Negative strange quark-chemical potential
A necessary and sufficient observable of the
deconfinement phase transition in a finite-baryon equilibrated state*


Apostolos D. Panagiotou and Panayiotis Katsas


University of Athens, Physics Department
Nuclear & Particle Physics Division
Panepistimiopolis, GR-157 71 Athens, Hellas



Abstract


We consider the variation of the strange quark-chemical potential in the phase


diagramme of nuclear matter, employing the order parameters and mass-scaled partition


functions in each domain and enforcing flavour conservation. Assuming the region beyond the


hadronic phase to be described by massive, correlated and interacting quarks, in the spirit of


lattice and N-J-L calculations, we find the strange quark-chemical potential to attain large

negative values in this domain. We propose that this change in the sign of the strange quark-


chemical potential, from positive in the hadronic phase to negative in the partonic, to be a


unique, concise and well-defined indication of the quark-deconfinement phase transition in


nuclear matter. We propose also that, for a chemically equilibrated state in the deconfined


region, which follows an isentropic expansion to hadronization via a second order phase


transition, the fugacities of the equilibrated quark flavours, once fixed in the primordial state,


remain constant throughout the hadronization process.


PACS codes: 25.75.+r,12.38.Mh,24.84.+p
Key words: Quark-chemical potential, phase transition, deconfined quark matter, HG, QGP
e-mail: apanagio@phys.uoa.gr



* Work supported in part by the Research Secretariat of the University of Athens.


1. Introduction


It is generally expected that ultra-relativistic nucleus-nucleus collisions will provide the


basis for strong interaction thermodynamics, which will lead to new physics. Quantum


Chromodynamics for massless quarks free of dimensional scales contain the intrinsic potential


for the spontaneous generation of two scales: one for the confinement force, coupling quarks to


form hadrons and one for the chiral force, binding the collective excitations to Goldstone


bosons. In thermodynamics, these two scales lead to two possible consecutive phase transitions,


deconfinement and chiral symmetry restoration, characterized by corresponding critical


temperatures: Td and T, [1]. At temperatures above Td, hadrons dissolve into quarks and gluons,

whereas at T, chiral symmetry is fully restored and quarks become massless (current-mass),

forming the ideal Quark - Gluon Plasma (QGP). A priori it is not evident that both non-


pertubative transitions have to take place at the same temperature. At finite net baryon number


density, Td < T would correspond to a regime of unbound, massive, correlated and interacting,

`constituent-like' quarks [2], as they appear in the additive quark model for hadron-hadron and


hadron-lepton interactions [3]. Thus, the consecutive appearance of these two transitions,


deconfinement and chiral symmetry restoration, forms an intermediate region on the phase


diagramme, the Deconfined Quark Matter (DQM), in-between the Hadron Gas (HG) and the


ideal QGP domains. Therefore, we define a QCD phase diagramme of strongly interacting


matter at finite baryon number density with three regions: HG - DQM - QGP [4-6].


The necessity for such an intermediate region with massive quarks and gluons is


conjectured also from recent lattice calculations [7], which show that (2Td) ~ 0.85SB and

3P(2Td) ~ 0.66SB, where SB is the Stefan-Boltzmann ideal QGP value for the energy density. In

addition, the running coupling constant s(T~300 MeV) ~ 0.3. These observations substantiate




2


quantitatively the need for an intermediate domain with 1 >
s > 0, where massive and


correlated- interacting quarks are found. This domain does not have a defined upper border, but


goes asymptotically into the ideal QGP region with increasing temperature and quark-chemical


potential.


Such a 3-state phase diagramme could be described by the variation of thermodynamic


quantities from one region to the other. The task is to establish a well-defined quantity, which


changes concisely (and measurably) as nuclear matter changes phases, thus indicating


deconfinement and / or chiral symmetry restoration. If this quantity can be expressed in


functional form of other thermodynamic parameters within an Equation of State (EoS), one may


then describe its variation throughout the phase diagramme. We propose and show that the


equilibrated strange quark-chemical potential of the strongly interacting system is the sought-for


thermodynamic quantity. A similar suggestion was put forth earlier [5], but confident


experimental data to substantiate it were not available at the time.


In sections 2, 3 and 4 we discuss the partition functions in the HG, ideal QGP and DQM


phases for the strange hadron sector, respectively. Taking into account in an approximate way


the dynamics of the DQM phase, we construct an empirical Equation of State for this domain.


We employ it, together with the known EoS of the HG and QGP phases, to obtain the strange


quark-chemical potentials in functional form of the variables temperature and light quark-


chemical potential throughout the phase diagramme, s = f(T,q). With this relation we study the


variation of s in the 3-region phase diagramme, attributing the changes in the sign and

magnitude of the strange quark-chemical potentials to the changes of phase of nuclear matter.


