arXiv:nucl-th/9802021v1 6 Feb 1998 o C tfiietmeaueso hs transition phase a temperature show critical do temperature calculations a but finite at lattice transition, at The phase QCD dynamics. the for hadron to close describe QGP not prop- transport the lat- the of of on erties results concentrate We ob- the from directly calculations. precisely is tice par- more mass of QCD, dependent evolution from medium tained the the for where equation tons, Vlasov Lagrangians. effective a on propose based We theories transport the data. to the of interpretation the strongly in excited helpful are highly very approaches phenomenological a manifestation the of system, their evolution interacting and space-time symmetry) the chiral aspects in certain (e.g. study QCD to of based permit ones Lagrangians the effective While on merits. their are have approaches they sections. Both cross input or and/or experimental functions (NJL)) with string-fragmentation [3] [2] like type Jona-Lasinio cascade Monte-Carlo Nambu de- of Friedberg or either by [1] approximate those are (like Lee of They Lagrangians number effective from a proposed. rived Instead been which have goal achieved. formulations much a yet di- QCD-Lagrangian, not very be the is are should from equations phase derived and transport rectly the hadronic phase Ideally, the deconfined for. of called into the descriptions transition at of accelerators, its reactions evolution LHC space-time ion and the heavy RHIC high-energy SPS, the in (QGP) plasma an nu- Lee The Friedberg established. by are that Jona-Lasinio . like Nambu theories calculations confinemen effective display lattice to equation Vlasov Relations from the one of solutions of the syst merical range to quasi-particle a the identical of for is density and the energy by equilibrium the determined thermal temperatures is selfconsis in form calculated whose that At equation, are requirement gap masses a masses. from the dependent tently point medium func space-time distribution with each the quasiparticles for of equation tion Vlasov a from calculated nrpdtmeauevrain around themselves variations manifest temperature aspects chiral They of Two rapid restoration in hadrons. confinement. transition: the and phase of symmetry the one to the related from are QGP the of h prahpeetdi hsppri lsri spirit in closer is paper this in presented approach The quark-gluon- a of creation possible the of view In 12.39.Ba 12.38.Mh, 25.75.-q, numbers: PACS h pc-iedvlpeto ur-lo lsais plasma quark-gluon a of development space-time The rnpr hoywt efcnitn ofieetrelated confinement self-consistent with theory Transport nttt fTertclPyis edlegUniversity, Heidelberg Physics, Theoretical of Institute .INTRODUCTION I. T c hc eaae h regime the separates which T .Bo˙zekP. c ftechiral the of Spebr2,2018) 29, (September ∗ ..H n .H¨ufner J. and He Y.B. , em t. d - - 1 rbto ucin,wihdpn ntequasi-particle the on depend energies which functions, tribution rc factor eracy hr h u usoe h atnspecies parton the over runs sum the where edn o: to leading h enfil otiuint h iprinrlto [4]. density relation potential dispersion The the to contribution mean-field the atcepcue iha ffciemass effective an with picture, particle of ”data”) called (often results calculations. the lattice as- in Both contained are respectively. pect loop, Poliakov the and condensate httetemdnmcptnil()hsaminimum, so a chosen has (1) be potential to thermodynamic the have that which parameters nomenological hroyai eain hnvrtemse r tem- are masses dependent. the perature whenever relations thermodynamic ra h qiiru aei e.I forpprand paper our of II Sec. in how case show, equilibrium the treat iecluain.Sc I el ihtenon-equilibrium the with where deals III case, Sec. calculations. tice ( time