Hadron production by quark combination in central Pb+Pb collisions at s NN − − − − √ =17.3 GeV
aa r X i v : . [ h e p - ph ] J u l Hadron production by quark combination in central Pb + Pb collisions at √ s NN = . GeV
Chang-en Shao, Jun Song, Feng-lan Shao, and Qu-bing Xie Department of Physics, Qufu Normal University, Shandong 273165, People’s Republic of China Department of Physics, Shandong University, Shandong 250100, People’s Republic of China
The quark combination mechanism of QGP hadronization is applied to nucleus-nucleus collisions at top SPSenergy. The yields, rapidity and transverse momentum distributions of identified hadrons in most central Pb + Pbcollisions at √ s NN = . √ s NN = PACS numbers: 25.75.Dw, 25.75.Ld, 25.75.Nq, 25.75.-q
I. INTRODUCTION
Lattice QCD predicts that at extremely high temperatureand density, the confined hadronic matter will undergo aphase transition to a new state of matter called quark gluonplasma (QGP) [1, 2]. The relativistic heavy ion collisionscan provide the condition to create this deconfined partonicmatter [3]. In general, two approaches are used to studythe properties of the deconfined hot and dense quark mat-ter produced in AA collisions. One is studying the high p T hadrons from initial hard jets, in which one can recur tothe perturbative QCD to a certain degree [4]. The other isinvestigating the properties of thermal hadrons frozen outfrom the hot and dense quark matter. For the latter, thehadronization of the hot and dense quark matter (a typi-cal non-perturbative process) is of great significance. Onlythrough a reliable hadronization mechanism, can we re-versely obtain various information of QGP properties fromthe final state hadrons measured experimentally. The abun-dant experimental data [5, 6] and phenomenological stud-ies [7, 8, 9, 10, 11, 12] at RHIC energies suggest thatquark combination mechanism is one of the most hopefulcandidates. The two most noticeable results are the suc-cessful explanation of the high baryon / meson ratios andthe constituent quark number scaling of the hadronic el-liptic flow in the intermediate transverse momentum range[8, 9], which can not be understood at all in the partonicfragmentation picture. Recently, the NA49 Collaborationhave measured the elliptic flow of identified hadrons at topSPS energy [13], and found that the quark number scal-ing of elliptic flow was shown to hold also. It immedi-ately gives us an inspiration of the applicability for thequark combination at top SPS energy. On the other hand,the NA49 collaboration have found three interesting phe-nomena around 30A GeV [14], i.e. the steepening of theenergy dependence for pion multiplicity, a maximum inthe energy dependence of strangeness to pion ratio and acharacteristic plateau of the e ff ective temperature for kaonproduction. These phenomena are indicative of the onsetof the deconfinement at low SPS energies. One can es-timate via Bjorken method that the primordial spatial en-ergy density in Pb + Pb collisions at top SPS energy is about 3.0 GeV / f m [15], exceeding the critical energy density(about 1 GeV / f m ) predicted by Lattice QCD. Therefore,the deconfined hot and dense quark matter has been prob-ably created, and we can extend the quark combinationmechanism to SPS energies.As is well known, hadron yield is one of the most ba-sic and important observables which can help us to test theunderstanding of the hadronization mechanism for the hotand dense quark matter created in the relativistic heavy ioncollisions. In most of recombination / coalescence models,the hadron wave function is necessary to get the hadronyield. As the wave functions for almost all hadrons are un-known at present, it is di ffi cult for these models to studythis issue quantitatively [7, 8, 16]. In addition, these mod-els do not satisfy the unitarity which is important to theissue as well [17]. Di ff erent from those models, the quarkcombination model [18, 19] uses the near-correlation inphase space and SU f (3) symmetry, instead of hadron wavefunction, to determine the hadron multiplicity. In addi-tion, the model satisfies unitarity as well and has repro-duced many experimental data at RHIC [20, 21, 22, 23].Therefore, we apply it in this paper to systematically studythe yields, rapidity and transverse momentum distributionsof various hadrons in most central Pb + Pb collisions at √ s NN = . ff er-ence in collective flow between light and strange quarks,which occurs at RHIC energies [23, 26]. The other is thestrangeness enhancement, a significant property of QGP[27].The paper is arranged as follows. In the next section, wemake a brief introduction to the quark combination model.In section III, we calculate the yields and rapidity distri-butions of identified hadrons in most central Pb + Pb col-lisions at √ s NN = . II. AN INTRODUCTION OF THE QUARKCOMBINATION MODEL
