Negative strange-quark chemical potential

A concise, well-defined indication of the quark-deconfinement phase

transition



Apostolos D. Panagiotou, Panayotis Katsas, Elpida Gerodimou



University of Athens, Department of Physics

Nuclear & Particle Physics Division

Panepistimiopolis, GR-157 71 Ilissia, Athens, Hellas





Abstract



We have studied the variation of a thermodynamic quantity, the strange-quark

chemical potential, in the phase diagramme of nuclear matter, by employing the

partition function in each domain and enforcing strangeness conservation. We propose

that the change in the sign of the strange-quark chemical potential, from positive in the

hadronic phase to negative in the deconfined quark-gluon phase, to be a new, unique,

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

nuclear matter. This signature is independent of model parameters and interaction

mechanisms. We propose also that, for a state in the deconfined region following an

isentropic path 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. This would enable the knowledge of the

thermodynamic quantities of states situated beyond the hadronic phase. Data from

nucleus-nucleus interactions at AGS and SPS give support to our proposals.



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






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

the basis for strong interaction thermodynamics of nuclear and hadronic matter. In normal

matter, colour confinement prohibits the quarks from ever escaping the hadronic `bag' and

be free outside. However, under certain conditions of high temperature, prevailing in the

very early moments of the cosmos, or density, existing in the core of neutron stars, both now

approached in ultra-relativistic nucleus-nucleus interactions in the laboratory, we expect

strongly interacting hadronic matter to undergo two phase transitions beyond the Hadronic

Gas (HG) phase: quark deconfinement, giving rise to a continuous QCD state of (initially)

constituent-mass quarks and gluons in an extended volume of nuclear size, and chiral

symmetry restoration, denoting the diminishing of the quark mass to current-mass.

At finite baryon density, lattice calculations suggest that deconfinement should take

place earlier, followed by the weakening of the q-q correlation-interaction1 and the decrease

of the quark mass with increasing temperature, thus defining the Deconfined Quark Matter

(DQM) domain. Full chiral symmetry restoration is achieved asymptotically with the

formation of the ideal Quark - Gluon Plasma (QGP) of non-interacting, current-mass quarks.
We may thus define a phase diagramme of strongly interacting matter at finite baryon

density with three regions: HG - DQM - (ideal) QGP [1-3]
Such a 3-region phase diagramme should be described by a systematic variation of

thermodynamic quantities, as one proceeds from one domain to the other. The task is to

establish a discrete and well-defined quantity, which changes concisely and observably as

nuclear matter changes phase. We propose and show that the strange-quark chemical

potential, s, created in an interacting system with finite baryon number density, is the
sought-for thermodynamic quantity. It is a unique, concise and well-defined observable-

signature, whose behaviour in the various phases can be precisely known, derived from the

corresponding Equation of State (EoS).

Many observables-signatures of the formation of deconfined matter have been

proposed and searched for, including the suppression of J/ [4], the broadening and shift of

hadronic resonance mass [5], the enhancement of strangeness [6], the formation of

disoriented chiral condensates (DCC) [7]. These theoretical observables are being tested and

compared against experimental data from nucleus-nucleus interactions at the SPS, however


1 Lattice calculations show that s (T=300 MeV) = 0.3 and that /T4 (T ~ 2Td ~ 350 MeV) ~ 0.85SB, 3P/T4 (T ~
2Td) ~ 0.65SB, where SB is the Stefan-Boltzmann ideal gas limit [31].



2


without conclusive results so far [8-10], due to known (and unknown) interfering effects,

stemming from the various possible interaction mechanisms, statistical fluctuations of basic

quantities measured, as well as model dependence and model uncertainties. On the other

hand, the change in the sign of the strange-quark chemical potential, a distinct and well-

defined characteristic thermodynamic quantity in each phase, may provide an unambiguous,

unique signature of the deconfinement phase transition, which is independent of models and

interaction mechanisms. In the following we shall derive and discuss the EoS and the

functional form of the strange-quark chemical potential in each of the three regions of the

phase diagramme.