space a qiiru eed ntetemperature the on depends equilibrium mal qainas equation ofieet h eaint te oesi ie in given is models other to IV. shows relation which Sec. plasma, The parton a of confinement. expansion the for lation o h rsuedniy(hroyai oeta)for potential) ansatz [4]: (thermodynamic an partons make density the we pressure QCD the of best for case is the system For original the approximated. of thermodynamics the that so neatosaogte srpae yasse fnon- of system masses a whose by A quasi-particles replaced and interacting is mass definite them system. with among particles interactions body of many consists which interacting system an to proximations hlspewg1,D610Hiebr,Germany Heidelberg, D-69120 19, Philosophenweg EOFNMN HS RNIINI A IN TRANSITION PHASE DECONFINEMENT P h ai supino u prahi h quasi- the is approach our of assumption basic The h masses The h us-atcemdli n ftems ipeap- simple most the of one is model quasi-particle The qp ( m 1 I HROYAISO THE OF THERMODYNAMICS II. E m , US-ATCEAPPROACH QUASI-PARTICLE i m ( m 2 g p x ( m T , . . . , i ∂m = ) ( , T ∂P T n where and ( t m a eotie rmrslso h lat- the of results from obtained be can ) ,t x, ,i h o-qiiru iuto.We situation. non-equilibrium the in ), .W eotrslso ueia calcu- numerical a of results report We ). i i p scluae rmtesm gap- same the from calculated is ) pern ntepesr r phe- are pressure the in appearing T = ) p 0 = 2 V + − X oteltiedata lattice the to i f otiue otiilyt the to nontrivially contributes V i , m i ( ( g i 2 E m i . i Z 1 = 1 ( p m , V )aeBs rFridis- Fermi or Bose are )) (2 , ( d 2 2 m π 3 . . . , , . . . , p ) 1 3 m , 3 m ) E m 2 , p hc nther- in which , i . . . , 2 ( ( T p i ) r chosen are ) ihdegen- with describes ) f T i ( n on and , E i ( p )) (2) (1) 3 ∂V d p mi masses mi(T ) or equivalently the mean-field energy den- + gi 3 fi(Ei)=0 . (3) ∂mi Z (2π) Ei sity V (T ). To date, the most detailed information about the QCD phase transition comes from the lattice cal- These equations have the form of a gap equation culations which give various thermodynamic functions, for the masses mi(T ) provided the potential density among them the energy density ǫlat(T ). We define our V (m1,m2,...) is given. Eq. (3) is equivalent to the quasi-particle approach by the requirement consistency relations [4,5] allowing to obtain the energy density from the pressure (1), in the form as for the ideal ǫqp(T )= ǫlat(T ) . (5) gas : Since Eq. (5) is only one relation, only one function m(T ) d3p ǫqp(T )= gi Eifi(Ei)+ V (T ) can be determined. We therefore have to assume that our Z (2π)3 Xi quasi-particle system consists of just one kind of massive partons (i = 1 in Eqs. (1) to (4)). We will assume a = ǫkin(m,T )+ V (T ) , (4) Boltzmann type distribution function f(E,T ) = exp−βE. where V (T )= V (m1(T ),m2(T ),...). The right hand side in Eq. (5) being given, this equation is a constraint on the unknown function m(T ) in ǫqp(T ). We note that the quasi-particle model constrained by Eq. (5) describes exactly the energy density of lattice QCD, but may reproduce other thermodynamic function only approximately [4,5] which is not a surprise since lattice QCD is not a quasi-particle gas. The functional form for m(T ) can be obtained by solving the differential equation 20 dm(T ) dǫ (T ) ∂ǫ (m,T ) = lat − kin dT dT ∂T 15 ∂ǫ (m,T ) dV / kin + , (6)
4 ∂m dm