The starting point of the model is a color singlet sys-tem which consists of constituent quarks and antiquarks.All kinds of hadronization models demand that they sat-isfy rapidity or momentum correlation for quarks in theneighborhood of phase space. The essence of this corre-lation is in agreement with the fundamental requirementof QCD [28]. According to QCD, a qq may be in a coloroctet or a singlet. The color factors h ( q ¯ q ) | − λ a · λ a | ( q ¯ q ) i = and h ( q ¯ q ) | − λ a · λ a | ( q ¯ q ) i = − , which means a repulsive oran attractive interaction between them. Here λ a are theGell-Mann matrices. If they are close with each other inphase space, they can interact with su ffi ciently time to bein the color singlet and form a meson. Similarly, a qq can be in a sextet or an anti-triplet, and the color factors h ( qq ) | λ a · λ a | ( qq ) i = and h ( qq ) ¯3 | λ a · λ a | ( qq ) ¯3 i = − . If itsnearest neighbor in phase space is a q , they form a baryon.If the neighbor is a q , because the attraction strength of thesinglet is two times that of the anti-triplet, qq will win thecompetition to form a meson and leave a q alone to com-bine with other quarks or antiquarks. Based on the aboveQCD and near-correlation in phase space requirements,we had proposed a quark combination rule(QCR) [18, 28]which combines all these quarks and antiquarks into initialhadrons. When the transverse momentum of quarks arenegligible, all q and q can always line up stochastically inrapidity. The QCR reads as follows:1. Starting from the first parton ( q or q ) in the line;2. If the baryon number of the second in the line is ofthe di ff erent type from the first, i.e. the first two par-tons are either qq or qq , they combine into a mesonand are removed from the line, go to point 1; Other-wise they are either qq or q q , go to the next point;3. Look at the third, if it is of the di ff erent type from thefirst, the first and third partons form a meson and areremoved from the line, go to point 1; Otherwise thefirst three partons combine into a baryon or an anti-baryon and are removed from the line, go to point1. The following example shows how the above QCRworks q q q q q q q q q q q q q q q q q q q q → M ( q q ) B ( q q q ) M ( q q ) B ( q q q ) M ( q q ) M ( q q ) B ( q q q ) B ( q q q )If the quarks and antiquarks are stochastically arrangedin rapidity, the probability distribution for N pairs ofquarks and antiquarks to combine into M mesons, B baryons and B anti-baryons is X MB ( N ) = N ( N !) ( M + B − N )! M !( B !) M − δ N , M + B . (1)Hadronization is the soft process of the strong interac-tion and is independent of flavor, so the net flavor numberremains constant during the process. In the quark com-bination scheme, this means that the quark number foreach certain flavor prior to hadronization equals to thatof all initially produced hadrons after it. Obviously thequark number conservation is automatically satisfied in themodel. It is di ff erent from the non-linear algebraic methodin ALCOR model [24] where normalization factor for eachquark flavor is introduced with the constraint of the quarknumber conservation.The average number of initially produced mesons M ( N )and baryons B ( N ) are given by h M ( N ) i = X M X B MX MB ( N ) , (2) h B ( N ) i = X M X B BX MB ( N ) . (3)Then the multiplicity of various initial hadrons is obtainedaccording to their production weights h M initialj i = C M j h M ( N ) i , h B initialj i = C B j h B ( N ) i , (4)where C M j and C B j are normalized production weights forthe meson M j and baryon B j , respectively. If three quarkflavors are considered only, we can obtain the productionweights using the SU f (3) symmetry with a strangenesssuppression factor λ s [18, 19], which are listed in TableI. The extension of the symmetry to excited states, ex-otic states and more quark flavors is also straightforward[19, 22, 29].Considering the decay contributions from the reso-nances, we can obtain the yields of final state hadrons h h finali i = h h initiali i + X j B r ( j → i ) h h j i , (5)where the B r ( j → i ) is the weighted decay branching ratiofor h j to h i [30].In principle, the hadron production probability shouldbe calculated from the matrix element h qq | M i for me-son or h qqq | B i for baryon. However, the wave functionsfor almost all hadrons which are governed by the non-perturbative QCD are unknown at present. It is di ffi cult TABLE I: The normalized production weight for baryons andmesons in the SU f (3) ground state. r i is the number of strangequarks in hadron. The ratio of the vector ( J P = − ) to pseu-doscalar ( J P = − ) meson follows the spin counting, while thatof the decouplet ( J P = + ) to octet ( J P = + ) baryon su ff ers aspin suppression e ff ect; see Ref. [19, 29] for details. C M C M i = J i + + λ s ) λ r i s , except C η = J η + + λ s ) + λ s C η ′ = J η ′ + + λ s ) + λ s C B i = + λ s ) (2 J i + λ r i s , except C B C Λ = C Σ = C Σ ∗ = C Λ (1520) = + λ s ) λ s to study the production of hadrons quantitatively throughtheir wave functions. In view of this, the hadron pro-duction probability in our model is determined by theSU f (3) symmetry with a strangeness