Phase transitions in quantum chromodynamics

TL;DR

This paper reviews the current understanding of finite temperature phase transitions in QCD, discussing theoretical predictions, experimental implications, and the importance of nonequilibrium effects in heavy-ion collisions.

Contribution

It provides a comprehensive review of phase transition criteria, analytical tools beyond mean-field models, and explores nonequilibrium approaches in QCD thermodynamics.

Findings

Comparison of theoretical predictions with experimental data

Identification of areas needing more reliable QCD equation of state data

Discussion of nonequilibrium effects in heavy-ion collision thermodynamics

Abstract

The current understanding of finite temperature phase transitions in QCD is reviewed. A critical discussion of refined phase transition criteria in numerical lattice simulations and of analytical tools going beyond the mean-field level in effective continuum models for QCD is presented. Theoretical predictions about the order of the transitions are compared with possible experimental manifestations in heavy-ion collisions. Various places in phenomenological descriptions are pointed out, where more reliable data for QCD's equation of state would help in selecting the most realistic scenario among those proposed. Unanswered questions are raised about the relevance of calculations which assume thermodynamic equilibrium. Promising new approaches to implement nonequilibrium aspects in the thermodynamics of heavy-ion collisions are described.

Tables

Table 1. TABLE I.: Criteria to distinguish between 1st and 2nd order transitions in a large but finite volume

criterion	1st order	2nd order
$P_L\left(E\right)$	double peak	single peak
$c_{Lmax}$	$\propto L^d$	$\propto L^{\alpha /\nu }$
${\chi }_{max}$	$\propto L^d$	$\propto L^{\gamma /\nu }$
$\beta \left(c_{Lmax}\right)-{\beta }_c$	$\propto L^{-d}$	$\propto L^{-1/\nu }$
$B_{min}$	$>0$	$\to 0$
$U{4}_{min}$	$\to 1$	$>1$

Table 2. TABLE III.: Order of the chiral transition as function of the number of flavors ( N f subscript 𝑁 𝑓 N_{f} ) and the anomaly strength ( g 𝑔 g ), partly conjectural

$N_f$	g=0	g=const	g=g(T)
	$N_c=\infty$	of order 1	g(T)$\sim d_I\left(T\right)$
1	2nd order	no transition	no transition
	O(2)-exponents
2	1st order	2nd order	1st order
		O(4)-exponents	O(2)$\times$ O(4)-symmetry
3	1st order	1st order	1st order
			$U_A\left(1\right)\times SU\left(3\right)\times SU\left(3\right)$
			symmetry
$\ge 4$	1st order	1st order	1st order

Table 3. TABLE IV.: Infinite volume behavior of the mass gap, the tunneling mass m ( unp ) superscript 𝑚 unp m^{\rm(unp)} and their corresponding correlation lengths.

phase	1st transition	2nd transition
symmetric	$m^{\left(p\right)}$ stays finite as $\beta \to {\beta }_c^{-}$	$\begin{array}{c}{m}^{\left(p\right)}\to {\left(\beta -{\beta }_c\right)}^{\nu }\to 0\mathrm{as}\beta \to {\beta }_c^{-},\\ {\xi }^{\left(p\right)}\to \infty \end{array}$
broken	$m^{\left(p\right)}$ and ${\xi }^{\left(p\right)}$ are finite	$m^{\left(p\right)}$ and ${\xi }^{\left(p\right)}$ are finite
symmetric	$m^{\left(\mathrm{unp}\right)}$ stays finite as $\beta \to {\beta }_c^{-}$	$\begin{array}{c}{m}^{\left(\mathrm{unp}\right)}\sim {\left({\beta }_c-\beta \right)}^{\nu }\to 0\mathrm{as}\beta \to {\beta }_c^{-},\\ {\xi }^{\left(t\right)}\to \infty \end{array}$
broken	$m^{\left(\mathrm{unp}\right)}=0$, ${\xi }^{\left(t\right)}=\infty$	$m^{\left(\mathrm{unp}\right)}=0$, ${\xi }^{\left(t\right)}=\infty$

Table 4. TABLE V.: Critical exponents 1 / β ​ δ 1 𝛽 𝛿 1/\beta\delta and 1 / δ 1 𝛿 1/\delta for 2-flavor QCD in comparison with O ​ ( 4 ) 𝑂 4 O(4) , O ​ ( 2 ) 𝑂 2 O(2) and mean-field critical exponents.

	2-flavor QCD	$O\left(4\right)$	$O\left(2\right)$	mean-field
$1/\left(\beta \delta \right)$	$0.77\pm 0.14$	0.55(2)	0.60(1)	0.67
$1/\delta$		0.208(2)	0.2080(3)	0.33

Table 5. TABLE VI.: Mass parameters and results for 2 and 3 flavors.

$N_f$	$m_{u,d}\cdot a$	$m_s\cdot a$	results of Brown et al. (1990)
2	0.01	$\infty$	no transition
2	0.025	$\infty$	no transition
3	0.025	0.025	1st order
3	0.025	0.1	no transition

Equations

$$Z(T,H,V)=\int \mathcal{D}U{e}^{-\beta \mathcal{H}(U)}$$

$$F=-T\ln Z$$

$$E=-\frac{\partial \ln Z}{\partial \beta }S=-{\left(\frac{\partial F}{\partial T}\right)}_{H}M=-{\left(\frac{\partial F}{\partial H}\right)}_T$$

$$c_H={\left(\frac{\partial E}{\partial T}\right)}_H{c}_{H,M}=T{\left(\frac{\partial S}{\partial T}\right)}_{H,M}{\chi }_T=\frac{1}{V}{\left(\frac{\partial M}{\partial H}\right)}_T.$$

$$F\left\{\varphi (x)\right\}=\int d^{d}x\{a{\left(\nabla \varphi (x)\right)}^2+\frac{r}{2}{\varphi }^2(x)$$

$$+\frac{\lambda }{4}{\varphi }^4(x)-h\varphi (x)\}.$$

$$r=\tilde{r}(T-T_c).$$

$$\varphi (x)=\frac{1}{L^d}\sum _{i\in L^d(x)}{s}_i.$$

$$e^{-\beta {\mathcal{H}}_{CG}[\varphi (x)]}=\int \mathcal{D}{s}_{i}P\left([\varphi (x)],[s_i]\right)\cdot e^{-\beta \mathcal{H}[s_i]},$$

$$F=-\frac{1}{\beta }\ln Z=-\frac{1}{\beta }\ln \int \mathcal{D}\varphi \exp \left\{-\beta F_{mf}[\varphi (x)]\right\}.$$

$$F=F_{mf},$$

$$F=-\frac{1}{\beta }\ln \int \mathcal{D}{s}_i\exp \left\{-\beta \mathcal{H}[s_i]\right\}.$$

$$⟨{\left(\delta \varphi \left(x\right)\right)}^2⟩\ll {{\varphi }_0}^2,$$

$$1\ll R^d(1-T/T_c{)}^{(4-d)/2},$$

$$f(g)=G(g)+b^{-d}f(g^{(1)})$$

$$\xi (g^{(1)})=b^{-1}\xi \left(g\right).$$

$$\xi (g^{\ast })=b^{-1}\xi (g^{\ast }).$$

$$\mathcal{H}(g)=\mathcal{H}(g^{\ast })+\sum _i{q}_i{O}_i+o\left({\left(\sum _i{q}_i^2\right)}^{1/2}\right).$$

$$q_1$$

$$q_2$$

$$t$$

$$R_b\mathcal{H}(g)=\mathcal{H}(g^{\ast })+\sum _i{q}_i{b}^{{\lambda }_i}{O}_i$$

$$f(t,h=0)\approx G(0)+A_f^{(\pm )}{\left|t\right|}^{d/{\lambda }_1},$$

$$2-\alpha =d/{\lambda }_1,$$

$$\xi (t,h=0)\approx A_{\xi }^{(\pm )}{\left|t\right|}^{-1/{\lambda }_1},|t|\ll 1.$$

$$\nu =1/{\lambda }_1.$$

$$M(t,0)={\frac{\partial f}{\partial h}|}_{h=0}\approx M_0+A_M^{(-)}{\left(-t\right)}^{d/{\lambda }_1-{\lambda }_2/{\lambda }_1},t<0$$

$$\chi (t,0)={\frac{{\partial }^{2}f}{\partial h^2}|}_{h=0}\approx {\chi }_0+A_{\chi }^{(\pm )}{\left|t\right|}^{d/{\lambda }_1-2{\lambda }_2/{\lambda }_1},|t|\ll 1$$

$$P_{\infty }(t)\sim A_P^{\pm }{\left|t\right|}^{-\mu },$$

$$P_L(t)={\left|t\right|}^{-\mu }{Q}_P^{(\pm )}\left(L/{\xi }_{\infty }(t)\right),$$

$$f(k_{1}t,$$

$${\chi }_L(t)$$

Full text

Contents

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I INTRODUCTION

Phase transitions are found over a wide range of temperatures. They start near absolute zero with the Mott transitions (Mott, 1968) from superconductor to insulator, and are common between $10^{2}$ K and $10^{3}$ K. Nine orders of magnitude higher one finds the phase transitions discussed here. At a temperature typical of the QCD scale ($\Lambda_{\overline{MS}}\sim 200$ MeV) nuclear matter melts. At this temperature quarks and gluons cease to be confined inside hadrons, and begin moving freely. Three orders of magnitude higher is the next phase transition, at which the electromagnetic and weak interactions are unified.

These two transitions may seem exotic, but according to the big bang theory have occurred at least once in the universe. The electroweak transition is predicted to have taken place at $10^{-10}\rm{s}$, and the deconfinement transition at $10^{-6}\rm{s}$ after the big bang. The deconfinement transition occurred, when the temperature of the universe dropped to the order of tera degrees, i.e. to a scale $T_{c}\sim(2.32\pm 0.6)\cdot 10^{12}\rm{K}$ or $200\pm 50$ MeV. (The conversion from natural energy units [MeV] to degrees Kelvin [K] is determined by the Boltzmann constant $k_{B}\sim 8.6\cdot 10^{-5}\rm{[eV/K]}$ or $(100~{}\rm{MeV}\sim 1.16\cdot 10^{12}K)$.) These temperatures with time scales of $10^{-23}\rm{s}$ and typical distances of 1fm may now be achievable in the laboratory. At CERN one reaches energy densities around $1.5-2.5~{}\rm{GeV/fm}^{3}$, and higher densities are expected when RHIC (Relativistic Heavy Ion Collision) experiments come on-line in Brookhaven (Stachel, 1994).

We have glibly mentioned the QCD transition, as if it were unique. However, there are really two, and they take place if either the temperature or the density crosses some critical point.

Each transition is well defined in some limit of QCD. If the quark masses are infinite, one has the deconfinement transition. If the quark masses are zero, one has the chiral (symmetry restoring) transition. It is not yet clear whether these transitions persist for physical quark masses nor whether they will occur together. If they take place at different temperatures, one expects the chiral transition to occur after the deconfinement transition (Banks and Casher, 1980; Campbell et al., 1990). Lattice results indicate, however, that both transitions coincide at the same critical point. One of them can then be regarded as driving the other, and the nature of the transition will be determined by the driving symmetry. Here it is assumed that both happen simultaneously, unless stated otherwise.

As mentioned above, a transition is expected for both increasing density as well as temperature. This review concentrates on the effects of temperature, as this is the area in which the lattice is strong and many partial or preliminary results are available.

One usually has many length, or equivalently energy, scales playing a role at a phase transition. This makes it impossible, in most cases, to find a suitable expansion parameter for a perturbative treatment. In addition, QCD is asymptotically free in the UV-limit, but has a coupling $g(T)\approx 1$ near the phase transition. So a nonperturbative approach must be used; the lattice regularization may be the only means of calculating the properties of the phase transition from first principles.

One expects, naively, a phase transition in QCD if the density rises to the point at which the hadrons begin to overlap. This will be around $0.5~{}\rm{GeV/fm}^{3}$, the energy density in a proton.

Similarly, for temperatures above the QCD scale parameter, $T>\Lambda_{\overline{MS}}=200~{}\rm{MeV}$, one expects a phase transition. At extremely high temperatures, $T\gg\Lambda_{\overline{MS}}$, with typical momenta $Q\approx T$, the coupling $Q^{2}/\Lambda^{2}_{\overline{MS}}$ becomes weak, and resummed finite temperature perturbation theory (Braaten and Pisarski, 1990 and 1992) may be applied.

There are three natural length scales in the QCD plasma, $l_{p}={1\over T}<l_{e}={1\over gT}<l_{m}={1\over g^{2}T}$ (DeGrand and DeTar, 1986). The first gives the range over which perturbation theory may be applied, the second corresponds to the scale set by the chromoelectric mass, and the third to the scale set by the chromomagnetic mass. The latter two scales reflect the increasingly nonperturbative nature of long-range interactions in the plasma, and the lack of the screening in the magnetic sector.