which is obtained from Eq. (5) by differentiating both (T)/T 10 ε sides with respect to T and where dV /dm is given by the gap equation (3). Eq. (6) is a first order differential equation which determines m(T ) for a given energy den- 5 sity ǫlat(T ) and for an initial value m(T0). Then V (T ) can be obtained from V (T ) = ǫlat(T ) − ǫkin(m(T ),T ). The lattice data are normalized so that ǫlat(T ) → 0 for 0 T → 0. It requires that V (T ) → 0 at low temperatures. If m(T ) is not obtained from Eq. 6 but is given from some other considerations, the gap equation can be integrated 0 1 2 3 4 to obtain the potential V (T ): T/Tc m(T ) d3p m V (T )= −g dm f(E(p)) + V (T0) , Z Z (2π)3 E(p) FIG. 1. The energy density of the four-flavor QCD divided m(T0) 4 by T as a function of the scaled temperature T/Tc. The (7) points are from the lattice data [6]. The solid line is the re- sult of the quasiparticle model described in the text with the where g = i gi. chiral symmetry restoration (case (b) and solid line in Fig. 2). We applyP the above methods to the lattice data from The short-dashed and the dotted lines are the corresponding [6] shown in Fig. 1. These data are calculated for 4 kinetic energy density and potential energy density respec- flavors, corrected for finite lattice size and extrapolated tively. The dashed line represents the energy density obtained to massless fermions. We use g = 62.8 corresponding to as a parameterization of the data points and leading to the the noninteracting limit of QCD with 4 massless flavors. parton mass increasing at high temperature, (case (a) and We draw the attention to the fact that the lattice data dashed line in Fig. 2). The dashed-dotted line indicates the in Fig. 1 for ǫ (T )/T 4 do not approach the Stefan- Stefan-Boltzmann limit. lat Boltzmann limit for T ≫ Tc. This may have two reasons. (a) The parton mass m(T ) never approaches the chiral We want to model the deconfinement phase transition limit m = 0. Perturbative arguments (whose valid- of QCD in a quasi-particle approach and therefore have ity is questionable around T ) suggest that m(T ) ∼ to set a criterion how to determine the quasi-particle c
2 T for large T . If we attribute the discrepancy be- T >Tc. The mass ma(T ) increases linearly with T for 4 tween ǫlat(T )/T and the Stefan-Boltzmann limit large temperatures while mb(T ) is set to reach the chiral to this reason, one finds the minimal mass m(T ) ≃ limit at high temperatures. Fig. 1 also shows also the en- 2.1Tc right above Tc and m(T ) ≃ 1.1T for large T . ergy density corresponding to the two cases. We observe that in the case (a) ǫ (T ) follows the lattice points as it (b) Chiral restoration requires that m(T ) → 0 above qp should. In particular it has the same high temperature the phase transition at least for the fermions. Then limit which is different from the Stefan-Boltzmann limit. the deviation from the Stefan-Boltzmann limit has to be attributed to a mechanism which is outside the scope of the quasi-particle approach.
10
8 10
6 8 4 4 c
2 6 U(T)/T c 0
m(T)/T 4 -2
-4 2 0 0.5 1 1.5 2 2.5 T/T 0 c 0 1 2 3 4
T/Tc FIG. 3. The potential energy density V (T ) as a function of the temperature for the case (a) and (b) of the temperature dependence of the mass, dashed and solid line respectively. FIG. 2. The dashed line represents the temperature depen- dence of the quasiparticle mass as obtained from the lattice data (case (a)). The short dashed line represents the extrap- The energy density for the case (b) is different for T >Tc olation to the low temperature region. It is taken to be con- and approaches the Stefan-Boltzmann limit. Fig. 1 also stant, and suffices to effectively confine the partons (an abso- shows the densities for the kinetic energy ǫkin(m(T ),T ) lute confinement would require that the mass goes to infinity and for the potential energy V (T ) for the case (b). Fig. at low temperatures). The solid line represents the assumed 3 shows the potential density functions V (T ) for both temperature dependence of the parton mass with zero limit cases. In the case (b), one can identify the large tem- at high temperature (case (b)). perature limit