients in ultra-relativistic heavy-ioncollisionsrdquo Nuclear Physics A vol 435 no 3-4 pp 826ndash8431985

[135] A Dobado and S N Santalla ldquoPion gas viscosity at low tem-perature and densityrdquo Physical Review D vol 65 no 9 ArticleID 096011 2002

[136] M Prakash M Prakash R Venugopalan and G Welke ldquoNon-equilibrium properties of hadronic mixturesrdquo Physics Reportvol 227 no 6 pp 321ndash366 1993

[137] J-W Chen Y-H Li Y-F Liu and E Nakano ldquoQCD viscosityto entropy density ratio in the hadronic phaserdquo Physical ReviewD vol 76 Article ID 114011 8 pages 2007

[138] J-W Chen and E Nakano ldquoShear viscosity to entropy densityratio of QCD below the deconfinement temperaturerdquo PhysicsLetters B vol 647 pp 371ndash375 2007

[139] K Itakura O Morimatsu and H Otomo ldquoShear viscosity of ahadronic gas mixturerdquo Physical Review D vol 77 no 1 ArticleID 014014 2008

[140] A S Khvorostukhin V D Toneev and D N VoskresenskyldquoViscosity coefficients for hadron and quark-gluon phasesrdquoNuclear Physics A vol 845 no 1ndash4 pp 106ndash146 2010

[141] A Muronga ldquoShear viscosity coefficient from microscopicmodelsrdquo Physical Review C vol 69 no 4 Article ID 0449012004

[142] S Muroya and N Sasaki ldquoA calculation of the viscosity toentropy ratio of a hadronic gasrdquo Progress of Theoretical Physicsvol 113 no 2 pp 457ndash462 2005

[143] N Demir and S A Bass ldquoShear-viscosity to entropy densityratio of a relativistic hadron gasrdquo Physical Review Letters vol102 Article ID 172302 2009

[144] L P Csernai J I Kapusta and L D McLerran ldquoStronglyinteracting low-viscosity matter created in relativistic nuclearcollisionsrdquo Physical Review Letters vol 97 no 15 Article ID152303 2006

[145] J Noronha-Hostler J Noronha and C Greiner ldquoTransportcoefficients of hadronicmatter near Tcrdquo Physical Review Lettersvol 103 no 17 Article ID 172302 2009

[146] R A Lacey N N Ajitanand J M Alexander et al ldquoHas theQCD critical point been signaled by observations at the BNLrelativistic heavy ion colliderrdquo Physical Review Letters vol 98Article ID 092301 2007

[147] A Dobado F J Llanes-Estrada and J M Torres-Rincon ldquo120578sand phase transitionsrdquo Physical Review D vol 79 no 1 ArticleID 014002 2009

[148] A Dobado F Llanes-Estrada and J M Torres-Rincon ldquoetasand phase transitionsrdquo Physical Review D vol 79 Article ID014002 2009

[149] S Pal ldquoShear viscosity to entropy density ratio of a relativisticHagedorn resonance gasrdquoPhysics Letters B vol 684 pp 211ndash2152010


[150] P Castorina J Cleymans D E Miller and H Satz ldquoThe speedof sound in hadronic matterrdquo European Physical Journal C vol66 no 1 pp 207ndash213 2010

[151] J Cleymans and D Worku ldquoThe Hagedorn temperature revis-itedrdquo Modern Physics Letters A vol 26 no 16 pp 1197ndash12092011

[152] R V Gavai and A Gocksch ldquoVelocity of sound in SU(3) latticegauge theoryrdquo Physical Review D vol 33 no 2 pp 614ndash6161986

[153] K Redlich and H Satz ldquoCritical behavior near deconfinementrdquoPhysical Review D vol 33 no 12 pp 3747ndash3752 1986

[154] F Karsch ldquoRecent lattice results on finite temperature anddensity QCDmdashpart Irdquo Proceedings of Science CPOD07 0262007

[155] D Prorok and L Turko ldquo119869120595 absorption in a multicomponenthadron gasrdquo in Proceedings of the AIP Conference vol 1038 p10 2008

[156] M Chojnacki andW Florkowski ldquoTemperature dependence ofsound velocity and hydrodynamics of ultra-relativistic heavy-ion collisionsrdquo Acta Physica Polonica B vol 38 pp 3249ndash32622007

[157] M I Gorenstein M Hauer and O N Moroz ldquoViscosity in theexcluded volume hadron gas modelrdquo Physical Review C vol 77no 2 Article ID 024911 2008

[158] E M Lifschitz and L P Pitaevski Physical Kinetics PergamonPress Oxford UK 2nd edition 1981

[159] G Policastro D T Son and A O Starinets ldquoShear viscosityof strongly coupled N = 4 supersymmetric Yang-Mills plasmardquoPhysical Review Letters vol 87 no 8 Article ID 081601 2001

[160] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 pp 231ndash252 1998

[161] Z Qiu C Shen and U Heinz ldquoHydrodynamic elliptic andtriangular flow in Pb-Pb collisions atradic119904119873119873 = 276ATeVrdquoPhysicsLetters B vol 707 no 1 pp 151ndash155 2012

[162] J Letessier and J Rafelski Hadrons and Quark-Gluon PlasmaCambridge University Press Cambridge UK 2004

[163] L D Landau Izvestiya Akademii Nauk vol 17 p 51 1953[164] E Schnedermann J Sollfrank and U Heinz ldquoThermal phe-

nomenology of hadrons from 200A GeV S+S collisionsrdquo Physi-cal Review C vol 48 no 5 pp 2462ndash2475 1993

[165] E Schnedermann and U Heinz ldquoCircumstantial evidence fortransverse flow in 200A GeV S+S collisionsrdquo Physical ReviewLetters vol 69 no 20 pp 2908ndash2911 1992

[166] S Q Feng and Y Zhong ldquoBaryon production and collectiveflow in relativistic heavy-ion collisions in the AGS SPS RHICand LHC energy regions (radic119904119873119873 le 5 GeV to 55 TeV)rdquo PhysicalReview C vol 83 Article ID 034908 2011

[167] S Q Feng and X B Yuan ldquoThe feature study on the pion andproton rapidity distributions at AGS SPS and RHICrdquo Science inChina G vol 52 p 198 2009

[168] S Uddin J S Ahmad W Bashir and R A Bhat ldquoA unifiedapproach towards describing rapidity and transverse momen-tum distributions in a thermal freeze-out modelrdquo Journal ofPhysics G vol 39 Article ID 015012 2012

[169] THiranoKMorita SMuroya andCNonaka ldquoHydrodynam-ical analysis of hadronic spectra in the 130GeVnucleonAu+Aucollisionsrdquo Physical Review C vol 65 no 6 Article ID 0619022002

[170] K Morita S Muroya C Nonaka and T Hirano ldquoComparisonof space-time evolutions of hot dense matter in radic119904119873119873 =17 and 130 GeV relativistic heavy ion collisions based on ahydrodynamical modelrdquo Physical Review C vol 66 Article ID054904 2002

[171] J Manninen E L Bratkovskaya W Cassing and O LinnykldquoDilepton production in p+p Cu+Cu and Au+Au collisions atradics = 200GeVrdquo European Physical Journal C vol 71 article 16152011

[172] S A Bass M Belkacem andM Bleicher ldquoMicroscopic modelsfor ultrarelativistic heavy ion collisionsrdquo Progress in Particle andNuclear Physics vol 41 pp 255ndash369 1998