Although these long-range interactions are strong, and may lead to a type of dynamical confinement in the plasma, the bulk properties, the equation of state etc., are still dominated by short-range interactions. So perturbation theory may well be valid at high temperatures for these quantities.

More evidence that the chromomagnetic, long-range sector of QCD may have a complicated, nonperturbative structure comes from the phenomenon of dimensional reduction. In $3+1$ dimensional Euclidean field theory the imaginary time dimension is proportional to the inverse temperature, $l_{\tau}\propto 1/T$. Thus the theory reduces, in the high temperature limit, to three dimensional QCD (Reisz, 1992; Reisz and Petersson, 1991; Kärkkäinen et al., 1992, 1993 and 1994).

Deriving the properties of the QCD phase transition is clearly difficult. After all, it is analogous to deriving the properties of boiling water directly from the quantum Hamiltonian of water molecules. So the thermodynamics of the phases, and the order of the transition between them, have only recently been settled for pure gauge theory, despite nearly two decades of computer simulations.

The order of a phase transition is one of the basic thermodynamic classifications. A phase transition is set to be of first order, if there is at least one finite gap in the first derivatives of a suitable thermodynamic potential in the thermodynamic limit. A finite latent heat goes often along with a gap in the order parameter. A transition is set to be of second order if there is a power like singularity in at least one of the second derivatives of the potential. If the thermodynamic potential is analytic over the whole temperature range (for temperature driven transitions), the phase conversion is called a crossover phenomenon.

In general the order of the QCD transitions depends on the number of colors ($N_{c}$), the number of flavors ($N_{f}$), the current quark masses, and on more subtle effects related to the IR- and UV-cut-offs. The deconfinement transition is now believed to be of first order for three colors (Fukugita et al., 1989; Gavai et al., 1989), and the equation of state is known in the limit of infinite quark masses (Engels et al., 1990b; Laermann et al., 1995). Similarly the chiral transition is believed to be of second order for 2 massless flavors (Pisarski and Wilczek, 1984; Karsch, 1994) and of first order for three or more massless flavors (Pisarski and Wilczek, 1984; Gavai and Karsch, 1985; Gavai et al., 1987; Brown et al., 1990; Iwasaki, 1995; Kanaya et al., 1995). These results do not contradict each other. They just indicate the sensitivity of the order to the involved approximations and should be taken as a warning not to jump to conclusions.

Ultimately one is interested in the physically relevant case of 2 light quarks (up and down), and one heavier quark (the strange quark). This case is still under discussion. It is an open question as to whether the chiral transition and the deconfinement transition persist for realistic quark masses.

The order of a phase transition has far-reaching phenomenological consequences. Phenomenological implications of QCD transitions are supposed to be visible in relativistic heavy-ion collisions and astrophysics. For a first order transition one expects metastabilities with latent heat, interfaces, supercooling and overheating etc.. The experimental consequences of a first order transition make it relatively easy to see, especially if the plasma “explodes” into the hadronic phase (Cleymans et al., 1986). A second order transition, lacking a jump in the energy density, may be less easy to see experimentally. However, it has divergent correlation lengths in the thermodynamic limit. These may also lead to observable consequences (Bialas and Peschanski, 1988; Wilczek, 1992; Rajagopal and Wilczek, 1993; Bjorken et al., 1993; Gavin et al., 1994a and b), in analogy to the well-known phenomenon of critical opalescence.

Perhaps most likely is a smooth crossover, as in the transition to an electric plasma from a molecular or atomic gas. One will have a smooth transition from a pion gas at low temperatures to a quark gluon plasma at high temperatures, with a highly nontrivial mixture of excitations in the neighborhood of the crossover. If one has a sharp crossover phenomenon with a rapid change in thermodynamic quantities over a small temperature interval (say of the order of 10 MeV), there is still some chance for measurable effects in experiments. For example, double $\Phi$-peaks in the dilepton invariant mass spectrum are still predicted as a signature for the phase conversion, as long as the crossover phenomenon is rapid enough (Ko and Asakawa, 1994; Ko, 1995).

The situation if further complicated by the nature of experiments. The ultrarelativistic heavy-ion collisions have a finite volume expanding in time. The finite volume smooths out the non-analyticities in the free energy. The high temperature matter created in the collisions may not reach equilibrium either, and does have different processes dominating in different regions of phase space. It becomes a nontrivial task to find “thermometers”, to tune and measure the temperature of the transient hot plasma. There is thus no clear signal of the phase transition known, although a number of signals has been proposed, and the quantities measured so far can usually be modelled by both a hot hadron gas and a quark-gluon-plasma.

In the early universe we have different competing scales. The expansion of the universe is rather slow in units of QCD. If a typical time scale of QCD is taken as $1/T_{c}$ ($\sim 1\rm{fm}/c\sim 10^{-23}\rm{s}$), the Hubble time is of the order of $10^{19}/T_{c}$. Thus there is enough time for equilibration before and after the transition. The spatial volume $V$ of the universe, $10^{-6}\rm{s}$ after the big bang, appears as almost infinite in units of QCD. In units of $T_{c}$ it is given as $V\cdot T_{c}^{3}=7.1\times 10^{55}({200~{}\rm{MeV}\over T_{c}})^{3}$, if the physical correlation length $\xi$ is taken as $\xi(T_{c})\cdot T_{c}=1.38\pm 0.24$ in the deconfinement phase. This leads to a rather small upper bound on the amount of supercooling during a (hypothetical) first order deconfinement transition (Banerjee and Gavai, 1992). The effect of supercooling is an important ingredient in the reasoning, why one should see remnants of the early QCD transition even nowadays. A first order deconfinement transition might have led to inhomogeneities in the baryon number density in the early universe. If these inhomogeneities had survived until the epoch of primordial nucleosynthesis, they may have influenced the light element abundances and led to deviations from values obtained in the standard scenario (without a phase transition) (Schramm et al., 1992; Fuller et al., 1988; Applegate et al., 1987). It is these deviations in the light element abundances which have been predicted as a visible remnant nowadays. Meanwhile this prediction has become rather questionable. Initial inhomogeneities which are compatible with the rather small supercooling are most likely insufficient to induce well separated proton and neutron rich regions. Pronounced inhomogeneities in the proton and neutron distributions are a precondition for a considerable change in the initial conditions of primordial nucleosynthesis.

Lattice Monte Carlo simulations get rather time consuming, when one tries to simulate the realistic situation with small but nonzero quark masses. The CPU-time of a typical simulation with dynamical quarks is of the order of months or even years. From the viewpoint of one less familiar with the lattice approach, it is difficult to understand, why, for example, extensive Monte Carlo calculations have been performed to simulate a strongly interacting system with eight nearly massless flavors, whereas only two flavors are light in nature. Without a deeper understanding of the lattice methods one is left with the impression that the field of QCD computer simulations has decoupled from the field of heavy-ion collisions. The connection between both areas of research seems sometimes to be lost.

It would be overly ambitious to attempt, in this article, to close the gaps between the various approaches to understanding the finite temperature transitions of QCD. This review will focus on the progress made in determining the nature of the phase transitions, and the development of sophisticated tools for this purpose. The complexity of the task, and the time needed to turn the hints of today into firm answers, will be explained.

The organization of the paper is as follows. Section II A gives the background from statistical physics. We summarize the main steps in a renormalization group approach, both in the infinite and finite volume, and recall some concepts to distinguish first and second order transitions, independently of the specific QCD dynamics. In section II B we deal with a renormalization group analysis applied to QCD. Such an analysis provides a guide for a more detailed investigation of the phase diagram.

In view of the anticipated nonperturbative nature of the transition and the need for simulating QCD rather than a simplified model, it is natural to start with the lattice approach (section III). From numerous papers we select a few to explain the type of progress which has been made. Section III A serves mainly to give the basic ideas, to fix the notations, and to introduce some calculational tools of lattice gauge theory.

Section III.B deals with the pure gauge theory without matter fields. We describe the controversial discussion about the order of the deconfinement transition in the $SU(3)$ gauge theory (section II B 1). Thermodynamics on the lattice is the topic of section III B 2. Over a long time it has been taken for granted that the effect of matter on the transition is only to weaken the strength of its order. Hence various interface or surface tensions have been calculated as further characteristics of the transition dynamics. Interface tensions refer to free energies associated with interfaces between different realizations of the plasma phase, between the phases of broken and restored chiral symmetry, or the phases of deconfinement (plasma phase) and confinement (hadronic phase). A reliable estimate of their values at the transition point could play a selective role in the phase transition scenarios. A measurement of the interface tension in the $SU(3)$ gauge theory is presented in section III B 3. Intimately related to the topic of interface tensions are phenomenological implications of the QCD transition in the early universe. We summarize the line of arguing for finding relics of the early QCD transition in section III B 4.

Section III C is devoted to the inclusion of dynamical fermions. Here we restrict the discussion mostly to the staggered fermion formulation. In section III C 1 we list the pitfalls on the lattice, including infrared, ultraviolet and finite-mass artifacts. We describe a finite size scaling analysis in the presence of fermions in section III C 2. Finite-mass artifacts (analyzed in a finite-mass scaling analysis) refer to the necessity of extrapolating results to the chiral limit from finite mass calculations (section III C 3). Examples for the subtleties of UV-artifacts are given in section III C.3 and 4. In section III C 5 we present results for the cases of two and three light fermion flavors, which come closest to the physical current quark masses. Recent results on the equation of state for two-flavor QCD are explained in section III C 6. Wilson fermions are the topic of section III C 7.

A brief review of lattice simulations at finite baryon density is presented in section III D.

The lattice approach plays an important role for QCD. It has the advantage of starting from first principles. But even if lattice calculations finally succeed in providing reliable predictions, there is some need for further alternatives. In a typical Monte Carlo calculation QCD’s full partition function is simulated at once (if we disregard the involved approximations for a moment). From such a simulation one does not get an intuitive insight why a result comes out in a particular way. Effective models try to fill this lack. They are usually not derived from QCD in a strict sense, but share some important symmetries with QCD. Accordingly we consider models for quark and gluonic degrees of freedom separately (section IV.A and B).

In section IV A we concentrate on theoretical tools which go beyond the mean-field level: the renormalization group approach (section IV A 1), chiral perturbation theory (section IV A.2 and 3), and a $1/N$-expansion (section IV A 4). The renormalization group leads to specific predictions for the critical indices and the temperature dependence of meson masses close to $T_{c}$. These predictions may serve as working hypothesis for lattice simulations (Wilczek, 1992; Rajagopal and Wilczek, 1993).

Chiral perturbation theory is well established for describing QCD at low energies and small temperatures. We outline the derivation of chiral condensates as a function of temperature in section IV A 2. Of particular interest are the influence of finite current quark masses and an estimate of the contribution of heavier mesons to thermodynamic quantities. Heavier mesons turn out to be nonnegligible in the phase transition region in contrast one’s naive guess (Gerber and Leutwyler, 1989). Although chiral perturbation theory fails in the vicinity of the transition region, it leads to an upper bound for the latent heat, when Clausius-Clapeyron relations are applied to QCD in a hypothetical first order transition (Leutwyler, 1992) (section IV A 3). The bound excludes strong supercooling scenarios, thus it may be of particular interest for phenomenologists.

In section IV A 4 we apply a $1/N$-expansion to the linear $SU(3)\times SU(3)$ sigma model. Here large $N$ refers to the number of flavors. The topics are the mass sensitivity of the chiral transition and the thermodynamics for physical values of (pseudo)-scalar meson masses. An attempt is made to locate the phase boundary between first order chiral transitions and crossover phenomena in meson mass space (Meyer-Ortmanns and Schaefer, 1995).

As an example for a simplified physical picture of the interaction dynamics in a pure gauge theory we consider Patel’s color flux tube models (Patel 1984a; 1984b) in section IV B 1. In the flux tube models the $SU(3)$-deconfinement transition occurs, when the network of color flux tubes gets infinitely connected. The models abandon the symmetry breaking as a driving mechanism for the deconfinement transition. Section IV B 2 deals with dual Ginzburg-Landau models, explaining the confinement property of QCD in terms of a dual Meissner effect.

If quark and gluonic degrees of freedom are included in an effective Lagrangian for QCD, their interplay in the transition dynamics may be investigated. We give an outlook for such attempts, and further directions in section IV B 3.

Section V is devoted to heavy-ion collisions. Although the application of a thermodynamic concept is not fully established, throughout the review we adopt the viewpoint that a thermodynamic description of nucleus nucleus collisions is meaningful. After a recapitulation of basic concepts in relativistic hydrodynamics (section V B) we point out possibilities for measuring thermodynamical observables (section V C 1). Substructures in dilepton spectra and enhanced strangeness production, which could be sensitive to the underlying transition dynamics, are the topic of sections V C.2 and 3. Pion interferometry, measurements of multiplicity fluctuations, and intermittency are powerful experimental tools in heavy-ion collisions. They may be also utilized for inferring the order of the QCD transition (section V C.4,5,6). The common difficulty to all of these experimental devices is to find a unique signature for a certain type of transition. Thus a flattening in the $\langle p_{T}\rangle$-distribution of charged pions as a function of multiplicity distributions in rapidity space is compatible with a first order transition, but lack of available phase space serves as an alternative and much simpler explanation as well (van Hove, 1985).

