# Hadron production by quark combination in central Pb+Pb collisions at
$\sqrt{s_{NN}}=17.3$ GeV

Chang-en Shao  Department of Physics, Qufu Normal University, Shandong 273165,
People’s Republic of China  Jun Song  Department of Physics, Shandong
University, Shandong 250100, People’s Republic of China  Feng-lan Shao
Department of Physics, Qufu Normal University, Shandong 273165, People’s
Republic of China  Qu-bing Xie  Department of Physics, Shandong University,
Shandong 250100, People’s Republic of China

###### Abstract

The quark combination mechanism of QGP hadronization is applied to nucleus-
nucleus collisions at top SPS energy. The yields, rapidity and transverse
momentum distributions of identified hadrons in most central Pb+Pb collisions
at $\sqrt{s_{NN}}=17.3$ GeV are systematically studied. The calculated results
are in agreement with the experimental data from NA49 Collaboration. The
longitudinal and transverse collective flows and strangeness of the hot and
dense quark matter produced in nucleus-nucleus collisions at top SPS energy
are investigated. It is found that the collective flow of strange quarks is
stronger than light quarks, which is compatible with that at RHIC energies,
and the strangeness is almost the same as those at $\sqrt{s_{NN}}=$ 62.4, 130,
200 GeV.

###### pacs:

25.75.Dw, 25.75.Ld, 25.75.Nq, 25.75.-q

##  I Introduction

Lattice QCD predicts that at extremely high temperature and density, the
confined hadronic matter will undergo a phase transition to a new state of
matter called quark gluon plasma (QGP) Susskind1979 ; qpg2004 . The
relativistic heavy ion collisions can provide the condition to create this
deconfined partonic matter Gyulassy200530 . In general, two approaches are
used to study the properties of the deconfined hot and dense quark matter
produced in AA collisions. One is studying the high $p_{T}$ hadrons from
initial hard jets, in which one can recur to the perturbative QCD to a certain
degree WangXN1998 . The other is investigating the properties of thermal
hadrons frozen out from the hot and dense quark matter. For the latter, the
hadronization of the hot and dense quark matter (a typical non-perturbative
process) is of great significance. Only through a reliable hadronization
mechanism, can we reversely obtain various information of QGP properties from
the final state hadrons measured experimentally. The abundant experimental
data abelev:152301 ; abelev:2007prl  and phenomenological studies
Fries22003prl ; Greco2003prl ; Fries:2003prc ; Greco2003prc ; Hwa:2004prc ;
Hwa:2008prc  at RHIC energies suggest that quark combination mechanism is one
of the most hopeful candidates. The two most noticeable results are the
successful explanation of the high baryon/meson ratios and the constituent
quark number scaling of the hadronic elliptic flow in the intermediate
transverse momentum range Greco2003prl ; Fries:2003prc , which can not be
understood at all in the partonic fragmentation picture. Recently, the NA49
Collaboration have measured the elliptic flow of identified hadrons at top SPS
energy Alt2007prc , and found that the quark number scaling of elliptic flow
was shown to hold also. It immediately gives us an inspiration of the
applicability for the quark combination at top SPS energy. On the other hand,
the NA49 collaboration have found three interesting phenomena around 30A GeV
Alt08onset , i.e. the steepening of the energy dependence for pion
multiplicity, a maximum in the energy dependence of strangeness to pion ratio
and a characteristic plateau of the effective temperature for kaon production.
These phenomena are indicative of the onset of the deconfinement at low SPS
energies. One can estimate via Bjorken method that the primordial spatial
energy density in Pb+Pb collisions at top SPS energy is about 3.0 GeV/$fm^{3}$
Stock2008 , exceeding the critical energy density (about 1 GeV/$fm^{3}$)
predicted by Lattice QCD. Therefore, the deconfined hot and dense quark matter
has been probably created, and we can extend the quark combination mechanism
to SPS energies.

As is well known, hadron yield is one of the most basic and important
observables which can help us to test the understanding of the hadronization
mechanism for the hot and dense quark matter created in the relativistic heavy
ion collisions. In most of recombination/coalescence models, the hadron wave
function is necessary to get the hadron yield. As the wave functions for
almost all hadrons are unknown at present, it is difficult for these models to
study this issue quantitatively Fries22003prl ; Greco2003prl ; Molnar03 . In
addition, these models do not satisfy the unitarity which is important to the
issue as well yang04 . Different from those models, the quark combination
model Xieqb:1988 ; Shao2005prc  uses the near-correlation in phase space and
SUf(3) symmetry, instead of hadron wave function, to determine the hadron
multiplicity. In addition, the model satisfies unitarity as well and has
reproduced many experimental data at RHIC shao2007prc ; Yao:2006fk ; WangYF08
; Yao08prc . Therefore, we apply it in this paper to systematically study the
yields, rapidity and transverse momentum distributions of various hadrons in
most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. On one hand, one
tests the applicability of the quark combination mechanism at this collision
energy. We note that the first attempt of the mechanism at SPS energies, for
hadron yields alone, was the ALCOR model ALCOR95 ; Levai2000 . Now, the rich
experimental data of hadron multiplicities and momentum spectra provide an
opportunity to make a further, systematical and even decisive test of the
mechanism at SPS energies. On the other hand, the parton momentum
distributions at hadronization, which carry the information on the evolution
of the hot and dense quark mater, are extracted from the final hadrons at top
SPS energy and compared with those at RHIC energies. We concentrate the
comparison on two properties related to strange hadron production. One is the
difference in collective flow between light and strange quarks, which occurs
at RHIC energies ChenJH2008 ; WangYF08 . The other is the strangeness
enhancement, a significant property of QGP Rafelski1982 .

The paper is arranged as follows. In the next section, we make a brief
introduction to the quark combination model. In section III, we calculate the
yields and rapidity distributions of identified hadrons in most central Pb+Pb
collisions at $\sqrt{s_{NN}}=17.3$ GeV. In section IV, the results of
transverse momentum distributions of various hadrons are shown. In section V,
we firstly make a detailed analysis of the longitudinal and transverse
collective flow of the hot and dense quark matter at top SPS energy. Secondly,
the energy dependence of the strangeness in the hot and dense quark matter is
extracted and analyzed. Section VI presents summary.