In section 5 we summarize several thermal model analyses of experimental particle yield


data from nucleus-nucleus interactions at AGS and SPS. Among the latter, the S+A interactions




3


at 200 AGeV indicate the existence of negative strange quark-chemical potential. We compare


these results with our proposal. In section 6 we predict the values for certain strange particle


ratios and suggest experiments at lower energies at RHIC, where the baryon number density is


finite. Finally, in section 7 we discuss our proposals and come to conclusions whether the S+A


and Pb+Pb interactions at the SPS have entered the deconfined phase.




2. Hadron Gas phase


In the HG phase, the mass spectrum is given by the partition function, lnZHG, in the

Boltzmann approximation. We assume that the hadronic state has attained thermal and chemical


equilibration of three quark flavors (u,d,s):


lnZ (T, V, , ) = Z + Z ( 3 + -3
) + Z ( -1 + -1
) + Z ( 2 + -2 1
- )
HG q s m n q q K q s s q Y s q q s





+ Z (2 + -2 1- )+ Z (3 + -3
(q



) = u,
(1)


d)
s q s q s s





The one particle Boltzmann partition function Zk is given by


2
VT3 m m
Z (V, T) = g j K j (2)






k j 2

2 2 j T T


The fugacity k
i controls the quark content of the k-hadron; i = s, b for strangeness and baryon

number, 3
b = q = exp(3q/T), respectively. The summation in Eq. (2) runs over the resonances


of each hadron species with mass mj, and the degeneracy factor gj counts the spin and isospin

degrees of freedom of the j-resonance. For the strange hadron sector, kaons with masses up to


2045 MeV/c2, hyperons up to 2350 MeV/c2 and cascades up to 2025 MeV/c2 are included, as


well as the known - states at 1672 MeV and 2252 MeV. For simplicity, we assume isospin


symmetry in Eq. (1), u = d = q.




4


Strangeness neutrality in strong interactions necessitates:


T
< N - N >= =
s s [lnZ (V,T, , )
HG s q ] (3)




0
V
s

which reduces to


Z ( -1 - 1
- ) + Z ( 2 - -2-1) + 2Z (2 - -2 1-) + 3Z (3 - -3 ) = (4)


0
K s q q s Y s q q s s q s q s s




This is an important condition as it defines the relation between light- and strange-quark


fugacities in the equilibrated primordial state. Eq. (4) can be used to derive the true (transverse-


flow independent) temperature of the state, once the fugacities q and s are known from

experimental strange particle yield ratios [4].


In the HG phase with finite net baryon number density, the chemical potentials q and s

are coupled through the production of strange hadrons. Due to this coupling, strangeness


conservation does not necessitate s = 0 everywhere in this phase. In fact, s > 0 in the hadronic

domain. The condition  -1
s = 0 requires s = s = 1, and Eq. (4) becomes [8],

-1 -1
[ ZY (q + q ) - ZK + 2Z ] (q - q ) = 0

The first factor gives the curve  as a function of T in the phase diagramme along which 
q s = 0:


q(T) = T cosh-1(Z - Z ) (5)
K /2ZY /ZY

A more elegant and concise formalism describing the HG phase is the Strangeness-


including Statistical Bootstrap model (SSBM) [9]. It includes the hadronic interactions through


the mass spectrum of all hadron species, in contrast to all other ideal hadron gas formalisms. The


SSBM is valid and applicable only within the hadronic phase, defining in a determined way the


limits of this phase. The boundary of the hadronic domain is given by the projection on the 2-


dimentional (T,q) phase diagramme of the intersection of the 3-dimentional bootstrap surface





5


with the strangeness-neutrality surface (s = 0). Note that the vanishing of the strange quark-


chemical potential on the HG borderline does not apriori suggest that s = 0 everywhere beyond.

It only states that this zero value for the strange quark-chemical potential characterizes the end


of the hadronic phase.




3. Ideal QGP Phase


In the ideal QGP region, the EoS for the current-mass u,d,s-quarks and gluons has the


form:

4 2
V 37
q g m 0 T 2 0
-
lnZ (T, V, , ) = 2T 4 + 2T 2 s s
+ + +
QGP q s q 2 2 ( 1
s s ) m
K s
(6)


2
T 90
2 2 T



where m o
s and gs is the current-mass and degeneracy of the s-quark, respectively. Strangeness


conservation gives -1
s = s = 1, or



QGP (T, ) = (7)







0
s q



throughout this phase. In this region the two order parameters, the average thermal Wilson loop


< L > and the scalar quark density < > , have reached their asymptotic values.