We will not cover the issue of preequilibrium production, although it is likely to be relevant in various aspects. Part V D concerns with off-equilibrium aspects. Estimates of the nucleation rate and transport coefficients are two examples for attempts to incorporate the knowledge of equilibrium QCD in situations slightly off-equilibrium. An alternative off-equilibrium process to nucleation is large domain coarsening. Under certain conditions it is more likely the relevant mechanism for phase separation at the transition point (section V D 3). A plasma evolution far out of equilibrium is the topic of the final section V D 4. Methods of nonequilibrium physics and the powerful concept of dynamical universality classes become applicable, if the hot plasma is quenched via almost instantaneous cooling (Rajagopal and Wilczek, 1993). Large correlation volumes may be created in spite of a small equilibrium correlation length, which is hampered in growing due to the finite pion mass. In realistic heavy-ion experiments the hypothetical plasma presumably cools neither adiabatically nor instantaneously, and the physical masses are likely not light enough for inducing large correlation volumes with the aid of nonequilibrium amplification. Nevertheless, we conclude section V in sketching ideas about dynamical universality as it may be challenging and stimulating to consider a quench of a hot quark gluon plasma as some kind of gedanken-experiment.

In section VI we summarize the most important points of the preceding sections. We list the main results, the open questions, and interesting perspectives for further studies.

It has been unavoidable to select papers out of numerous others related to the same topic to keep the article readable. If I do not refer to a paper in spite of its relevance, this may be a result of ignorance.

II GENERALITIES

A Phase transitions in statistical systems

1 First and second order transitions

in the infinite volume limit

The order of a phase transition is one of the basic thermodynamic classifications. It concerns the thermodynamic potential and its derivatives at the transition. The thermodynamic potential $\Omega$ is the free energy $F$ in case of a ferromagnet or the Gibbs free energy $G=F+pV$ for a fluid. In Eqs. (II A 1) we recall the basic thermodynamic formulas for a magnet.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here ${\cal H}$ denotes the spin Hamiltonian, $\beta$ is the inverse temperature, the Boltzmann constant $k_{B}$ has been set 1 everywhere, $H$ is an external magnetic field, $V$ is the volume and $\int{\cal D}U$ stands for the sum over all spin configurations $\{U\}$, weighted with the Boltzmann factor $\exp\left\{-\beta{\cal H}\right\}$. Eq. (3) defines the free energy $F$, depending on $T$, $H$, and $V$, of a system in a finite volume $V$. In the large volume limit the free energy density $f$ is given as $f=\lim_{V\to\infty}F(V)/V$ with $F(V)\equiv\int_{V}d^{d}xf(x)$ in $d$ space-dimensions. The free energy is assumed to depend on the scaling fields $T$ and $H$.In view of a finite size scaling analysis we keep also the dependence on $V$. The parameters $T$ and $H$ are on an equal footing, both may drive a phase transition. One speaks of a temperature or a field driven transition. In QCD we focus our discussion on temperature driven transitions (although density- or “mass-driven” transitions may be considered as well). First derivatives of the free energy with respect to $T$ or $H$ lead to the internal energy $E$, the entropy $S$ or the magnetization $M$ according to Eq. (4). The magnetization is the conjugate variable to the external field and plays the role of an order parameter in the case of a magnet. More generally it plays the role of an order parameter, if the external field explicitly breaks the symmetry which may be spontaneously broken at a phase transition.

On the second level of derivatives (Eqs. (5)) we have the specific heat c (at constant $H$ and / or $M$, respectively) and the isothermal susceptibility $\chi_{T}$. For a fluid the isothermal susceptibility would be replaced by the compressibility $\kappa$.

In the infinite volume limit a phase transition is signalled by a singularity (in the sense of non-analyticity) in the thermodynamical potential $\Omega$. If there is a finite discontinuity in at least one of the first derivatives of $\Omega$, the transition is called first order. In the case of a ferromagnet there is a jump in the magnetization, if one passes through the transition temperature from the phase of broken symmetry to the symmetric phase. This gives $M$ the name of an order parameter, $M$ indicates the order of spins. This way it tells us the phase in which the system is encountered at a given temperature.

The remaining first derivatives of $F$ with respect to $T$ (the internal energy and entropy) usually also show a discontinuity at the transition point. A gap in the entropy is associated with a finite latent heat $\Delta Q=T_{c}\cdot\Delta S$, but there need not be such a gap. (Consider a transition in a ferromagnet between states of magnetization opposite in sign but equal in magnitude. The latent heat would vanish in this case, while the magnetization jumps between values of opposite sign.) Vice versa there may be a finite latent heat without a gap in the order parameter at the transition point.

The second derivatives of the thermodynamic potential at a first order transition are typically $\delta$-function singularities (corresponding to the discontinuities in the first derivatives) or finite.

According to the original Ehrenfest classification of phase transitions, n-th order transitions are defined by the occurrence of discontinuities (rather than divergences) in the n-th order derivative of the appropriate thermodynamical potential. In M.E.Fisher’s terminology one distinguishes between first order and continuous (or higher order) transitions. In continuous transitions the first derivatives of $\Omega$ are continuous, whereas second derivatives are either discontinuous or divergent. In a second order transition at least one of the second derivatives of $\Omega$ is divergent. (If there are at most finite discontinuities in the second derivatives, the transition is of higher than second order.) Hence the order parameter $M$ will vanish continuously at the transition point.

The susceptibility $\chi_{T}$ and the specific heat $c$ typically both diverge in a second order transition. (Again it is not necessary that both of them diverge.) Here the divergences are power law singularities. They are characterized by critical indices. For example, in the infinite volume limit the susceptibility scales close to $T_{c}$ according to $\chi_{T}\equiv-(\partial^{2}f/\partial H^{2})_{T}\propto{|1-T/T_{c}|}^{-\gamma}$ with an index $\gamma$, $f$ as defined above, and the specific heat according to $c_{H}\equiv-T({\partial^{2}}F/\partial T^{2})_{H}\propto{|1-T/T_{c}|}^{-\alpha}$ with critical index $\alpha$.

Landau’s free energy. A first criterion on the order of the phase transition is given by Landau’s theory (Landau and Lifschitz, 1958; Aizu, 1970; Michel, 1980; Toledano, 1981). It consists in an expansion of the free energy in powers of the order parameter. The allowed terms in this expansion are further selected by symmetry arguments. Phase transitions can be classified according to their transformation behavior of the order parameter under a symmetry transformation. In this introductory section we discuss only an order parameter described by a scalar field $\phi$. In QCD applications the scalar $\phi$ will be replaced by an $O(N)$-vector with $N$ components or an $SU(3)$ matrix parametrized by two independent directions of possible “ordering” (see section IV).

The ansatz of a free energy functional for a scalar order parameter $\phi$ in $d$ space-dimensions is given as

[TABLE]

For vanishing $h$ this is the simplest form that admits spontaneous symmetry breaking. Although $F$ is reflection invariant (if $h=0$), the ground state need not be so. $F$ may take its minimum for nonvanishing values $\pm\phi_{0}\neq 0$ (later denoted as $\left<\phi\right>$), depending on the values of $a$, $r$, and $\lambda$. The “couplings” $a$, $r$, $\lambda$ and $h$ should be considered as parameters, where $a$, $\lambda>0$. In the example of a magnet the condition $\lambda>0$ corresponds to the physical condition that the magnetization is bounded. Later $r$ has the meaning of a mass squared, $\lambda$ of a coupling strength of the interaction and $h$ of an external field. In the vicinity of a second order transition the order parameter is small (more generally, if fluctuations in the field are allowed, its average expectation value is small), hence one drops higher powers of $\phi$. A further assumption is that $\phi$ is slowly varying in space (thus there are no higher derivative terms than $(\nabla\phi)^{2}$). A $\phi^{3}$-term is missing, if a symmetry under sign inversion $\phi\to-\phi$ is required for vanishing $h$. For $h=0$ it is easily verified that Eq. (7) predicts a second order phase transition. For $r<0$ two stable states are predicted with magnetizations $\pm\phi_{0}$. The condition $r=0$ defines the critical temperature, thus one may write for $r$

[TABLE]

Figs. 1(a) and 1(b) display the typical signatures of a second order transition. The nontrivial minima at $\pm\phi_{0}$ move continuously inwards as the temperature is increased towards $T_{c}$, where the $Z(2)$-symmetry is restored (Fig. 1(a)). Fig. 1(b) illustrates the vanishing of the order parameter as function of $T$ and the power law divergence in the susceptibility.

For fixed temperature $T<T_{c}$ a field driven transition can be considered as function of $h$. In this case the transition is of first order, $\phi$ jumps from $\sqrt{(-r/\lambda)}$ to $-\sqrt{(-r/\lambda)}$ as $h$ changes sign.

Although the ansatz (7) has been originally proposed for second order transitions, temperature driven first order transitions can be described as well. We mention two possibilities.

• $\bullet$

$\lambda<0$ in Eq. (7). The coupling $\lambda$ can play the role of a renormalized coupling. In later applications of QCD $\lambda$ occurs as a renormalized coupling in an effective description of QCD. It varies as a function of the value which is chosen for the strange quark mass $m_{s}$. For certain values of $m_{s}$, $\lambda$ becomes negative. If $\lambda<0$, one has to include a term $\propto\phi^{6}$ with positive coefficient to stabilize the free energy functional (see section IV A 1). Adding a term $(\kappa/6)\cdot\phi^{6}(x)$ to Eq. (7) $(\kappa>0)$, $F$ has now two local minima over a certain temperature interval $T_{0}<T<T_{1}$, where $T_{0}$ and $T_{1}$ have the meaning of stability limits of the disordered phase in the ordered and the ordered in the disordered phase, respectively. At $T_{c}$ the minima are equally deep, the order parameter jumps from $\phi_{0}=\pm(3\lambda/4\kappa)^{1/2}$ to zero.

In general $\lambda$ may change its sign as function of an external parameter $P$ ($P\equiv m_{s}$ in our example of section IV).

When $\lambda$ changes its sign at some value $P^{*}$, a line $T_{c}(P)$ of second order transitions ends at a so called tricritical point $T_{t}=T_{c}(P^{*})$ and continues as a line of first order transitions. Tricritical behavior is predicated to occur for QCD in certain limiting cases (see section IV).

• $\bullet$

a cubic term in Eq. (7). If such a term is not suppressed by a symmetry argument, it admits a first order temperature driven transition. For $\lambda>0$, $F$ has two minima at $\phi_{0}=0$ and at $\phi_{0}\neq 0$ over a temperature interval $T_{0}<T<T_{1}$. If the cubic term is written as $(\alpha/3)\phi^{3}(x)$, the nontrivial minimum at $T_{c}$ occurs for $\phi_{0}=-9r/\alpha$.

The very existence of a cubic term in an effective potential for the electroweak phase transition has been under much debate in the last few years. In the simplest models for the electroweak transition the order parameter is an $O(N)$-vector field, $r$, $\lambda$ and $\alpha$ are renormalized parameters.

Here we add a warning. A cubic term in the classical potential of Eq. (7) does not guarantee the first order of the transition. If the transition is weakly first order in the sense that the maximum between zero and the nontrivial minimum is not high, the transition may be washed out by fluctuations of the order parameter field. Such fluctuations will be discussed below.

Landau’s concept for the free energy and the criterion for the order of the transition is widely used in applications of particle physics. Two caveats should be mentioned at least. The first one concerns convexity properties, the second the validity range of the mean-field approximation.

$\bullet$ Convexity properties. It is known from general thermodynamic principles that thermodynamic potentials in thermal equilibrium are convex functions of their variables. The non-convex shape of Fig. 1 and the coexisting minima for a first order transition are obviously at odds with the general expectation. Landau’s free energy is a macroscopic concept. Its non-convex and physical realization may be understood as a coarse grained free energy arising in an intermediate step from a microscopic to a macroscopic scale.

To be specific, let us start with a microscopic spin Hamiltonian ${\cal H}(\{s_{i}\})$ depending on spin variables $s_{i}$ associated to sites $i$ of a hypercubic lattice. As a first step short-wavelength fluctuations are eliminated by dividing the lattice into cells of linear dimensions $L$ and introducing new variables $\phi(x)$ in a block spin transformation according to

[TABLE]

Here $x$ is a site on the block lattice, the block spin variable $\phi$ is identified with an order parameter field as it enters the ansatz (7). This way we make the connection to Landau’s “mean field” free energy $F=F_{mf}$. The dynamics of the coarse grained block lattice with field variables $\phi(x)$ is determined by the so called coarse grained Hamiltonian ${\cal H}_{CG}$. It is obtained as

[TABLE]

where the path integral extends over all spin configurations $[s_{i}]$ under the constraint that a particular configuration $[\phi(x)]$ on the block lattice (determined according to Eq. (9)) is kept fixed. The constraint is denoted by $P(\ldots)$. Such a block spin transformation may be iterated. The general folklore is then (although it is extremely hard to be proven rigorously) that after a sufficient number of iterations, ${\cal H}_{CG}[\phi(x)]$ has the form of Landau’s free energy $F=F_{mf}$, if one is close to a second order phase transition. At such a transition the correlation length $\xi$ diverges, the condition $L\ll\xi$ on the linear cell size $L$ is easily satisfied.