##  II An introduction of the Quark Combination Model

The starting point of the model is a color singlet system which consists of
constituent quarks and antiquarks. All kinds of hadronization models demand
that they satisfy rapidity or momentum correlation for quarks in the
neighborhood of phase space. The essence of this correlation is in agreement
with the fundamental requirement of QCD Xie:ap . According to QCD, a
$q\overline{q}$ may be in a color octet or a singlet. The color factors
$\langle(q\bar{q})_{8}|\frac{-\lambda^{a}\cdot\lambda^{a}}{4}|(q\bar{q})_{8}\rangle=\frac{1}{6}$
and
$\langle(q\bar{q})_{1}|\frac{-\lambda^{a}\cdot\lambda^{a}}{4}|(q\bar{q})_{1}\rangle=-\frac{4}{3}$,
which means a repulsive or an attractive interaction between them. Here
$\lambda^{a}$ are the Gell-Mann matrices. If they are close with each other in
phase space, they can interact with sufficiently time to be in the color
singlet and form a meson. Similarly, a $qq$ can be in a sextet or an anti-
triplet, and the color factors
$\langle(qq)_{6}|\frac{\lambda^{a}\cdot\lambda^{a}}{4}|(qq)_{6}\rangle=\frac{1}{3}$
and
$\langle(qq)_{\bar{3}}|\frac{\lambda^{a}\cdot\lambda^{a}}{4}|(qq)_{\bar{3}}\rangle=-\frac{2}{3}$.
If its nearest neighbor in phase space is a $q$, they form a baryon. If the
neighbor is a $\overline{q}$, because the attraction strength of the singlet
is two times that of the anti-triplet, $q\overline{q}$ will win the
competition to form a meson and leave a $q$ alone to combine with other quarks
or antiquarks. Based on the above QCD and near-correlation in phase space
requirements, we had proposed a quark combination rule(QCR) Xieqb:1988 ;
Xie:ap  which combines all these quarks and antiquarks into initial hadrons.
When the transverse momentum of quarks are negligible, all $q$ and
$\overline{q}$ can always line up stochastically in rapidity. The QCR reads as
follows:

  1. 1.

Starting from the first parton ($q$ or $\overline{q}$) in the line;

  2. 2.

If the baryon number of the second in the line is of the different type from
the first, i.e. the first two partons are either $q\overline{q}$ or
$\overline{q}q$, they combine into a meson and are removed from the line, go
to point 1; Otherwise they are either $qq$ or $\overline{q}\,\overline{q}$, go
to the next point;

  3. 3.

Look at the third, if it is of the different type from the first, the first
and third partons form a meson and are removed from the line, go to point 1;
Otherwise the first three partons combine into a baryon or an anti-baryon and
are removed from the line, go to point 1.

The following example shows how the above QCR works

$\displaystyle
q_{1}\overline{q}_{2}\overline{q}_{3}\overline{q}_{4}\overline{q}_{5}q_{6}\overline{q}_{7}q_{8}q_{9}q_{10}\overline{q}_{11}q_{12}q_{13}q_{14}\overline{q}_{15}q_{16}q_{17}\overline{q}_{18}\overline{q}_{19}\overline{q}_{20}$
$\displaystyle\rightarrow
M(q_{1}\overline{q}_{2})\;\overline{B}(\overline{q}_{3}\overline{q}_{4}\overline{q}_{5})\;M(q_{6}\overline{q}_{7})\;B(q_{8}q_{9}q_{10})\;M(\overline{q}_{11}q_{12})\;$
$\displaystyle
M(q_{13}\overline{q}_{15})\;B(q_{14}q_{16}q_{17})\;\overline{B}(\overline{q}_{18}\overline{q}_{19}\overline{q}_{20})$

If the quarks and antiquarks are stochastically arranged in rapidity, the
probability distribution for $N$ pairs of quarks and antiquarks to combine
into $M$ mesons, $B$ baryons and $B$ anti-baryons is

$X_{MB}(N)=\frac{2N(N!)^{2}(M+2B-1)!}{(2N)!M!(B!)^{2}}3^{M-1}\delta_{N,M+3B}.$
(1)

Hadronization is the soft process of the strong interaction and is independent
of flavor, so the net flavor number remains constant during the process. In
the quark combination scheme, this means that the quark number for each
certain flavor prior to hadronization equals to that of all initially produced
hadrons after it. Obviously the quark number conservation is automatically
satisfied in the model. It is different from the non-linear algebraic method
in ALCOR model ALCOR95  where normalization factor for each quark flavor is
introduced with the constraint of the quark number conservation.

The average number of initially produced mesons $M(N)$ and baryons $B(N)$ are
given by

$\displaystyle\langle M(N)\rangle$ $\displaystyle=$
$\displaystyle\sum_{M}\sum_{B}MX_{MB}(N)\;,$ (2) $\displaystyle\langle
B(N)\rangle$ $\displaystyle=$ $\displaystyle\sum_{M}\sum_{B}BX_{MB}(N)\;.$ (3)

Then the multiplicity of various initial hadrons is obtained according to
their production weights

$\langle M^{initial}_{j}\rangle=C_{M_{j}}\langle M(N)\rangle,\hskip
22.76228pt\langle B^{initial}_{j}\rangle=C_{B_{j}}\langle B(N)\rangle,$ (4)

where $C_{M_{j}}$ and $C_{B_{j}}$ are normalized production weights for the
meson $M_{j}$ and baryon $B_{j}$, respectively. If three quark flavors are
considered only, we can obtain the production weights using the SUf(3)
symmetry with a strangeness suppression factor $\lambda_{s}$ Xieqb:1988 ;
Shao2005prc , which are listed in Table 1. The extension of the symmetry to
excited states, exotic states and more quark flavors is also straightforward
excit95 ; Shao2005prc ; Yao08prc .