4. DQM Phase


In formulating our description beyond the hadronic phase, we use the following picture:


The thermally and chemically equilibrated primordial state with finite baryon number density,


produced in nucleus-nucleus interactions, consists of the deconfined valance quarks of the


participant nucleons, as well as of q- q pairs (q=u,d,s), created by quark and gluon interactions.


Beyond but near the HG boundary, T > Td, the correlation-interaction between q-q is near


maximum, s(T) 1, a prelude to confinement into hadrons upon hadronization. With increasing



6


temperature, the correlation-interaction of the deconfined quarks gradually weakens, s(T) 0,

as colour mobility and colour charge screening increase. The masses of all (anti)quarks depend


on the temperature of the state and scale according to a prescribed way. The initially constituent


mass decreases and as the DQM region goes asymptotically into the ideal QGP domain, as T

T, quarks attain current-mass. In this formulation, the equation of state in the DQM region

should lead to the EoS of the hadronic phase, Eq. (1), at T < Td and to the EoS of the ideal QGP,

Eq.(6), at T ~ T.

To construct the empirical EoS in the DQM phase, we use the two order parameters (for


details see Appendix A):


(a) The average thermal Wilson loop, < L > = exp(-Fq/T) ~ Rd(T) = 0 1, as T = Td T,

describing the quark deconfinement and subsequent colour mobility, Fq being the free

quark energy.


(b) The scalar quark density, < > ~ R(T) = 1 0, as T = Td T, denoting the scaling


of the quark mass with temperature.


We assume that above Td the deconfined quarks retain a degree of correlation,

resembling "hadron-like" states, since 1 > s > 0. The diminishing of this correlation-interaction,

as a result of progressive increase of colour mobility, is approximated by the factor: [1-Rd(T)] =


1 0, as T = Td T. Note that effectively [1-Rd(T)] ~ s(T) in the DQM region, the

temperature dependence of the running coupling constant [10]. To account for the ``effective


mass'' of the state as a function of temperature, we assume the mass of the quarks to decrease


from the constituent value and reach the current-mass as T T. The quark mass scales with

temperature as:





7


m * o o
q (T) = R(T)(mq - mq ) + mq , (8)


where m o o o
q and mq are the constituent and current quark masses, respectively, (mu ~ 5 MeV, md


~ 9 MeV, m o
s ~ 170 MeV). Similarly, the `effective hadron' mass scales as:


m * o o
i (T) = R(T)(mi - mi ) + mi , (9)


where m o
i is the hadron mass in the hadronic phase and mi is equal to the sum of the hadron's



quarks current-mass (m o o
K ~ 175 MeV, mY ~ 185 MeV, m ~ 350 MeV and m ~ 510 MeV).


In the EOS, the former scaling is employed in the mass-scaled partition function *
ln ZQGP ,

whereas the latter in the mass-scaled partition function *
ln Z HG , which accounts for the produced

hadron species. Note that this mass-scaling is effectively equivalent to the one given in the


Nambu - Jona-Lasinio (N-J-L) formalism [11] (see Appendix A).

Employing the described dynamics, we construct the empirical EoS of the DQM phase:


lnZ (V, T, , ) = [1 - R (T)]lnZ (V, T, , ) + R (T)lnZ (V, T,  ,  (10)


)
DQM q s d HG q s d QGP q s



The factor [1-Rd(T)] describes the weakening of the correlation-interaction of the deconfined

quarks constituting the "hadron-like" entities and *
ln Z HG gives the mass-scaling of these entities

with increasing temperature. In the second term, the factor Rd(T) defines the rate of colour

mobility, whereas *
ln ZQGP represents the state as it approaches the QGP region. Thus, at T = Td,

the EoS of the DQM region goes over to the corresponding in the HG phase and at T ~ T, to the

EoS in the ideal QGP region. In this calculation we have taken T = 3.5Td.

Strangeness neutrality in the DQM phase leads to:

[1- R (T) - - - + - - - + - - -
d ] [Z
( 1 1 ) Z ( 2 1 2 ) 2Z ( 2q 2 1)
K s q q s Y s q s q s s q


2 m
+ 3Z (3 - -3 )] + R (T)g m* K s ( - 1
- ) = (11)





0
s s d s s 2 s s

T




8


For given q, Eq. (11) defines the variation of the strange quark-chemical potential with

temperature in the DQM domain. Combining Eq's (4,11) we obtain the variation of the strange


quark-chemical potential with temperature in the entire phase diagramme.


