Extreme states of nuclear matter

Nuclau Phydcs A374(1982)489o-502c_ O North-Holland Publla}tiug Co., Amatecdam Not to ba rcproduced by photoprint or microfilm without written pavnission from the publisher.

EXTREME STATES OF NUCLEAR MATTER Johann Rafelski Institut für Theoretische Physik der Johann Wolfgang Goethe-Universität Frankfurt am Main, Germany Abstract : A theoretical description of highly excited nuclear matter is presented . Two physically different domains are treated : the hadronic gas phase in which individual nucleons and mesons coexist as separate entities, and that consisting of one large hadronic cluster of quarks and gluons . Possible characteristic observable phenomena of the quark plasma are considered . 1 . Introduction A theory of hot hadronic matter is being developed ) based on the present knowledge about strong interactions . Physically most important is the phase transition in which baryons and mesons are dis

solved into the quark-gluon plasma 2) . Essential quantity controlling the occurrence of this transition is the energy density of hadronic matter . Hence the heat and pions created in central high energy heavy ion collisions or low energy p annihilations on nuclei 3) lower the necessary nuclear density to about 1-3 times normal nucleon density at temperatures of the order of the pion mass . For lack of time and space I can only discuss the nuclear collisions

here ; for a discussion of p annihilations the reader may consult ref . (3) . The theoretical descriptions of both phases are entirely diffe-

rent : In the hadronic gas phase ) the hadron-nucleon cross section is being dominated by the formation of hadronic resonances, the essen tial physical input is the particle mass spectrum, here derived in the statistical bootstrap model . In the dense domain of the matter it is essential to take into account the finite size and the clustering of individual quark bags . In the quark-gluon plasma, we have to deal with a many body gauge field theory at moderate inter-

action strength 2) . An insight into the behaviour of the plasma can be gathered studying the Fermi-Bose interacting quantum gases in a large quark bag4) . In order to test our understanding of hot hadronic matter in kinetic and chemical equilibrium and to show the existence of the

49 0c

J . RAFELSKI

plasma phase, different experiments are necessary . As an initial step I consider the temperatures of particles emitted in a hypothetical ~'S) . central fireball created in high energy nuclear collisions Suitable care must be taken in an experiment to eliminate the con-

tributions of projectile and target nuclei - here we do not know the model dependent internal excitation . Among the results I discuss here,a substantial entropy production in the explosion of the

fireball is due primarily to the production of new particles . Under certain conditions I also anticipate that strangeness would equilibrate - the relative yields of (anti) strange particles can be

used as a measure of the size of the reaction zone to be confronted

with correlation experiments . If the phase transition occurs at moderate energy densities (500 MeV/fm 3 ), then the relative Ä to p yields can be used as a measure of the relative antistrange quark

abundance in the quark phase for heavy ions energies between 2 and 5 GeV/N~ ) . Other phenomena will be studied at higher energies : prompt photons, leptons and eventually heavy flavours (charm etc) . it is important to appreciate that all this depends sensitively

on the existence of cooperative phenomena in which many nucleons from the projectile and target participate . This behaviour was not

so apparent in hadron-nucleus collisions . However, we note that in a first nucleon-nucleon collision the first participants are slowed down and other nucleons can run into the reaction zone . Some of the reaction products acquire significant transverse moments . Thus the assumption of kinetic equilibrium may be correct in a slightly restricted sense : the center of the energy and momentum distribu-

tion will be at a certain mean value - while the low and the important high energy tails will be underpopulated . The chemical equili-

brium,in which the relative population of different particle states is also governed by the statistical distribution. i s more difficult

to achieve . It is controlled by partial reaction cross sections which are much smaller than total cross sections . Hence for example at low energies (e .q . 1-2 GeV/N kinetic energy) or large impact parameters strangeness may be decoupled from the presumed equilibrium state of hadronic matter . cenI would like to emphasize that only in very high energy tral collisions of heavy nuclei we will be able to study the properties of the quark-gluon plasma at high temperatures . Is this worth the effort? Let us consider as an example the production of heavy quark flavours in e+e

annihilations and in nuclear collisions :

EXTREME STATES OF NUCLEAR MATTER

49 1c

while the first kind of experiment is almost certainly much clearer, a complex state like 8) ccss could only be produced when high quark density is reached at high plasma temperature Ts~1 GeV . [This process may be seen as an analogy to the production of â in pp-ISR collisions .] Another very fundamental aspect of experiments with high energy nuclei is the exploration of the phase transition be-

tween hadron gas and quark plasma states of hadronic matter and the determination of the critical energy density .