[173] U Mayer and U Heinz ldquoGlobal hydrodynamics with continu-ous freeze-outrdquo Physical Review C vol 56 no 1 pp 439ndash4521997

[174] UHeinz ldquoPrimordial hadrosynthesis in the little bangrdquoNuclearPhysics A vol 661 pp 140ndash149 1999

[175] F Becattini and J Cleymans ldquoChemical equilibrium in heavyion collisions rapidity dependencerdquo Journal of Physics G vol34 no 8 pp S959ndashS963 2007

[176] F Becattini J Cleymans and J Strumpfer ldquoRapidity variationof thermal parameters at SPS and RHICrdquo Proceeding of ScienceCPOD07 012 2007

[177] B Biedron and W Broniowski ldquoRapidity-dependent spectrafrom a single-freeze-out model of relativistic heavy-ion colli-sionsrdquo Physical Review C vol 75 Article ID 054905 2007

[178] W Broniowski and B Biedron ldquoRapidity-dependent chemicalpotentials in a statistical approachrdquo Journal of Physics G vol 35Article ID 044018 2008

[179] J Cleymans J Strumpfer and L Turko ldquoExtended longitudinalscaling and the thermal modelrdquo Physical Review C vol 78 no1 Article ID 017901 2008

[180] W Broniowski andW Florkowski ldquoGeometric relation betweencentrality and the impact parameter in relativistic heavy-ioncollisionsrdquo Physical Review C vol 65 no 2 Article ID 0649052002

[181] C Adler Z Ahammed and C Allgower ldquoMeasurement ofinclusive antiprotons from Au+Au collisions at radic119904119873119873 = 130GeVrdquo Physical Review Letters vol 87 Article ID 262302 2001

[182] K Adcox S S Adler N N Ajitanand et al ldquoSingle identifiedhadron spectra from radic119904119873119873 = 130 GeV Au+Au collisionsrdquoPhysical Review C vol 69 Article ID 024904 2004

[183] A Adare C Aidala and N N Ajitanand ldquoCold-nuclear-matter effects on heavy-quark production in d+Au collisions atradic119904119873119873 = 200 GeVrdquo Physical Review Letters vol 109 Article ID242301 7 pages 2012

[184] P Bozek ldquoViscous evolution of the rapidity distribution ofmatter created in relativistic heavy-ion collisionsrdquo PhysicalReview C vol 77 Article ID 034911 2008

[185] Y B Ivanov Physical Review C vol 87 Article ID 064905 2013[186] P Huovinen P F Kolb and U Heinz ldquoIs there elliptic flow

without transverse flowrdquo Nuclear Physics A vol 698 no 1ndash4p 475 2002

[187] Y B Ivanov V N Russkikh and V D Toneev ldquoRelativis-tic heavy-ion collisions within three-fluid hydrodynamicshadronic scenariordquo Physical Review C vol 73 no 4 Article ID044904 2006

[188] BMohanty and J Alam ldquoVelocity of sound in relativistic heavy-ion collisionsrdquo Physical Review C vol 68 Article ID 0649032003


[189] P K Netrakanti and B Mohanty ldquoWidth of the rapidity distri-bution in heavy-ion collisionsrdquo Physical Review C vol 71 no 4Article ID 047901 2005

[190] F Cooper and G Frye ldquoSingle-particle distribution in thehydrodynamic and statistical thermodynamic models of mul-tiparticle productionrdquo Physical Review D vol 10 no 1 pp 186ndash189 1974

[191] D Adamova G Agakichiev H Appelshauser et al ldquoUniversalpion freeze-out in heavy-ion collisionsrdquo Physical Review Lettersvol 90 Article ID 022301 2003

[192] I G Bearden ldquoCharged meson rapidity distributions in centralAu+Au collisions at radic119904119873119873 = 200 GeVrdquo Physical Review Lettersvol 94 Article ID 162301 2005

[193] S S Adler S Afanasiev and C Aidala ldquoIdentified chargedparticle spectra and yields in Au+Au collisions at radic119904119873119873 = 200GeVrdquo Physical Review C vol 69 Article ID 034909 2004

[194] M Floris and ALICE collaboration ldquoIdentified particles in ppand Pb-Pb collisions at LHC energies with the ALICE detectorrdquoJournal of Physics G vol 38 Article ID 124025 2011

[195] C Shen U Heinz P Huovinen and H Song ldquoRadial andelliptic flow in Pb+Pb collisions at energies available at theCERN Large Hadron Collider from viscous hydrodynamicsrdquoPhysical Review C vol 84 no 4 Article ID 044903 2011

[196] P K Srivastava and C P Singh ldquoA hybrid model for QCDdeconfining phase boundaryrdquo Physical ReviewD vol 85 ArticleID 114016 2012

[197] P K Srivastava S K Tiwari and C P Singh ldquoQCD criticalpoint in a quasiparticle modelrdquo Physical Review D vol 82 no1 Article ID 014023 2010

[198] P K Srivastava S K Tiwari and C P Singh ldquoOn locating thecritical end point in QCD phase diagramrdquo Nuclear Physics Avol 862-863 pp 424ndash426 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


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Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


BeamBeam Space

Preequilibrium

Equilibrated QGP

Mixed phase

Hadron gas

Freeze-outTime

Figure 1 A schematic diagram of space-time evolution of ultrarel-ativistic heavy-ion collisions

labeled as ldquoPreequilibriumrdquo in Figure 1 processes of parton-parton hard scatterings may predominantly occur in theoverlap region of two colliding nuclei thus depositing a largeamount of energy in the medium The matter is still notin thermal equilibrium and perturbative QCD models candescribe the underlying dynamics as a cascade of freely collid-ing partons The time of the preequilibrium state is predictedto about 1 fmc or less (ii) After the short preequilibriumstage the QGP phase would be formed in which parton-parton andor string-string interactions quickly contributeto attain thermal equilibrium in the medium The energydensity of this state is expected to reach above 3ndash5GeVfm3equivalent to the temperature of 200ndash300MeV The volumethen rapidly expands and matter cools down (iii) If thefirst order phase transition is assumed the ldquomixed phaserdquo isexpected to exist between the QGP and hadron phases inwhich quarks and gluons are again confined into hadrons atthe critical temperature 119879

119888 In the mixed phase the entropy

density is transferred into lower degrees of freedom andtherefore the system is prevented from a fast expansionThis leads to a maximum value in the lifetime of themixed phase which is expected to last for a relatively longtime (120591 gt 10 fmc) (iv) In the hadronic phase the systemkeeps collective expansion via hadron-hadron interactionsdecreasing its temperature Then the hadronic interactionsfreeze after the system reaches a certain size and temperatureand hadrons that freely stream out from the medium are tobe detected There are two types of freeze-out stages Wheninelastic collisions between constituents of the fireball do notoccur any longer we call this a chemical freeze-out stageLater when the elastic collisions also cease to happen in thefireball this stage specifies the thermal freeze-out

Since many experiments running at various places mea-sure the multiplicity ratios and so forth of various hadronsit is necessary to know to which extent the measured hadronyields indicate equilibration The level of equilibration ofparticles produced in these experiments is tested by analyzingthe particle abundances [22 23 29ndash57] or their momentumspectra [22ndash27 37 38 46 47 58] using thermal modelsNow in the first case one establishes the chemical compo-sition of the system while in the second case additional

information on dynamical evolution and collective flow canbe extracted Furthermore study of the variations of multi-plicity of produced particles with respect to collision energythe momentum spectra of particles and ratios of various par-ticles have led to perhaps one of the most remarkable resultscorresponding to high energy strong interaction physics [6]