Considering the decay contributions from the resonances, we can obtain the
yields of final state hadrons

$\langle{h_{i}^{final}}\rangle=\langle{h_{i}^{initial}}\rangle+\sum\limits_{j}B_{r}(j\rightarrow
i)\langle{h_{j}}\rangle,$ (5)

where the $B_{r}(j\rightarrow i)$ is the weighted decay branching ratio for
$h_{j}$ to $h_{i}$ pdg08p355 .

In principle, the hadron production probability should be calculated from the
matrix element $\langle{q}\overline{q}|M\rangle$ for meson or
$\langle{qqq}|B\rangle$ for baryon. However, the wave functions for almost all
hadrons which are governed by the non-perturbative QCD are unknown at present.
It is difficult to study the production of hadrons quantitatively through
their wave functions. In view of this, the hadron production probability in
our model is determined by the SUf(3) symmetry with a strangeness suppression.
This symmetry has been supported by many experiments, particularly by the
coincidence of the observed $\lambda_{s}$ obtained from various mesons and
baryons Hofmann:1988gy . Therefore, the model can quantitatively describe many
global properties for the bulk system by virtue of the Monte Carlo method
Shao2005prc ; shao2007prc ; Yao:2006fk ; WangYF08 ; Yao08prc ; excit95 .

Table 1: The normalized production weight for baryons and mesons in the $\texttt{SU}_{f}(3)$ ground state. ${r_{i}}$ is the number of strange quarks in hadron. The ratio of the vector ($J^{P}=1^{-}$) to pseudoscalar ($J^{P}=0^{-}$) meson follows the spin counting, while that of the decouplet ($J^{P}=\frac{3}{2}^{+}$) to octet ($J^{P}=\frac{1}{2}^{+}$) baryon suffers a spin suppression effect; see Ref. excit95 ; Shao2005prc  for details.  $C_{M}$ |  $C_{M_{i}}=\frac{2J_{i}+1}{4(2+\lambda_{s})^{2}}\lambda_{s}^{r_{i}}$, except
---|---
$C_{\eta}=\frac{2J_{\eta}+1}{4(2+\lambda_{s})^{2}}\frac{1+2\lambda_{s}^{2}}{3}$
$C_{\eta^{\prime}}=\frac{2J_{\eta^{\prime}}+1}{4(2+\lambda_{s})^{2}}\frac{2+\lambda_{s}^{2}}{3}$
|  $C_{B_{i}}=\frac{4}{(2+\lambda_{s})^{3}(2J_{i}+1)}\lambda_{s}^{r_{i}}$,
except
$C_{B}$ | $C_{\Lambda}=C_{\Sigma^{0}}=C_{\Sigma^{\ast 0}}=C_{\Lambda(1520)}=\frac{3}{2\,(2+\lambda_{s})^{3}}\lambda_{s}$

When applying the model to describe the hadronizaton of the hot and dense
quark matter produced in heavy ion collisions, the net-baryon quantum number
of the system perplexes the analysis formula of Eq. 1 but it can be easily
evaluated in Monte Carlo program. On the other hand, the the transverse
momentum of quarks is not negligible due to the strong collective flow of
quark matter. In principle, we should define the QCR in three-dimensional
phase space, but it is quite complicated to have it because one does not have
an order or one has to define an order in a sophisticated way so that all
quarks can combine into hadrons in a particular sequence. In practice, the
combination is still put in rapidity and meanwhile the maximum transverse
momentum difference $\Delta_{p}$ between (anti)quarks are constrained as they
combine into hadrons. The transverse spectra of hadrons have a relationship
with the quark spectra as follows (e.g. for meson)

$\displaystyle\frac{dN_{M}}{d^{2}\mathrm{\textbf{p}_{T}}}\varpropto$
$\displaystyle\int
d^{2}\mathrm{\textbf{p}_{1,T}}d^{2}\mathrm{\textbf{p}_{2,T}}f_{q}(\mathrm{\textbf{p}_{1,T}})f_{\overline{q}}(\mathrm{\textbf{p}_{2,T}})\delta^{2}(\mathrm{\textbf{p}_{T}}-\mathrm{\textbf{p}_{1,T}}-\mathrm{\textbf{p}_{2,T}})$
(6)
$\displaystyle\times\,\Theta(\Delta_{p}-|\mathrm{\textbf{p}^{\ast}_{1,T}}-\mathrm{\textbf{p}^{\ast}_{2,T}}|),$

where $f_{q/\overline{q}}(\mathrm{\textbf{p}_{T}})$ is the transverse momentum
distribution of the quark/antiquark, assumed to be rapidity-independent in
present work. The superscript asterisk denotes the quark momentum in the
center-of-mass frame of formed hadron. The limitation $\Delta_{p}$ is treated
as parameter in our study, and fixed to be $\Delta_{p}=0.3$ GeV for mesons and
$\Delta_{p}=0.6$ GeV for baryons both at RHIC and SPS energies. Note that the
spectrum normalization is determined by the multiplicity in Eq. 4, i.e. the
constraint of the parameter $\Delta_{p}$ on the hadron yield is neglected.

One issue that is often questioned is the energy and entropy conservation in
quark combination process. As the non-perturbative QCD is unsolved, there is
no rigorous theory which can incorporate the partonic phase as well as
hadronic phase, thus it is difficult to justify or condemn this issue in
essence at the moment. As we know, a lot of the experimental phenomena in
intermediate transverse momentum range at RHIC can be explained beautifully
only in the quark combination scenario. It suggests that maybe this ’puzzling’
issue does not exist. As far as the quark combination itself is concerned,
there is no difference for the combination occurred in the different
(intermediate or low) transverse momentum range. Therefore, whether the
properties of low $p_{T}$ hadrons can be reproduced or not is also a
significant test of the quark combination mechanism, as the vast majority of
hadrons observed experimentally are just these with low transverse momentum.

##  III Hadron yields and rapidity distributions

In high energy nucleus-nucleus collisions, the energy deposited in the
collision region excites large numbers of newborn quarks and antiquarks from
the vacuum. Subsequently, the hot and dense quark matter mainly composed of
these newborn quarks will expend hydrodynamically until hadronization. The
net-quarks from the colliding nuclei still carry a fraction of beam energy,
thus their evolution is different from the newborn quarks. One part of net-
quarks are stopped in the hot and dense quark matter, and hadronize together
with it. The other part of net-quarks penetrate the hot quark matter, and run
up to the forward rapidity region. The latter, together with small amount
newborn quarks, form the leading fireball. Their hadronization should be
earlier than that of the hot and dense quark matter with a prolonged expansion
stage, and the hadronization outcomes consist of nucleons and small mount of
mesons.