2, Hot Hadronic Gas Phase The basic assumption will be the conceptual validity of the quark

bag model4) - what I propose below is practically thermodynamics of finite size bags 9) interacting through creation or destruction of

new bag states . I will neglect quantum statistics - this approximation turns out to be permissible when the temperatures of individual hadrons are above 40-50 MeV . From the partition function Z(V,k,T) all physically interesting

quantities, such as energy, pressure, entropy can be derived as they are simply first derivatives of Z . Here V is the total reaction vo lume, u the baryonic chemical potential and T = 1/B the temperature . We will also use often the baryon fugacity a = exp(u/T) . a or reap . u are introduced to allow the conservation of the baryon number . The partition function can be expressed in terms of the hadronic mass -

spectrum T(p Z ,b) : T(mZ ,b)dm~ is the number of hadronic resonances of baryon number b in the mass interval (m 2 ,m 2 + dmz), We obtain ) : ln Z -- -

ae

(z,r)'H

~(ß .a) = H

£~ b = -m

? a(a,a) aß

ab Je-9PT(PZ~b)d~P

(1a) (1b)

The bootstrap constant H has been introduced here mainly for dimensional convenience : the dimensionless quantity ~ will be derived from the statistical bootstrap model . e is the available volume in which hadrons are free to move after subtraction of their proper volumes . e = V - £ Vh~i

i ' Here the sum over all individual hadronic volumes

(2) Vh~ i in V is

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J . RAFELSKI

taken . We know that in the bag model of massless relativistic quarks 4) we have Vh ~ i = Mi/~B and hence the sum in Eq .

(3)

(2) may be easily carried out :

where is the statistical average of the total energy . We emphasize that the quantity ~ must always remain positive ; hence the total energy (mass) of the hadronic gas phase cannot exceed 4BV .

When this limiting value is approached, a tendency of quark bags to cluster together and to form one large bag is found . A necessary condition for this to occur is that the energy density of the (hypothetical) pointlike hadrons

2 8z E . - ~ 8 In Z = ~(B~a) p ~ aßz (2n)sH 2ß

(5)

diverges . The energy density E _ < B>/V in terms of E p is just e = D/VE and hence with Eq . (4) P E =

E

P 1+EP/4B

(6)

-~4B . eP+

/4B)" 1 represents the part of the van der Waals The factor (1+eP effect introduced by the finite siae of individual hadrons . In order to obtain a quantitative description of the hadronic phase we must derive an expression for the bootstrap function Eq . (1b) . By requiring that in the limit ~ + O the hadronic state

as described by Eq . (1) is again jnst another particle contained in the mass spectrum r a nonlinear equation for T is obtained~~. in terms of the function ß it has the convenient form~~ )

The input function ~ is defined in terms of the basic hadronic states [q q and q q q ] as

= 2nHT[3m nK i~

m

T

~ + 8 cos h (u/Z)mNRl ~ ~

~

(8)

EXTREME STATES OF NUCLEAR MATTER We recall that Eq . <

~°

(7)

has a real solution ~ = G(~)

49 3c only for

= ln(4/e) . At this point there is a root singularity . Further (1b) then requires an exponential growth of

study shows that Eq . T" rem/T° .

The highest hadronic gas temperature T ° has the analogous

physical meaning as the boiling point of water - for T > T ° the phase containing individual hadrons cannot exist and the constituents - quarks - are liberated hadronic volume Vh , Eq . shown following from Eq .

[in the sense that they can move within the (5)] . (8)

In Fig .

1a the dependence T(u)

is

at the critical point ~_~° . T(u=0)=T°

has been chosen to be 190 MeV

( H = .724 GeV ~ ) . This choice leads

to agreement with the slope of inclusive cross sections : The observed low pion temperatures T

6 = 1GeV/ .14GeV .

120-140 MeV are so small be-

cause pious are emitted mainly fran the less dense and cooler domains of hadronic fireballs 5) . We note in Fig .