Fig. 1 maps the QCD phase diagramme as a function of T and the strange quark-


chemical potential. It exhibits the behaviour of s with temperature for fixed q = 0.45T. The

strange quark-chemical potential attains positive values in the HG phase, as a result of the


coupling between q and s in strange hadrons. It approaches zero as the hadron density reaches

its asymptotic Hagedorn limit [12] at the end of the hadronic phase, where a phase transition to


partonic matter takes place, at the deconfinement temperature Td. At this temperature, s(Td) = 0,

signifying the vanishing of this coupling in hadrons. In the DQM region it grows strongly


negative, where an effective coupling - progressively weakening - remains in effect among the

deconfined, but correlated and interacting quarks with non-current mass. Finally s returns to

zero as the ideal QGP phase is approached with s ~ 0 and current-mass quarks.


Fig. 2 exhibits the variation of s/T as a function of T throughout the 3-region phase


diagramme for q = 0.45T. We have assumed that the deconfinement temperature is Td ~ 176

MeV at q ~ 80 MeV, as given by the SSBM hadronic phase boundary [18]. We observe that:

) If upon deconfinement quarks attain instantaneously current mass and s(T>Td) = 0, the

ideal QGP phase follows immediately after the HG phase and the EoS beyond the HG domain is


given by Eq. (6). In this case s(T) = 0 for all T > Td.

b) If 0< 0 *
s(T>Td)<1 and mq <mq (T>Td)<mq, then the EoS (10), which includes mass-scaled


"hadron-like" states and variable s(T) ~ 1 - Rd(T), as well as (anti)quarks with finite (scaled)


mass, gives large negative values for s/T in the DQM phase, which, after reaching a minimum,

returns asymptotically to zero as the ideal QGP phase is approached.



9


c) Scaling alone of the effective hadron masses produces large negative strange quark-


chemical potential beyond the HG phase, saturating at high temperature without ever returning


to zero at the ideal QGP phase.


On the basis of the above observations, we propose that the change of sign of the strange


quark-chemical potential from positive to negative defines uniquely and precisely the phase


transition to quark-deconfinement. We note that the sign of s: positive in the HG, negative in

the DQM and zero in the ideal QGP domains, is independent of the particular form of the


parameters Rd and R used in the EoS and unique in each region (see Appendix A). It is also

independent of assumptions and ambiguities regarding interaction mechanisms and matter media


effects, as well as weakness and uncertainties of models, as is the case with other proposed


signatures for deconfinement: J/ suppression, resonance shift and broadening, strangeness


enhancement, etc. In the present calculation, however, the magnitude of the negative chemical


potential should be taken in a qualitative manner, due to the empirical treatment of the dynamics


in the DQM phase. A detailed, quantitative treatment of the EoS in the DQM phase will require


the use of a three-flavour effective Lagrangian in the N-J-L formalism [13].




5. Experimental Data


Data from nucleus-nucleus interactions, obtained by experiments E802 [14] at AGS and


NA35 [15], NA49 [16] at SPS have been analyzed in terms of several statistical-thermal models:


the Strangeness-including Statistical Bootstrap Model, SSBM [9,17-20] and others employing


the canonical and grand-canonical formalisms [21-25]. Table 1 summarizes the results of these


analyses, from which the quantities T, q and s have been deduced. Fig. 3 is a plot of the mean

values of the temperature and strange quark-chemical potential, obtained from these




10


calculations. Observe that the interactions Si+Au, Au+Au and Pb+Pb at both energies have


positive s, whereas the interactions S+S and S+Ag at 200 AGeV exhibit negative values. This

is the first confident [5] experimental confirmation of negative values for the strange-quark-


chemical potential. It should be noted that earlier thermal model fits [23] to preliminary Pb+Pb


data of NA49 [26], with larger and particle yields than the final values [16], produced a high


temperature of 193 MeV and s = - 72 MeV for the equilibrated state. These values, although

wrong, show that the strange quark-chemical potential could attain even larger negative values at


higher temperatures, a systematic trend.


In contrast to other thermal-statistical models, the Statistical Bootstrap Model (SBM) of


Hagedorn [27] incorporates the hadronic interactions in the EoS through the bootstrap equation


and thus gives, in a definitive way, the limits of the hadronic phase as a result of the branch


point of this equation. The development of the SBM with the inclusion of Strangeness (SSBM)


[9,17-19] has been employed in the analysis of nucleus-nucleus interactions at the SPS. The


SSBM analysis for the S+S interaction [18] has shown that this equilibrated state is situated


mostly (75%) outside the hadronic phase, whereas in the case of S+Ag [19], it is just on the


deconfinement line. In addition, the calculations have pointed out a large (~30%) entropy


enhancement of the experimental data compared to the model for both systems, an effect


observed also by other calculations [21-23]. This enhancement may be attributed to


contributions from the DQM phase with many liberated new partonic degrees of freedom. The


fact that beyond the HG phase the deconfined quarks retain a degree of correlation resembling


"hadron-like" states, allows the thermal models [21-23] to describe the state adequately using a


hadron gas EoS. These models do not sense the HG phase limit and treat the deconfined state as





11


consisting of colourless hadrons. This may not lead to erroneous results for a domain in the


phase diagramme not too far away from the deconfinement line.