of Vb(T ) with the bag constant B, since Vb(T = 0) = 0. Indeed for large T (in the deconfined We will explore both possibilities and call the correspond- phase) Eqs. (1) and (4) take the form of the free mass- less gas with a bag constant B = Vb(T = ∞). For the ing masses ma(T ) and mb(T ) respectively. First we solve case (a) one can define the bag constant as the value of the Eq. (6) to obtain ma(T ) for the case (a), where we set the potential density Va(T ) for the temperature where ma(T0)=9.5Tc for T0 =0.8Tc. For the case (b) we take the mass ma(T ) is the smallest (slightly above Tc). We ma(T ) obtained in (a) for T
3 rect, but in order to compare it to the usual bag constant This comparison clearly favors case (b), where m(T ) → 0 1/4 (B = 135 − 200 MeV) one should use the lattice re- above Tc. In what follows, we will mainly work with this sults for the energy density and the critical temperature solution. of the two-flavor QCD as an input for the quasi-particle Before we treat the nonequilibrium case we apply the parton model. gap equation to the case of finite density and calculate In order to distinguish between the two solutions for the density dependence of the parton mass. We introduce m(T ) we investigate the temperature dependence of the variables which will be also useful for the discussion of the chiral condensate in the quasi-particle picture nonequilibrium case. First let us write the gap equation in the form d3p m hΨΨ¯ i (T )= hΨΨ¯ i (T )+ g f(E) , (9) qp vac qq¯ (2π)3 E dV dρ Z = V ′(ρ) = −ρ , (12) dm dm where gqq¯ counts fermion and anti-fermion degrees of freedom and the vacuum part of the chiral condensate where we define the scalar density is given by an expression including a cutoff in momen- tum d3p m ρ = g f(E(p)) . (13) 3 (2π)3 E(p) ¯ d p m Z hΨΨivac(T )= gqq¯ 3 . (10) Z|p|<Λ 2(2π) E
10
1 8 0.8
0.6 6 c >(0) )/T f
ΨΨ 0.4 m(p 4
>(T)/< 0.2 ΨΨ < 2 0
-0.2 0 0 1 2 3 4 5 6 0.8 1 1.2 1.4 1.6 1.8 2 pf/Tc T/Tc
FIG. 5. The quasiparticle mass as a function of the scaled FIG. 4. The chiral condensate as a function of the tempera- Fermi momentum pf /Tc. The solid and dashed lines corre- ture for the case (a) and (b) of the temperature dependence of spond to two different temperature dependences of the parton the mass, dashed and solid line respectively. The data points mass in Fig. 2. are for the lattice data in the four-flavor QCD [7].
It allows us to use the functions V (ρ) and m(ρ) instead of The value of the cutoff Λ = 430 MeV is fixed to reproduce V (T ) and m(T ) for the parameterization of the potential the zero temperature value of the chiral condensate which energy density and the mass respectively. Assuming that we take twice the usual value the energy density depends on temperature T and chemi- 4 hΨΨ¯ ivac(0) = 2(250MeV) , (11) cal potential µ only through ρ(T,µ) we can generalize the finite temperature case also to finite density. In principle because we are modeling the four-flavor lattice QCD. the potential V (ρ) could depend on other quantities, e.g. Fig. 4 shows a comparison between the condensate func- the baryon density. Any such more general case cannot tions for the quasi-particle picture and the lattice data.
4 be discussed using only the lattice data at finite temper- is conserved by the evolution according to the Vlasov Eq. ature. As a support for our assumption we note that for (15), i.e. the energy of the system is constant. the NJL model the potential density V depends only on We have numerically solved the Vlasov equation to- the density ρ (Sec. IV). gether with the gap equation for the two cases of the We write the gap equation (3) at finite density functional dependence of the mass m(T ) on the temper- ature discussed in Sec. II, but most of the results shown dV