1a that the maximal temperature of hadronic gas

phase decreases with increasing chemical potential - at u = O the

50 100 ß0 ' 200 T(MeV) T(MeV) Fiq . 1 : The critical relationship between temperature T and a) chemical potential y ; b) baryon density v . Curves 1-5 of constanttshown)projectile kinetic energy per baryon indicate the hypothetical evolution of central fireballs at selected kinetic energies ;v° _ .56 4B/mN corresponds to normal nuclear baryon density v° _ .14/fm 3 for the old value of the bag constant : B ~" = 145 MeV . large number of mesons generated by high T is sufficient to induce the change to the quark plasma . As a consequence of the Van der Waals effect discussed the energy density along the phase boundary is equal to 4B while pressure vanishes . With Bra (170 MeV)" = 110 MeV/fm~ we expect the phase transition at about 0 .5 GeV/fm' .

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J . RAFELSKI

To test these ideas about the hadronic gas phase the temperatures (slope parameters) of inclusive particle cross sections expected in relativistic heavy ion collisions have been computed . These calculations require that : 1)

small impact parameter collisions are identified, 2) only particles from the central fireball are counted . Under these circumstances the initial kinetic energy of the projectile nucleus per nucleon,

defines the available excitation Ekip, energy per participating baryon in the fireball E/b = m N

1

+ Ek p 2mN

(9)

The equations of state distribute this energy among kinetic and chemical degrees of freedom neglected] .

[collective motion and its energy is

The temperature-density relationship of the exploding

fireball is shown for the LBL-DUBNA energies in Fig .

1b . Averaging the temperatures of emitted particles along these cooling curves gives the results shown in Fig . 2 . We record that the nucleon temperature TN is significantly higher than that of pious . [Hagedorn

is presently finetuning the parameters H and B and a ±10 MeV change in the results shown is anticipated] . The substantial rise of the temperature with the kinetic energy shown in Fiq . 2 is in good agree-

ment with experiment (BEVALAC-ISR) 12) .

. . teo E 100

Fig . 2 : The dependence of temperatures of nucleons and pious on initial projectile kinetic energy . T is the highest temperature of the hadronic g~~ phase . In Fiq .

3,i show another interesting result : Along the cooling

lines of constant energy per baryon (see Fig .l)the (specific) entropy per baryon rises significantly fron the (high) valve computed at the critical line . This is due to the onset of stra®q pionization

EXTREME STATES OF NUCLEAR MATTER

49 5c

of the available energy . The entropy at the critical line is found recalling that when pressure vanishes (S/b)/P=0 =

E

/T-u /P=0

h~ g

E/bTe .

(10)

We thus keep in mind that hadronic fireballs do not expand adiabatically and that substantial cooperative entropy production is expected . We will return to this point again below .

10 5

2

150

100

150 T(MeV]

Fi 3 " Specific entropy per baryon along cooling 1 nes " or given projectile kinetic energy as function of T . The boundary is the specific entropy at the critical line .

3 . The Quark-Gluon Plasma When hadrons have coalesced into a large quark bag at the critical curve, we must change the theoretical model underlyiaq the descrip-

tion of hot hadronic matter . The new central assuaption, valid strictly only at very high energies, is the weakness of the quark quark interaction : only in this case a descriptiaa of interacting

quantum Fermi-Bose gases t) may be successful . l'he quark passes in the relativistic quark-bag model are small : mq~ 10-3Q l1eV, while gluons are massless . [These masses should not be confounded with the nonrelativistic quark model masses fair which correspond to the kinetic energies of the bag model : nr~ 2 .04 ßic/Rbag 400 MeV .) Hence in the region of interest to us of W and T, the quark chemical potential

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J . RAFELSKI

uq = 3 u

ber 1/3 .

> mq .

The factor 1/3 arises in view of the quark baryon num-

As long as this condition is satisfied, the Fermi -Boxe gas

with interaction O(as ) Quarks :

~

T ln Z q = ~ 3 2n

can be integrated analytically l3)

and we find : (11a)

as ~ ~ u n 4"3a 4

_ _2 C\1

+

u2

2~3 z

(nT) ~ +

_ _50 \1

~ (nTl 21n as/ 60 +~

Gluons : T ln Z

g

--

SV 45n 2

C1

-

15 4n

as~

(nT) "

(11b)

Vacuum : T ln Z V = -BV where

Z

= Zq Zg ZV

(11c)

is the total partition function .