suppression. Thissymmetry has been supported by many experiments, par-ticularly by the coincidence of the observed λ s obtainedfrom various mesons and baryons [31]. Therefore, themodel can quantitatively describe many global propertiesfor the bulk system by virtue of the Monte Carlo method[19, 20, 21, 22, 23, 29].When applying the model to describe the hadronizatonof the hot and dense quark matter produced in heavy ioncollisions, the net-baryon quantum number of the systemperplexes the analysis formula of Eq. 1 but it can be eas-ily evaluated in Monte Carlo program. On the other hand,the the transverse momentum of quarks is not negligibledue to the strong collective flow of quark matter. In princi-ple, we should define the QCR in three-dimensional phasespace, but it is quite complicated to have it because onedoes not have an order or one has to define an order ina sophisticated way so that all quarks can combine intohadrons in a particular sequence. In practice, the combi-nation is still put in rapidity and meanwhile the maximumtransverse momentum di ff erence ∆ p between (anti)quarksare constrained as they combine into hadrons. The trans-verse spectra of hadrons have a relationship with the quarkspectra as follows (e.g. for meson) dN M d p T ∝ Z d p , T d p , T f q ( p , T ) f q ( p , T ) δ ( p T − p , T − p , T ) × Θ ( ∆ p − | p ∗ , T − p ∗ , T | ) , (6)where f q / q ( p T ) is the transverse momentum distribution ofthe quark / antiquark, assumed to be rapidity-independent inpresent work. The superscript asterisk denotes the quarkmomentum in the center-of-mass frame of formed hadron.The limitation ∆ p is treated as parameter in our study, andfixed to be ∆ p = . ∆ p = . ∆ p on thehadron yield is neglected.One issue that is often questioned is the energy and en-tropy conservation in quark combination process. As thenon-perturbative QCD is unsolved, there is no rigorous theory which can incorporate the partonic phase as well ashadronic phase, thus it is di ffi cult to justify or condemn thisissue in essence at the moment. As we know, a lot of theexperimental phenomena in intermediate transverse mo-mentum range at RHIC can be explained beautifully onlyin the quark combination scenario. It suggests that maybethis ’puzzling’ issue does not exist. As far as the quarkcombination itself is concerned, there is no di ff erence forthe combination occurred in the di ff erent (intermediate orlow) transverse momentum range. Therefore, whether theproperties of low p T hadrons can be reproduced or not isalso a significant test of the quark combination mechanism,as the vast majority of hadrons observed experimentallyare just these with low transverse momentum. III. HADRON YIELDS AND RAPIDITYDISTRIBUTIONS
In high energy nucleus-nucleus collisions, the energydeposited in the collision region excites large numbers ofnewborn quarks and antiquarks from the vacuum. Sub-sequently, the hot and dense quark matter mainly com-posed of these newborn quarks will expend hydrodynami-cally until hadronization. The net-quarks from the collid-ing nuclei still carry a fraction of beam energy, thus theirevolution is di ff erent from the newborn quarks. One partof net-quarks are stopped in the hot and dense quark mat-ter, and hadronize together with it. The other part of net-quarks penetrate the hot quark matter, and run up to theforward rapidity region. The latter, together with smallamount newborn quarks, form the leading fireball. Theirhadronization should be earlier than that of the hot anddense quark matter with a prolonged expansion stage, andthe hadronization outcomes consist of nucleons and smallmount of mesons.The current version of quark combination model sim-ulates only the hadronization of the hot and dense quarkmatter and subsequently decays of resonances. One in- y c.m. d N / d y newborn lightquarksstrange quarksnet-quarks FIG. 1: (Color online) Rapidity spectra of newborn quarks andnet-quarks at hadronization in most central Pb + Pb collisions at √ s NN = . dispensable input is the momentum distributions of ther-mal quarks and antiquarks at hadronization, which are theresults of the hydrodynamic evolution in partonic phase.In order to focus attentions on the test of quark combi-nation mechanism in this and next sections, we reverselyextract the quark distributions by fitting the experimentaldata in the model. A detailed analysis of quark distribu-tions at hadronization will be in section V. The solid anddashed lines in Fig. 1 show the rapidity distributions ofnewborn light and strange quarks at hadronization respec-tively, obtained from the π − and K + data [32].The dotted-dashed line is the rapidity distribution of net-quarks in thehot and dense quark matter, which is extracted from net-proton data [33].Firstly, we calculate the yields and rapidity densities atmidrapidity of various hadrons in most central Pb + Pb col-lisions at √ s NN = . φ are shown to be about twice as high as the exper-imental data. The results of other hadrons are basically inagreement with the experimental data, but slight deviationsexist also. The overpredictions of φ meson may be asso-ciated with the exotic particle f (980), which has a possi-ble tetraquark structure containing a strange quark and astrange antiquark [40]. As a bond state containing strangecomponents, it has a slightly lower mass than φ meson butis not included in the SU f (3) ground states. In the presentwork, we consider only the production of 36 − plets of me-son and 56 − plets of baryon in the SU f (3) ground states,and the excited states and exotic states are not taken intoaccount. The f (980) multiplicity is found to be nearly thesame as φ meson in the e + e − annihilations [30]. The m T distribution of f (980) measured by STAR Collaborationin Au + Au collisions at √ s NN =