d3p m relate to the case where the chiral condensate vanishes = −gf 3 Θ(pf − |p|) , (14) dm Z (2π) E(p) at high temperatures (case (b) in the previous section). where gf counts the number of fermion degrees of free- We have used the test particle method for the solution of dom. Note that here we are using the Fermi distribution the Vlasov equation, i.e. we have made the ansatz for the fermions at finite density and zero temperature N and V (m) from the lattice data. Fig. 5 shows the mass 3 3 f(x,t,p)= δ (x − xj (t))δ (p − pj (t)) , (18) of the partons as a function of the Fermi momentum pf . The behavior is similar as in the finite temperature case. Xj=1 For low density the mass increases leading to the effective where the trajectories xj (t) and pj (t) of the N test par- confinement. At high density the mass is proportional to ticles satisfy Hamilton’s equations with pf for the case (a) and goes to zero in the case (b). H(x, p)= p2 + m2(x, t) . (19) p III. TRANSPORT THEORY OF THE EFFECTIVE The initial conditions xj (0) and pj (0) are chosen so that CONFINING MODEL given initial densities for matter and momentum are re- produced. The initial density is chosen spherically sym- In this section we will discuss the nonequilibrium evo- metric. Also in the solution of the gap equation the lution of the parton densities. In the semiclassical limit spherical symmetry is imposed by angle averaging. the collisonless plasma of quasi-particle partons can be described by the Vlasov equation for the phase-space dis- tribution function f(x,t,p): p ∂tf(x,t,p)+ ∇xf(x,t,p) 1.6 E(p,x,t) t=0 fm/c m(x, t) 2 fm/c − ∇xm(x, t)∇pf(x,t,p)=0 . (15) 1.4 6 fm/c E(p,x,t) )
-3 10 fm/c Eq. (15) has to be supplemented by an equation for the 1.2 space-time dependent mass m(x, t) . A sufficient condi- tion for the requirement, that the Vlasov equation de- 1 scribes the same physics at thermal equilibrium as pre- sented in the previous section is that m(x, t) satisfies the 0.8 same gap equation dV d3p m(x, t) 0.6 = −g f(x,t,p)= −ρ(x, t) , (16) dm Z (2π)3 E(p,x,t) 0.4 where the thermal distribution function f(E) in Eq. (3) particle number density (fm has been replaced by the nonequilibrium solution of the 0.2 Vlasov equation f(x,t,p) and where V (m) is the same as in equilibrium. The solution of the nonequilibrium gap 0 equation is then a function m(x, t) of space and time. 0 1 2 3 4 5 Eq. (16) is however not the most general equation which r (fm) reduces to the equilibrium finite temperature gap Eq. (3), but one can add to it terms which depend on the FIG. 6. The parton density distribution at different times, spatial and time derivatives of m(x, t). We will come as obtained from the nonequilibrium evolution of the initial back to this question in Sec. IV. Here we note that the fireball (t = 0). choice in Eq. (16) guarantees that the total energy of the system 3 At time t = 0 the system is described by the density 3 d p profile shown in Fig. 6 and with a momentum distribu- E(t)= g d x 3 E(p,x,t)f(x,t,p) Z Z (2π) tion corresponding to a temperature T = 1.3Tc = 180 MeV. Fig. 6 shows the density for different times. The + d3x/V (ρ(x, t)) (17) Z system expands for t = 2 and 6 fm/c, but then comes
5 back (t = 10 fm/c). The dependence of m(x, t) on the radius at different times is shown in Fig. 7. As expected
400
300
1600 t=0 fm/c 200 1400 2 fm/c 6 fm/c 100 1200 10 fm/c (MeV) x 0 1000 P
800 -100
m(r,t) (MeV) 600 -200 400 -300 200 0 10 20 30 40 t (fm/c) 0 0 1 2 3 4 5 FIG. 8. The component px of the momentum of a particu- r (fm) lar test-particle taken from the simulation of the time evolu- tion of the region of deconfined plasma. FIG. 7. The parton mass distribution at different times, as obtained from the nonequilibrium evolution of the fireball.