Recently various approaches have been proposed for theprecise formulation of a proper equation of state (EOS) forhot and dense HG Lacking lattice QCD results for the EOSat finite baryon density 119899

119861 a common approach is to con-

struct a phenomenological EOS for both phases of stronglyinteractingmatter Among those approaches thermalmodelsare widely used and indeed are very successful in describingvarious features of the HG These models are based on theassumption of thermal equilibrium reached in HG A simpleform of the thermal model of hadrons is the ideal hadron gas(IHG) model in which hadrons and resonances are treatedas pointlike and noninteracting particles The introductionof resonances in the system is expected to account for theexistence of attractive interactions among hadrons [59] Butin order to account for the realistic behaviour of HG a shortrange repulsion must also be introduced The importance ofsuch correction is more obvious when we calculate the phasetransition using IHG picture which shows the reappearanceof hadronic phase as a stable configuration in the simpleGibbs construction of the phase equilibrium between the HGand QGP phases at very high baryon densities or chemi-cal potentials This anomalous situation [60ndash62] cannot beexplained because we know that the stable phase at any given(119879 120583119861) is the phase which has a larger pressure Once the

system makes a transition to the QGP phase it is expectedto remain in that phase even at extremely large 119879 and 120583

119861due

to the property of asymptotic freedom of QCD Moreover itis expected that the hadronic interactions become significantwhen hadrons are densely packed in a hot and dense hadrongas One significant way to handle the repulsive interactionbetween hadrons is based on a macroscopic descriptionin which the hadrons are given a geometrical size andhence they will experience a hardcore repulsive interactionwhen they touch each other and consequently a van-derWaals excluded-volume effect is visible As a result theparticle yields are essentially reduced in comparison to thatof IHG model and also the anomalous situation in thephase transitionmentioned above disappears Recentlymanyphenomenological models incorporating excluded-volumeeffect have been widely used to account for the propertiesof hot and dense HG [63ndash76] However these descriptionsusually suffer from some serious shortcomings First mostlythe descriptions are thermodynamically inconsistent becauseone does not have a well-defined partition function orthermodynamical potential (Ω) fromwhich other thermody-namical quantities can be derived for example the baryondensity (119899

119861) = 120597Ω120597120583

119861 Secondly for the dense hadron gas

the excluded-volumemodel violates causality (ie velocity ofsound 119888

119904in the medium is greater than the velocity of light)

So although some of the models explain the data very wellsuch shortcomings make the models mostly unattractiveSun et al [76] have incorporated the effect of relativisticcorrection in the formulation of an EOS for HG However




120596119903) which



119899119861 Thus 119880











1198941198810

119894and 1198810

119894is the



ln119885 (119879 119881 120582) = ln1198850 (119879 Δ 120582) (1)




1198941198810

119894= 1198644119861 =


120598ex119894=

1205980

119894

1 + 12059804119861 (2)



119899ex119894=

1198990

119894

1 + 12059804119861

119875ex119894=

1198750

119894

1 + 12059804119861

(3)

Here 1198990119894and 1198750




1198941198731198941198810

119894 here 1198810




119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

(4)

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

(5)

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

(6)

where 1198990119894 1198750119894 and 1205980



1198941198810

119894



119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198990

119894

1 + sum1198990

1198941198810

119894

+ 1198990

119872

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

+ 1198750

119872

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+ 1205980

119872

(7)

where 1198990119894and 1198990



119894) and 1198750

119894(1198750

119894)


119872 1198750119872 and 1205980








119866for


119862as follows

1198850

119866(119879 120583 119881) =

infin

sum

119873=0

exp(120583119873119879)119885119862 (119879119873119881) (8)



119885ex119866(119879 120583 119881 minus 119881

0119873)

=

infin

sum

119873=0

exp(120583119873119879)119885119862(119879119873119881 minus 119881

0119873)

times 120579 (119881 minus 1198810119873)

(9)


119885119875= int

infin

0


0119873) (10)


or

119885119875= int

infin

0


119909] (11)


119875ex(119879 120583) = 119875

0(119879 120583) (12)

where

120583 = 120583 minus 1198810119875ex(119879 120583) (13)


119899ex(119879 120583) =

1205971198750(119879 120583)

120597120583

120597120583

120597120583=

1198990(119879 120583)

1 + 11988101198990 (119879 120583)

119904ex1(119879 120583) =

1205971198750(119879 120583)

120597119879=

1199040

1(119879 120583)

1 + 11988101198990 (119879 120583)

120598ex(119879 120583) =

1205980(119879 120583)

1 + 11988101198990 (119879 120583)

(14)


119875ex(119879 1205831 120583

ℎ) =

ℎ

sum

119894=1

1198750

119894(119879 120583119894) (15)

where

120583119894= 120583119894minus 1198810

119894119875ex(119879 120583119894) (16)


119899ex119894(119879 120583119894) =

1198990

119894(119879 120583119894)

1 + sumℎ

119895=11198810

1198951198990

119895(119879 120583119895)

(17)


119861= 0



ln119885ex119894=119892119894120582119894

61205872119879int

119881minussum1198951198731198951198810119895

1198810119894

119889119881

times int

infin

0

1198964119889119896

radic1198962 + 1198982

119894


2

119894

119879)

(18)

where 119892119894and 120582

119894= exp(120583



119894


1198951198731198951198810


119895


ln119885ex119894= 119881(1 minussum

119895

119899ex1198951198810

119895)119868119894120582119894 (19)


119868119894=119892119894

21205872(119898119894

119879)

2

11987931198702(119898119894

119879) (20)


119894) can be

obtained as

119899ex119894=120582119894

119881(120597 ln119885ex

119894

120597120582119894

)

119879119881

(21)


119899ex119894= (1 minus 119877) 120582119894119868119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

(22)

Here 119877 = sum119894119899ex1198941198810






119899ex=1

119881

int120582


(23)


119877 = (1 minus 119877)sum

119894

1198681198941198810

119894120582119894minussum

119894

1198681198941198810

1198941205822

119894

120597119877

120597120582119894

(24)


120582119894 (119905) =

1

(119886119894+ 1198681198941198810

119894119905) (25)



119877 = 1 minus

intinfin

119905[exp (minus1199051015840) 119866 (1199051015840)] 1198891199051015840

exp (minus119905) 119866 (119905) (26)



119889120582119894(119905)

119889119905= minus1198681198941205822

1198941198810

119894

119866 (119905) = 119905

ℎ

prod

119894=2

(119886119894+ 1198681198941198810

119894119905)

(27)




111988111205821 here







120597119877

120597120582119894

=

120597sum119895119899ex1198951198810

119895

120597120582119894

= (120597119899

ex119894

120597120582119894

)1198810

119894 (28)


120597119899ex119894

120597120582119894

+ 119899ex119894(1

1198681198941198810

1198941205822

119894

+1

120582119894

) =1

1205821198941198810

119894

(1 minus sum

119894 = 119895

119899ex1198951198810

119895)

(29)


119899ex119894=

119876119894(1 minus sum

119895 = 119894119899ex1198951198810

119895)

1205821198941198810

119894

exp( 1

1198681198941198810

119894120582119894

) (30)

where

119876119894= int

120582119894

0

exp(minus 1

1198681198941198810

119894120582119894

)119889120582119894 (31)


119877 = sum

119895

119899ex1198951198810

119895=119883

1 + 119883 (32)

where

119883 =sum119894119899ex1198941198810

119894

1 minus sum119894119899ex1198941198810

119894

(33)