The bulk free energy $F$ of the system which is governed by the Hamiltonian $H_{CG}$ is then obtained upon the remaining path integration over all configurations $[\phi]$ according to

[TABLE]

If it is justified to evaluate Eq. (11) in a saddle point approximation, i.e. to drop $\int{\cal D}\phi$ and take $F_{mf}\{~{}\}$ at its minimum $\phi_{0}$, the result is

[TABLE]

the free energy coincides with its mean-field value, i.e. Landau’s free energy. Landau’s free energy is inherently a mean-field approximation. For a spin system, the order parameter is the magnetization. Landau’s free energy is an expansion in terms of mean values of spins. It does not include a summation over all spin configurations according to

[TABLE]

$\bullet$ The validity of the mean-field approximation. Next we address the question when one is allowed to ignore the fluctuations in the order parameter field, i.e. to drop $\int{\cal D}\phi$ in Eq. (11) and take $F_{mf}$ for $F$. On a microscopic level it amounts to a replacement of spin-spin interactions by some average background represented by $\phi\equiv M$. The validity of the mean field approximation is guaranteed, if the fluctuations $\delta\phi\equiv\phi(x)-\phi_{0}$ in the order parameter field are small compared to the order parameter itself, i.e.

[TABLE]

where the average $\left<\ldots\right>$ should be taken over all cells of the coarse grained lattice. Upon using the fluctuation dissipation theorem, Eq. (14) translates to (see e.g. Binder, 1987)

[TABLE]

where $R$ is the interaction range. Mean-field theory becomes exact, if either the dimensionality, the range of interactions or the number of interacting neighbors (large $N$ approximation, where $N$ is the number of order parameter components) becomes infinite.

Mean-field theory often gives correct qualitative predictions for phase diagrams of three-dimensional systems. Three-dimensional systems play some role as effective models for high temperature QCD. As the dimensionality increases, mean-field theory improves, while numerical calculations get harder. The critical dimension depends on the form of Landau’s free energy expansion. Symmetry considerations are essential for constructing Landau’s free energy. As both finite temperature transitions of QCD are supposed to be driven by symmetry breaking, it is natural to construct effective actions for QCD by using Landau’s concept as a guide-line.

If the condition (15) is violated, one should use the renormalization group approach of (Wegner, 1972; Wilson and Kogut, 1974; Fisher, 1974; Ma, 1976) to describe critical phenomena.

The renormalization group approach in the infinite

volume limit

In this section we recall the main steps of the renormalization group approach in the infinite volume limit as a tool to describe critical phenomena for second order transitions. The approach leads to predictions of critical indices and scaling relations between them.

In the renormalization group approach one attempts to solve the path integral (13) in iterated steps. (It is useful to visualize these steps as block spin transformations in a spin system.) One step refers to one application of a renormalization group transformation $R_{b}$ in coordinate space. The transformation increases the scale by a factor $b>1$, $b$ is called the scale factor. A series of Hamiltonians ${\cal H}^{(n)}$ is generated along with the iterated application of $R_{b}$, ${\cal H}^{(n)}$ depends on the fields $\phi^{(n)}$ and couplings $g^{(n)}$ after $n$ steps. For simplicity let us denote the set of couplings after $n$ steps $g$ and after $(n+1)$ steps $g^{(1)}$. The relation between ${\cal H}^{(n+1)}$ and ${\cal H}^{(n)}$ implies for the coarse grained free energy density $f(g)$ after one application of $R_{b}$

[TABLE]

with the same functional dependence $f$ on both sides of the equality sign. The rescaling factor $b^{-d}$ is necessary to account for the reduction of the effective degrees of freedom after one step by $N^{(1)}=b^{-d}N$. To keep the free energy $F=\int d^{d}xf(x)$ constant under the operation $R_{b}$, one has to compensate the reduced volume on the block lattice by a factor $b^{-d}<1$. The term $G(g)$ denotes the contribution to $f$ which is regular in $g$.

Similarly the correlation length on the ‘block lattice’ is reduced by a factor $b^{-1}$ in one step

[TABLE]

At criticality ($T=T_{c}$), ${\cal H}^{(n\to\infty)}$ is supposed to converge to a nontrivial fixed point Hamiltonian ${\cal H}(g^{*})$, which is invariant under the transformation $R$. The set of fixed point couplings $g^{*}$ which is invariant under $R$ implies

[TABLE]

Eq. (18) is only compatible with a vanishing or a diverging correlation length, since $b>1$ by assumption.

Singularities of thermodynamic functions arise as the critical point is approached. Their power law divergences are characterized by critical exponents. Critical exponents can be derived in the renormalization group approach, when the transformation $R$ acting in the parameter space of couplings is linearized in the vicinity of the fixed point. Let ${\cal H}(g)$ denote a Hamiltonian “close” to the fixed point Hamiltonian, close in the sense that $g\approx g^{*}$, such that we can write

[TABLE]

The scaling operators $O_{i}$ are taken as eigenfunctions of $R$, the coefficients $q_{i}$ are the scaling fields, measuring the distance in coupling parameter space between $g$ and $g^{*}$. In a typical second order transition we have

[TABLE]

where $k_{1}$ and $k_{2}$ are constants, $t$ is the reduced temperature, $h$ is an external field. The eigenvalues $\Lambda_{i}$ of the scaling operators $O_{i}$ may be written as $b^{\lambda_{i}}$ due to an imposed semigroup-property of $R$. Thus we have

[TABLE]

under the action of $R$. Scaling fields for which $\lambda_{i}>0$ are called relevant, those with $\lambda_{i}=0$ marginal and with $\lambda_{i}<0$ irrelevant. In the linearized version Eq. (23) the relevant and marginal scaling fields (or couplings) have to vanish (e.g. $T\to T_{c}$ and $h\to 0$) for ${\cal H}(g)$ converging to ${\cal H}(g^{*})$.

The free energy density $f$ and the correlation length $\xi$ can be expressed as functions of the scaling fields by using Eqs. (16)-(17). As we are interested in a temperature driven second order phase transition in the presence of an external field $h$ we choose $q_{1}$ and $q_{2}$ as the only relevant scaling fields according to Eq. (21). A frequently used notation is $\lambda_{1}\equiv y_{t}$ for the thermal exponent and $\lambda_{2}\equiv y_{h}$ for the magnetic exponent. Here we only indicate the subsequent steps. For a concrete application one has to specify the relevant scaling fields. If one is interested in temperature driven transitions in the presence of an external field (later we are interested in this case in applications to QCD, cf. section III C 3), it is sufficient to choose two fields $q_{1}$ and $q_{2}$ according to Eq. (21). A frequently used notation is $\lambda_{1}\equiv y_{t}$ for the thermal exponent and $\lambda_{2}\equiv y_{h}$ for the magnetic exponent. Further fields $q_{\alpha}$ refer to marginal and irrelevant scaling fields. The singular parts of $f$ and $\xi$ are written in terms of the reduced temperature times a scaling amplitude. The scaling amplitudes are assumed to admit an analytic expansion in their arguments. This implies that in the vicinity of $T_{c}$ ($|t|\ll 1$) contributions of irrelevant scaling fields may be neglected. We explicitly mention this assumption entering the final formulas, because the condition $|t|\ll 1$ is easily violated in numerical applications.

The asymptotic behavior of $f$ is then given as

[TABLE]

where $A_{f}^{\pm}$ denotes the scaling amplitude for $f$, the signs $(\pm)$ refer to the approach of $T_{c}$ from above or below. This behavior implies the scaling relation

[TABLE]

$\alpha$ is the critical exponent characterizing the scaling behavior of the specific heat close to $T_{c}$. For the correlation length one obtains

[TABLE]

From Eq. (26) it is seen that the critical exponent $\nu$ characterizing the divergence of the correlation length may be identified as

[TABLE]

Finally we denote the relations for the magnetization $M$ and the susceptibility $\chi$ in the zero field limit ($h=0$)

[TABLE]

[TABLE]

with an analogous splitting to Eq. (24) in singular parts and additive analytic contributions $M_{0}$ and $\chi_{0}$, originating in derivatives of the analytic part $G$. The exponent $\lambda_{1}$ is distinguished as long as we consider temperature driven transitions.

Note that only the leading singular terms in the expansion of $A_{f}^{\pm}|_{h=0}$ and $A^{\pm}_{\xi}|_{h=0}$ in powers of ($q_{\alpha}|t|^{-\lambda_{\alpha}/\lambda_{1}}$) with $\lambda_{\alpha}/\lambda_{1}<0$ have been kept in Eqs. (27), (26), (24). As emphasized above this is justified as long as $|t|\ll 1$. Due to critical slowing down it is inherently difficult in a Monte Carlo simulation to satisfy the condition $|t|\ll 1$. If one observes deviations in measured critical indices from theoretical expectations, one should keep in mind that contributions from irrelevant terms are a possible explanation among others.

2 Finite size scaling analysis

Monte Carlo simulations are necessarily performed in a finite volume. Effects of the finite size are certainly important, if either the volume is small compared to physical length scales or the physical scales are given by long-wavelength fluctuations or massless modes. Massless modes feel the finite volume however large the size of the system.

It is well known that finite size effects render it difficult to infer the order of a phase transition. Even qualitative conclusions may be misleading. Thus a truly second order transition may look like first order showing a double peak structure in the probability distribution of the internal energy or magnetization. The double peak structure is a typical signal for the coexistence of phases at a first order transition. In this case, however, it is a finite size artifact. It would eventually vanish, if the volume is further increased.

Vice versa large correlation lengths in a finite volume suggest a second order transition. One possible reason is that the correlation length is truly large, but finite. The finite value would be visible, if the volume is further increased. A further increase may involve a practical problem of computer time. A second reason for a large correlation length may be a particle-like excitation in the spectrum which gets its mass from the volume and decouples in the infinite volume limit. In such a case the associated correlation length does not correspond to a bulk correlation length. Moreover one has to be careful about the order of the critical ($T\to T_{c}$) and the thermodynamic ($V\to\infty$) limits. If $T\to T_{c}$ is taken first, the correlation length diverges also in case of a (truly) first order transition. A misinterpretation of data for a correlation length was at the origin of the controversy about the order of the deconfinement transition in the pure $SU(3)$ gauge theory (cf. section III B 1).

Fortunately there are characteristic signatures in a finite volume, which anticipate the behavior in the thermodynamic limit. A careful analysis of these signatures allows one to determine the order of the transition. The measurements should be performed as function of a varying volume. A measurement for a single size is not conclusive in general.

In the following we discuss the finite size scaling behavior of singularities in thermodynamic functions. We consider the case of second order transitions first.

Finite size scaling analysis for second order transitions. The renormalization group approach is an appropriate framework to discuss finite size effects (see e.g. Barber, 1983). It puts the size of the system on an equal footing with other relevant scaling fields like temperature, masses, magnetic fields.

The central conjecture of a finite size scaling analysis refers to a thermodynamic function $P$, which admits a critical singularity in the infinite volume limit, characterized by an index $\mu$

[TABLE]

where the reduced temperature $t$ has been defined in Eq. (21), and $A^{\pm}$ are the amplitudes obtained in the limit $T\to T_{c}$ with $t>0(+)$ or $t<0(-)$. Consider a system, which is finite at least in one dimension with linear size $L$. The conjecture about the finite size behavior of $P$ is the following

[TABLE]

where $L$ is assumed to be large and $T$ close to $T_{c}$, $L\gg 1$, $\left|t\right|\ll 1$, $\xi_{\infty}$ denotes the correlation length in the infinite volume limit. The derivation of Eq. (32) in a renormalization group approach is essentially the same as in the infinite volume limit. The additional ingredients are arguments, why $L$ can be considered as a scaling field. There may be a conflict in principle. Already in a single renormalization transformation non-local terms are generated in the Hamiltonian. Thus the system size should be sufficiently large to account for all non-local couplings which one wants to keep.

The assumption is that close to the critical point one needs to keep only a few couplings. Once this set of couplings has been fixed, $L$ can be chosen sufficiently large compared to the typical interaction range. In further distinction to the infinite volume limit the renormalization group transformations can be iterated only a finite number of times, as the scale increases by a factor $b>1$ in each step. Here the allowed number of steps should be sufficiently large. It is also taken for granted that the stability of a set of fixed point couplings $g^{*}=R_{b}g^{*}$ and the semigroup property of $R$ are preserved in a finite volume. Under these assumptions $L^{-1}$ may be treated as an additional scaling field.