Figure 1: (Color online) Rapidity spectra of newborn quarks and net-quarks at
hadronization in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV.

The current version of quark combination model simulates only the
hadronization of the hot and dense quark matter and subsequently decays of
resonances. One indispensable input is the momentum distributions of thermal
quarks and antiquarks at hadronization, which are the results of the
hydrodynamic evolution in partonic phase. In order to focus attentions on the
test of quark combination mechanism in this and next sections, we reversely
extract the quark distributions by fitting the experimental data in the model.
A detailed analysis of quark distributions at hadronization will be in section
V. The solid and dashed lines in Fig. 1 show the rapidity distributions of
newborn light and strange quarks at hadronization respectively, obtained from
the $\pi^{-}$ and $K^{+}$ data Afanasiev2002prc .The dotted-dashed line is the
rapidity distribution of net-quarks in the hot and dense quark matter, which
is extracted from net-proton data Appelshauser1999prl .

Firstly, we calculate the yields and rapidity densities at midrapidity of
various hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV.
The results are shown in Table 2. From the energy dependence of the rapidity
density for net-baryon bearden04stop , one can see that the nucleus-nucleus
collisions at SPS energies exhibit a strong stopping power. Therefore, the
leading particles contribute little to yields and rapidity distributions of
various hadrons.

The calculated yields and rapidity densities of vector meson $\phi$ are shown
to be about twice as high as the experimental data. The results of other
hadrons are basically in agreement with the experimental data, but slight
deviations exist also. The overpredictions of $\phi$ meson may be associated
with the exotic particle $f_{0}(980)$, which has a possible tetraquark
structure containing a strange quark and a strange antiquark Hirar07prc . As a
bond state containing strange components, it has a slightly lower mass than
$\phi$ meson but is not included in the SUf(3) ground states. In the present
work, we consider only the production of $36-plets$ of meson and $56-plets$ of
baryon in the SUf(3) ground states, and the excited states and exotic states
are not taken into account. The $f_{0}(980)$ multiplicity is found to be
nearly the same as $\phi$ meson in the $e^{+}e^{-}$ annihilations pdg08p355 .
The $m_{T}$ distribution of $f_{0}(980)$ measured by STAR Collaboration in
Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV is also shown to be comparable to
that of $\phi$ Fachini2003f980 ; Adams2005phi . Therefore, the overprediction
of $\phi$ meson can be removed by incorporating the $f_{0}(980)$ production.

Table 2: The yields (left) and rapidity densities at midrapidity (right) of identified hadrons in central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The experimental data are taken from Ref. Afanasiev2002prc ; Antinorik2005 ; Alt2008phi ; Alt2006prc ; Alt08Xi ; Mischke03Lam  | yield | $\frac{dN}{dy}|_{y=0}$
---|---|---
| data | model | data | model
$\pi^{+}$ | $619\pm 17\pm 31$ | $566$ | $170.1\pm 0.7\pm 9$ | 168.2
$\pi^{-}$ | $639\pm 17\pm 31$ | $630$ | $175.4\pm 0.7\pm 9$ | 183.5
$K^{+}$ | $103\pm 5\pm 5$ | $92.5$ | $29.5\pm 0.3\pm 1.5$ | 27.3
$K^{-}$ | $51.9\pm 1.6\pm 3$ | $45.3$ | $16.8\pm 0.2\pm 0.8$ | 15.7
$K^{0}_{s}$ | $75\pm 4$ | 66.7 | $26.0\pm 1.7\pm 2.6$ | 20.7
$\phi$ | $8.46\pm 0.38\pm 0.33$ | $15.2$ | $2.44\pm 0.1\pm 0.08$ | 5.26
$p$ |  | 120 | $29.6\pm 0.9\pm 2.96$ | 25.9
$\overline{p}$ |  | 3.2 | $1.66\pm 0.17\pm 0.17$ | 1.53
$\Lambda$ | $44.9\pm 0.6\pm 8$ | 52.9 | $9.5\pm 0.1\pm 1.0$ | 13.3
$\overline{\Lambda}$ | $3.07\pm 0.06\pm 0.31$ | 2.88 | $1.24\pm 0.03\pm 0.13$ | 1.35
$\mathrm{\Xi^{-}}$ | $4.04\pm 0.16\pm 0.57$ | 4.9 | $1.44\pm 0.1\pm 0.15$ | 1.43
$\mathrm{\overline{\Xi}^{\,{}_{+}}}$ | $0.66\pm 0.04\pm 0.08$ | 0.58 | $0.31\pm 0.03\pm 0.03$ | 0.26

Subsequently, we will calculate the longitudinal rapidity distributions of
various hadrons. Due to the deviations in hadron yields, it is difficult to
directly compare the calculated hadron spectra with the experimental data. In
order to focus attentions on the property of hadron momentum spectra, we will
scale the calculated rapidity densities to the center value of the
experimental data when we show the hadronic rapidity and $p_{T}$ spectra in
Fig. 2 and 4 respectively, thereby removing these deviations in hadron yields.

Figure 2: (Color online) The scaled rapidity distributions of identified
hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The
contributions from leading particles are not included. The open circles of
$\phi$ data in the second panel are the latest results measured by NA49
Collaboration Alt2008phi , and filled circles are the previous ones
Afanasiev2000plb . Other experimental data are taken from Ref.
Afanasiev2002prc ; Mischke03Lam ; Alt08Xi . The open symbols of $K_{s}^{0}$,
$\Lambda$ and $\Xi$ show data points reflected around midrapidity.