For the Pb+Pb interaction, the SSBM analysis [20] has shown the system to be located


well within the hadronic phase. In addition, thermal model calculations [20,24] find no entropy


enhancement for this interaction. These results are corroborating the observation of positive


strange quark-chemical potential for the Pb+Pb and negative for the S+A interactions,


positioning the former within and the latter beyond the HG phase, in the DQM domain. Fig. 4


shows the phase diagramme with the SSBM maximally extended1 deconfinement line and the


location (T, q mean value) of several interactions, Table 1. Observe that the sulfur-induced

interactions at 200 AGeV, having negative s, are situated beyond the deconfinement line,

whereas all others with positive s are located well within the hadronic phase.




6. Predictions for particle yield ratios at RHIC


New data obtained at RHIC with the Au+Au interaction at s = 130 AGeV (see a

compilation of data in [28]) show that at midrapidity the light quark fugacity is q = 1.08,

indicating very small quark-chemical potential. This has an effect on the strange quark-chemical


potential, making it very small and close to zero throughout the 3-region phase diagramme. In


such case of minimal net baryon density, our model cannot be applied. We have, therefore,


calculated the particle yield ratios +/- and +/- for a finite baryon density region, obtained


in Au+Au interactions at s = 20 - 90 AGeV. The quantities needed for these calculations (q/T

and T) were obtained by fitting all available corresponding values, obtained for equilibrated




1The maximally extended HG phase limit is defined for T0(q=0)~183 MeV, which is the maximum temperature for
non-negative s in the HG domain. Recent lattice QCD calculations give T0 ~175 MeV for 2+1 quark flavours [7].




12


states in nucleus-nucleus collisions from SIS to SPS energies [24,25] and extrapolating to RHIC


energies. The data were fitted as a function of (s/participant). Table 2 contains the extrapolated

values for T and q/T, corresponding to equilibrated primordial states (not chemical freeze out),

as well as predictions of our empirical EoS for s/T and for the two particle ratios. Fig. 5 shows

the predicted particle ratios and the maximum values of the ratios in the case s does not become

negative (s=0).


Note that negative s means the particle yield ratio +/- = exp(-6s/T) becomes larger


than unity. We find, within the error of the extrapolation, that at about s = 50 AGeV both ratios


attain their largest values ~ (2.5  5) and ~ (1  1.5) for +/- and +/-, respectively, compared


to the maximum value of 1 and 0.55 for the case s = 0, respectively.




7. Summary and Discussion


We have shown that the existence of an intermediate region of deconfined, massive and


correlated quarks, in-between the hadronic and ideal quark-gluon phases, is realistic and


necessary to explain some first experimental observations and theoretical conjectures. We have


constructed an empirical EoS for this DQM phase, in terms of the order parameters and the


mass-scaled partition functions of the HG and QGP phases, from which a relation for the strange


quark-chemical potential is obtained in terms of q and T. This empirical EoS includes mass-

scaled strange hadrons, modified by appropriate factors of the order parameters and


approximating the effects of the progressive decrease of the interactioncorrelation with


increasing temperature, as well as a mass-scaled QGP term. In effect, it describes realistically


the gross features of the dynamics and characteristics of the DQM phase. The EoS indicates that


an unambiguous and concise characteristic observable of this phase is the large negative strange




13


quark-chemical potential. Analysis of data from sulfur-induced interactions at 200 AGeV give


clear indications that the strange quark-chemical potential does assume large, negative values


beyond the HG phase, in the deconfined domain.


Negative chemical potential appears also in condensed matter systems. For example, in a


transition between weekly coupled Cooper pairs, with > 0 and the usual BCS superconducting


gap |k|, and the strongly coupled diatomic pairs, with < 0 and the corresponding gap

( 2
k + 2)1/2, representing an insulating system [29]. The analogy with the baryon-dense nuclear


matter case is rather the opposite. The positive strange quark-chemical potential (s > 0)

corresponds to a colour insulator (hadron gas state), whereas s < 0 to a colour (super)conductor

(deconfined parton state).