In the HG phase, the hadron-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):


ln Z (T ,V , , ) Z Z
HG q s = m + ( 3 3
-
q + )
q n


+ Z ( 1
- 1
-

K q s + )
s q
+ Z ( 2 -2 1
-

Y s q + )
q s

+ Z ( 2 -2 1
-

s q + )
s q

+ Z ( 3 -3

s +
) (1)

s



The one particle Boltzmann partition function Zk is

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


where the summation runs over the resonances of each hadron species with mass mj and gj,
the degeneracy factor, counts the spin and isospin degrees of freedom of the j-resonance.

i = exp(i/T) is the fugacity of the i-quark, which controls the quark content of the k-hadron,
i being the quark chemical potential (i=u,d,s). Hadron species with mass up to 2.5 GeV/c2
have been used.

Strangeness conservation in strong interactions necessitates:

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



0
V 
s



3


which reduces to


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


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




This is an important equation, as it defines the relation between the temperature and

the light-, strange-quark fugacities, q, s (q = u, d), hence the relation between the quark
chemical potentials q, s and T in the equilibrated primordial state. For given q, Eq. (2)
defines the variation of the strange-quark chemical potential with temperature in the HG

phase. Eq. (2) can be used to derive the true (transverse flow velocity-independent)

temperature of the equilibrated state, once the fugacities q and s are known from
experimental strange particle yield ratios [1-3].

In the ideal QGP region, the EoS for the non-interacting, current-mass u,d,s-quarks

and gluons has the form:

V 37 4 2
0 2 0
q g m T m
s s - s
ln Z (T ,V ,  ,  ) 2 4 2 2

QGP q s = T + T
q + + s + K
s
2 ( 1
2 ) (3)


T 90 2 2 2
T



where m o
s is the current-mass of the s-quark.


Strangeness conservation: (T/V)[lnZ -1
QGP/s] = 0, gives s = s = 1, or


 QGP (T,  ) =
0
(4)

s q


independent of temperature.

To describe the deconfined-quark state in the DQM region, we use the following

picture: Beyond but near the HG boundary, at T > Td, Td being the deconfinement
temperature, the correlation-interaction between the deconfined quarks is near maximum

(s(T) < 1), and the quark-correlations resemble "hadron-like" states. With increasing

temperature, this correlation-interaction weakens as colour mobility increases and s 0.
The masses of (anti)quarks vary with the temperature of the state and scale according to a

prescribed way. The initially constituent-mass of the quarks - at T = Td - decreases and, as

the DQM domain approaches asymptotically the QGP one, as T Tqgp, Tqgp being the
asymptotic temperature in the ideal QGP region, the current-mass of quarks is attained (full

chiral symmetry restoration).



4


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. (3), at T ~ Tqgp. To construct
the EoS in the DQM phase, we use the two order parameters:

(a) The average thermal Wilson loop: <L> = exp(-Fq/T) ~ Rd(T) = 0 1, as T = Td Tqgp,
describing the quark deconfinement and subsequent colour mobility, Fq being the free quark
energy.

(b) The scalar quark density: < > ~ Rch(T) = 1 0, as T = Td Tqgp, denoting the

scaling of the quark mass with temperature.

We assume that above Td "hadronic language" is still appropriate to some extent and
that the quarks have a degree of correlation-interaction resembling "hadron-like" entities,

since 1 > s > 0. The diminishing of this correlation-interaction is approximated by the factor

(1-Rd) = 1 0, as T = Td Tqgp. Note that effectively (1-Rd) ~ s(T) in the DQM region.
To account for the "effective mass" of the state as a function of temperature, we assume the

mass of the quarks to decrease and reach the current-mass value as T Tqgp. The quark
mass scales with temperature as:

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

where m o
q and mq are the constituent and current quark masses, respectively (q=u,d,s).