Pion is the lightest and most abundant hadron produced in AA collisions, and
its momentum distribution can best reflect the global evolution property of
the hot and dense quark matter. In various models of high energy heavy ion
collisions, the reproduction of pion meson is always taken as a paramount test
of models. In Landau hydrodynamic model Landau , the rapidity distribution of
pion can be well described and the sound velocity (which is an important
physical quantity standing for the property of the hot and dense quark matter)
can be extracted from the pion distribution. For other hadrons, such as kaons,
protons, $\Lambda$, $\Xi$ and so on, the Landau model can not describe their
rapidity distributions with the same sound velocity or freeze-out temperature
Mohanty2003prc ; Satarov2007prc ; Sarkisyan:2006 . For a systematic
description of the rapidity distributions of various hadrons, the detailed
longitudinal dynamics, e.g. the evolution of net-baryon density which will
result in the yield and spectrum asymmetry between hadron and antihadron,
should be included. In addition, the hadronization mechanism is especially
important to describe the differences in the yield and momentum distribution
of various hadron species. Using the extracted quark distributions in Fig. 1,
we calculate the rapidity distributions of pions, kaons,
$\Lambda(\overline{\Lambda})$, $\Xi^{-}(\overline{\Xi}^{{}_{+}})$ and $\phi$
in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The results are
shown in Fig. 2 and are compared with the experimental data. The calculated
rapidity spectrum of $\phi$ meson is narrow than the latest data of NA49
Collaboration (open circles in the second panel), but is in good agreement
with previous data (filled circles). The rapidity spectra of other hadrons are
well reproduced. One can see that the quark combination mechanism is
applicable for describing the longitudinal distributions of various hadrons at
top SPS energy.

##  IV Hadron transverse momentum distributions

In this section, we calculate the transverse momentum distributions of various
hadrons in the midrapidity range. In this paper, we only consider the
hadronization of the hot and dense quark matter. The transverse momentum
invariant distribution of constituent quarks at hadronization is taken to be
an exponential form $\exp(-m_{T}/T)$, where $T$ is the slope parameter which
is also called effective production temperature. Fig. 3 shows the midrapidity
$p_{T}$ spectra of constituent quarks at hadronization in most central Pb+Pb
collisions at $\sqrt{s_{NN}}=17.3$ GeV. The spectra of newborn light and
strange quarks are extracted from the data of $\pi^{+}$ and $K^{+}$
respectively Alt2008hpt . The net-quark distribution is fixed by the data of
$K^{-}/K^{+}$ ratio as a function of $p_{T}$ Alt2008hpt . A detailed analysis
of the quark $p_{T}$ spectra will be shown in section V.

Figure 3: (Color online) The transverse momentum distributions of constituent
quarks in the midrapidity region at hadronization in most central Pb+Pb
collisions at $\sqrt{s_{NN}}=17.3$ GeV.

Fig. 4 shows the calculated $p_{T}$ spectra of pions, kaons, protons,
$\Lambda(\overline{\Lambda})$, $\Xi^{-}(\overline{\Xi}^{{}_{+}})$ and $\phi$
in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. For kaons,
protons, $\Lambda$ and $\Omega$, the spectral slopes of antihadrons measured
experimentally are all steeper than those of hadrons Alt2008hpt ; Anticic2004L
; Anticic2005O . However, the spectrum of $\Xi^{-}$ is abnormally steeper than
that of $\overline{\Xi}^{{}_{+}}$ Alt08Xi . Our predictions are in good
agreement with all the data except $\Xi^{-}$.

Figure 4: (Color online) The scaled transverse momentum distributions of
identified hadrons at midrapidity in most central Pb+Pb collisions at
$\sqrt{s_{NN}}=17.3$ GeV. Only the combination of thermal quarks is taken into
account. Solid lines are the calculated results of hadrons and dashed lines
for antihadrons. The experimental data are from Ref. Alt2008hpt ; Alt08Xi ;
Alt2008phi

The exponential function $\exp(-m_{T}/T)$ is often used experimentally to fit
the transverse momentum distributions of identified hadrons in the low $p_{T}$
range, and to extract the effective production temperature $T$ of various
hadrons. It is found at top SPS energy that all final-state hadrons except
pion meson have much higher $T$ than the critical temperature Alessandro2003 ,
which indicates a strong collective flow at this collision energy. It is
regarded in Ref. XuNu1998  that this observed flow mainly develops in the late
hadronic rescattering stage. But results in Fig. 2 and Fig. 4 all show that
both longitudinal and transverse spectra of various hadrons can be coherently
explained by the same quark distributions, respectively. It suggests that the
observed flow should mainly come from the expansive evolution stage of the hot
and dense quark matter before hadronization, but not from the post-
hadronization stage. In addition, the same constituent quark spectra contained
in light, single- and multi- strange hadrons also imply that the hot quark
matter hadronize into these initial hadrons almost at the same time, i.e. the
hadronization is a rapid process.

Figure 5: (Color online) The transverse momentum distributions of identified
hadrons at midrapidity in most central Au+Au collisions at $\sqrt{s_{NN}}=200$
GeV. Only the combination of thermal quarks is taken into account. Solid lines
are the calculated results of hadrons and dashed lines for antihadrons. The
experimental data are from Ref. abelev:152301 ; Adams07hyperon ; Abelev07phiv2
Figure 6: (Color online) The ratios of $\overline{p}/\pi^{-}$,
$\Lambda/K_{s}^{0}$ and $\Omega/\phi$ at midrapidity in most central Pb+Pb
collisions at $\sqrt{s_{NN}}=17.3$ GeV and Au+Au collisions at 200 GeV. Only
the combination of thermal quarks is taken into account. Solid lines are the
calculated results in Au+Au collisions and dashed lines for Pb+Pb collisions.
The experimental data are from Ref. abelev:152301 ; Abelev06ks0 ;
Abelev07phiv2 ; Alt2008hpt ; Andr06ksL

In Fig. 5, we calculate the $p_{T}$ spectra of pions, kaons, protons,
$\Lambda(\overline{\Lambda})$, $\Xi^{-}(\overline{\Xi}^{{}_{+}})$, $\phi$ and
$\Omega(\overline{\Omega})$ in most central Au+Au collisions at top RHIC
energy. The momentum distributions of constituent quarks at hadronization are
taken to be $\exp(-m_{T}/0.375)$ for strange quarks and $\exp(-m_{T}/0.34)$
for light quarks. The numbers and rapidity spectra of the light and strange
quarks and antiquarks have been obtained in the study of the longitudinal
hadron production songjun09 . Here, the result of $\phi$ meson is multiplied
by a factor 0.5. One can see that the $p_{T}$ spectra of various hadrons are
well reproduced.