200 GeV is also shown tobe comparable to that of φ [41, 42]. Therefore, the over-prediction of φ meson can be removed by incorporating the f (980) production.Subsequently, we will calculate the longitudinal rapiditydistributions of various hadrons. Due to the deviations inhadron yields, it is di ffi cult to directly compare the calcu-lated hadron spectra with the experimental data. In orderto focus attentions on the property of hadron momentumspectra, we will scale the calculated rapidity densities tothe center value of the experimental data when we showthe hadronic rapidity and p T spectra in Fig. 2 and 4 respec-tively, thereby removing these deviations in hadron yields.Pion is the lightest and most abundant hadron producedin AA collisions, and its momentum distribution can bestreflect the global evolution property of the hot and densequark matter. In various models of high energy heavy ioncollisions, the reproduction of pion meson is always takenas a paramount test of models. In Landau hydrodynamicmodel [43], the rapidity distribution of pion can be well TABLE II: The yields (left) and rapidity densities at midrapid-ity (right) of identified hadrons in central Pb + Pb collisions at √ s NN = . dNdy | y = data model data model π + ± ±
31 566 170 . ± . ± π − ± ±
31 630 175 . ± . ± K + ± ± . . ± . ± . K − . ± . ± . . ± . ± . K s ± . ± . ± . φ . ± . ± .
33 15 . . ± . ± .
08 5.26 p
120 29 . ± . ± .
96 25.9 p . ± . ± .
17 1.53 Λ . ± . ± . ± . ± . Λ . ± . ± .
31 2.88 1 . ± . ± .
13 1.35 Ξ − . ± . ± .
57 4.9 1 . ± . ± .
15 1.43 Ξ + . ± . ± .
08 0.58 0 . ± . ± .
03 0.26 described and the sound velocity (which is an importantphysical quantity standing for the property of the hot anddense quark matter) can be extracted from the pion distri-bution. For other hadrons, such as kaons, protons, Λ , Ξ and so on, the Landau model can not describe their rapid-ity distributions with the same sound velocity or freeze-out temperature [44, 45, 46]. For a systematic descrip-tion of the rapidity distributions of various hadrons, thedetailed longitudinal dynamics, e.g. the evolution of net-baryon density which will result in the yield and spectrumasymmetry between hadron and antihadron, should be in-cluded. In addition, the hadronization mechanism is espe-cially important to describe the di ff erences in the yield andmomentum distribution of various hadron species. Usingthe extracted quark distributions in Fig. 1, we calculate therapidity distributions of pions, kaons, Λ ( Λ ), Ξ − ( Ξ + ) and φ in most central Pb + Pb collisions at √ s NN = . φ meson is narrow than the latest data of NA49 Collaboration(open circles in the second panel), but is in good agreementwith previous data (filled circles). The rapidity spectra ofother hadrons are well reproduced. One can see that thequark combination mechanism is applicable for describingthe longitudinal distributions of various hadrons at top SPSenergy. IV. HADRON TRANSVERSE MOMENTUMDISTRIBUTIONS
In this section, we calculate the transverse momentumdistributions of various hadrons in the midrapidity range.In this paper, we only consider the hadronization of the hotand dense quark matter. The transverse momentum invari-ant distribution of constituent quarks at hadronization is y c.m. d N / d y p - K + · - · y c.m. K s0 f · y c.m. LL – · y c.m. X - X –+ · FIG. 2: (Color online) The scaled rapidity distributions of identified hadrons in most central Pb + Pb collisions at √ s NN = . φ data in the second panel are the latest results measuredby NA49 Collaboration [35], and filled circles are the previous ones [47]. Other experimental data are taken from Ref. [32, 37, 38].The open symbols of K s , Λ and Ξ show data points reflected around midrapidity. taken to be an exponential form exp( − m T / T ), where T isthe slope parameter which is also called e ff ective produc-tion temperature. Fig. 3 shows the midrapidity p T spec-tra of constituent quarks at hadronization in most centralPb + Pb collisions at √ s NN = . π + and K + respectively [48]. The net-quark distri-bution is fixed by the data of K − / K + ratio as a function of p T [48]. A detailed analysis of the quark p T spectra willbe shown in section V. -3 -2 -1 T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] newborn lightquarksstrange