Similarly, the `effective hadron' mass scales as:

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

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

mass of the hadron's quarks. In the EOS, the former scaling is employed in the mass-scaled

partition function in the ideal QGP phase, *
ln Z , whilst the latter in the mass-scaled
QGP

partition function in the HG phase, *
ln Z , which also accounts for the hadron species. Note
HG

that this mass-scaling is effectively equivalent to the one given by the Nambu-Jona-Lasinio
formalism [11].

Employing the described dynamics, we construct the EoS of the DQM region:

lnZ (V,T, , ) = [1- R (T)]lnZ (V,T, , ) + R (T)lnZ (V,T,  ,  (5)

)
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, forming the "hadron-like" entities and the *
ln Z gives the mass-scaling
HG

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


5


rate of colour mobility, whilst the *
ln Z represents the state as it proceeds towards the
QGP

ideal QGP region. Thus, at T = Td, the EoS of the DQM region goes over to the EoS in the

HG phase, and at T Tqgp, to the EoS in the ideal QGP region.
Strangeness conservation 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-) = (6)





0
s s d s s 2 T s s



Eq. (4) defines, for given q, the variation of the strange-quark chemical potential with

temperature in the DQM domain (T = Td Tqgp).
Combining Eq's (2,4,6) we obtain the variation of the strange-quark chemical

potential with temperature in the entire phase diagramme. Fig. 1 exhibits the well-defined

behaviour of s(T) for given light-quark chemical potential, say q = 0.45T. s attains
positive values in the HG phase, due to the positive change of the Gibbs free energy, s =

(G/Ns)|T as the strange-hadron number density, Ns, increases with increasing temperature.
It approaches zero as the hadron density reaches its asymptotic Hagedorn limit [12] on the

end of the hadronic phase at the deconfinement temperature Td, where a phase transition to

quark-gluon matter takes place. At this temperature, s(Td) = 0. Then, s grows negative in

the DQM phase, due to the negative change of the Gibbs energy of the deconfined-quark -

gluon state, s = (G/ns)|T, ns being the strange-quark number density, giving rise to quark
mass and interaction energy. Finally it approaches zero as the ideal QGP phase is reached

asymptotically. We note that the sign of s(T) - 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, Rch used in the EoS and unique in each region. A detailed, quantitative treatment of the
EoS in the DQM phase from `first principles' will require the use of a three-flavour effective

Lagrangian in the Nambu - Jona-Lasinio formalism [work in progress].
Systematic experimental studies at relativistic energies, aiming at the understanding

of nucleus-nucleus interactions and the possibility of producing the deconfined partonic

phase, have been performed at the AGS accelerator at BNL and the SPS at CERN, spanning

the energy range between 11 and 200 GeV per nucleon. The data obtained by the

experiments E802 [13,14] and NA35 [15,16], NA49 [17-19] at AGS and SPS, respectively,

have been analyzed in terms of several statistical-thermal models: The Statistical Bootstrap



6


Model [12], as developed with the inclusion of the strange quantum number and isospin

asymmetry, the so-called SSBM [20-23], as well as others employing the canonical and

grand-canonical formalisms [24-26]. In Table 1 we summarize the results of these analysis

for several nucleus-nucleus interactions, from which the thermodynamic quantities T, q and
s have been deduced.
Fig. 2 is a plot of the mean values of the temperature and strange-quark chemical

potential, obtained from these calculations, Table 1. We observe that the interactions Si+Au

at 14.6 AGeV, Au+Au at 11.6 AGeV and Pb+Pb at 40 and 158 AGeV have positive s,
whilst the interactions S+S and S+Ag, both at 200 AGeV, exhibit negative values2. This is

the first experimental confirmation of negative values for the strange-quark chemical

potential.