The ansatz for the free energy density in terms of scaling fields and eigenvalues of $R$ takes then a similar form as in the infinite volume analysis. It is given as

[TABLE]

where $b$ has been chosen large enough that $b^{\lambda_{1}}|t|\sim 1$, $\phi_{2}$ stands for $\lambda_{2}/\lambda_{1}$. Eq. (34) implies the finite size scaling behavior of various derived quantities. For the zero-field susceptibility we find

[TABLE]

Irrelevant terms have been dropped on the right hand side of Eq. (36). The analytic part $\chi_{0}$ stays finite as $T\to T_{c}$. It may be dropped for large volumes, if the second term is singular as $T\to T_{c}$. If the argument of $A_{\chi}^{(\pm)}$ is expressed in terms of $\chi_{\infty}(t)$, it is easily shown that the scaling behavior of $\chi$ takes the form of Eq. (32) with $P\equiv\chi$ and $\mu\equiv\gamma$.

Let us see, what Eqs. (31) and (32) imply for the explicit $L$-dependence of $P$. In order to guarantee that

[TABLE]

the amplitude $Q^{(\pm)}_{P}$ has to satisfy

[TABLE]

where $x\equiv L/\xi_{\infty}$. The limit $x\rightarrow 0$ is realized, if $T\rightarrow T_{c}$ for fixed $L<\infty$. This is the limit we are interested in. In this case $P_{L}(t=0)$ should be finite. Hence the singularity of $P_{\infty}(t)$ has to be compensated by the scaling function $Q^{(\pm)}_{P}(x)$ according to

[TABLE]

where we have used $\xi_{\infty}(t)\sim|t|^{-\nu}$ for $t\rightarrow 0$. It follows from Eqs. (32) and (39) that

[TABLE]

The last relation (40) predicts for the susceptibility

[TABLE]

and for the specific heat

[TABLE]

Finally we state the scaling behavior of the correlation length in a finite volume

[TABLE]

Refined criteria in Monte Carlo calculations for QCD account for the finite lattice size according to formulas (41), (42), (43), when they test on a second order transition. Note that these formulas have been derived under the conditions $L\gg 1$ and $|t|\ll 1$. One should make sure that these conditions are satisfied in the actual simulations, otherwise subleading corrections must be kept.

An important feature of the finite size scaling analysis in the renormalization group approach is the similar treatment of the system size and other scaling fields. The treatment on an equal footing is manifest in Eq. (34). The inverse extension $1/L$ appears as a relevant scaling field with eigenvalue $\lambda_{i}=1$, $i\equiv 1/L$. To obtain the critical limit, all scaling fields which are relevant with respect to the fixed point set of couplings have to vanish: $t\rightarrow 0\,(T\rightarrow T_{c})$, $h\rightarrow 0$ and $1/L\rightarrow 0$. As long as $\xi_{\infty}(t)\ll L$, finite size effects are negligible. If $|t|$ is so small that $(\xi_{\infty}(t)/L)\sim 0(1)$, or, if $L$ is not sufficiently large, finite size corrections may drastically alter the thermodynamic behavior.

Thus the formal treatment of $1/L$ is the same as the treatment of an external field. The external field may be a mass. From physical applications it is well known that the finite volume can act like a mass. An obvious manifestation is the finite correlation length as result of a finite volume or of a nonvanishing mass. The analogy goes even further. Consider an Ising model in four dimensions in a finite volume. In the broken phase there is always a finite probability in a finite volume for tunneling between degenerate states with magnetization $\pm 1$. Below $T_{c}$ the Ising model can be approximated by the effective potential of an anharmonic oscillator, where the mass in the effective action is proportional to the volume. The corresponding “particle” decouples in the infinite volume limit. The kink describing the tunneling between states of opposite magnetization is associated with a so called vacuum tunneling energy. This energy goes to infinity along with the volume (spin flips of the entire system are rather unlikely in a very large volume) (see e.g. Meyer-Ortmanns, 1989).

Finite size scaling analysis for first order transitions. The finite size scaling analysis for first order transitions is inherently more difficult than for second order transitions. The correlation length stays finite even in the infinite volume limit as $T\to T_{c}$ after $L\to\infty$, $\xi_{\infty}/L$ is no longer a sensible scaling variable as in Eq. (39). In the second order case it is the thermal eigenvalue $1/\nu$ of the renormalization group transformation that controls rounding and shifting of the algebraic singularities, both being of the order of $L^{-1/\nu}$.

The goal of the finite size scaling analysis for first order transitions is to predict the rounding and shifting of $\delta$-function singularities in the second derivatives of a thermodynamic potential due to the finite volume (e.g. in the specific heat due to a latent heat, or in the susceptibility due to a jump in the order parameter).

The volume has to be finite at least in one direction. This time the rounding of the singularity, the width of the sharp crossover region, and the shift in the transition temperature are predicted to be of the order $L^{-d}$, where $d$ is the dimension of the system.

Before we enter the subtleties of refined criteria, which are based on the precise form of the rounding and shifting as function of $L$, we summarize some qualitative signatures of a first order transition in a typical Monte Carlo simulation.

Qualitative signatures are essentially the large or infinite volume signatures, which we call ‘naive’ criteria for inferring the order of the transition.

Naive criteria

• $\bullet$

Some thermodynamic quantities (e.g. the internal energy) are almost discontinuous at the transition.

• $\bullet$

A starting configuration half-ordered, half-disordered relaxes to very different equilibrium states on both sides of $T_{c}$ (rather than frequently tunneling between both sides). In a second order transition the system relaxes to an equilibrium configuration independently of the initial condition.

• $\bullet$

At infinite volume tunneling between both phases is completely suppressed, at small volumes tunneling may be mixed up with fluctuations of statistical origin. For large volumes tunneling events are clearly visible in a Monte Carlo simulation. The system is in the ordered phase with a probability $\exp\{-L^{d}\,F_{0}(\beta)\}$, and in the disordered phase with $\exp\{-L^{d}\,F_{d}(\beta)\}$, where $F_{0}$ and $F_{d}$ denote the free energies in the ordered and disordered phases, respectively. If the time history is followed over a number of Monte Carlo iterations, it shows flip-flops between states of different “magnetizations”. The frequency of flip-flops decreases with increasing volume.

• $\bullet$

Another manifestation of tunneling shows up in the probability distribution $P_{E}$ for the internal energy $E(\beta)$. For large volumes it is sharply peaked at the energy values of the ordered ($E_{0}$) and disordered ($E_{d}$) phases. The deep valley between these peaks reflects the rare number of tunneling events. If the initial condition is an ordered start, the probability is large to find $E_{0}$ for $E$, for a disordered start it is large to find $E_{d}$. This is nothing but a sign for metastability.

• $\bullet$

Hysteresis effects are observed even if one is less close to the transition temperature.

The qualitative signature of a pronounced double peak structure can be made more quantitative, when it is analyzed as a function of the lattice size. This way we are led to refined criteria.

Refined criteria. Different approaches have been studied to determine the precise form of the finite size scaling. One possibility is the incorporation of first order transitions in the renormalization group approach. First order transitions are then associated with a discontinuity fixed point at zero temperature (Nienhuis and Nauenberg, 1975; van Leeuwen, 1975). In this approach one would derive the finite size scaling as special cases of Eq. (34). For example Eq. (41) for the susceptibility $\chi_{L}\sim L^{(\gamma/\nu)}=L^{(2\lambda_{2}-d)}$ gives a scaling proportional to the volume $\chi_{L}\sim L^{d}$ with $\lambda_{2}=d$. Similarly Eq. (42) for the specific heat $c_{L}\sim L^{\alpha/\nu}=L^{2\lambda_{1}-d}$ gives $c_{L}\sim L^{d}$ with $\lambda_{1}=d$. The dimensionality $d$ is the only eigenvalue of the discontinuity fixed point.

Scaling behavior at a first order transition can be discussed also independently of the existence of such a discontinuity fixed point. A phenomenological approach has been developed by (Imry, 1980; Fisher and Berker, 1982; Privman and Fisher, 1983; Binder and Landau, 1984; Challa et al., 1986).

More recently exact results have been obtained by Borgs and Kotecky (1990) and Borgs et al. (1991) for models which can be represented by a contour expansion with small activities like the $q$-state Potts model for large $q$. The partition function $Z(\beta,L)$ in a finite lattice of volume $L^{d}$ with periodic boundary conditions is expanded according to

[TABLE]

The free energy is the minimum of $f_{d}$ and $f_{o}$, where the indices $o$ and $d$ stand for ordered and disordered, $b$ is a constant larger than zero. In what follows we discuss predictions of the phenomenological two Gaussian peak model for the energy probability distribution $P_{L}(E)$ which has been introduced by Binder and Landau (1984) and Challa et al. (1986). This model is related to the rigorous expansion (45) by an inverse Laplace transform. Therefore results of the phenomenological model may be compared to exact results and improved in a more systematic finite size scaling analysis (Billoire et al., 1990 a and b; Billoire, 1990). We present part of their results in the following.

The two Gaussian peak model. The basic quantity for temperature (energy $E$) driven transitions in the first order case is the probability distribution $P_{L}(E)$ of the internal energy. (For field driven transitions $E$ is replaced by the magnetization $s$, (Binder and Landau, 1984).) For large volumes and away from the transition point $P_{L}(E)$ is given by a simple Gaussian. This is seen, when the exponent is expanded around its maximum as function of the internal energy. The double peak structure of $P_{L}(E)$ in the vicinity of a first order transition is then easily understood by noticing that the exponent is extremal for two values $\widetilde{E_{o}}(\beta)$ and $\widetilde{E_{d}}(\beta)$, one for each of the coexisting phases. As a phenomenological ansatz the energy probability density $P_{L}(E)$ is now replaced by a sum of two Gaussian distributions. It is plausible that their width should be determined by the distance from $T_{c}$. Expanding $\widetilde{E}(\beta)$ around $T_{c}$ leads to

[TABLE]

Here $E_{o,d}$ and $c_{o,d}$ are the infinite volume energy and specific heat in the pure ordered and disordered phases, respectively, $A$ is a normalization factor, $\delta T$ is defined as $\delta T=(\beta_{c}-\beta)/\beta^{2}_{c}$, and $a_{o}$, $a_{d}$ are weight factors for each phase

[TABLE]

The factor $q$ corresponds to $q$ ways of realizations of the broken phase in the $q$-state Potts model, $q=2$ for the ($\pm$) magnetizations in the Ising model. Finally $\Delta$ is given as

[TABLE]

Eqs. (47)-(50) predict energy moments correctly up to order $L^{1/d}$.

A specific feature for a first order transition is the sharpening of the double peak structure as the volume increases. The tunneling between both phases is less likely the larger the volume is. The spin configurations corresponding to mixed states lead to an energy $E$, which lies between the locations of the peaks at $E_{o}$ and $E_{d}$. (The distribution of these “intermediate” energies is not correctly represented by the superposition of Gaussians according to Eq. (47).) Contributions of mixed states vanish in the infinite volume limit, as long as they are equilibrium configurations. Nothing is said about the phase conversion itself, which may proceed as an off-equilibrium process with “droplet” formation.

The double peak structure and the criteria we present in the following are essentially based on the double valued internal energy $E_{o}$ and $E_{d}$ at the transition point. The free energies are degenerate in this case.

Let us first consider the rounding in the specific heat and the shifting of the transition point. The specific heat in the volume $L^{d}$ is as usual obtained as

[TABLE]

where

$\displaystyle\langle E\rangle_{L}=\int_{-\infty}^{+\infty}P_{L}(E)\,E\,dE$

with $P_{L}(E)$ given by Eqs. (47)-(50). Its maximum $c_{Lmax}$ occurs for an inverse temperature

[TABLE]

Eq. (52) shows that the shift of the critical temperature in the finite volume compared to the infinite volume value is of the order of $1/L^{d}$. The height of the maximum in the specific heat $c_{Lmax}$ is derived to be proportional to the volume

[TABLE]

Thus Eqs. (52)-(53) display the anticipated volume dependence of shifting and rounding in a first order temperature driven transition . Subleading corrections of $0(1/L^{2d})$ in Eq. (52) and of $0(1)$ in Eq. (53) have been determined by Lee and Kosterlitz (1990).

Binder’s cumulant. Rather than calculating the $P_{L}(E)$ distribution itself, certain moments and cumulants of the energy probability distribution have been proven to be suitable indicators of the order of the transition in a finite volume. An important example is Binder’s cumulant (Challa et al., 1986) defined as

[TABLE]

It vanishes in the infinite volume limit for all temperatures apart from the transition, where $B$ gets a finite value from tunneling between coexisting phases of a first order transition. It also vanishes in the case of a second order transition. Calculating the moments $\langle E^{4}\rangle$ and $\langle E^{2}\rangle$ from Eqs. (47)-(50) with use of Eq. (51), $B$ takes its minimum $B_{min}$ at (Billoire et al., 1990a and b)

[TABLE]

at an inverse temperature

[TABLE]

The shifting in the critical temperature in Eq. (56) is again of order $1/L^{d}$. A nonvanishing value of $B_{min}$ signals a first order transition . Analogous formulas can be derived for the spin probability distribution $P_{L}(s)$, implying that the maximum of the susceptibility grows proportional to the volume.