Here g = 2 " 2 " 3 = 12 is the number of distinct quark modes with spin, isospin and colour . as is the QCD colour coupling constant

[a s ra .5

for space-like qz and a s ~ .2 for time-like qs - but we ignored the qz dependence of a s ] . In Eq .

(11) we show separately the contributions of quarks and

gluons ; the vacuum term is a phenomenological supplement at this stage of the discussion and has been chosen so that the bag energy density is B inside the region of the plasma aad that an inside pointing pressure P ~ -B acts on the surface of the bag region . Contributions of heavy and strange quark flavours have been neglected . The vacuum contribution, Eq . (11c) is na postulated in the quark bag model .

However, at finite temperatures additional difficulty

arises not shown by the perturbative expression, Eq .

(11) . We re

call that the vacuum structure terms originates presumably in the absence of the true vacuum gluonic structure fraan the region of space containing quarks . Therefore B should be calculated non-perturbatively together with the glue term, Eq . (11 b) to yield some T-dependent quantity . In particular, this gauge pressure should vanish at some high temperature Tcrx 1 .5B ~~ , ~rhen the structure of the true vacuum is destroyed l4) . Only above Tar is the expression (11 b) valid -while Eq . (11 .c) is only correct for T ~+ O . For T < Tcr there is only a partial restoration of symt~try and not all 8 " 2 gluonic degrees of freedom can be excited . Another aspect of this point is that gluons may not be able to exist as independent par-

49 7c

EXTREME STATES OF NUCLEAR MATTER

tides in the plasma region and are rapidly absorbed on the surface by the true vacuum . This glue dissipation may be at the origin of a high instability of gluonic states, In order to estimate the boundary of the quark-gluon plasma phase in the u-T plane, c .f . Fig . 1a, we search for the line of zero pressure PV = T In Z, as given by Eq . (11), though omittïnq the little important, but obscure glue term at relatively small temperatures . The result is very similar to the bootstrap line, Fiq . with Tox B 1~" at u = O :

1a,

now it is the quark-antiquark pair pressure

that balances the vacuum pressure . This perhaps not accidental coincidence of the critical lines leads to the conclusion that both phases described here are directly adjacent to each other . Adjusting slightly H, B and a(T) we can achieve exact coincidence of the critical lines along which in both phases the energy density is 4 B

and pressure vanishes . However, we can have a discontinuity

of the baryon density : a s in nuclear collisions baryon number should be conserved,

the hadronic volume mould be discontinuous . Of course

this will not be the case - instead we have a Van der Waals transition shown schematically in Fig .

4 : on the hadronic gas side we have

to construct a new state consisting of a mixture between plasma and hadronic cluster .

We will not enter here further into the discussion

of this subject .

Yt V,n

V-11 v

V2

Fi 4 : Pressure va Volume ( 1/v), qua " itatively with the Maxwell construction Finally let us note that as in the hadronic gas phase gyre can construct in the u-T plane curves of given available energy per baryon . In a qualitative picture,

Fig .

5, the most important aspects are

illustrated ; there is a highest temperature that can be reached :

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J . RAFELSKI

1 GeV

Fig .

5:

Cooling lines in the u-T plane in the quark-gluon plasma (qualitatively)

Tmax rs~A . Furthermore, at hadronization we have very small chemical potentials for ISR energies - hence we find free Eq . (10) that the expected specific entropy in a nucleus-nucleus collision at ISR could reach .100 . In other words, following Boltzaann, we find that there are 2ioo different final states available to each 15 GeV

We conclude that this enormous amount of entropy must be produced in the nonadiabatic explosions of highly compressed quark baryon . plasma .

4 . Strangeness in Nuclear Collisions Unlike p-p collisions, strangeness nay be close to kinetic equilibrium in nuclear collisions : it is the large reaction volume with typical lengths exceeding the Compton wavelength of the strange mass

that is of great importance here 6) . Let me explain why we should not expect the usual kinetic equilibrium result in p-p collisions : in generalwhen a strangeness-antistrangeness pair~is made in hadronic

reactions at is

:

a

given temperature T, the expected number of pairs

EXTREME STATES OF NUCLEAR MATTER

499c

where ms s 280 MeV is the relativistic strange quark mass . Instead, when we consider production of particles in a hot star we find