In contrast to all other thermal-statistical models, the SSBM [20,21] 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 borderline of the hadronic phase with zero strangeness is the projection on the

(T, q) plane of the intersection of the 3-dimentional (T, q, s) critical surface of the
bootstrap equation and the strangeness conservation surface. The analysis with this model

has shown that the S+S interaction is situated mostly (75%) outside the hadronic phase,

whilst the S+Ag is on the deconfinement line. In addition, the analysis has identified a large

(~ 30%) entropy enhancement of the experimental data compared to the model, which effect

was also observed by other calculations [24,26]. This enhancement may be attributed to

contributions from the DQM phase with the many more partonic degrees of freedom. For the

Pb+Pb interaction, the SSBM analysis [27] has shown that this system is located well within

the hadronic phase. In addition, the SSBM and the other thermal models do not find any

entropy enhancement for this interaction. All these results are corroborating the observations

of positive strange-quark chemical potential for the Pb+Pb and negative for the sulfur-

induced interactions, positioning the former within and the later beyond the HG domain, in

the DQM phase. Fig. 3 shows the phase diagramme with the SSBM maximally extended3


2 The T, q, s values for the sulfur-induced interactions were obtained from fits to particle yields with thermal
hadronic models [24-26], which cannot distinguish the onset of the DQM phase. However, beyond but near the
deconfinement line, the state may still be assumed `hadron-like' and hadronic models may describe the state
adequately.
3 The maximally extended HG phase (deconfinement line) is defined for T0 = 183 MeV (corresponding to B1/4 =
235 MeV) which is the maximum temperature at q = 0 for non-negative strange-quark chemical potential in
the HG domain. This T0 will decrease slightly if charm-hadron resonances are included in the mass spectrum.
Recent lattice QCD calculations give T0 ~ 175 MeV for three quark flavours [31].


7


deconfinement line, as well as the location (average T, q values) of the equilibrated states of
several nucleus-nucleus interactions, deduced from the analysis of experimental data with

thermal models, Table 1. We observe that only the sulfur-induced interactions at 200 AGeV,

having negative s, are situated beyond the deconfinement line, whilst all other interactions
with positive s are located well within the hadronic phase (see fig. 2).
Fig. 4 shows the theoretical variation of the strange-quark chemical potential

throughout the phase diagramme, obtained from Eq's (2,4,6) for q = 82 MeV, together with
the mean T, s values given by thermal model fits to the sulfur-induced interactions. In this
figure there is no intention to show a fit to the points, since the EoS used in the DQM region

is constructed with the empirical dynamics of the deconfined region without adjusting any

parameters (see also footnote 2), but rather to exhibit the qualitative correspondence between

the experimentally deduced thermodynamic quantities T, s of the (considered as
deconfined) states and our proposed signature of deconfinement, the negative values of the

strange-quark chemical potential.

The experimental observation of negative strange-quark chemical potential values in

sulfur-induced interactions at 200 AGeV, together with the proposed notion that it indicates

deconfinement, suggest that thermodynamic quantities of equilibrated primordial states,

situated beyond the HG phase, may indeed be measured. 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, we should have knowledge

of these quantities only on this line. That is, always s = 0 and, for the sulfur-induced
interactions with q ~ 82 MeV, T ~ 175 MeV [22,23].
To overcome this apparent difficulty, we propose that the conservation of the quark

fugacities, i = exp(i/T), (i = u, d, s), is a characteristic property of strong interactions and
thermodynamic equilibration in general, affecting all thermally and chemically equilibrated

states throughout the phase diagramme. That is, the fugacities i, determining the quark
number density ni, once fixed in an equilibrated primordial state with finite baryon number
density located beyond the HG phase, are constants of the entire sequent evolution process.

This is contingent on an isentropic path and hadronization via a second order phase transition

(fast hadronization without mixed phase). Both of these notions are now fairly accepted.

This statement has far-reaching consequences for defining a primordial state situated in the

deconfined partonic region, since the thermodynamic quantities T, q, s of the state may
indeed be determined from the experimental hadron yields and an appropriate EoS.