Binder’s cumulant depends on the choice of an arbitrary additive constant which may be added to the energy. This has led to consider the quantity $U4$ (Billoire et al., 1992)

[TABLE]

This quantity can be shown to be larger than 1 apart from a first order transition point $U4$ can be shown to be larger than 1. Its minimum is given as

[TABLE]

The power law corrections to $C_{Lmax}$, $B_{min}$ and $U4_{min}$ should be contrasted with exponential corrections to bulk quantities like the average internal energy $E$ which are directly obtained as derivatives of $Z$. According to Eq. (45) $Z$ itself has only exponential corrections from the finite volume.

In Table I we summarize criteria to distinguish first and second order transitions in a finite volume. The indicated volume dependence should be understood as leading term in a large volume expansion. Similarly the numbers [math] or $1$ indicate limiting values which are approached as $T\rightarrow T_{c}$. Recall that $\alpha/\nu$ and $\gamma/\nu\leq d$ for a second order transition. Thus a typical test in a Monte Carlo simulation could be a calculation of $c_{Lmax}/L^{d}$. If this ratio as function of $L^{d}$ goes to a nonvanishing constant for large values of $L$, a first order transition is signalled, if it approaches zero, the transition must be of second or higher order.

The two-phase coexistence at $T_{c}$ refers to the Ising-model. The analytic results for leading and subleading finite size scaling behavior have been derived by Billoire et al. (1992) for the more general case of $d$-dimensional q-state Potts models which include the familiar Ising model for $q=2$. Potts models have served as testing ground for the above criteria.

$q$-state Potts models. The Hamiltonian of a $d$-dim $q$-state Potts model is given as

[TABLE]

The spin variables $s_{i}$ are associated to sites of a $d$-dim hypercubic lattice. They can take $q$ different integer values. The sum extends over nearest neighbor pairs $\langle ij\rangle$. The symmetry group leaving $H$ invariant is the permutation group of $q$ elements. The $q=3$-Potts model in three dimensions plays a distinguished role in view of QCD. It shares the global $Z(3)$-symmetry with the pure SU(3)-gauge theory. The restoration of the spontaneously broken $Z(3)$-symmetry at finite temperature is assumed to drive the deconfinement transition in the pure gauge theory.

Beyond this application to QCD, Potts models provide a suitable testing ground for numerical methods. Exact results are available for comparison in certain special cases of $q$ and $d$. The strength of the first order varies as a function of $q$. The $q=5$-model has a weakly first order transition accompanied by a tiny latent heat of $\Delta Q=E_{d}-E_{o}=0.05292$ in lattice units and a large correlation length of 2000 in lattice units at the transition point (Billoire, 1991). The transition in the $q=10$-model is strongly first order, $\Delta Q=0.69605$ and $\xi\sim 6$, both quantities measured in lattice units (Billoire, 1990). Obviously the transition in the $q=5$-Potts model is easily misinterpreted as being of second order. The linear lattice size has to exceed the large value of $\xi\sim 2000$.

Simulations of the $2d$-$q$=10-Potts model have been performed by Billoire et al. (1992) to check the limiting behavior of $(c_{Lmax}/L^{d})$, $B_{min}$, $U4_{min}$ and $c(\beta_{t})$ ($\beta_{t}$ being the infinite volume transition point). According to the above predictions these quantities should scale as (const1 + const$2/L^{d}$). Deviations have been observed, which do not behave in a simple way as function of $L$. Several possible reasons for the discrepancies are mentioned by the authors: $q=10$ may not be in the large $q$-limit, for which Eq. (45) has been derived, $L$ may not be sufficiently large to be in the asymptotic regime.

Therefore a $2d$-$q$=20-Potts model has been simulated in addition (Billoire et al., 1992). The first order transition there is yet stronger than in the $q=10$-case. Thus the asymptotic behavior in $L$ should earlier set in than for $q=10$, ’earlier’ means for smaller volumes. (The ratio which counts is $\xi/L$.) Here the agreement with the theoretical predictions is reasonable.

Let us summarize so far. Computer simulations (as well as real experiments) are always performed in a finite volume, whereas the order of a phase transition is usually formulated in the infinite volume limit. Although this formulation is very convenient, it is not the only appropriate way to proceed. The good message comes from finite size scaling analysis. In a finite volume it is prevailed in a specific way, whether the transition is going to be of first or second order in the thermodynamic limit. (Likewise a small mass can play the role of a scaling field. A finite mass scaling analysis is suited for an extrapolation to the zero mass limit.) From a practical point of view a finite size scaling analysis may be less useful, since it is derived for finite but large volumes. The original hope was to disentangle first and second order signatures already for moderate lattice sizes by using refined criteria. The results for the $2d$-$q$=10-Potts model involve a somewhat discouraging message. One has to go to so large volumes to verify the predictions of a finite size scaling analysis that the ‘naive’ criteria are equally well applicable for inferring the order of the transition. This message refers to weakly first order transitions. Otherwise the asymptotic region of large volumes is easily realized as $\xi/L\ll 1$ for moderate values of $L$ when $\xi$ is sufficiently small.

Typical volumes in Monte Carlo calculations for QCD are marginal in large finite size. For QCD transitions it seems to be rather difficult to ever get into the asymptotic regime of large volumes. Right from the beginning one is restricted to much smaller volumes when simulating full QCD rather than a spin model. The final efficiency of refined criteria is, however, a question of size, which has to be answered in the concrete model of interest. The value of the correlation length in a first order transition depends on details of the dynamics. For QCD the largest correlation length is likely not small compared to the typical lattice size.

This concludes our section about concepts and results of statistical mechanics. As we turn now (and for the remainder of this review) to applications in QCD, we give a dictionary between thermodynamic quantities in statistical physics and their pendants in QCD in Table II.

The second and third columns refer to transitions in liquid/gas or ferromagnet systems. Two magnet-fluid analogies are seen, the first one between the set $(T,p,V)$ of a fluid and $(T,H,-M)$ in a magnet, and the second one between $(T,\varrho,\mu)$ in a fluid and $(T,M,H)$ in a magnet (Stanley, 1971). The external field $H$ is the thermodynamic variable conjugate to the order parameter $M$. Likewise the chemical potential is conjugate to the density $\varrho$, $\varrho$ being the order parameter for a fluid/gas transition. A substitution of $V$ by $(-M)$ and $p$ by $H$ transforms almost all equations for a fluid/gas system in the corresponding equations for a magnet.

In the two last rows various response functions are listed, describing a response of the system to a stimulus in the temperature or in an external field. They are second derivatives of the thermodynamic potential $\Omega$, while the order parameter is obtained as a first derivative of $\Omega$ with respect to the conjugate field. The last column shows the associated critical exponents, characterizing the singular behavior of thermodynamic functions in the case of the second order transition.

The QCD transitions refer to limiting cases of vanishing quark masses (chiral symmetry) or infinite quark masses (Z(3) symmetry). A further column is devoted to the chiral transition described on an effective level (here in the linear sigma model), where the quark- and gluonic substructures of mesons are disregarded. If one is interested in the evolution of a QCD-plasma “fluid”, (T, p, V) is an appropriate set of thermodynamic variables entering the equation of state.

Formally the current quark masses $m_{q}$ will be shown to play the same role as an external magnetic field $H$ in a ferromagnet. On an effective level this analogy is more manifest, as nonvanishing quark masses will be described by external fields $\varepsilon_{0},\varepsilon_{8}$, see section IV A 4. The analogy is extensively utilized, when effects of finite quark masses on the order of the QCD transitions are conjectured from results in the massless or pure gauge limits.

The most popular order parameters in the QCD case are the quark condensate $\langle\bar{q}q\rangle$ (or $\langle\bar{q}q\rangle$ and $\langle\bar{s}s\rangle$), or mesonic condensates $\langle\sigma_{0}\rangle$, $\langle\sigma_{8}\rangle$ on the mesonic level for the chiral transition, and the expectation value of the Wilson line $\langle L\rangle$ for the deconfinement transition. The possibility to predict the sigma- and pion-meson mass from a critical equation of state (section IV A 1) sounds less surprising, if one keeps this dictionary in mind.

The indices $b$ and $s$ at $T_{c}$ refer to the approach of $T_{c}$ from the symmetric or the broken phase. For the deconfinement transition the phase of broken Z(3)-symmetry is realized above $T_{c}$, thus the commonly used ($\pm$) signs to indicate the approach from above or below $T_{c}$ would be misleading in this case. Note that the singular behavior of the “magnetization”, i.e. the order parameter, is characterized by a thermal exponent $\beta$ as $T\rightarrow T_{c}^{b}$ and by a “magnetic” exponent $1/\delta$ as “H”$\rightarrow 0$ at $T_{c}$. Further explanations to the QCD-part will be given in the succeeding sections.

B Phase transitions in QCD

In a continuum notation the QCD-Lagrangian is given as

$\displaystyle L=L_{g}+L_{m}$

where

[TABLE]

and

[TABLE]

Here $A^{a}_{\mu}$ are the gauge fields, $\psi_{\alpha,f,c}$, $\bar{\psi}_{\alpha,f,c}$ denote the quark fields, where $\alpha$ is a Dirac index, $f=1,\ldots,N_{f}$ labels the flavors, $c=1,\ldots,N_{c}$ labels the colors ($N_{f}$ is the number of flavors and $N_{c}$ the number of colors). The gauge coupling constant is denoted as $g$, the structure constants as $f_{abc}$, and $\lambda_{a}$ are the generators of the fundamental representation of $SU(N_{f})$, $a=1\ldots 8$ for $N_{f}=3$, where $\lambda_{a}$ are the Gell-Mann matrices. The quark mass matrix is denoted as $M$.

As a first step in the investigation of the phase structure one may consider certain limiting cases of Eq. (63). One limit is the pure gauge theory, where $L=L_{g}$. This is the limit of infinitely heavy quark masses and will be discussed in section II B 2. The topic of the following section is the limit of massless quarks, which is called the chiral limit. In the chiral limit the QCD Lagrangian is invariant under global $U(1)_{V}\times SU(N_{f})_{L}\times U(1)_{A}\times SU(N_{f})_{R}$ transformations. The $U(1)_{V}$-invariance corresponds to the baryon number conservation. The invariance under axial $U(1)_{A}$-transformations is only classically preserved. On the quantum level it is broken via the axial anomaly. Even in the presence of an anomaly there remains axial $Z_{A}(N_{f})$ symmetry (Callan et al., 1976). At zero temperature the invariance under $Z_{A}(N_{f})\times SU(N_{f})_{L}\times SU(N_{f})_{R}$ chiral transformations is assumed to be spontaneously broken by the QCD vacuum to the vector $SU(N_{f})_{V}$-symmetry. For $N_{f}=2$ this isospin symmetry is realized in the hadronic spectrum to a very good approximation, for $N_{f}=3$ the realization of the $SU(3)_{V}$-symmetry is a little more questionable, nevertheless it is also frequently considered as approximate symmetry of QCD.

1 Renormalization group analysis in the chiral limit

Rather than directly studying the QCD Lagrangian (63) one can analyze chiral symmetry breaking in an effective Lagrangian which shares the chiral symmetry properties of QCD. The Lagrangian is formulated in terms of a self-interacting $N_{f}\times N_{f}$ matrix field $\phi$. Chiral symmetry breaking is parametrized in terms of $\phi_{ij}$ according to $\phi_{ij}\sim\langle\bar{q}_{i}(1+\gamma_{5})q_{j}\rangle$, where $\phi$ transforms under transformations of $G_{f}\equiv U_{A}(1)\times SU(N_{f})_{L}\times SU(N_{f})_{R}$ according to

[TABLE]

Here $U_{+}$ and $U_{-}$ are arbitrary and independent $SU(N_{f})$ matrices, $\alpha$ generates a $U_{A}(1)$ transformation. The most general renormalizable Lagrangian in terms of $\phi$ which is consistent with the chiral symmetry properties of QCD is given as (Pisarski and Wilczek, 1984)

[TABLE]

As a necessary condition for stability at large values of $\phi$, $f_{2}$ and ($f_{1}+f_{2}/N_{f}$) have to be larger than zero. The determinant-term accounts for the anomaly. It vanishes in the pure gauge case ($N_{f}=0$) and in the limit of infinite colors $N_{c}=\infty$ (Witten, 1979). At zero temperature the vacuum expectation value $\langle\phi\rangle$ is $SU(N_{f})$-symmetric and different from zero, $m^{2}<0$. The spontaneous symmetry breaking is associated with an $SU(N_{f})$ massless multiplet of Goldstone bosons and a massive flavor singlet $\eta^{\prime}$.

In Table III we summarize results of a renormalization group analysis of Pisarski and Wilczek (1984) about the order of chiral phase transitions. These transitions are driven by chiral symmetry restoration, as the temperature is raised.