119899ex119894=(1 minus 119877)

1198810

119894

119876119894

120582119894exp (minus1119868

1198941198810

119894120582119894) minus 119876119894

(34)




119861gt



119868119894=119892119894

61205872119879int

infin

0

1198964119889119896

radic1198962 + 1198982

119894

1

[exp (119864119894119879) + 120582

119894] (35)


119899ex119894= (1 minus 119877) 119868119894120582119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

+ 1205822

119894(1 minus 119877) 119868

1015840

119894 (36)



119894 We


119877 = 1198771+ Ω119877 (37)

where 1198771= 1198770(1 + 119877

0) with 1198770 = sum1198990

1198941198810

119894+ sum 1198681015840

1198941198810

1198941205822

119894 1198990119894is


Ω = minus1

1 + 1198770sum

119894

1198990

1198941198810

119894120582119894

120597

120597120582119894

(38)


119877 = 1198771+ Ω1198771+ Ω21198771 (39)


119875ex= 119879 (1 minus 119877)sum

119894

119868119894120582119894+sum

119895

119875meson119895 (40)






119894119878119894(119899119904

119894minus119899119904

119894) = 0 where 119878







120598ex119894=1198792

119881

120597 ln119885ex119894

120597119879+ 120583119894119899ex119894 (41)


119904 =120598ex119894+ 119875

exminus 120583119861119899119861minus 120583119878119899119878

119879 (42)






119885ex119866(119881 119879 120583) =

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(43)

where

119911 (119879) =119892

21205872int

infin

0

1198962119889119896 exp[

[

minus

(1198962+ 1198982)12

119879

]

]

(44)


120597119885ex119866

120597120583=

infin

sum

119873=0

119873

119879exp(

120583119873

119879)

(119881 minus 1198810119873)119873

119873120579 (119881 minus 119881

0119873)119911119873

(45)


119879

119885ex119866

120597119885ex119866

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(46)


⟨119873⟩ =1

119885ex119866

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

119873 = 119879120597 ln119885ex

119866

120597120583=119879

119885ex119866

120597119885ex119866

120597120583

(47)


119873 = ⟨119873⟩ (48)


119864 = ⟨119864⟩ (49)


119885ex(119881 119879 120583) =

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873 (50)


120597119885ex

120597120583=

infin

sum

119873=0

119873

119879exp (

120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(51)



119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)


119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)


⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)


119861= 400MeV We


60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV



119861








119879 (MeV) = 119879lim (1 minus1


120583119861=

119889

1 + 119890radic119904119873119873

(56)







120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120596119903) which



119899119861 Thus 119880











1198941198810

119894and 1198810

119894is the



ln119885 (119879 119881 120582) = ln1198850 (119879 Δ 120582) (1)




1198941198810

119894= 1198644119861 =


120598ex119894=

1205980

119894

1 + 12059804119861 (2)



119899ex119894=

1198990

119894

1 + 12059804119861

119875ex119894=

1198750

119894

1 + 12059804119861

(3)

Here 1198990119894and 1198750




1198941198731198941198810

119894 here 1198810




119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

(4)

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

(5)

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

(6)

where 1198990119894 1198750119894 and 1205980



1198941198810

119894



119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198990

119894

1 + sum1198990

1198941198810

119894

+ 1198990

119872

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

+ 1198750

119872

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+ 1205980

119872

(7)

where 1198990119894and 1198990



119894) and 1198750

119894(1198750

119894)


119872 1198750119872 and 1205980








119866for


119862as follows

1198850

119866(119879 120583 119881) =

infin

sum

119873=0

exp(120583119873119879)119885119862 (119879119873119881) (8)



119885ex119866(119879 120583 119881 minus 119881

0119873)

=

infin

sum

119873=0

exp(120583119873119879)119885119862(119879119873119881 minus 119881

0119873)

times 120579 (119881 minus 1198810119873)

(9)


119885119875= int

infin

0


0119873) (10)


or

119885119875= int

infin

0


119909] (11)


119875ex(119879 120583) = 119875

0(119879 120583) (12)

where

120583 = 120583 minus 1198810119875ex(119879 120583) (13)


119899ex(119879 120583) =

1205971198750(119879 120583)

120597120583

120597120583

120597120583=

1198990(119879 120583)

1 + 11988101198990 (119879 120583)

119904ex1(119879 120583) =

1205971198750(119879 120583)

120597119879=

1199040

1(119879 120583)

1 + 11988101198990 (119879 120583)

120598ex(119879 120583) =

1205980(119879 120583)

1 + 11988101198990 (119879 120583)

(14)


119875ex(119879 1205831 120583

ℎ) =

ℎ

sum

119894=1

1198750

119894(119879 120583119894) (15)

where

120583119894= 120583119894minus 1198810

119894119875ex(119879 120583119894) (16)


119899ex119894(119879 120583119894) =

1198990

119894(119879 120583119894)

1 + sumℎ

119895=11198810

1198951198990

119895(119879 120583119895)

(17)


119861= 0



ln119885ex119894=119892119894120582119894

61205872119879int

119881minussum1198951198731198951198810119895

1198810119894

119889119881

times int

infin

0

1198964119889119896

radic1198962 + 1198982

119894


2

119894

119879)

(18)

where 119892119894and 120582

119894= exp(120583



119894


1198951198731198951198810


119895


ln119885ex119894= 119881(1 minussum

119895

119899ex1198951198810

119895)119868119894120582119894 (19)


119868119894=119892119894

21205872(119898119894

119879)

2

11987931198702(119898119894

119879) (20)


119894) can be

obtained as

119899ex119894=120582119894

119881(120597 ln119885ex

119894

120597120582119894

)

119879119881

(21)


119899ex119894= (1 minus 119877) 120582119894119868119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

(22)

Here 119877 = sum119894119899ex1198941198810






119899ex=1

119881

int120582


(23)


119877 = (1 minus 119877)sum

119894

1198681198941198810

119894120582119894minussum

119894

1198681198941198810

1198941205822

119894

120597119877

120597120582119894

(24)


120582119894 (119905) =

1

(119886119894+ 1198681198941198810

119894119905) (25)



119877 = 1 minus

intinfin

119905[exp (minus1199051015840) 119866 (1199051015840)] 1198891199051015840

exp (minus119905) 119866 (119905) (26)



119889120582119894(119905)

119889119905= minus1198681198941205822

1198941198810

119894

119866 (119905) = 119905

ℎ

prod

119894=2

(119886119894+ 1198681198941198810

119894119905)

(27)




111988111205821 here







120597119877

120597120582119894

=

120597sum119895119899ex1198951198810

119895

120597120582119894

= (120597119899

ex119894

120597120582119894

)1198810

119894 (28)


120597119899ex119894

120597120582119894

+ 119899ex119894(1

1198681198941198810

1198941205822

119894

+1

120582119894

) =1

1205821198941198810

119894

(1 minus sum

119894 = 119895

119899ex1198951198810

119895)

(29)


119899ex119894=

119876119894(1 minus sum

119895 = 119894119899ex1198951198810

119895)

1205821198941198810

119894

exp( 1

1198681198941198810

119894120582119894

) (30)

where

119876119894= int

120582119894

0

exp(minus 1

1198681198941198810

119894120582119894

)119889120582119894 (31)


119877 = sum

119895

119899ex1198951198810

119895=119883

1 + 119883 (32)

where

119883 =sum119894119899ex1198941198810

119894

1 minus sum119894119899ex1198941198810

119894

(33)



119899ex119894=(1 minus 119877)