quarksnet-quarks FIG. 3: (Color online) The transverse momentum distributionsof constituent quarks in the midrapidity region at hadronizationin most central Pb + Pb collisions at √ s NN = . Fig. 4 shows the calculated p T spectra of pions, kaons,protons, Λ ( Λ ), Ξ − ( Ξ + ) and φ in most central Pb + Pb col-lisions at √ s NN = . Λ and Ω , the spectral slopes of antihadrons measured experimen-tally are all steeper than those of hadrons [48, 49, 50].However, the spectrum of Ξ − is abnormally steeper thanthat of Ξ + [37]. Our predictions are in good agreementwith all the data except Ξ − .The exponential function exp( − m T / T ) is often used ex-perimentally to fit the transverse momentum distributionsof identified hadrons in the low p T range, and to extract the e ff ective production temperature T of various hadrons.It is found at top SPS energy that all final-state hadronsexcept pion meson have much higher T than the criticaltemperature [51], which indicates a strong collective flowat this collision energy. It is regarded in Ref. [52] thatthis observed flow mainly develops in the late hadronicrescattering stage. But results in Fig. 2 and Fig. 4 allshow that both longitudinal and transverse spectra of vari-ous hadrons can be coherently explained by the same quarkdistributions, respectively. It suggests that the observedflow should mainly come from the expansive evolutionstage of the hot and dense quark matter before hadroniza-tion, but not from the post-hadronization stage. In addi-tion, the same constituent quark spectra contained in light,single- and multi- strange hadrons also imply that the hotquark matter hadronize into these initial hadrons almost atthe same time, i.e. the hadronization is a rapid process.In Fig. 5, we calculate the p T spectra of pions, kaons,protons, Λ ( Λ ), Ξ − ( Ξ + ), φ and Ω ( Ω ) in most central Au + Aucollisions at top RHIC energy. The momentum distribu-tions of constituent quarks at hadronization are taken tobe exp( − m T / . − m T / . φ meson is multiplied by a factor 0.5.One can see that the p T spectra of various hadrons are wellreproduced.The baryon / meson ratio as a function of p T is sen-sitive to the hadronization mechanism. As we know,the observed high baryon / meson ratios in the interme-diate p T range at RHIC energies [5] can not be under-stood at all in the scheme of parton fragmentation, butcan be easily explained in the quark combination mech-anism. Fig. 6 shows the model predictions of p /π − , Λ / K s and Ω /φ at midrapidity in both central Pb + Pb colli-sions at √ s NN = . + Au collisionsat √ s NN =
200 GeV. In the intermediate p T range wherethe hadron production is dominated by the combination ofthermal quarks, the baryon / meson ratios increase with the -7 -5 -3 -1 T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] p + · p - · -1 -7 -5 -3 -1 T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] k + · k - · -1 -6 -4 -2 T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] pp – -2 -1
110 0 1 2m t -m [ GeV ] d N / ( m t d m t d y ) [ ( G e V ) - ] LL - -3 -2 -1
110 0 1 2m t -m [ GeV ] d N / ( m t d m t d y ) [ ( G e V ) - ] X - X –+ -2 -1
110 0 1 f m t -m [ GeV ] d N / ( m t d m t d y ) [ ( G e V ) - ] FIG. 4: (Color online) The scaled transverse momentum distributions of identified hadrons at midrapidity in most central Pb + Pbcollisions at √ s NN = . -6 -4 -2 P T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] p + · p - · -1 -5 -3 -1 P T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] K + · K - · -1 K S0 -6 -4 -2 P T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] p · p – · -1 -5 -3 -1
10 0 1 2 3 4 5 P T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] L· L - · -1 -6 -4 -2 P T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] X - · X –+ · -1 -5 -4 -3 -2 -1 P T [ GeV/c ] d N / ( p P T d P T d y ) [ ( G e V / c ) - ] W – + W – + f · FIG. 5: (Color online) The transverse momentum distributions of identified hadrons at midrapidity in most central Au + Au collisions at √ s NN =
200 GeV. Only the combination of thermal quarks is taken into account. Solid lines are the calculated results of hadrons anddashed lines for antihadrons. The experimental data are from Ref. [5, 53, 54] increasing p T . One can see that the experimental data inthis region are well reproduced. The falling tendency ofmeasured baryon / meson ratios after peak position is owingto the abundant participation of jet quarks, which is beyondthe concern of the present paper. Besides the hadronizationmechanism, the baryon / meson ratio in the intermediate p T region is also influenced by other two factors. One is thenuclear stopping power in collisions. Comparing with the strong collision transparency at top RHIC energy [39], thestrong nuclear stopping at top SPS energy causes the de-tention of abundant net-quarks in the midrapidity region.These net-quarks significantly suppress the production ofanti-baryons and enhance that of the baryons. Therefore,the p /π − ratio at top SPS energy is much lower than that attop RHIC energy while the Λ / K s ratio at top SPS energyis higher than that at top RHIC energy. The other is the -2 -1 P T [ GeV/c ] p (cid:190) / p- r a t i o NA49, 17.3GeVSTAR, 200GeV P T [ GeV/c ] L / K s0 r a t i o NA49, 17.3GeVSTAR, 200GeV -1 P T [ GeV/c ] W / f r a t i o STAR, 200GeV