For each number of flavors three cases are distinguished: a vanishing anomaly corresponding to a vanishing number of flavors or an infinite number of colors, $g$=const of order 1, where ‘const’ refers to the assumed temperature independence of $g$, and $g=g(T)$. Here $g(T)$ is taken to be approximately equal to the instanton density $d_{I}$, which is supposed to vanish at high temperatures. In the following we summarize, how the conjectures of Table III arise in a renormalization group analysis.

In the previous section we have mentioned the real space renormalization group approach as a tool to describe critical phenomena. An alternative approach for the construction of renormalization group equations which is utilized here is the $\varepsilon$-expansion performed in momentum space. Starting from a Hamiltonian in terms of an order parameter field $\phi$, large momenta are integrated out (large momenta in contrast to short length scales in the real space approach). The new Hamiltonian in terms of the new field variables is arranged to have the same form as the old one. The new couplings should be understood as renormalized couplings. They are obtained in a perturbative expansion. The coefficients of the expansion depend analytically on the dimension $d$. For our applications the appropriate small expansion parameter is $\varepsilon=d-4$.

Normally a change in the scale of momenta induces intricate changes in the action and in derived quantities like correlation functions, unless there are special instances which we know from the previous section. Simple scaling behavior is recovered, if the set of couplings reaches a value such that any further change in the scale of momenta does not affect them. This is the point in coupling parameter space, where the Hamiltonian approaches a fixed point Hamiltonian.

In the present framework fixed points occur as zeros of the $\beta$-functions. In the case of one coupling the $\beta$-function gives the change in the renormalized coupling under a change in momentum scale. Equivalently the $\beta$-function gives the change of the dimensionless renormalized coupling $u$ under a change of the dimensionless bare coupling $u_{0}$ according to

[TABLE]

where the coefficients $a_{n}$ depend on $\varepsilon$. Given a set of bare couplings in parameter space at a certain momentum scale, the question arises as to whether, and under what conditions the renormalized couplings $v$ flow into the fixed point $v^{*}$ (leading this way to a second order phase transition). The answer is given as a condition on the occurrence of an IR-stable fixed point $v^{*}$. The stability criterion is that the matrix

[TABLE]

should have real and positive eigenvalues at $v=v^{*}$. Here $v$ and $v^{*}$ stand for sets of couplings, $\beta_{i}$ are the associated $\beta$-functions.

The $\beta$-functions for a chiral $SU(N_{f})\times SU(N_{f})$ linear sigma model in the absence of an anomaly (case $g=0$ in Table III) have been derived within an $\varepsilon$-expansion (Paterson, 1981). When the two functions $\beta_{1}$ and $\beta_{2}$ for $f_{1}$ and $f_{2}$ are inserted in Eq. (68), the stability criterion may be applied to search for IR-stable fixed points. The results are summarized in the first column of Table III. For $0\leq N_{f}<\sqrt{2}$ the IR-stable fixed point has $f^{*}_{2}=0$ with $O(2N_{f})$ critical exponents. No infrared stable fixed point occurs for $N_{f}>\sqrt{3}$, if $f_{1}$, $f_{2}$ are of order $\varepsilon$.

For some time it has been taken for granted that the absence of an IR-stable fixed point implies a first order phase transition (Bak et al., 1976). The very absence or existence of an IR stable fixed point is, however, less conclusive than it was originally supposed. The existence of an IR-stable fixed point does not exclude a first order phase transition (there may be a region in coupling parameter space, which is not attracted by the fixed point). Vice versa does the absence of an IR-stable fixed point not exclude a second order phase transition. Instead of the fixed point criterion precise conditions for the occurrence of a first order phase transition have been specified by Iacobson and Amit (1981). The framework is again the renormalization group approach, realized in the perturbative tool of the $\epsilon$-expansion. The prediction of first order phase transitions applies to all multi-component $\phi^{4}$-theories with more than one dimensionless coupling constant. The interaction term of a multi-component $\phi^{4}$-theory can be written as

[TABLE]

where $i,\,j,\,k,\,l\ \in\{1,\ldots,N\}$. The quartic terms of the linear sigma model Eq. (66) can be recast in this form. Explicit expressions can be found e.g. in (Paterson, 1981). As long as the quartic terms proportional to $f_{1}$ and $f_{2}$ in Eq. (66) are independent of each other (as it is the case e.g. for $N_{f}=3$), the conditions of the criteria of Iacobson and Amit are satisfied by the linear sigma model, and a first order chiral transition is predicted for $N_{f}\geq 2$.

Note that these predictions are neither of the type of conjectures nor rigorous statements. They are not only based on the absence of an IR-stable fixed point, but derived within the perturbative framework with $\varepsilon\ll 1$. Non-perturbative features are not attainable in this approach. One may try to extrapolate the results from four-to three-dimensional models by setting $\varepsilon=1$ (although $\varepsilon\ll 1$ has been assumed in deriving the $\varepsilon$-expansion). Examples are known (Bak et al., 1976), where the results to leading order in $\varepsilon$ remain a good guide for $\varepsilon=1$.

The second column of Table III subsums the suggestions for the order of the chiral transition in the presence of a temperature independent strength of the anomaly, $g$ is assumed to be of the order of the other couplings $f_{1}$ and $f_{2}$. For two flavors the $\det$-term in Eq. (66) acts itself like a mass term. Thus it may change the order of the transition depending on the magnitude of $g$. For three flavors, the $\det$-term is trilinear in the matrix elements of $\phi$. A cubic term on the classical level is usually regarded as sufficient for inducing a first order phase transition (although we know from the discussion above (section II A) that a weakly first order transition, predicted on the classical level, may be wiped out by fluctuations; hence the classical cubic term makes the first order transition likely, but is no guarantee for its occurrence).

For $N_{f}=4$ Pisarski and Wilczek (1984) argue again with the absence of an IR-stable fixed point in favor of a first order transition. (The $g\neq 0$ case is not covered by the analysis of Iacobson and Amit). As claimed by Paterson (1981), the fixed point structure of the $SU(4)\times SU(4)$ linear sigma model is unchanged, when a term $\propto g(\det\phi+\det\phi^{+})$ is included. For $N>4$ the $\det$-term is an irrelevant operator and should not change the critical behavior.

The effective symmetry of the linear sigma model Eq. (66) may change as function of temperature, if the anomaly strength is temperature dependent and determined by the density of instantons $d_{I}(T)$. Since $d_{I}(T)\rightarrow 0$ as $T\rightarrow\infty$ (Gross et al., 1981), $g$ could be small at the transition temperature compared to the $T=0$-value. Thus the axial $U_{A}(1)$-symmetry would be partially restored. Predictions accounting for this partial symmetry restoration as result of decreasing $g$ are listed in the third column of Table III. For further details about consequences of an approximate $U_{A}(1)$-restoration we refer to the original references.

The effective Lagrangian (66) can be extended to include nonzero bare (meson) masses. The most simple ansatz has the form (tr M $\phi$), it is linear in the mass matrix $M$. Formally it acts as a background magnetic field. The formal analogy is evident, if we recall that the linear sigma model can be rewritten as a multi-component $\phi^{4}$-theory with an additional (multi-component) $\phi^{3}$-term (the $\det$-term) and a symmetry breaking mass term, which is linear in $\phi$. This way it takes the form of Landau’s free energy functional Eq. (7) for a multicomponent order parameter field $\phi$ in an external field.

For later comparison we notice that the magnetic field term is proportional to the mass and vanishes in the chiral limit. In the other extreme case of infinitely heavy quark masses, the effective magnetic field of pure gauge theories will be shown to be proportional to ($e^{-m}$), thus vanishing in the infinite mass limit ($m\rightarrow\infty$). We come back to this point at the end of the next section.

To summarize so far, conjectures about the order of the chiral transition as function of $N_{f}$ are based on a perturbative renormalization group analysis in momentum space. The analysis has been performed for an effective $SU(N_{f})\times SU(N_{f})$ linear sigma model sharing the chiral symmetry properties with QCD. None of the conjectures is rigorously proven. Some of them ($g=0$) are verified within an $\varepsilon$-expansion, others ($g\neq 0$) are based on the absence of an IR-stable fixed point, when a first order transition is predicted, a criterion which should be taken with care.

From statistical physics it is known that a first order transition remains first order when a background field is introduced and the field is sufficiently weak. Otherwise the transition may be washed out completely. The relative size of the latent heat compared to the strength of the external field, i.e. the values of the quark masses, decides whether the $N_{f}=3$ chiral transition is preserved under realistic QCD conditions or not. This question cannot be answered within a renormalization group analysis, but only by detailed calculations. The renormalization group approach is nevertheless a good guideline to start with.

2 The limit of a pure $SU(N_{c})$ gauge theory

In this section we deal with the quenched limit of QCD. The quenched limit is obtained as the number of flavors goes to zero or the quark masses are sent to infinity. The gluonic vacuum in a background of infinitely heavy quarks may be probed by test quarks. In perturbation theory this means that virtual quark loops are suppressed.

In the quenched limit the theory has an extra global symmetry, which results from the periodicity of the gauge fields in the temperature direction. The partition function may be represented as functional integral over gauge fields, which are periodic in Euclidean time with period $\beta=1/T$. The periodicity condition arises as a consequence of the trace in the definition of the thermodynamical potential

[TABLE]

(We still use a continuum notation for the path integral over all gauge fields $A_{\mu}$ with Yang Mills field strength $F_{\mu\nu}$.) Gauge transformations which are compatible with the periodicity condition need only be periodic up to an element $c_{N}$ of the center $Z(N)$ of the gauge group $SU(N)$. Thus a gauge transformation must obey

[TABLE]

with $c_{N}\in Z(N)$. The $n$-th element $c_{N}^{n}$ of $Z(N)$ is given as $\exp(2\pi in/N)$. It is easily checked that the generalized gauge transformation (71) leaves all topologically trivial Wilson loops invariant (in particular those appearing in the action) and thus the action itself.

This means that the extra symmetry is generated by the action of local gauge transformations, which are periodic up to an arbitrary element of the center modulo strictly periodic local gauge transformations. The essential ingredient, which ensures the invariance of topologically trivial loops under the additional symmetry, is the property of center elements to commute with all elements of the group.

The issue of the transition from a confinement phase at low temperatures to the deconfinement phase at high temperatures can be related to the issue of whether the pure glue vacuum of QCD is $Z(N)$-invariant like the action or not. As it turns out, the transition from the confinement- to the deconfinement phase may be explained as a spontaneous breaking of the extra $Z(N)$-symmetry at finite temperature.

Qualitatively the quark gluon plasma can be probed by a heavy test quark. The free energy of this test quark should be infinite in the confinement phase, but finite in the deconfinement phase. It may be computed as the expectation value of the Wilson line (synonymously for Polyakov loop or thermal Wilson loop). The Wilson line $L$ is defined as the spatially local operator

[TABLE]

where $P$ stands for the path ordered product. The Wilson line is a topologically nontrivial loop. It is closed due to the periodic boundary conditions and transforms nontrivially under the center transformations

[TABLE]

Thus $L$ can have a vanishing expectation value. Vice versa a nonvanishing expectation value can be taken as signal of a spontaneous breakdown of the global $Z(N)$-symmetry of the action. We have

[TABLE]

Here $F(\vec{x})$ denotes the free energy of an isolated test quark. This behavior qualifies the Wilson line expectation value as an order parameter for the confinement/deconfinement phases of QCD in the absence of dynamical quarks.

It is natural to look for an effective action of the $SU(N)$ gauge theory in terms of the order parameter field, this means in term of $L$. Such an effective action could simplify the investigation of the phase structure of pure QCD, if universality arguments may be used. Svetitsky and Yaffe (1982a, 1982b) have given plausible arguments that a suitable candidate for such an effective action is a $Z(N)$-spin theory.

The result of an integration over spatial gauge fields is an effective action in terms of Wilson lines. The Wilson lines (originally given in terms of elements of the $SU(N)$-Lie algebra), are represented by $Z(N)$ variables. The action is argued to be short ranged. The high temperature behavior of the $3+1$-dimensional theory is determined by the dynamics of the three-dimensional $SU(N)$ gauge theory. Three dimensional $SU(N)$ gauge theories show an area-law for Wilson loops (Feynman, 1981) with short range correlations. The range of interactions is supposed to stay finite over the entire temperature range.

The effective action is invariant under $Z(N)$ transformations. The $Z(N)$ symmetry can be spontaneously broken at high temperatures, but is restored at low temperatures and ensures confinement.

The path integral representation of the original $SU(N)$ gauge theory is then replaced by

[TABLE]

Here $s_{i}\in Z(N)$, the sum extends over all $Z(N)$-spin configurations, $\Omega$ is the thermodynamic potential.