8


Thermal model fits to sulfur-induced interactions gave negative s values, which is
not a characteristic of the hadronic phase, as well as temperatures in the range of 180-190

MeV, which are 5-15 MeV higher than the maximum temperature for deconfinement, Td ~
175 MeV at q ~ 82 MeV, as given by the SSBM. Both of these observations point to the
likelihood of our suggestions. Note also that negative strange-quark chemical potential

means that the particle yield ratio: +/- = exp(-6s/T) is greater than one4. For the Pb+Pb

interaction at 158 AGeV, situated in the hadronic phase (s > 0), it was found experimentally

that +/- = 0.383, [28].

To scan the region of negative s values, in a finite baryon number density DQM
phase, an `excitation function' should be performed at the Relativistic Heavy Ion Collider,

RHIC at Brookhaven with Au+Au collisions at energies 20< s <100 AGeV. Higher

energies give very small baryochemical potential at mid-rapidity4 [29,30], hence almost zero

s throughout the phase diagramme. If the observation of negative s values, together with
temperatures in excess of Td, is confirmed at these energies, it will be a profound
observation, indicating that the negative strange-quark chemical potential is indeed a unique,

concise and well-defined signature of the deconfinement phase transition, identifying the

partonic phase. Of equal importance will be the possibility to determine the thermodynamic

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

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











4 For Au+Au interactions at RHIC (s = 130 AGeV) we predict +/- ~ 1, since s ~ 0, for small light-quark
chemical potential (q ~ 14 MeV). From the experimental data compiled in ref. [32] we calculate: q/T = 0.073
 0.006 and s/T = 0.005  0.01.


9


References

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

in Hadronic Matter'', pg. 449, Tucson, January, 1995, (American Institute of

Physics, New York Press).

[2] A. D. Panagiotou, G. Mavromanolakis and J. Tzoulis, Phys. Rev. C53, 1353 (1996).

[3] A. D. Panagiotou, G. Mavromanolakis and J. Tzoulis, "Strangeness '96" Conference,

May 1996, Budapest, Heavy Ion Physics 4 (1996) 347.

[4] T. Matsui, H. Satz, Phys. Lett. B178, 416 (1986)

[5] W. Weise, Nucl. Phys. A610, 35c (1996).

[6] P. Koch, B. Muller, J. Rafelski, Phys. Rep. 142, 167 (1986)

[7] B. J. Bjorken, Acta Phys. Polon. B28, 2773 (1997)

[8] J. Qiu, Quark Matter 2001 Conference Proceedings

[9] K. Ozawa, Quark Matter 2001 Conference Proceedings

[10] K. Redlich, Quark Matter 2001 Conference Proceedings

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

[12] R. Hagedorn, Riv. Nuov. Cim. 6, 1 (1983), R. Fiore, R. Hagedorn and F. d' Isep,

Nuov. Cim. A88, 301 (1985)

[13] L.
Ahle
et al, E-802 Collaboration, Phys. Rev. C57, 466 (1998).

[14] L.
Ahle
et al, E-802 Collaboration, Phys. Rev. C60, 044904 (1999).

[15] J.
Bachler
et al, NA35 Collaboration, Z. Phys. C58, 367 (1993).

[16] T.
Alber
et al, NA35 Collaboration, Z. Phys. C64 195 (1994).

[17] R. A. Barton et al, NA49 Collaboration, J. Phys. G27, 367 (2001)

[18] F.
Sikler
et al, NA49 Collaboration, Nucl. Phys. A661, 45 (1999)

[19] C. Blume, Quark Matter 2001 Conference Presentation

[20] A. S. Kapoyannis, C. N. Ktorides and A. D. Panagiotou, J. Phys. G23, 1921 (1997)

[21] A. S. Kapoyannis, C. N. Ktorides, A. D. Panagiotou, Phys. Rev. D58, 034009 (1998).

[22] A. S. Kapoyannis, C. N. Ktorides, A. D. Panagiotou, Phys. Rev. C58, 2879 (1998).