The baryon/meson ratio as a function of $p_{T}$ is sensitive to the
hadronization mechanism. As we know, the observed high baryon/meson ratios in
the intermediate $p_{T}$ range at RHIC energies abelev:152301  can not be
understood at all in the scheme of parton fragmentation, but can be easily
explained in the quark combination mechanism. Fig. 6 shows the model
predictions of $\overline{p}/\pi^{-}$, $\Lambda/K_{s}^{0}$ and $\Omega/\phi$
at midrapidity in both central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV
and central Au+Au collisions at$\sqrt{s_{NN}}=200$ GeV. In the intermediate
$p_{T}$ range where the hadron production is dominated by the combination of
thermal quarks, the baryon/meson ratios increase with the increasing $p_{T}$.
One can see that the experimental data in this region are well reproduced. The
falling tendency of measured baryon/meson ratios after peak position is owing
to the abundant participation of jet quarks, which is beyond the concern of
the present paper. Besides the hadronization mechanism, the baryon/meson ratio
in the intermediate $p_{T}$ region is also influenced by other two factors.
One is the nuclear stopping power in collisions. Comparing with the strong
collision transparency at top RHIC energy bearden04stop , the strong nuclear
stopping at top SPS energy causes the detention of abundant net-quarks in the
midrapidity region. These net-quarks significantly suppress the production of
anti-baryons and enhance that of the baryons. Therefore, the
$\overline{p}/\pi^{-}$ ratio at top SPS energy is much lower than that at top
RHIC energy while the $\Lambda/K_{s}^{0}$ ratio at top SPS energy is higher
than that at top RHIC energy. The other is the momentum distribution of
constituent quarks at hadronization. This can be illustrated by $\Omega/\phi$
ratio because the production of these two hadron species is less influenced by
the net-quarks. The calculated $\Omega/\phi$ ratio shows a weak dependence on
the collision energy in the intermediate $p_{T}$ range. The well description
of various baryon/meson ratios in such a wide energy range is an indication of
the universality for the quark combination mechanism.

##  V Analysis of parton distributions at hadronization

The constituent quark distributions at hadronization carry the information on
the evolution of the hot and dense quark matter in partonic phase. In this
section, we focus attentions on the longitudinal and transverse collective
flows and strangeness enhancement of the hot and dense quark matter produced
at top SPS energy.

###  V.1 The longitudinal and transverse collective flow

Due to the thermal pressure, the hot and dense quark matter created in high
energy heavy ion collisions will expand during the evolution before
hadronization. The longitudinal and transverse collective flow of final
hadrons measured experimentally is the exhibition of this early thermal
expansion in the partonic phase. Utilizing the relativistic hydrodynamic
evolution of the hot and dense quark matter, one can obtain the collective
flow in quark level from the extracted quark momentum distributions, and
compare it with that at RHIC energies.

There are two well known hydrodynamic models for the description of the space
time evolution of the hot and dense quark matter produced in heavy ion
collisions. One is Bjorken model Bjorken84  which supposes that the collision
is transparent, and it is appropriate to extremely high energy collisions,
such as LHC. The other is Landau model Landau  with an assumption of the full
stopping for nucleus-nucleus collisions. The longitudinal evolution result is
equivalent to the superposition of a set of thermal sources in rapidity axis,
with a (Bjorken) uniform or (Landau) Gaussian weight. In general, when
applying the model to describe the hadron rapidity distributions, different
parameter values are required to make a good fit of different hadron species
Mohanty2003prc . In this paper, we apply the hydrodynamic description to the
evolution in quark level, thus the collective flow of various hadrons can be
coherently explained.

One can see from the energy dependence of the net-baryon rapidity distribution
bearden04stop  that the collisions at top SPS energy are neither full
transparent nor full stopping. The suppositions of nuclear stopping power in
the two models are inappropriate to the nucleus-nucleus collisions at top SPS
energy. For the description of the rapidity distribution for constituent
quarks, one can limit the boost invariance into a finite rapidity range in the
framework of Bjorken model. This modification is often used to analyze the
longitudinal collectivity in hadronic level Heinz:1993 ; Mohanty2003prc . The
rapidity distribution in a isotropic, thermalized fluid element moving with a
rapidity $\eta$ is

$\displaystyle\frac{dN_{th}}{dy}(y-\eta)$ $\displaystyle=A\,T_{f}^{3}\exp\
\bigg{(}-\frac{m}{T_{f}}cosh\,(y-\eta)\bigg{)}\times$ (7)
$\displaystyle\bigg{(}\frac{m^{2}}{T_{f}^{2}}+\frac{m}{T_{f}}\frac{2}{cosh\,(y-\eta)}+\frac{2}{cosh^{2}(y-\eta)}\bigg{)}.$

The rapidity distribution of constituent quarks in the hot and dense quark
matter is the longitudinal boost-invariant superposition of multiple
isotropic, thermalized fluid elements

$\frac{dN}{dy}=\int_{-\eta_{max}}^{\eta_{max}}\frac{dN_{th}}{dy}(y-\eta)\,d\eta,$
(8)

$\eta_{max}$ is the maximal boot rapidity of fluid elements. The average
longitudinal collective velocity is taken to be
$<\beta_{L}>=\tanh(\eta_{max}/2)$.

Here, $T_{f}$ is the temperature of the locally-thermalized hot and dense
quark matter at hadronization. It is taken to be $T_{f}=170$ MeV. $m$ is the
constituent mass of quarks when they evolve to the transition point. It is
taken to be $340$ 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 a more complex evolution than hydrodynamic
expansion in longitudinal axis Wolschin04RDM . Therefore, we extract the
longitudinal collective flow from the rapidity distribution of newborn quarks.
Since most of the data are measured in the rapidity range about [-1.5, 1.5],
the rapidity spectra of constituent quarks extracted from experimental data
are valid only in this region. Using above equations to fit the rapidity
distribution of newborn constituent quarks in Fig. 1, we obtain
$<\beta_{L}>=0.58$ for light quarks and $<\beta_{L}>=0.65$ for strange quarks.
It is interesting to find that the average longitudinal collective velocity of
strange quarks is obviously greater than that of light quarks.