FIG. 6: (Color online) The ratios of p /π − , Λ / K s and Ω /φ at midrapidity in most central Pb + Pb collisions at √ s NN = . + Au collisions at 200 GeV. Only the combination of thermal quarks is taken into account. Solid lines are the calculated results inAu + Au collisions and dashed lines for Pb + Pb collisions. The experimental data are from Ref. [5, 48, 54, 56, 57] momentum distribution of constituent quarks at hadroniza-tion. This can be illustrated by Ω /φ ratio because the pro-duction of these two hadron species is less influenced bythe net-quarks. The calculated Ω /φ ratio shows a weakdependence on the collision energy in the intermediate p T range. The well description of various baryon / meson ra-tios in such a wide energy range is an indication of theuniversality for the quark combination mechanism. V. ANALYSIS OF PARTON DISTRIBUTIONS ATHADRONIZATION
The constituent quark distributions at hadronizationcarry the information on the evolution of the hot and densequark matter in partonic phase. In this section, we focus at-tentions on the longitudinal and transverse collective flowsand strangeness enhancement of the hot and dense quarkmatter produced at top SPS energy.
A. The longitudinal and transverse collective flow
Due to the thermal pressure, the hot and dense quarkmatter created in high energy heavy ion collisions will ex-pand during the evolution before hadronization. The longi-tudinal and transverse collective flow of final hadrons mea-sured experimentally is the exhibition of this early thermalexpansion in the partonic phase. Utilizing the relativistichydrodynamic evolution of the hot and dense quark matter,one can obtain the collective flow in quark level from theextracted quark momentum distributions, and compare itwith that at RHIC energies.There are two well known hydrodynamic models forthe description of the space time evolution of the hot anddense quark matter produced in heavy ion collisions. Oneis Bjorken model [58] which supposes that the collisionis transparent, and it is appropriate to extremely high en-ergy collisions, such as LHC. The other is Landau model[43] with an assumption of the full stopping for nucleus- nucleus collisions. The longitudinal evolution result isequivalent to the superposition of a set of thermal sourcesin rapidity axis, with a (Bjorken) uniform or (Landau)Gaussian weight. In general, when applying the modelto describe the hadron rapidity distributions, di ff erent pa-rameter values are required to make a good fit of di ff erenthadron species [44]. In this paper, we apply the hydro-dynamic description to the evolution in quark level, thusthe collective flow of various hadrons can be coherentlyexplained.One can see from the energy dependence of the net-baryon rapidity distribution [39] that the collisions at topSPS energy are neither full transparent nor full stopping.The suppositions of nuclear stopping power in the twomodels are inappropriate to the nucleus-nucleus collisionsat top SPS energy. For the description of the rapidity dis-tribution for constituent quarks, one can limit the boost in-variance into a finite rapidity range in the framework ofBjorken model. This modification is often used to analyzethe longitudinal collectivity in hadronic level [44, 59]. Therapidity distribution in a isotropic, thermalized fluid ele-ment moving with a rapidity η is dN th dy ( y − η ) = A T f exp (cid:18) − mT f cosh ( y − η ) (cid:19) × (cid:18) m T f + mT f cosh ( y − η ) + cosh ( y − η ) (cid:19) . (7)The rapidity distribution of constituent quarks in the hotand dense quark matter is the longitudinal boost-invariantsuperposition of multiple isotropic, thermalized fluid ele-ments dNdy = Z η max − η max dN th dy ( y − η ) d η, (8) η max is the maximal boot rapidity of fluid elements. Theaverage longitudinal collective velocity is taken to be <β L > = tanh( η max / T f is the temperature of the locally-thermalizedhot and dense quark matter at hadronization. It is takento be T f =
170 MeV. m is the constituent mass of quarkswhen they evolve to the transition point. It is taken to be340 MeV for light quarks and 500 MeV for strange quarks.We have mentioned in above section that the net-quarks,still carrying a fraction of initial collision energy, have amore complex evolution than hydrodynamic expansion inlongitudinal axis [60]. Therefore, we extract the longitu-dinal collective flow from the rapidity distribution of new-born quarks. Since most of the data are measured in therapidity range about [-1.5, 1.5], the rapidity spectra of con-stituent quarks extracted from experimental data are validonly in this region. Using above equations to fit the rapid-ity distribution of newborn constituent quarks in Fig. 1, weobtain < β L > = .