An important argument of this work is that the light and strange quark fugacities, q, s

and the temperature are attributed to the equilibrated primordial state, to which Eq's (1,10) refer.


Only in this case one would expect for a state in the deconfined phase to observe negative


strange quark-chemical potential and temperatures in excess of those corresponding to


deconfinement. This appears at first as `impossible', since hadronization always takes place on


the deconfinement-hadronization line, separating the HG phase from the DQM one and,


therefore, these quantities should attain values only on this line. That is, always s = 0, and for

the sulfur-induced interactions, for example, which have q ~ 80 MeV, a temperature T ~ 176

MeV [18,19].


To overcome this apparent difficulty, we propose that the conservation of fugacities i

(i=u,d,s), is a characteristic property of strong interactions and thermodynamic equilibration in


general, affecting thermally and chemically equilibrated states throughout the phase diagramme.


That is, the quark number densities ni (i=u,d,s) and hence i, since ni ~ i, once fixed in a




14


chemically equilibrated primordial state, are constants of the entire sequent evolution process.


This is contingent on an isentropic expansion and hadronization via a second order


deconfinement phase transition (fast hadronization without mixed phase). The relation between

i and T, and hence between i and T in the primordial state2 is defined by the strangeness

neutrality equation, obtained from the partition function by imposing conservation of the S-


quantum number.


The above statements have far-reaching consequences for defining and understanding the


thermodynamic characteristics of the primordial state. They suggest that, if nuclear interactions


form equilibrated states beyond the HG phase in the deconfined region with finite net baryon


number density, the expansion to hadronization is isentropic and the confinement phase


transition is of second order, one may determine the thermodynamic quantities of the state from


the strange hadron yields, by employing an appropriate EoS. The observation of large negative


strange quark-chemical potential in sulfur-induced interactions, which is not a characteristic of


the hadronic phase, as well as temperatures in the range of 180  190 MeV (at q ~ 80 MeV),

which are higher than the maximum temperature for deconfinement (Td ~ 176 MeV at q ~ 80

MeV as given by SSBM), together with the proposed notion that negative strange quark-


chemical potential indicates deconfinement, suggest that the thermodynamic quantities T, i may

indeed be deduced and attributed to the primordial state in the partonic phase.


If negative s-values, together with temperatures in excess of Td are confirmed at the


proposed RHIC energies 20<s<100 AGeV, it will be a profound observation, indicating that

negative strange quark-chemical potential is indeed a unique and well-defined signature of




2 The quantities ,
s q and T are the Lagrange multipliers of strangeness, baryon number and energy,
attaining their values in the equilibrated state.




15


deconfinement, identifying the partonic phase. Of equal importance will be the possibility to


determine the quantities q, s and T, hence, the energy density and entropy of equilibrated

primordial states situated beyond the hadronic phase, in the deconfined-quark region.


On the basis of the present analysis we also conclude that the S+A interactions at 200


AGeV at the SPS are the only interactions to have produced an equilibrated partonic state


beyond the hadronic phase. On the other hand, the thermodynamic parameters of the primordial


state of the Pb+Pb interaction at 158 AGeV strongly indicate that its location is well within the


hadronic phase.





Appendix A.


1. Order Parameters


In order to construct the EoS in the DQM region it is necessary to employ an analytic


functional form of the order parameters. We have approximated the Wilson loop, obtained from


lattice calculations, with two different functions of temperature:


1
R (T) = (A.1)






d 1 + e-a(T-b)



T - T
and d
R (T) = c (A.2)
d T

In Eq.(A.1) a Fermi  type function is used, where the parameters {a, b} control the


difference of the critical temperatures Td and T. They can be chosen arbitrarily, in order to

obtain realistic compatibility with the lattice QCD results. In Eq.(A.2) the order parameter is


described using a more common  in the theory of critical phenomena  function near the critical





16


point Td, where the critical exponent can be chosen according to universality class arguments

or arbitrarily. The parameter c simply controls the values of Rd(T), so that asymptotically Rd(T)

1 as T T. Figures 6a,b show the approximated order parameter for different values of

{a,b} and of the critical exponent , respectively. In Fig. 7 we show the two approximated order


parameters, where we have taken R(T) 1-Rd(T) and used the Fermi-type function, Eq.(A.1),

with {a = 0.03, b = 268}.