the mass is small in the interior and increases towards the surface. This increase is responsible for the confine- ment. Indeed one can show that the equation of motion 2.5 of one particle in the mean-field of the other particles conserves the energy of the particle p2 + m2. Thus a 2 particle cannot leave the region of deconfinedp plasma, if its initial momentum p satisfies: 1.5 2 2 2 p The vacuum mass should be infinite, if the confinement x (fm) 0 is absolute. In our calculation we take the vacuum mass equal to ∼ 9.4Tc (see Fig. 2), which effectively confines -0.5 the partons at the temperatures discussed here. Thus the partons cannot leave the hot fireball if their initial -1 momentum is smaller than ∼ 9.4Tc, which is the case for -1.5 most of the partons at our initial temperature. Figs. 8 and 9 show the time development of the mo- -2 mentum px and the coordinate x of a particular test par- 0 10 20 30 40 ticle. The particle oscillates between the borders of the t (fm/c) fireball and at the border its momentum is reduced and eventually reversed by the action of the force: FIG. 9. The x coordinate of the same test-particle as in Fig. 8. dp m = − ∇ m . (21) dt E x In contrast to the simple particle motion, partons trav- the internally created field leads to small masses inside eling in bunches may leave into the vacuum region, since the bunch. This possibility is excluded in our method of 6 solution, since we require at each time step that the sys- ton density inside and thus a small mass of partons. This tem stays spherically symmetric. This mechanism then fragmentation mechanism cannot be studied in our spher- generates collective vibrations of the surface of the fire- ically symmetric mean-field theory. It would require a ball, when particles are trying to leave the hot region description including the fluctuations of the density. 9 Case (a) 80 Case (b) Etot 3×V 8 75 70 7 ) 2 65 6 (t)> (fm 2 60 4 50 3 45 0 10 20 30 40 0 10 20 30 40 t (fm/c) t (fm/c) FIG. 10. The time evolution of the mean square radius FIG. 11. The time dependence of the total and potential of the region of deconfined plasma for the case (a) and (b), energy of the parton plasma, solid and dashed lines respec- dashed and solid line respectively tively simultaneously. Fig. 10 shows the time dependence for We state the main result of this section: the Vlasov the mean square radius of the fireball : equation together with the gap equation based on lattice 3 data shows confinement. For not too high initial tem- 2 3 2 d p hr (t)i = g d x x 3 f(x,t,p) . (22) peratures the fireball stays compact. Of course the sur- Z Z (2π) face of the fireball can oscillate, but the system remains The oscillation which can be seen on the figure reflect bounded. This behavior is in contrast to the free expan- the collective monopole oscillations of the parton density, sion of the parton gas, where the parton fireball would with partons leaving the fireball and reflected back when start to expand and cool down faster. In our picture the their mass grows. Analogous collective oscillations have partons are confined, since no hadronization is included been observed in the Vlasov evolution of the nucleon in (see Fig. 10). the Friedberg-Lee model [8]. As the volume of the fireball oscillates, its potential energy will also oscillate growing for large volumes and IV. RELATION TO OTHER EFFECTIVE decreasing when the system is compressed. The period of MODELS the collective oscillations is basically determined by the time which a particle needs to travel from one border of The effective models of the QCD using parton degrees the fireball to the other (Fig. 9). For massless deconfined of freedom are mostly restricted to the fermion sector. partons (case (b) in Sec. II) this time is twice the radius Thus NJL type models [2] have only fermionic degrees of of the system divided by the speed of light. The total en- freedom with four-fermion interaction. At high temper- ergy is of course conserved (see Fig. 11), to the accuracy ature and/or density the quarks are massless or almost of our numerical solution, and the kinetic (and then also and at low densities due to the nonzero quark-condensate the potential energy) has oscillations of the same period they acquire a finite mass with value around 350 MeV. as the oscillations of the mean square radius in Fig. 10. The quarks are not confined in this theory. The NJL gap We mention the possibility of fragmentation of the fire- equation for the quark mass can also be written in the ball into smaller pieces, each of them having large par- form of Eq. (3) where dV/dm is defined by the expression 7 3 dV (m − m0) d p m Of course as in the NJL model the degrees of freedom ≡ − 6Nf 3 , (23) dm 2G Z|p|<Λ (2π) E are restricted to the fermions. The Lagrangian of the Friedberg-Lee model can be written as: where G is the four fermion coupling constant, Λ is the µ 1 µ infrared cutoff, m0 is the current quark mass and Nf L = Ψ(iγ ∂µ)Ψ − gσΨΨ + ∂µσ∂ σ − U(σ) , (24) is the number of flavors. Fig. 12 