~ e -m s /T (12b) ~n~pair The fact that a pair is produced seems not to matter here . The crucial question is where the transition from case a) to b) occurs . The quantitative result is that in the hadronic volume Vh = 4n/3 (1 fm) 3 the pair mass matters, case a), while already for V p 6-8 Vh we are in the limit b) - assuming a typical T a150 MeV . The suppréssion of the kaon yield in p-p collisions as compared with thermodynamic models has so found an explanation . It is expected that in nuclear collisions sufficiently large hadronic volumes are found . Furthermore, I have found from this study6) that there is no information about the hadronic aggregate state that can be extracted from the total strangeness yields . However, it is the antistrangeness that may be very useful in the search for quark matter : below the ATI threshold [at 9-5 GeV

taking account of Fermi-motion in both nuclei] the antistrangeness

that must balance the strangeness produced in hadronic reactions will generally be found only in kaons, eacept if a quark plasma state were formed . In that case the kinematic limit is of no importance and it is the relative abundance of antiquarks in the plasma that will control the antibaryon yields . Consider for example the 7[/N ratio as a

measure of the relative s and u or d abundances : on theoretical grounds I expect s to be as abundant as û and a taken together, supposing the typical T and u values at 4-5 GeV/N . Hence if quark plasma is formed at these energies, Â/N ratio could become an important observable . A preliminary search for  at 2 .1 GeV/N has been negative ly) . As the energy density reached at 2 .1 GeV/N - projectile kinetic energy is perhaps only 250 MeV/fm3 , this result fulfils expectations and sets a lower limit on the value of the critical energy density . It

is quite conceivable [say 508-508] that at 4-5 GeV/N in some collisions the quark plasma state is formed . With direct  production still strongly suppressed,I am awaiting impatiently future results on Â/N ratios at these energies .

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J . RAFELSKI 5.

Conclusions

In order to obtain a theoretical description of the hadronic gas and quark plasma phases,I have used some 'common' knowledge and plausible interpretations of the currently available experimental observations . In particular, the strongly attractive hadronic interactions are included through the rich, exponentially growing hadronic mass spectrum T(m Z ,b)

while the introduction of the finite volume of each

hadron is responsible for an effective short range repulsion . Aside from these manifestations of strong interactions, I only satisfy the usual conservation laws of energy, momentum and baryon number . 2 neglect quantum statistics since quantitative study has revealed that this is allowed above T = 50 MeV . But particle production is allowed, which introduces a quantum physical aspect into the otherwise 'classical'

theory of Boltzmann particles .

The study of the properties of hadronic matter has just begun and it is too early to say if the features of strong interactions that have been included in these considerations are the most relevant ones . It is important to observe that the currently predicted pion and nucleon mean transverse moments and temperatures show the required substantial rise, see Fiq . 2, as required by the experimental results at Ek,lab~A = 2 GeV [BEVALAC] and at 1000 GeV [ISR] . Further comparisons involving, in particular, particle multiplicities available~ 3)

and strangeness production discussed above are under consideration . I wish to emphasize the internal consistency of the two-fold approach . With the proper interpretation the description of hot hadronic matter leads us,

in a straightforward

fashion, to the postu

late of a phase transition to the quark-gluon plasma . This new phase is treated by a quite different method ; in addition to the standard

many-body theory of weakly interacting particles at finite temperature and density, we also introduce the phenomenological vacuum pressure and energy density B . Perhaps the most interesting and far reaching aspect of this work is the realization that the transition to quark matter will occur at much lower baryon density for highly excited hadronic matter in the (T = 01 . The precise baryon density of the phase transition depends somewhat on the phenomenological value of the bag ground state

constant ; we estimate it to be at about 2-4 v o at T ~ 150 MeV . The detailed study of the different aspects of this phase transition, as

EXTREME STATES OF NUCLEAR MATTER

50 1c

well as of possible characteristic signatures of quark matter, must still be carried out . I have given here only a very preliminary report on the status of my present understanding . The occurrence of the quark plasma phase will certainly be an observable phenomenon . Here I have discussed a measurement of the A/p relative yield at about 4-5 GeV/N kinetic energy nuclear collisions . In the quark plasma phase we expect a significant enhancement of  production which will be most likely visible in the

7Ï/p relative rate . Thinking ahead a decade from now, I can foresee

colliding nuclear beams with energies of the order of 100 GeV/N : the anticipated temperatures of several GeV in the quark plasma may lead to the formation of very exotic heavy flavour states involving c and b quarks at the same time .