58 for light quarks and < β L > = .
65 forstrange quarks. It is interesting to find that the average lon-gitudinal collective velocity of strange quarks is obviouslygreater than that of light quarks.For the transverse expansion of the hot and dense quarkmatter, we adopt a blast-wave model proposed by Heniz[59] within the boost-invariant scenario. The quarks andantiquarks transversely boost with a flow velocity β r ( r ) asa function of transverse radial position r . β r ( r ) is param-eterized by the surface velocity β s : β r ( r ) = β s ξ n , where ξ = r / R max , and R max is the thermal source maximum ra-dius (0 < ξ < ρ = tanh − β r dN π p T d p T = A Z ξ d ξ m T × I (cid:18) p T sinh ρ T f (cid:19) K (cid:18) m T cosh ρ T f (cid:19) . (9)Here, I and K are the modified Bessel functions. m T = √ p T2 + m is the transverse mass of the constituent quark.The average transverse velocity can be written as h β r i = R β s ξ n ξ d ξ R ξ d ξ = n + β s . (10)With fixed parameter n = .
3, the average transverse ve-locity h β r i is able to characterize the transverse collectiveflow of the hot and dense quark matter. Using the aboveequations to fit the transverse momentum distributions ofthe newborn quarks in Fig. 3, we obtain h β r i = .
41 forstrange quarks and h β r i = .
36 for light quarks. One cansee that the h β r i of strange quarks is obviously greater thanthat of light quarks.Both longitudinal and transverse results at top SPS en-ergy show that the strange constituent quarks get a strongercollective flow than the light quarks in the hydrodynamicevolution of partonic matter. By analyzing the data ofmulti-strange hadrons [26] and primary charged hadrons[23], the same property is found also at top RHIC energy.It suggests that the hot and dense quark matter producedat top SPS energy undergoes a similar hydrodynamic evo-lution to that at RHIC energies. It is generally believedthat the decoupled quark and gluon plasma (QGP) has beencreated at RHIC energies [61]. This similarity of collective flow property in quark level may be regarded as a signal ofQGP creation at top SPS energy. B. The enhanced strangeness
An interesting phenomenon in high energy heavy ioncollisions is the enhanced production of strange hadrons,which is absent in elementary particle collisions. In rela-tivistic heavy ion collisions, enormous amounts of energyare deposited in the collision region to create a deconfinedhot and dense quark matter. The multiple scatterings be-tween partons in the hot and dense quark matter will causethe large production rate of strangeness by gg → s ¯ s [27].The high strangeness of the hot and dense quark matter, af-ter hadronization, finally leads to the abundant productionof the strange hadrons. This phenomenon is regarded asa signal of QGP creation. As we know, the enhancementof strangeness production at top RHIC energy is quite ob-vious [62], and it is generally believed that the QGP hasbeen created at RHIC energies. When the collision energydrops to the SPS and AGS energies, it is found that thestrangeness production peaks at about 30A GeV and turnsinto a plateau at higher collision energies [14]. It is an in-dication of the onset of deconfinement.In our model, the strangeness of the hot and dense quarkmatter is characterized by the suppression factor λ s = N ¯ s : N ¯ u = N ¯ s : N ¯ d , i.e. the ratio of s quark number to new-born u (or d ) quark number. By fitting the experimentaldata of identified hadrons, we use the model to extract the λ s of hot and dense quark matter at midrapidity in cen-tral AA collisions at four energies √ s NN = .