1198810

119894

119876119894

120582119894exp (minus1119868

1198941198810

119894120582119894) minus 119876119894

(34)




119861gt



119868119894=119892119894

61205872119879int

infin

0

1198964119889119896

radic1198962 + 1198982

119894

1

[exp (119864119894119879) + 120582

119894] (35)


119899ex119894= (1 minus 119877) 119868119894120582119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

+ 1205822

119894(1 minus 119877) 119868

1015840

119894 (36)



119894 We


119877 = 1198771+ Ω119877 (37)

where 1198771= 1198770(1 + 119877

0) with 1198770 = sum1198990

1198941198810

119894+ sum 1198681015840

1198941198810

1198941205822

119894 1198990119894is


Ω = minus1

1 + 1198770sum

119894

1198990

1198941198810

119894120582119894

120597

120597120582119894

(38)


119877 = 1198771+ Ω1198771+ Ω21198771 (39)


119875ex= 119879 (1 minus 119877)sum

119894

119868119894120582119894+sum

119895

119875meson119895 (40)






119894119878119894(119899119904

119894minus119899119904

119894) = 0 where 119878







120598ex119894=1198792

119881

120597 ln119885ex119894

120597119879+ 120583119894119899ex119894 (41)


119904 =120598ex119894+ 119875

exminus 120583119861119899119861minus 120583119878119899119878

119879 (42)






119885ex119866(119881 119879 120583) =

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(43)

where

119911 (119879) =119892

21205872int

infin

0

1198962119889119896 exp[

[

minus

(1198962+ 1198982)12

119879

]

]

(44)


120597119885ex119866

120597120583=

infin

sum

119873=0

119873

119879exp(

120583119873

119879)

(119881 minus 1198810119873)119873

119873120579 (119881 minus 119881

0119873)119911119873

(45)


119879

119885ex119866

120597119885ex119866

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(46)


⟨119873⟩ =1

119885ex119866

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

119873 = 119879120597 ln119885ex

119866

120597120583=119879

119885ex119866

120597119885ex119866

120597120583

(47)


119873 = ⟨119873⟩ (48)


119864 = ⟨119864⟩ (49)


119885ex(119881 119879 120583) =

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873 (50)


120597119885ex

120597120583=

infin

sum

119873=0

119873

119879exp (

120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(51)



119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)


119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)


⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)


119861= 400MeV We


60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV



119861








119879 (MeV) = 119879lim (1 minus1


120583119861=

119889

1 + 119890radic119904119873119873

(56)







120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





ln119885 (119879 119881 120582) = ln1198850 (119879 Δ 120582) (1)




1198941198810

119894= 1198644119861 =


120598ex119894=

1205980

119894

1 + 12059804119861 (2)



119899ex119894=

1198990

119894

1 + 12059804119861

119875ex119894=

1198750

119894

1 + 12059804119861

(3)

Here 1198990119894and 1198750




1198941198731198941198810

119894 here 1198810




119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

(4)

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

(5)

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

(6)

where 1198990119894 1198750119894 and 1205980



1198941198810

119894



119899ex=

sum1198941198990

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198990

119894

1 + sum1198990

1198941198810

119894

+ 1198990

119872

119875ex=

sum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

minussum1198941198750

119894

1 + sum1198941198990

1198941198810

119894

+ 1198750

119872

120598ex=

sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+sum1198941205980

119894

1 + sum1198941198990

1198941198810

119894

+ 1205980

119872

(7)

where 1198990119894and 1198990



119894) and 1198750

119894(1198750

119894)


119872 1198750119872 and 1205980








119866for


119862as follows

1198850

119866(119879 120583 119881) =

infin

sum

119873=0

exp(120583119873119879)119885119862 (119879119873119881) (8)



119885ex119866(119879 120583 119881 minus 119881

0119873)

=

infin

sum

119873=0

exp(120583119873119879)119885119862(119879119873119881 minus 119881

0119873)

times 120579 (119881 minus 1198810119873)

(9)


119885119875= int

infin

0


0119873) (10)


or

119885119875= int

infin

0


119909] (11)


119875ex(119879 120583) = 119875

0(119879 120583) (12)

where

120583 = 120583 minus 1198810119875ex(119879 120583) (13)


119899ex(119879 120583) =

1205971198750(119879 120583)

120597120583

120597120583

120597120583=

1198990(119879 120583)

1 + 11988101198990 (119879 120583)

119904ex1(119879 120583) =

1205971198750(119879 120583)

120597119879=

1199040

1(119879 120583)

1 + 11988101198990 (119879 120583)

120598ex(119879 120583) =

1205980(119879 120583)

1 + 11988101198990 (119879 120583)

(14)


119875ex(119879 1205831 120583

ℎ) =

ℎ

sum

119894=1

1198750

119894(119879 120583119894) (15)

where

120583119894= 120583119894minus 1198810

119894119875ex(119879 120583119894) (16)


119899ex119894(119879 120583119894) =

1198990

119894(119879 120583119894)

1 + sumℎ

119895=11198810

1198951198990

119895(119879 120583119895)

(17)


119861= 0



ln119885ex119894=119892119894120582119894

61205872119879int

119881minussum1198951198731198951198810119895

1198810119894

119889119881

times int

infin

0

1198964119889119896

radic1198962 + 1198982

119894


2

119894

119879)

(18)

where 119892119894and 120582

119894= exp(120583



119894


1198951198731198951198810


119895


ln119885ex119894= 119881(1 minussum

119895

119899ex1198951198810

119895)119868119894120582119894 (19)


119868119894=119892119894

21205872(119898119894

119879)

2

11987931198702(119898119894

119879) (20)


119894) can be

obtained as

119899ex119894=120582119894

119881(120597 ln119885ex

119894

120597120582119894

)

119879119881

(21)


119899ex119894= (1 minus 119877) 120582119894119868119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

(22)

Here 119877 = sum119894119899ex1198941198810






119899ex=1

119881

int120582


(23)


119877 = (1 minus 119877)sum

119894

1198681198941198810

119894120582119894minussum

119894

1198681198941198810

1198941205822

119894

120597119877

120597120582119894

(24)


120582119894 (119905) =

1

(119886119894+ 1198681198941198810

119894119905) (25)



119877 = 1 minus

intinfin

119905[exp (minus1199051015840) 119866 (1199051015840)] 1198891199051015840

exp (minus119905) 119866 (119905) (26)



119889120582119894(119905)

119889119905= minus1198681198941205822

1198941198810

119894

119866 (119905) = 119905

ℎ

prod

119894=2

(119886119894+ 1198681198941198810

119894119905)

(27)




111988111205821 here







120597119877

120597120582119894

=

120597sum119895119899ex1198951198810

119895

120597120582119894

= (120597119899

ex119894

120597120582119894

)1198810

119894 (28)


120597119899ex119894

120597120582119894

+ 119899ex119894(1

1198681198941198810

1198941205822

119894

+1

120582119894

) =1

1205821198941198810

119894

(1 minus sum

119894 = 119895

119899ex1198951198810

119895)

(29)


119899ex119894=

119876119894(1 minus sum

119895 = 119894119899ex1198951198810

119895)

1205821198941198810

119894

exp( 1

1198681198941198810

119894120582119894

) (30)

where

119876119894= int

120582119894

0

exp(minus 1

1198681198941198810

119894120582119894

)119889120582119894 (31)


119877 = sum

119895

119899ex1198951198810

119895=119883

1 + 119883 (32)

where

119883 =sum119894119899ex1198941198810

119894

1 minus sum119894119899ex1198941198810

119894

(33)