2. Order parameters and the phase diagramme



Since an empirical partition function for the DQM phase is used, Eq. (10), containing the

parameters Rd(T) and R(T) 1-Rd(T), it is important to study the behaviour of the strange

quark-chemical potential in the DQM region for the different approximations of Rd(T). Figure 8

is a plot of s vs. T, obtained for q = 0.45T and the two different functions for Rd(T). It is clear

that the DQM region is characterized by negative strange quark-chemical potential

independently of which Rd(T) function is used in the EoS. Therefore, the change in the sign of

the strange quark-chemical potential - from positive to negative - is a unique and well-defined

indication of the quark-deconfinement phase transition and does not depend on the

phenomenological parameters of the model. For only a qualitative comparison with data, we also

show in the figure the two points of the sulphur-induced interactions. There is no adjustment of

the parameters of the EoS to fit the data.



3. Order parameters and the effective quark masses


The parameter R(T) is included in the expressions of the effective quark and hadron

masses. The use of temperature dependent quark masses is one of the essential aspects of the


model, since it is a dynamical term in the equation of state. On the other hand, more precise





17


theoretical models such as the N-J-L or the (non)linear sigma model, have also studied the

temperature dependence of the light and strange quark masses. Figures 9a,b plot the light and


strange quark masses as a function of temperature using the two different approximations of the


order parameter, whereas Fig. 10 exhibits the same quantities but within the N-J-L model. The

quark mass scaling used in our model is consistent with the results of an effective field theory


model, despite the difference for m *
s which can be diminished by choosing appropriate R(T) for


strange quarks, evidently giving strong support to our approximations and results.





18


References


[1] Marciano W, Pegels H, 1978 Phys. Rep. 36 137


[2] Shuryak E V 1981 Phys. Lett. B 107 103, Pisarski R D 1982 Phys. Lett. B 110 155


[3] Satz H 1977 Nuovo Cim. A 37 141


[4] Panagiotou A D, Mavromanolakis G, Tzoulis J Proc. Int. Conf. ``Strangeness in


Hadronic Matter'' (Tucson, Jan. 1995) (New York: AIP) p 449


[5] Panagiotou A D, Mavromanolakis G, Tzoulis J Strangeness '96 Conf. (Budapest, May


1996) Heavy Ion Physics 4 347


[6] Panagiotou A D, Mavromanolakis G, Tzoulis J 1996 Phys. Rev. C 53 1353


[7] Karsch F 2002 Nucl. Phys. A 698 199


[8] Asprouli M N and Panagiotou A D 1995 Phys Rev D 51 1086


[9] Kapoyannis A S, Ktorides C N and Panagiotou A D, 1997 J. Phys. G: Nucl. Part. Phys.


23 921


[10] Kajantie K and Kapusta J 1985 Ann. Phys.160 477


[11] Klevansky S.P. 1992 Rev. Mod. Phys. 64 649


[12] Cabbibo N, Parisi G 1975 Phys. Lett. B 59 67


[13] Katsas
P
et al, work under preparation


[14] Ahle L et al (E-802 Collaboration) 1998 Phys. Rev. C 57 466 ; 1999 Phys. Rev. C 60


044904


[15] Bachler J et al (NA35 Collaboration) 1993 Z. Phys. C 58 367, Alber T et al, (NA35


Collaboration) 1994 Z. Phys. C 64 207


[16] Afanasiev S V et al (NA49 Collaboration) 2000 Phys. Lett. B 491 59, Barton R A et al


(NA49 Collaboration) 2001 J. Phys. G: Nucl. Part. Phys. 27 367




19


[17] Kapoyannis A S, Ktorides C N and Panagiotou A D 1998 Phys. Rev. D 58 034009


[18] Kapoyannis A S, Ktorides C N and Panagiotou A D 1998 Phys. Rev. C 58 2879


[19] Kapoyannis A S, Ktorides C N and Panagiotou A D 2000 Eur. Phys. J. C 14 299


[20] Kapoyannis A S, Ktorides C N and Panagiotou A D to be published


[21] Becattini F 1997 J. Phys. G: Nucl. Part. Phys. 23 1933


[22] Sollfrank J 1997 J. Phys. G: Nucl. Part. Phys. 23 1903


[23] Becattini F, Gazdzicki M, Sollfrank J 1998 Eur. Phys. J. C 5 143


[24] Becattini F, Cleymans J, Redlich K 2001 Phys. Rev. C 64 024901


[25] Cleymans J, Redlich K arXiv: 


[26] Appelshauser H et al (NA49 Collaboration) 1998 Phys. Lett. B 444 523


[27] Hagedorn R, 1983 Riv. Nuov. Cim. 6 1; Proc. Advanced NATO workshop: Hot Hadronic


Matter. Theory & Experiment Divonne-les-Bains July 1994 and ref's therein.