For the transverse expansion of the hot and dense quark matter, we adopt a
blast-wave model proposed by Heniz Heinz:1993  within the boost-invariant
scenario. The quarks and antiquarks transversely boost with a flow velocity
$\beta_{r}(r)$ as a function of transverse radial position $r$. $\beta_{r}(r)$
is parameterized by the surface velocity $\beta_{s}$:
$\beta_{r}(r)=\beta_{s}\,\xi^{\,n}$, where $\xi=r/R_{max}$, and $R_{max}$ is
the thermal source maximum radius ($0<\xi<1$). The transverse momentum
distribution of constituent quarks in the hot and dense quark matter can be
equivalently described by a superposition of a set of thermalized fluid
elements, each boosted with transverse rapidity $\rho=tanh^{-1}\beta_{r}$

$\dfrac{dN}{{2\pi\hskip
2.84526pt\mathrm{{p}_{T}}d{p}_{T}}}=A\int_{0}^{1}\xi\,d\xi\,m_{T}\,\times{}I_{0}\bigg{(}\dfrac{{p}_{T}\,sinh\,\rho}{T_{f}}\bigg{)}K_{1}\bigg{(}\dfrac{m_{T}\,cosh\,\rho}{T_{f}}\bigg{)}.$
(9)

Here, $I_{0}$ and $K_{1}$ are the modified Bessel functions.
$m_{T}=\surd{\overline{{\mathrm{{p}_{T}}}^{2}+m^{2}}}$ is the transverse mass
of the constituent quark. The average transverse velocity can be written as

$\langle\beta_{r}\rangle=\dfrac{\int\beta_{s}\,\xi^{\,n}\xi\,d\xi}{\int\xi\,d\xi}=\dfrac{2}{n+2}\beta_{s}.$
(10)

With fixed parameter $n=0.3$, the average transverse velocity
$\langle\beta_{r}\rangle$ is able to characterize the transverse collective
flow of the hot and dense quark matter. Using the above equations to fit the
transverse momentum distributions of the newborn quarks in Fig. 3, we obtain
$\langle\beta_{r}\rangle=0.41$ for strange quarks and
$\langle\beta_{r}\rangle=0.36$ for light quarks. One can see that the
$\langle\beta_{r}\rangle$ of strange quarks is obviously greater than that of
light quarks.

Both longitudinal and transverse results at top SPS energy show that the
strange constituent quarks get a stronger collective flow than the light
quarks in the hydrodynamic evolution of partonic matter. By analyzing the data
of multi-strange hadrons ChenJH2008  and primary charged hadrons WangYF08 ,
the same property is found also at top RHIC energy. It suggests that the hot
and dense quark matter produced at top SPS energy undergoes a similar
hydrodynamic evolution to that at RHIC energies. It is generally believed that
the decoupled quark and gluon plasma (QGP) has been created at RHIC energies
Gyulassy05qcdMater . This similarity of collective flow property in quark
level may be regarded as a signal of QGP creation at top SPS energy.

###  V.2 The enhanced strangeness

An interesting phenomenon in high energy heavy ion collisions is the enhanced
production of strange hadrons, which is absent in elementary particle
collisions. In relativistic heavy ion collisions, enormous amounts of energy
are deposited in the collision region to create a deconfined hot and dense
quark matter. The multiple scatterings between partons in the hot and dense
quark matter will cause the large production rate of strangeness by
$gg\rightarrow s\bar{s}$ Rafelski1982 . The high strangeness of the hot and
dense quark matter, after hadronization, finally leads to the abundant
production of the strange hadrons. This phenomenon is regarded as a signal of
QGP creation. As we know, the enhancement of strangeness production at top
RHIC energy is quite obvious Abelev08enhan , and it is generally believed that
the QGP has been created at RHIC energies. When the collision energy drops to
the SPS and AGS energies, it is found that the strangeness production peaks at
about 30A GeV and turns into a plateau at higher collision energies Alt08onset
. It is an indication of the onset of deconfinement.