119899ex119894=(1 minus 119877)

1198810

119894

119876119894

120582119894exp (minus1119868

1198941198810

119894120582119894) minus 119876119894

(34)




119861gt



119868119894=119892119894

61205872119879int

infin

0

1198964119889119896

radic1198962 + 1198982

119894

1

[exp (119864119894119879) + 120582

119894] (35)


119899ex119894= (1 minus 119877) 119868119894120582119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

+ 1205822

119894(1 minus 119877) 119868

1015840

119894 (36)



119894 We


119877 = 1198771+ Ω119877 (37)

where 1198771= 1198770(1 + 119877

0) with 1198770 = sum1198990

1198941198810

119894+ sum 1198681015840

1198941198810

1198941205822

119894 1198990119894is


Ω = minus1

1 + 1198770sum

119894

1198990

1198941198810

119894120582119894

120597

120597120582119894

(38)


119877 = 1198771+ Ω1198771+ Ω21198771 (39)


119875ex= 119879 (1 minus 119877)sum

119894

119868119894120582119894+sum

119895

119875meson119895 (40)






119894119878119894(119899119904

119894minus119899119904

119894) = 0 where 119878







120598ex119894=1198792

119881

120597 ln119885ex119894

120597119879+ 120583119894119899ex119894 (41)


119904 =120598ex119894+ 119875

exminus 120583119861119899119861minus 120583119878119899119878

119879 (42)






119885ex119866(119881 119879 120583) =

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(43)

where

119911 (119879) =119892

21205872int

infin

0

1198962119889119896 exp[

[

minus

(1198962+ 1198982)12

119879

]

]

(44)


120597119885ex119866

120597120583=

infin

sum

119873=0

119873

119879exp(

120583119873

119879)

(119881 minus 1198810119873)119873

119873120579 (119881 minus 119881

0119873)119911119873

(45)


119879

119885ex119866

120597119885ex119866

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(46)


⟨119873⟩ =1

119885ex119866

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

119873 = 119879120597 ln119885ex

119866

120597120583=119879

119885ex119866

120597119885ex119866

120597120583

(47)


119873 = ⟨119873⟩ (48)


119864 = ⟨119864⟩ (49)


119885ex(119881 119879 120583) =

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873 (50)


120597119885ex

120597120583=

infin

sum

119873=0

119873

119879exp (

120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(51)



119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)


119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)


⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)


119861= 400MeV We


60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV



119861








119879 (MeV) = 119879lim (1 minus1


120583119861=

119889

1 + 119890radic119904119873119873

(56)







120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




or

119885119875= int

infin

0


119909] (11)


119875ex(119879 120583) = 119875

0(119879 120583) (12)

where

120583 = 120583 minus 1198810119875ex(119879 120583) (13)


119899ex(119879 120583) =

1205971198750(119879 120583)

120597120583

120597120583

120597120583=

1198990(119879 120583)

1 + 11988101198990 (119879 120583)

119904ex1(119879 120583) =

1205971198750(119879 120583)

120597119879=

1199040

1(119879 120583)

1 + 11988101198990 (119879 120583)

120598ex(119879 120583) =

1205980(119879 120583)

1 + 11988101198990 (119879 120583)

(14)


119875ex(119879 1205831 120583

ℎ) =

ℎ

sum

119894=1

1198750

119894(119879 120583119894) (15)

where

120583119894= 120583119894minus 1198810

119894119875ex(119879 120583119894) (16)


119899ex119894(119879 120583119894) =

1198990

119894(119879 120583119894)

1 + sumℎ

119895=11198810

1198951198990

119895(119879 120583119895)

(17)


119861= 0



ln119885ex119894=119892119894120582119894

61205872119879int

119881minussum1198951198731198951198810119895

1198810119894

119889119881

times int

infin

0

1198964119889119896

radic1198962 + 1198982

119894


2

119894

119879)

(18)

where 119892119894and 120582

119894= exp(120583



119894


1198951198731198951198810


119895


ln119885ex119894= 119881(1 minussum

119895

119899ex1198951198810

119895)119868119894120582119894 (19)


119868119894=119892119894

21205872(119898119894

119879)

2

11987931198702(119898119894

119879) (20)


119894) can be

obtained as

119899ex119894=120582119894

119881(120597 ln119885ex

119894

120597120582119894

)

119879119881

(21)


119899ex119894= (1 minus 119877) 120582119894119868119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

(22)

Here 119877 = sum119894119899ex1198941198810






119899ex=1

119881

int120582


(23)


119877 = (1 minus 119877)sum

119894

1198681198941198810

119894120582119894minussum

119894

1198681198941198810

1198941205822

119894

120597119877

120597120582119894

(24)


120582119894 (119905) =

1

(119886119894+ 1198681198941198810

119894119905) (25)



119877 = 1 minus

intinfin

119905[exp (minus1199051015840) 119866 (1199051015840)] 1198891199051015840

exp (minus119905) 119866 (119905) (26)



119889120582119894(119905)

119889119905= minus1198681198941205822

1198941198810

119894

119866 (119905) = 119905

ℎ

prod

119894=2

(119886119894+ 1198681198941198810

119894119905)

(27)




111988111205821 here







120597119877

120597120582119894

=

120597sum119895119899ex1198951198810

119895

120597120582119894

= (120597119899

ex119894

120597120582119894

)1198810

119894 (28)


120597119899ex119894

120597120582119894

+ 119899ex119894(1

1198681198941198810

1198941205822

119894

+1

120582119894

) =1

1205821198941198810

119894

(1 minus sum

119894 = 119895

119899ex1198951198810

119895)

(29)


119899ex119894=

119876119894(1 minus sum

119895 = 119894119899ex1198951198810

119895)

1205821198941198810

119894

exp( 1

1198681198941198810

119894120582119894

) (30)

where

119876119894= int

120582119894

0

exp(minus 1

1198681198941198810

119894120582119894

)119889120582119894 (31)


119877 = sum

119895

119899ex1198951198810

119895=119883

1 + 119883 (32)

where

119883 =sum119894119899ex1198941198810

119894

1 minus sum119894119899ex1198941198810

119894

(33)



119899ex119894=(1 minus 119877)

1198810

119894

119876119894

120582119894exp (minus1119868

1198941198810

119894120582119894) minus 119876119894

(34)




119861gt



119868119894=119892119894

61205872119879int

infin

0

1198964119889119896

radic1198962 + 1198982

119894

1

[exp (119864119894119879) + 120582

119894] (35)


119899ex119894= (1 minus 119877) 119868119894120582119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

+ 1205822

119894(1 minus 119877) 119868

1015840

119894 (36)



119894 We


119877 = 1198771+ Ω119877 (37)

where 1198771= 1198770(1 + 119877

0) with 1198770 = sum1198990

1198941198810

119894+ sum 1198681015840

1198941198810

1198941205822

119894 1198990119894is


Ω = minus1

1 + 1198770sum

119894

1198990

1198941198810

119894120582119894

120597

120597120582119894

(38)


119877 = 1198771+ Ω1198771+ Ω21198771 (39)


119875ex= 119879 (1 minus 119877)sum

119894

119868119894120582119894+sum

119895

119875meson119895 (40)






119894119878119894(119899119904

119894minus119899119904

119894) = 0 where 119878







120598ex119894=1198792

119881

120597 ln119885ex119894

120597119879+ 120583119894119899ex119894 (41)


119904 =120598ex119894+ 119875

exminus 120583119861119899119861minus 120583119878119899119878

119879 (42)