[28] Braun-Munzinger P, Magestro D, Redlich K and Stachel J 2001 Phys. Lett B 518 41


[29] Moulopoulos K and Ashcroft N W 1990 Phys. Rev. B 42 7855





20


Table 1. Deduced values for T, q, s from thermal  statistical model calculations and fits to
experimental data for several nucleus-nucleus interactions. (*)Thermodynamic quantities
extracted using several strange hadron yield ratios.


Interaction/Experiment

Si + Au (14.6 AGeV)/E802
Reference(*) Reference [24] Mean
T (MeV) 134  6 135  4 135  3
q (MeV) 176  12 194  11 182  5
s (MeV) 66  10 66  10

Au + Au (11.6 AGeV)/E802
Reference(*) Reference [24] Mean
T (MeV) 144  12 121  5 124  5
q (MeV) 193  17 186  5 187  5
s (MeV) 51  14 51  14

Pb + Pb (158 AGeV)/NA49
Reference(*) Reference [24] Reference [20] Mean
T (MeV) 146  9 158  3 156  4 156  3
q (MeV) 74  6 79  4 78  5 78  3
s (MeV) 22  3 26  4 24  2

Pb + Pb (40 AGeV)/NA49
Reference(*)
T (MeV) 147  3
q (MeV) 136  4
s (MeV) 35  4

S + S (200 AGeV)/NA35
Reference [21] Reference [23] Reference [22] Mean
T (MeV) 182  9 181  11 202  13 188  6
q (MeV) 75  6 73  7 87  7 78  4
s (MeV)  60  20  58  18  59  13

S + Ag (200 AGeV) NA35
Reference [21] Reference [23] Reference [22] Mean
T (MeV) 180  3 179  8 185  8 181  4
q (MeV) 79  4 81  6 81  7 80  3
s (MeV)  66  20  65  23  65  15





21


Table 2. The extrapolated values for the temperature T and the ratio q/T obtained for the
Au+Au interaction at RHIC, as well as predictions of the empirical EoS for s/T and two
strange-particle ratios.




s (AGeV) T (MeV) q / T s / T a +/- +/-

20 181.76  17.30 0.495  0.016 -0.03  0.17 1.20  1.22 0.42  0.28

30 204.40  19.35 0.418  0.014 -0.20  0.10 3.24  1.94 0.95  0.38

40 220.45  20.85 0.313  0.012 -0.21  0.05 3.50  1.05 1.23  0.25

50 232.91  22.02 0.299  0.010 -0.22  0.05 3.77  1.13 1.33  0.27

60 243.10  23.06 0.252  0.008 -0.19  0.03 3.18  0.57 1.30  0.16

70 251.70  23.81 0.213  0.007 -0.16  0.01 2.68  0.16 1.26  0.05

80 259.15  24.53 0.180  0.006 -0.14  0.02 2.29  0.19 1.21  0.07

90 265.72  24.16 0.152  0.005 -0.11  0.02 1.99  0.24 1.06  0.04



a Calculated from the empirical DQM EoS





22


Figure Captions


1. Variation of the strange quark-chemical potential with temperature in the 3-region phase
diagram. The curve s(T) intersects the (T,q)-plane at the intersection point of the
q=0.45T line and the deconfinement line.

2. Variation of s/T as a function of the temperature of the equilibrated primordial state. A
deconfinement temperature of Td ~ 176 MeV is assumed, whereas chiral symmetry is
restored at T ~ 3.5Td.

3. Plot of the mean values of the temperature and strange quark-chemical potential, as
obtained from different thermal model fits to experimental data.


4. Phase diagram with the SSBM deconfinement line and the location (T, q average
values) of several interactions, deduced from the analysis with thermal models.


5. Predicted particle yield ratios +/- and +/- for a finite baryon density region,
obtained in Au + Au interactions at s = 20 - 90 AGeV. The lines correspond to the case
s=0.

6 a,b. Approximated average thermal Wilson loop Rd(T), obtained from Eq.(A.1) and Eq.(A.2)
for different values of the model parameters {a, b} and v.


7. Plot of the approximated order parameters Rd(T) and R(T)=1-Rd(T) using Eq.(A.1) with
a=0.03 and b=268.


8. Plot of the strange quark-chemical potential as a function of the temperature and the two
different functions for Rd(T), obtained for {a=0.03, b=268} and v=0.2, respectively.

9 a,b. The effective quark masses mq*(T) and ms*(T) calculated using the expressions of
Eq.(A.1) and Eq.(A.2) for R(T) = 1-Rd(T) and the values a=0.03, b=268 and v=0.2, c=1,
respectively.


10. Results on the temperature dependence of the light and strange quark masses obtained
from the N  J-L model [11].





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