3, 62.4,130 and 200 GeV, and the results are shown in Table III.The data of midrapidity dN / dy and the calculated resultswith minimum deviations at di ff erent collision energies areshown also. The statistical uncertainty of λ s is fixed bythe twice minimum deviation. The model reproduces thehadron yield in reasonably good way, and the chi-squarefit seems to indicate that with increasing collision energythe agreement with data significantly improves.One can see that λ s in such a broad energy range isnearly unchanged within statistical uncertainties, exhibit-ing an obvious saturation phenomenon. The results of λ s are consistent with the grand canonical limit ( ≈ .
45) ofstrangeness [15]. Using the Bjorken model, one can esti-mate that the primordial spatial energy density of the hotand dense quark matter produced in collisions at top RHICenergy is about 6 . GeV / f m [15], double of that in Pb + Pbcollisions at top SPS energy. The di ff erence in primordialenergy density is large while the final strangeness is nearlyequal. The hot and dense quark matter created in heavy ioncollisions is shown to be very close to a perfect fluid [71].It means that the local relaxation time toward to thermalequilibrium is much shorter than the macroscopic evolu-tion time of the hot and dense quark matter. When the hotand dense quark matter evolves to the point of hadroniza-tion, the strangeness abundance should be mainly deter-mined by the current temperature, irrelevant to the initial TABLE III: The strange suppression factor λ s and the calculated hadron dN / dy at midrapidity in central AA collisions at di ff erentenergies. The experimental data are taken from Ref. [32, 36, 37, 53, 63, 64, 65, 66, 67, 68, 69, 70].Pb Pb 17.3 GeV Au Au 62.4 GeV Au Au 130 GeV Au Au 200 GeVdata model data model data model data model π + . ± . ± ± . ±
14 211 276 ± ± . . . ± . π − . ± . ± ± . ±
14 217 270 ± . ± . . . ± . K + . ± . ± . . ± .
15 36 . . ± . ± . . ± . K − . ± . ± . . ± .
76 29 . . ± . ± . . ± . p . ± . ± . ± . ± . .
17 19 . ± . ± . .
45 18 . ± . p . ± . ± .
16 1.53 11 . ± . ± . .
15 13 . ± . ± . .
63 13 . ± . Λ . ± . ± . ± . ± .
49 13 .
42 17 . ± . ± . .
99 16 . ± . ± . Λ . ± . ± .
13 1.35 8 . ± . ± . .
77 12 . ± . ± . . ± . ± . Ξ − . ± . ± .
15 1.43 1 . ± . ± .
014 1 .
63 2 . ± . ± . .
99 2 . ± . ± .
19 2.12 Ξ + . ± . ± .
03 0.26 0 . ± . ± .
057 0 .
96 1 . ± . ± .
17 1 .
67 1 . ± . ± . Ω + Ω . ± . ± .
014 0 .
369 0 . ± . ± .
05 0.551 0 . ± . ± .
04 0.539 χ / nd f / / . / . / λ s . ± .
09 0 . ± .
02 0 . ± .
04 0 . ± . energy density and temperature. The same strangeness isan indication of the universal hadronization temperaturefor the hot and dense quark matter with low baryon chem-ical potential. VI. SUMMARY
In this paper, we have systematically studied the lon-gitudinal and transverse production of various hadrons attop SPS energy in the scheme of quark combination. Us-ing the quark combination model, we firstly calculate theyields and rapidity distributions of various hadrons in mostcentral Pb + Pb collisions at √ s NN = . p T distributionsof various hadrons at top SPS energy are calculated andcompared with the data. It is found that the light, sin-gle and multi-strange hadrons are well reproduced by thesame quark distributions. It indicates that the hadroniza-tion of the hot and dense quark matter is a rapid process.The well reproduced baryon / meson ratios in the interme- diate p T range at di ff erent collision energies are indicativeof the universality for the quark combination mechanism.By fitting the extracted constituent quark distributions athadronization with the hydrodynamic scenario, we furtherobtain the longitudinal and transverse collective flow of thehot and dense quark matter produced at top SPS energy.It is found that the strange quarks get a stronger collec-tive flow than light quarks, which is consistent with that atRHIC energies. The strangeness in the hot and dense quarkmatter produced at √ s NN = .
3, 62.4, 130, 200 GeV areextracted. The almost unchanged strangeness may be as-sociated with a universal hadronization temperature for thehot and dense quark matter with low baryon chemical po-tential.
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