119885ex119866(119881 119879 120583) =

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(43)

where

119911 (119879) =119892

21205872int

infin

0

1198962119889119896 exp[

[

minus

(1198962+ 1198982)12

119879

]

]

(44)


120597119885ex119866

120597120583=

infin

sum

119873=0

119873

119879exp(

120583119873

119879)

(119881 minus 1198810119873)119873

119873120579 (119881 minus 119881

0119873)119911119873

(45)


119879

119885ex119866

120597119885ex119866

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(46)


⟨119873⟩ =1

119885ex119866

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

119873 = 119879120597 ln119885ex

119866

120597120583=119879

119885ex119866

120597119885ex119866

120597120583

(47)


119873 = ⟨119873⟩ (48)


119864 = ⟨119864⟩ (49)


119885ex(119881 119879 120583) =

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873 (50)


120597119885ex

120597120583=

infin

sum

119873=0

119873

119879exp (

120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(51)



119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)


119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)


⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)


119861= 400MeV We


60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV



119861








119879 (MeV) = 119879lim (1 minus1


120583119861=

119889

1 + 119890radic119904119873119873

(56)







120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119889120582119894(119905)

119889119905= minus1198681198941205822

1198941198810

119894

119866 (119905) = 119905

ℎ

prod

119894=2

(119886119894+ 1198681198941198810

119894119905)

(27)




111988111205821 here







120597119877

120597120582119894

=

120597sum119895119899ex1198951198810

119895

120597120582119894

= (120597119899

ex119894

120597120582119894

)1198810

119894 (28)


120597119899ex119894

120597120582119894

+ 119899ex119894(1

1198681198941198810

1198941205822

119894

+1

120582119894

) =1

1205821198941198810

119894

(1 minus sum

119894 = 119895

119899ex1198951198810

119895)

(29)


119899ex119894=

119876119894(1 minus sum

119895 = 119894119899ex1198951198810

119895)

1205821198941198810

119894

exp( 1

1198681198941198810

119894120582119894

) (30)

where

119876119894= int

120582119894

0

exp(minus 1

1198681198941198810

119894120582119894

)119889120582119894 (31)


119877 = sum

119895

119899ex1198951198810

119895=119883

1 + 119883 (32)

where

119883 =sum119894119899ex1198941198810

119894

1 minus sum119894119899ex1198941198810

119894

(33)



119899ex119894=(1 minus 119877)

1198810

119894

119876119894

120582119894exp (minus1119868

1198941198810

119894120582119894) minus 119876119894

(34)




119861gt



119868119894=119892119894

61205872119879int

infin

0

1198964119889119896

radic1198962 + 1198982

119894

1

[exp (119864119894119879) + 120582

119894] (35)


119899ex119894= (1 minus 119877) 119868119894120582119894 minus 119868119894120582

2

119894

120597119877

120597120582119894

+ 1205822

119894(1 minus 119877) 119868

1015840

119894 (36)



119894 We


119877 = 1198771+ Ω119877 (37)

where 1198771= 1198770(1 + 119877

0) with 1198770 = sum1198990

1198941198810

119894+ sum 1198681015840

1198941198810

1198941205822

119894 1198990119894is


Ω = minus1

1 + 1198770sum

119894

1198990

1198941198810

119894120582119894

120597

120597120582119894

(38)


119877 = 1198771+ Ω1198771+ Ω21198771 (39)


119875ex= 119879 (1 minus 119877)sum

119894

119868119894120582119894+sum

119895

119875meson119895 (40)






119894119878119894(119899119904

119894minus119899119904

119894) = 0 where 119878







120598ex119894=1198792

119881

120597 ln119885ex119894

120597119879+ 120583119894119899ex119894 (41)


119904 =120598ex119894+ 119875

exminus 120583119861119899119861minus 120583119878119899119878

119879 (42)






119885ex119866(119881 119879 120583) =

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(43)

where

119911 (119879) =119892

21205872int

infin

0

1198962119889119896 exp[

[

minus

(1198962+ 1198982)12

119879

]

]

(44)


120597119885ex119866

120597120583=

infin

sum

119873=0

119873

119879exp(

120583119873

119879)

(119881 minus 1198810119873)119873

119873120579 (119881 minus 119881

0119873)119911119873

(45)


119879

119885ex119866

120597119885ex119866

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(46)


⟨119873⟩ =1

119885ex119866

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

119873 = 119879120597 ln119885ex

119866

120597120583=119879

119885ex119866

120597119885ex119866

120597120583

(47)


119873 = ⟨119873⟩ (48)


119864 = ⟨119864⟩ (49)


119885ex(119881 119879 120583) =

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873 (50)


120597119885ex

120597120583=

infin

sum

119873=0

119873

119879exp (

120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(51)



119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)


119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)


⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)


119861= 400MeV We


60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV



119861








119879 (MeV) = 119879lim (1 minus1


120583119861=

119889

1 + 119890radic119904119873119873

(56)







120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119894119878119894(119899119904

119894minus119899119904

119894) = 0 where 119878







120598ex119894=1198792

119881

120597 ln119885ex119894

120597119879+ 120583119894119899ex119894 (41)


119904 =120598ex119894+ 119875

exminus 120583119861119899119861minus 120583119878119899119878

119879 (42)






119885ex119866(119881 119879 120583) =

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(43)

where

119911 (119879) =119892

21205872int

infin

0

1198962119889119896 exp[

[

minus

(1198962+ 1198982)12

119879

]

]

(44)


120597119885ex119866

120597120583=

infin

sum

119873=0

119873

119879exp(

120583119873

119879)

(119881 minus 1198810119873)119873

119873120579 (119881 minus 119881

0119873)119911119873

(45)


119879

119885ex119866

120597119885ex119866

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873

times 120579 (119881 minus 1198810119873)119911119873

(46)


⟨119873⟩ =1

119885ex119866

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

119873 = 119879120597 ln119885ex

119866

120597120583=119879

119885ex119866

120597119885ex119866

120597120583

(47)


119873 = ⟨119873⟩ (48)


119864 = ⟨119864⟩ (49)


119885ex(119881 119879 120583) =

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873 (50)


120597119885ex

120597120583=

infin

sum

119873=0

119873

119879exp (

120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(51)



119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)


119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)


⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)


119861= 400MeV We


60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV



119861








119879 (MeV) = 119879lim (1 minus1


120583119861=

119889

1 + 119890radic119904119873119873

(56)







120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119879

119885ex120597119885

ex

120597120583=1

119885ex

infin

sum

119873=0

119873 exp (120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

minus119879

119885ex

infin

sum

119873=0

exp (120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 1

times 1199111198731198810 120597119873

120597120583

(52)


119873 = ⟨119873⟩ minus119879

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873minus1

119873 minus 11199111198731198810 120597119873

120597120583

(53)

or

119899 = ⟨119899⟩ minus 119879⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩ (54)


⟨1198990

(1 minus 119877)1198810 120597119899

120597120583⟩

=1

119885ex

infin

sum

119873=0

exp(120583119873

119879)

(119881 minus 1198810119873)119873

119873119911119873

times (119873

(119881 minus 1198810119873)

1198810 120597119873

120597120583)

(55)


119861= 400MeV We


60 80 100 120 140 160 180 200

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

T (MeV)

120583B = 400MeV



119861








119879 (MeV) = 119879lim (1 minus1


120583119861=

119889

1 + 119890radic119904119873119873

(56)







120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




120406080

100120140160180200

0100200300400500600700800900

T(M

eV)