Many fruitful discussions with the GSI/LBL Relativistic Heavy

Ion group stimulated many of the ideas presented here . I would like to thank R. Bock and R . Stock for their hospitality at GSI . Parts of this work were performed in collaboration with R. Hagedorn, B . Müller, H .-Th . Elze, and M . Danos .

References 1) R. Hagedorn and J . Rafelski, manuscript in preparation for Physics Reports . See also 'From Hadron Gas to Quark Matter', CERN preprints TH 2947 and TH 2969, to appear in the Proceedings of the 'International Symposium on Statistical Mechanics of Quarks and Hadrons', Bielefeld, Germany, August 1980, H . Satz, editor, North Holland Publishing Company. 2) The many-body theory for QCD at finite temperatures has been discussed by : B,A . Freedman and L.D . McLerran, Phys . Rev . D16 (1977) 1169 ; S,A . Chin, Phys . Lett . 78B (1978) 552 ; P .D . MorleY and M.B . Rislinger, Phys . Rep . 51 (1979) 63 ; J .I . Kapusta, Nucl . Phys . B148 (1979) 461 ; E .V . Shuryak, Phys . Lett . x(1979) 65 and Phys .Lett .~61 (1980)71 ; O .K . Kalashnikov and V .V . Klimov, Phys . Lett, 888 (1979) 328 3)

J . Rafelski, H .-Th . Elze, and B. Hagedorn, 'Hot Hadronic Matter in ~-Annihilation on Nuclei', CERN preprint TH 2912, in Proceedings of the 5th European P-Symposium, June 1980, Bressanone, Italy . CLEUP, Edts ., Padova 1980 . See also J . Bafelski, Phys . Lett . 91B (198Qj28

4)

For review see, for example, R. Johnson, 'The lIIT Bag Model', Acta Phys . Polon. H6 (1975) 865

50 2c

J . RAFELSKI

5)

R . Hagedorn and J . Rafelski, Phys . Lett . 97B (1980) 136

6)

J . Rafelski and M . Danos, Phys . Lett . 97B (1980) 279

7)

J . Rafelski, 'Extreme States of Nuclear Matter', Frankfurt Preprint UFTP 52 (1981) in Proceedings of the Workshop on 'Future Relativistic Heavy Ion Experiments', R . Bock and R . Stock, eds . Darmstadt 1981, GSI Yellow Report 6-1981 .

8)

It has been suggested by P . Richard (private communication) that this would be a stable state .

9)

We record the first attempt by J . Baacke, Acta Phys . Pol B8 (1977) 625 to develop a thermodynamic description of a gas of quark bags .

10)

R. Hagedorn, I . Montvay, and J. Rafelski, 'Hadra~nic Matter at Extreme Energy Density', Proceedings of Erice Workshop, eds . N. Cabibbo and L . Sertorio, Plenum Press (New York 1980), p . 49

11)

J . Yellin, Nucl . Phys . B52 (1973) 583 ; see also E . Schröder, Zs . für Math . und Physik15 (1870) 361

12)

For LBL experiments see e .g . S . Nagamiya, 'Heavy Ion Collisions at Relativistic Energies', LBL preprint 9494 (1979) in Proceedings of the Symposium on Heavy-Ion Physics, Brookhaven National Laboratory, Upton, N .Y ., July 15-20, 1979 . The ISR inclusive pp results are summarized in G . Giacomelli and M. Jacob, Phys . Lett . C55 (1979) 1

13)

H .-Th . Elze, W . Grelner, and J. Rafelski, J . Phys . G6 (1980) L149 and in Ref . 3 above, H .-Th . Elae et al ., to be pnbl~shed .

14)

In the model of magnetic Sawidy vacuum the restoration of the perturbative vacuum has first been shown by B . lHiller and J. Rafelski, 'Temperature Dependence of the Bag Constant and the Effective Lagrangian for Gauge Fields at Finite Temperatures', CERN TH 2928, Phys . Lett . B 101 (1981) 111 . See also J . Kapusta, Nucl . Phys . B in print and W . Dittrich and V. Schanbacher, Phys . Lett . B .100. (1981) 415

15)

R. Stock, private communication.