[23] A. S. Kapoyannis, C. N. Ktorides, A. D. Panagiotou, Eur. Phys. J. C14, 299 (2000).

[24] J. Sollfrank, J. Phys. G23, 1903 (1997).

[25] F. Becattini, M. Gazdzicki, J. Sollfrank, Eur. Phys. J. C5, 143 (1998).

[26] F. Becattini, J. Cleymans, A. Kernen, E. Suhonen, K. Redlich, Phys.Rev. C64,

024901 (2001).

[27] A. S. Kapoyannis, C. N. Ktorides, A. D. Panagiotou, to be published



10


[28] R. Caliandro et al, WA97 Coll., J. Phys. G25, 171-180 (1999)

[29] C. Adler et al, STAR Coll., Phys. Rev. Lett. 86, 4778-4782 (2001)

[30] B. Back et al, PHOBOS Coll., Phys. Rev. Lett. 87, 102301 (2001).

[31] F. Karsch, Quark Matter 2001 Conference Presentation

[32] P. Braun-Munzinger, D.Magestro, K. Redlich, J. Stachel, Phys. Lett. B518, 41 (2001)

[33] A. D. Panagiotou, P. Katsas, E. Gerodimou, Proceedings SQM 2001 Conference,

Frunkfurt, October, 2001









11


Table 1. Deduced values for T, q, s from thermal  statistical model calculations and fits
to experimental data for several nucleus-nucleus interactions.





Interaction/Experiment Si + Au (14.6 AGeV) E802


Ref.[33]

Ref.[26]


Mean
T (MeV) 134  6 135  4 135  3
q
(MeV)
176
 12 194  11 182  5
s (MeV) 66  10 66  10


Interaction/Experiment Au + Au (11.6 AGeV) E802


Ref.[33]
Ref.[26]

Mean
T (MeV) 144  12 121  5 124  5
q
(MeV)
193
 17 186  5 187  5
s (MeV) 51  14 51  14


Interaction/Experiment Pb + Pb (158 AGeV) NA49


Ref.[33]
Ref.[26]
Ref.[27]


Mean
T (MeV) 146  9 158  3 157  4 157  3
q (MeV) 74  6 79  4 81  7 78  3
s (MeV) 22  3 25  4 23  2


Interaction/Experiment Pb + Pb (40 AGeV) NA49

Ref.[33]
T (MeV) 147  3
q
(MeV)
136
 4
s (MeV) 35  4


Interaction/Experiment S + S (200 AGeV) NA35

Ref.[33] Ref.[25] Ref.[24] Mean
T (MeV) <189  8> 181  11 202  13 189  8
q
(MeV)

95
 12 74  7 87  7 83  5
s (MeV) - 51  30 - 58  18 - 56 15


Interaction/Experiment S + Ag (200 AGeV) NA35


Ref.[33]
Ref.[25] Ref.[24]

Mean
T (MeV) <182  6> 179  8 185  8 182  6
q (MeV) 87  11 81  6 81  7 82  4
s (MeV) - 15  25 - 65  20 - 45 16



12


Figure Captions



1. Variation of strange-quark chemical potential with temperature (Eq's 2,4,6) in the 3-

region phase diagramme. The curve s(T) intersects the (T, q)-plane at the
intersection point of the q = 0.45T line and the SSBM deconfinement line.


2. Plot of the mean values of the deduced quantities, temperature and strange-quark

chemical potential, from thermalstatistical model fits to experimental data for

several interactions.



3. The phase diagramme with the SSBM deconfinement line and the location of the

states of nucleus-nucleus interactions in the two phases HG and DQM, deduced from

the analysis of thermal models (Table 1).



4. Variation of the strange-quark chemical potential throughout the phase diagramme

(Eq's 2,4,6) for q = 82 MeV, and the average points (T, s) corresponding to the
sulfur-induced interactions.







13




























































14




















































Figure 2







15








































Figure 3








16








































Figure 4





17