shows a compari- 2 son between the potential extracted from lattice calcu- where the fermion field operator Ψ also carries the fla- lations and the one from the NJL model. The difference vor and color indices. The effective quark mass gσ can is twofold: (i) Shape: While the potentials for Friedberg- include also a contribution from the current quark mass. Lee and NJL model show a clear minimum at the position When the expectation value of the σ field is infinity, of the constituent mass in the vacuum, the potential from the quarks are confined. The same is also effectively true our approach drops monotonically to zero, reflecting the if the vacuum expectation value of the σ field is very confinement (infinite vacuum mass). (ii) Magnitude: At large. The gap equation for the quark mass σ is given by m = 0 the potential from our approach differs by about a the classical equation of motion for the σ field: factor 5 from the other approaches. This may be partly ′ due to the different numbers of degrees of freedom. While ✷σ + U (σ)= −g < ΨΨ¯ > (x, t) . (25) Friedberg-Lee and NJL refer to a two-flavor quark the- ory (no gluons), the lattice calculation is performed for In the case of homogeneous systems ✷σ = 0 (e.g. in Nf = 4 and gluons. The remaining difference may be due the mean-field thermodynamics) the Friedberg-Lee gap to quantitative difference between the four-flavor and the equation is equivalent to the gap equation used in Sec. two-flavor QCD, which goes beyond a simple rescaling of II if V (m) = U(σ). In particular the thermodynamical the number of degrees of freedom. energy density discussed in Sec. II can be reproduced in the Friedberg-Lee model if not for the neglect of the gluon degrees of freedom. The difference between our approach and the one by Friedberg and Lee rests in the choice of the potential. While they assume certain forms, we let V (m) 400 be determined by the lattice data. One should note that Case (b) NJL in the homogeneous systems the confining Friedberg-Lee F.-L. model, with the fermion mass m = κ(σ) being a function of the field σ, is also equivalent to our approach after 300 a change of variables σ → m. The infinite value of the ) fermion mass means simply in the language of Eq. (24), 3 that the vacuum gap equation dV/dm = 0 has a solution at m = ∞. 200 The kinetic term for the σ makes a difference for the case of nonequilibrium or nonhomogeneous systems. In the dynamical evolution of a nonequilibrium system in V(m) (MeV/fm the Friedberg-Lee model, the σ field is another dynami- 100 cal field not related to the local value of the scalar den- sity ρ [8]. The inclusion of the kinetic term for the σ field however leads to the problem that the value of the σ field can go negative. The potential U(σ) cannot be ex- 0 tracted for the negative values of σ from the lattice data. 0 200 400 600 800 1000 1200 Moreover in the cases when the mass of the fermion field becomes negative, its evolution cannot be described by a m (MeV) semiclassical Vlasov equation. FIG. 12. The potential energy density V (m) as a function of the mass as extracted from lattice QCD (solid line), for the two-flavor NJL model (dashed line) and for the Friedberg-Lee V. DISCUSSION model (dotted line). We used Tc = 140 MeV in defining V (m). The large value of the bag constant B = V (m = 0) for the The description of the deconfinement transition in potential density from the lattice data can be due to a larger heavy ion collisions is a very important theoretical prob- number of degrees of freedom in the four-flavor model. lem. A realistic description could allow to define which observables are relevant for the observation of the quark- gluon plasma formation. A dynamical simulation is The Friedberg-Lee model describes fermions coupled to wished for in order to extract the properties of the plasma a scalar field σ(x, t) which plays the role of an effective from the experimental data. Any such approach meets mass and whose dynamics is driven by a potential U(σ). difficulties in the description of the confinement and of 8 the hadronization transition when the temperature of the deconfined region drops down. In the present work we have addressed only a part of this program, namely the influence of the confinement on the dynamics of partons. The confinement of partons below T is described in ∗ c on leave from: Institute of Nuclear Physics, Cracow, a quasi-particle gas model by the increase of the parton Poland mass at low energy densities. The temperature depen- [1] R. Friedberg and T.D. Lee, Phys. Rev. D 15, 1694 (1977); dence of the parton mass is extracted from the lattice Phys. Rev. D 16, 1096 (1977). L. Wilets, Nontopological data. In the range of temperatures analyzed (0.8Tc < Solitons, (World Scientific, Singapore, 1989). T ACKNOWLEDGMENTS One of the authors (P.B.) wishes to thank the Alexan- der von Humboldt Foundation for financial support. This work has been supported in part by the German Min- istry for Education and Research (BMBF) under contract number 06 HD 742. 9