120583B

(MeV

)1 10 102

1 10210




radicsNN (GeV)

radicsNN (GeV)










119861at various radic119904119873119873


4= 3120594











follows [103]

120583119861=

119886

1 + 119887radic119904119873119873

119879 = 119888 minus 1198891205832

119861minus 1198901205834

119861

(57)



1205752= sum

119894

(119877exp119894minus 119877

therm119894)2

(119877therm119894)2 (58)

where 119877exp119894

and 119877therm119894






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1198611205752

119879 120583119861

1205752

119879 120583119861

1205752

119879 120583119861

1205752

270 60 740 085 60 740 075 70 753 119 70 760 115332 80 670 089 78 680 034 89 686 075 90 670 045384 100 645 050 86 640 090 101 639 037 100 640 034432 101 590 070 100 590 098 109 600 017 105 600 023876 140 380 045 145 406 062 144 386 005 140 360 025123 148 300 031 150 298 071 153 300 003 150 276 020173 160 255 025 160 240 062 1586 228 063 155 206 027130 1723 3553 010 1655 38 054 1658 3584 015 1635 32 005200 1723 2353 0065 1655 25 060 1659 235 010 164 20 005

0

01

02

03

04

05

06

07

08

120583B

(GeV

)







0

20

40

60

80

100

120

140

160

180

200

100 200 300 400 500 600 700 800 900 1000

T(M

eV)

120583B (MeV)




119904as used in



1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1 100

005

01

015

02

025



102 103 104

Λ120587

minus

radicsNN (GeV)






119861with respect


119861


0

5

10

15



1 10 102 103 104

1000

120601120587

radicsNN (GeV)










119904(1198882119904= 120597119875120597120598 at





0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

02

04

06

08

1

12

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

KminusK

+

pp

ΛΛ

1 10 102 103 104 1 10 102 103

1 10 102 103 104 1 10 102 103 104





Ξ+Ξ









119904





0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

10

20

30

40

50

60

70

80

80 100 120 140 160 180 200

T (MeV)

snB







119861


0

05

1

15

2

25

80 100 120 140 160 180 200T (MeV)


120598 HG

(GeV

fm3)



119899 (119879 120583) =124

119881ℎ

[1 minus119899119861(119879 120583)

119899 (119879 120583)] +034

119881ℎ

119899119861(119879 120583)

119899 (119879 120583) (59)





05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




05

1

15

2

25

3

35

4

2 4 6 8 10 12 14 16 18 20


Tota

l had

roni

c pre

ssur

e (G

eVfm

3)

sn = 58 (times1000)sn = 9

sn = 112


119904[31]



01

02

03

04

05

06

Our modelRGSG model

2 4 6 8 10 12 14

Spee

d of

soun

d (cs)

(sn)


06

08

1

12

14


RGSG modelIHG model

EN

1 10 102

radicsNN (GeV)




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




4

5

6

7

8

9

10


RGSG modelIHG model

10 102

SN

radicsNN (GeV)




0

05

1

15

2

25

60 80 100 120 140 160 180 200

T (MeV)


120578s





120578 prop ln ⟨10038161003816100381610038161199011003816100381610038161003816⟩ (60)


120578 =5

64radic81199032sum

119894

⟨10038161003816100381610038161199011198941003816100381610038161003816⟩ times119899119894

119899 (61)



119888= 170ndash180MeV) 120578119904 shows a



0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 200 400 600 800 1000

107

106

105

104

103

102

10

1

120583B (MeV)

120578s










119861(=300MeV)


10

1

10minus1

10minus2

10minus3

10minus460 80 100 120 140 160 180

T (MeV)






120 130 140 150 160 170 180 190

10minus2

10minus3

10minus4

120578(G

eV3)

T (MeV)







0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

02

04

06

08

1

12

120578s

101 102


radicsNN (GeV)












119899ex119894=119892119894120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877) int

infin

0

1198893119896

[exp (119864119894119879) + 120582

119894]2]

(62)



119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)119864119894

[exp (119864119894119879) + 120582

119894]

minus 120582119894(1 minus 119877)

119864119894

[exp (119864119894119879) + 120582

119894]2]

(63)


119894120597119877120597120582

119894] However


(119889119873119894

119889119910)

th=119892119894119881120582119894

(21205872)[((1 minus 119877) minus 120582

119894

120597119877

120597120582119894

)

times int

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]

minus 120582119894(1 minus 119877)

timesint

infin

0

1198982

119879cosh119910119889119898

119879

[exp (119898119879cosh119910119879) + 120582

119894]2]

(64)


mass (119898119879= radic1198982 + 119901




119889119873119894

119889119910= int

120578max

minus120578max

(119889119873119894

119889119910)

th(119910 minus 120578) 119889120578 (65)



⟨120573119871⟩ = tanh(

120578max2) (66)





119889119873119894

119889119910= int

+infin

minusinfin

120588 (119910FB)119889119873119894

1(119910 minus 119910FB)

119889119910119889119910FB (67)



120588 (119910FB) =1

radic2120587120590exp(

minus1199102

FB21205902) (68)


119889119873119894

119898119879119889119898119879

=119892119894119881120582119894

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]1198981198791198701(119898119879

119879)

(69)


1198701(119898119879

119879) = int

infin

0


)]119889119910 (70)


119864119894

1198893119873119894

1198891198963=119892119894119881120582119894

(2120587)3[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]


119879)119896120582119889120590120582

(71)



119889119873119894

119898119879119889119898119879

=119892119894119881120582119894119898119879

(21205872)[(1 minus 119877) minus 120582

119894

120597119877

120597120582119894

]

times int

1198770

0

1199031198891199031198701(119898119879cosh 120588119879

) 1198680(119901119879sinh 120588119879

)

(72)


1198680(119901119879sinh 120588119879

) =1

2120587int

2120587

0

exp(119901119879sinh 120588 cos120601119879

)119889120601

(73)


chosen as120573119903= 120573119904(120585)119899 [37 38 164]120573



1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1000

2000

3000

4000

5000

0

2000

4000

6000

8000

10000

12000


ABC

D

A998400

B998400

C998400

D998400

dVdy

(fm3)

Tota

l fre

eze-

out v

olum

e (fm

3)

101 102

radicsNN (GeV)




⟨120573119903⟩ =int120573119904120585119899120585119889120585

int 120585119889120585= (

2

2 + 119899)120573119904 (74)





Rapidity (y)

0

50

100

150

200

250

300

350

400


dN120587dy

120587+




119871= 092 at radic119904119873119873 =



119879lt 03GeV We


119879except 119898







400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




400

300

200

100


Rapidity (y)

dNdy

RHIC 200 GeV 120590 = 218

SPS 172 GeV 120590 = 125

SPS 123 GeV 120590 = 105

SPS 87 GeV 120590 = 088









119904= 080 (120573



119904= 088

[58]

Our model


05 1 15 2 25 3 35 4 45 5

103

102

10

1

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

mT minus m0 (GeV)

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

p120587+








103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




103

102

10

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

(1(2120587

))(d

2N

(mTdmTdy

)) (1

(G

eV)2

)

1

05 1 15 2 25 3 35 4 45 5

120587+


p

Our model + flow

Our model


pT (GeV)





103

102

10

1

10minus1

10minus2

05 1 15 2 25 3 35 4 45 5

pT (GeV)

d2N(dpTdy)

(1G

eV)

Our model + flow

Our model + flow

120587minus


p







Acknowledgment


References

















































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Acknowledgment


References

















































































































































































































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



review article particle production in ultrarelativistic...

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