Table 3: The strange suppression factor $\lambda_{s}$ and the calculated hadron $dN/dy$ at midrapidity in central AA collisions at different energies. The experimental data are taken from Ref. Afanasiev2002prc ; Alt2006prc ; Alt08Xi ; Arsene08Kpi ; Abelev62GeV ; Taka05sqm ; Speltz04sqm ; Adcox130GeV ; Adcox02Lam ; Adams04Mults ; Adler04Light ; Adams07hyperon . | Pb Pb 17.3 GeV | Au Au 62.4 GeV | Au Au 130 GeV | Au Au 200 GeV
---|---|---|---|---
| data | model | data | model | data | model | data | model
$\pi^{+}$ | $170.1\pm 0.7\pm 9$ | 168.3 | $212\pm 5.8\pm 14$ | $211$ | $276\pm 3\pm 35.9$ | $268.7$ | $286.4\pm 24.2$ | 287.3
$\pi^{-}$ | $175.4\pm 0.7\pm 9$ | 183.5 | $204\pm 7.4\pm 14$ | $217$ | $270\pm 3.5\pm 35.1$ | $272.4$ | $281.8\pm 22.8$ | 288.3
$K^{+}$ | $29.6\pm 0.3\pm 1.5$ | 27.3 | $33.35\pm 2.15$ | $36.3$ | $46.7\pm 1.5\pm 7$ | $46.6$ | $48.9\pm 6.3$ | 48.35
$K^{-}$ | $16.8\pm 0.2\pm 0.8$ | 15.7 | $28.16\pm 1.76$ | $29.9$ | $40.5\pm 2.3\pm 6$ | $43.1$ | $45.7\pm 5.2$ | 46.48
$p$ | $29.6\pm 0.9\pm 2.9$ | 25.9 | $27\pm 1.8\pm 4.6$ | $26.17$ | $19.3\pm 0.6\pm 3.3$ | $16.45$ | $18.4\pm 2.6$ | 17.41
$\overline{p}$ | $1.66\pm 0.17\pm 0.16$ | 1.53 | $11.5\pm 1.5\pm 2.9$ | $11.15$ | $13.7\pm 0.7\pm 2.3$ | $11.63$ | $13.5\pm 1.8$ | 13.48
$\Lambda$ | $9.5\pm 0.1\pm 1$ | 13.3 | $14.9\pm 0.2\pm 1.49$ | $13.42$ | $17.3\pm 1.8\pm 2.7$ | $14.99$ | $16.7\pm 0.2\pm 1.1$ | 15.76
$\overline{\Lambda}$ | $1.24\pm 0.03\pm 0.13$ | 1.35 | $8.02\pm 0.11\pm 0.8$ | $6.77$ | $12.7\pm 1.8\pm 2$ | $11.4$ | $12.7\pm 0.2\pm 0.9$ | 12.6
$\mathrm{\Xi^{-}}$ | $1.44\pm 0.1\pm 0.15$ | 1.43 | $1.64\pm 0.03\pm 0.014$ | $1.63$ | $2.04\pm 0.14\pm 0.2$ | $1.99$ | $2.17\pm 0.06\pm 0.19$ | 2.12
$\mathrm{\overline{\Xi}^{\,{}_{+}}}$ | $0.31\pm 0.03\pm 0.03$ | 0.26 | $0.989\pm 0.057\pm 0.057$ | $0.96$ | $1.74\pm 0.12\pm 0.17$ | $1.67$ | $1.83\pm 0.05\pm 0.2$ | 1.72
$\mathrm{\Omega+{\overline{\Omega}}}$ |  | 0.17 | $0.356\pm 0.046\pm 0.014$ | $0.369$ | $0.56\pm 0.11\pm 0.05$ | 0.551 | $0.53\pm 0.04\pm 0.04$ | 0.539
$\chi^{2}/ndf$ | 10.7/7 | 6.2/8 | $1.6/8$ | $0.88/8$
$\lambda_{s}$ | $0.48\pm 0.09$ | $0.44\pm 0.02$ | $0.44\pm 0.04$ | $0.42\pm 0.025$

In our model, the strangeness of the hot and dense quark matter is
characterized by the suppression factor
$\lambda_{s}=N_{\bar{s}}:N_{\bar{u}}=N_{\bar{s}}:N_{\bar{d}}$, i.e. the ratio
of $s$ quark number to newborn $u$ (or $d$) quark number. By fitting the
experimental data of identified hadrons, we use the model to extract the
$\lambda_{s}$ of hot and dense quark matter at midrapidity in central AA
collisions at four energies $\sqrt{s_{NN}}=17.3$, 62.4, 130 and 200 GeV, and
the results are shown in Table 3. The data of midrapidity $dN/dy$ and the
calculated results with minimum deviations at different collision energies are
shown also. The statistical uncertainty of $\lambda_{s}$ is fixed by the twice
minimum deviation. The model reproduces the hadron yield in reasonably good
way, and the chi-square fit seems to indicate that with increasing collision
energy the agreement with data significantly improves.

One can see that $\lambda_{s}$ in such a broad energy range is nearly
unchanged within statistical uncertainties, exhibiting an obvious saturation
phenomenon. The results of $\lambda_{s}$ are consistent with the grand
canonical limit ($\approx 0.45$) of strangeness Stock2008 . Using the Bjorken
model, one can estimate that the primordial spatial energy density of the hot
and dense quark matter produced in collisions at top RHIC energy is about
$6.0\,GeV/fm^{3}$ Stock2008 , double of that in Pb+Pb collisions at top SPS
energy. The difference in primordial energy density is large while the final
strangeness is nearly equal. The hot and dense quark matter created in heavy
ion collisions is shown to be very close to a perfect fluid visHD08 . It means
that the local relaxation time toward to thermal equilibrium is much shorter
than the macroscopic evolution time of the hot and dense quark matter. When
the hot and dense quark matter evolves to the point of hadronization, the
strangeness abundance should be mainly determined by the current temperature,
irrelevant to the initial energy density and temperature. The same strangeness
is an indication of the universal hadronization temperature for the hot and
dense quark matter with low baryon chemical potential.

##  VI Summary

In this paper, we have systematically studied the longitudinal and transverse
production of various hadrons at top SPS energy in the scheme of quark
combination. Using the quark combination model, we firstly calculate the
yields and rapidity distributions of various hadrons in most central Pb+Pb
collisions at $\sqrt{s_{NN}}=17.3$ GeV. The calculated results are in
agreement with the experimental data. This indicates that the quark
combination mechanism is applicable in describing the longitudinal hadron
production at this collision energy. Secondly the $p_{T}$ distributions of
various hadrons at top SPS energy are calculated and compared with the data.
It is found that the light, single and multi-strange hadrons are well
reproduced by the same quark distributions. It indicates that the
hadronization of the hot and dense quark matter is a rapid process. The well
reproduced baryon/meson ratios in the intermediate $p_{T}$ range at different
collision energies are indicative of the universality for the quark
combination mechanism. By fitting the extracted constituent quark
distributions at hadronization with the hydrodynamic scenario, we further
obtain the longitudinal and transverse collective flow of the hot and dense
quark matter produced at top SPS energy. It is found that the strange quarks
get a stronger collective flow than light quarks, which is consistent with
that at RHIC energies. The strangeness in the hot and dense quark matter
produced at $\sqrt{s_{NN}}=17.3$, 62.4, 130, 200 GeV are extracted. The almost
unchanged strangeness may be associated with a universal hadronization
temperature for the hot and dense quark matter with low baryon chemical
potential.

### ACKNOWLEDGMENTS

The authors thank Q. Wang, Z. T. Liang and R. Q. Wang for helpful discussions.
The work is supported in part by the National Natural Science Foundation of
China under the grant 10775089 and the science fund of Qufu Normal University.

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