Effective vs physical temperatures

In the original action of Eq. (70) $g$ parametrizes the interaction strength. When the time dependence of the field is dropped, the $\int_{0}^{\beta}dt$ can be performed to yield a prefactor of the remaining 3-dim action $(-\beta/g^{2})$. The $\beta$-factor can be absorbed by a rescaling of the fields such that the action in three dimensions takes the same form as the original action in four dimensions. Now the prefactor plays the role of an inverse temperature of the 3-dim model. Thus $g^{2}$ is identified with $T_{eff}$, the effective temperature of a classical spin system described by the partition function of Eq. (75). For $T_{eff}$ small, the ‘spin system’ is in the ordered phase, we have $\left<L\right>\neq 0$ corresponding to deconfinement. Deconfinement is realized at high physical temperatures $T$, for which $g$ is small due to asymptotic freedom. A small value of $g$ means a small value of $T_{eff}$, which is consistent with the initial assumption. Thus a low effective temperature $T_{eff}$ corresponds to a high physical temperature $T$ and vice versa. This explains, why the order parameter vanishes in the low $T$-phase and signals “order” in the high $T$-phase.

The coupling $g$ becomes manifestly temperature dependent, when high frequency contributions are integrated out in passing from the $3+1$- to the 3-dimensional theory.

The phase structure of $Z(N)$-spin models

The phase structure of $Z(N)$-spin models has been studied in statistical physics for various values of $N$ and space dimension $d-1$. If a second order transition is predicted in the spin model, one could attempt to locate the renormalization group fixed point and consider the simpler spin model as fixed point theory in the universality class of the $SU(N)$-theory.

A table for various $Z(N)$-, $SU(N)$- and $U(1)$-models in dimensions $d=2,3$ and $\geq 4$ can be found in (Svetitsky and Yaffe, 1982b). Here we mention the cases of two and three colors.

The spin model, which is associated with a $(3+1)$-dimensional $SU(2)$ gauge theory at high temperatures, is the Ising model in three dimensions. The 3-d-Ising model is known to have a second order phase transition.

The case of $N_{c}=3$ in $d=3$ is particular. In the space of $3d$-$Z(3)$- symmetric theories no IR-stable renormalization group fixed point is known. A specific realization of a $Z(3)$-symmetric spin theory is the 3-state Potts model (cf. section II A), which is known to have a first order transition. The famous cubic term on the classical level, driving the transition to first order, is allowed by the $Z(3)$-symmetry. The potential of a $Z(3)$-symmetric theory of a single complex scalar field $L(\vec{x})$ may be written as a sum of a $U(1)$-symmetric term depending on $|L|^{2}$ and a term depending on $\hbox{Re }L^{N}$. In $d=3$ a $U(1)$ gauge theory has a second order transition (see e.g. Pfeuty and Toulouse, 1977). A term $\hbox{Re }[L(\vec{x})^{3}]$ explicitly breaks the $U(1)$-symmetry down to $Z(3)$. In the renormalization group sense it is relevant enough to affect the critical behavior of the $U(1)$-theory.

Thus the conjecture of Svetitsky and Yaffe is that in the absence of an IR stable fixed point the original (3+1)-dimensional $SU(3)$ gauge theory has a first order finite temperature phase transition as well. The prediction has been confirmed in early Monte Carlo calculations (see e.g. Celik et al., 1983a, 1983b), was doubted in the meantime (Bacilieri et al., 1988, see also the introduction and section III B below) and has been reestablished in more recent calculations by Fukugita et al. (1989). Before we report on this controversy in section III B 1, let us consider the influence of dynamical quarks on the deconfinement transition.

Inclusion of dynamical quarks

From spin systems we know that a first order transition stays first order, if the magnetic field is sufficiently small. The gaps in thermodynamic quantities are only continuously deformed for a perturbatively small external field. The magnetization, however, ceases to be a good order parameter, as it is always finite due to the presence of an external background field.

Similarly the Wilson line may be expected to fail as an order parameter when dynamical quarks are included. The physical reason is easily understood. The free energy of an isolated test quark does no longer diverge ($\left<L\right>\neq 0$ for all temperatures). When the flux tube between two test quarks is sufficiently stretched, a $q\bar{q}$-pair is popped out of the vacuum. The test quarks may always form finite energy bound states with dynamical quarks.

Since the deconfinement transition was supposed to be driven by the spontaneous symmetry breaking of the global $Z(3)$-symmetry, we consider the invariance of the QCD action under $Z(3)$-transformations in the presence of dynamical quarks. The generalized gauge transformations Eq. (71) are periodic only up to an element of the center of the gauge group. While the periodic boundary conditions on the gauge field are preserved under the generalized transformations, the antiperiodic boundary conditions on the quark fields are not. Imposing

[TABLE]

on the quark fields as before, Eq. (76) transforms under a generalized gauge transformation with $U(\vec{x},\beta)=c_{N}^{i}U(\vec{x},0)$ ($i$ labeling the center elements) according to

[TABLE]

Eq. (78) shows that the gauge transformed fields $U^{.}\psi$ (under the generalized gauge transformation) do no longer satisfy antiperiodic boundary conditions. If we compare the path integral over all gauge field configurations in the absence and presence of dynamical quarks satisfying Eq. (76), configurations differing by generalized gauge transformations get the same Boltzmann weight in the absence of quarks, but a different weight in their presence. Hence the effect of dynamical quarks is an explicit symmetry breaking of the $Z(N)$-symmetry. Its strength is determined by the values of the quark masses.

Finite quark masses and external fields

The analogy between an external field in a spin system and dynamical quarks in QCD becomes manifest, when the quarks are integrated out. The integration over quark degrees of freedom induces an external field on the effective level of spin models (Banks and Ukawa, 1983; DeGrand and DeTar, 1983). We refer to the work by DeGrand and DeTar (1983) and anticipate results from lattice gauge theory.

Consider a 4-d-$SU(3)$-lattice gauge theory with fermions at high temperatures. The original action consists of a gauge field part $S_{g}$ and a fermionic part $S_{F}$. The fermionic part may be written as $S_{F}(U)=\sum\bar{\psi}M(U)\psi$ (dropping all indices, $U$ is the $SU(3)$ gauge field and $M$ the fermion matrix, explicit expressions will be given later). When the Grassmann variables are integrated out, the effective action takes the form

[TABLE]

In the high temperature- and strong coupling limit of the $SU(3)$ gauge theory it can be shown that the gauge part $S_{g}$ transforms into the action of a 3d- 3-state-Potts model. (For the Potts model cf. section II A.) The fermionic part of Eq. (79) simplifies, when it is treated in a hopping parameter expansion. The hopping parameter $\kappa$ may be related to the bare quark mass $m$ of the original action according to

[TABLE]

where $m^{.}\tau$ is the bare quark mass in lattice units, $\tau$ being the lattice spacing in time direction. (Relation (80) only holds at strong coupling and for small values of $\kappa$). The result for the fermionic term together with the simplified gauge part leads to the following effective action (for zero chemical potential)

[TABLE]

Here $a$ is the lattice spacing in spatial directions. The sums go over all sites $n$ and nearest neighbor pairs ($n$,$n+\hat{\mu}$), $\hat{\mu}$ is the unit vector in $\mu$-th direction, $z_{n}$ are elements of $Z(3)$. The very coupling $h$ is the external field, which is a remnant of the dynamical fermions. The $\kappa$-dependence is explicitly known and approximately given as

[TABLE]

Together with Eq. (80) we see that $h$ vanishes with $\kappa$ for infinitely heavy quark masses, in the limit of a pure gauge theory. The strength of the external field grows with decreasing quark mass. Hence sufficiently light quark masses will completely wash out the first order transition in the 3-d-3-state-Potts model. DeGrand and DeTar (1983) obtain for the critical strength of the magnetic field $h_{cr}={2\over 3}(\ln 2-{2\over 3})$. The result has been obtained in a mean-field analysis. It holds for a cubic lattice in three dimensions. At the critical field (mass) the line of first order transitions terminates at a second order critical point and disappears for larger fields (smaller masses).

A particular virtue of the lattice “derivation” of the effective action Eq. (81) is that it allows a calculation of the effective coupling in terms of parameters of the original action (here $h=h(\kappa)$). In this aspect the effective Potts model differs from an effective Lagrangian of chiral perturbation theory (see section IV A 2), where the Lagrangian parameters have to be fixed from an experimental input. The price one has to pay for such a derivation on the lattice are the involved approximations (strong coupling and high temperature expansions). Strong coupling results on the lattice must be extrapolated to continuum results over a long way in coupling parameter space. It remains to be shown that the qualitative predictions of the effective $Z(N)$-Potts models survive the continuum limit.

In summary of section II B, the renormalization group analysis of Pisarski and Wilczek, and of Svetitsky and Yaffe refer to the idealized limits of vanishing or infinite quark masses, respectively. They have been performed in the $SU(N_{f})\times SU(N_{f})$ linear sigma model and in $Z(N)$-spin-models rather than directly in QCD. Studies in these limiting cases reveal a dependence of the order of QCD transitions on the number of flavors and the number of colors. The case of physical interest is included for three colors and two or three light flavors. The outcome of the renormalization group analysis is the message that one has to do hard work in the following sense. Model calculations in terms of a scalar field theory with an $N$-component order parameter field would suffice to model QCD, if the transition were conjectured to be of second order also in case of three light or, more realistically, two light and one heavier flavor. One would be free to choose as simple a model as possible within the conjectured universality class. Since the deconfinement and the chiral transitions are expected to be of first order (for three colors and three massless flavors) (see e.g. Karsch, 1990), the transition depends on details of the dynamics. Here this is the dynamics of full QCD.

Compared to the scale of $T_{c}$, two quarks (up, down) are light, three are heavy (charm, bottom, top), but the strange quark happens to be just of the order of $T_{c}$. In the thermodynamics of QCD the quark masses play the same role as external magnetic fields in temperature driven transitions of ferromagnets. From this analogy one must expect that the effect of finite masses depend on their actual values. Results in the idealized limit cases may even qualitatively change under their influence. Thus a challenge for further investigations of the phase structure of QCD is to find out, which of the alternatives displayed in Fig. 2(a)-(b) is realized. Partly conjectural diagrams are shown in the $(m,T)$-plane, where $m$ stands for a generic current quark mass, and $N_{f}=3=N_{c}$ is assumed. For comparison a $(H,T)$-diagram for a ferromagnet and a $(p,T)$-diagram for a liquid/gas system are shown in Fig.s 2(c)-(d). For $m=0$, a first order chiral transition is predicted at some temperature $T_{ch}$. The chiral transition continues to be of first order, but in a weakened form, as long as the mass is smaller than a critical value (corresponding to $h_{ch}<h_{cr}$). For $m=\infty$ or $\kappa=0$ the first order deconfinement transition occurs at some temperature $T_{d}$. It will persist as long as $h(\kappa)\equiv h_{d}$ is smaller than some $\tilde{h}_{cr}$ or $m>m_{cr}$. Both transitions may “meet” and coincide for intermediate mass values (Fig. 2(a)) or not (Fig. 2(b)). In the latter case the discontinuities disappear completely. The dynamical quark masses are then too large for the chiral transition and too small for the deconfinement transition to persist. The chiral symmetric deconfining high temperature region is then smoothly connected with the chiral symmetry broken, confined, low temperature world of daily life. To date lattice results suggest that Fig. 2(b) gives a more realistic description. Lattice calculations are the topic of the next chapter.

III THE LATTICE APPROACH TO THE QCD TRANSITION

A A primer to lattice gauge theory

This section serves mainly to fix the notation of lattice gauge theory. We list the basic definitions, summarize some tools which are relevant to the phase transitions, and point out sources of errors for misleading results. For pedagogical introductions we refer to the literature (see e.g. (Kogut, 1983; Creutz et al., 1983) or more recently (Rothe, 1992; Montvay and Münster, 1994)).

The partition function of an $SU(N)$ gauge theory interacting with matter fields can be written in the Euclidean path integral formulation as

[TABLE]

where the action depends on the gauge fields $A_{\mu}$, quark fields $\bar{\psi}$, $\psi$, the gauge coupling $g$ and the quark masses $m_{i}$. The action is given by

[TABLE]

where ${\cal L}$ is the QCD Lagrangian (Eq. (63)) for $N_{f}$ flavors and $N_{c}$ colors. The chemical potential is set to zero. At finite temperature the gauge- and matter fields have to satisfy the following boundary conditions

[TABLE]

One usually has periodic boundary conditions in the spatial directions in computer simulations. Note that the finite temperature enters only via the boundary conditions, $\beta$ denotes the inverse temperature.

In finite temperature physics a typical task is to evaluate the thermal expectation value of a physical observable $O$. The thermal expectation value, defined as

[TABLE]

takes the following form in the path integral formulation

[TABLE]

where $pbc$ stands for the periodic (antiperiodic) boundary conditions as specified in Eq. (86). The functional integral (89) is by itself not well defined. Many schemes may be used to perform the functional integration in an approximate way.

One possibility, which is particularly suited in view of the nonperturbative nature of the phase transition is the lattice formulation. The (3+1)-dimensional spacetime continuum is discretized commonly on a hypercubic lattice. The lattice provides a gauge invariant regularization scheme. Lattice results may be extrapolated to the continuum limit in the very end. Lattice artifacts can be controlled via renormalization group equations and should vanish in the continuum limit. In practice it may be quite subtle and intricate to disentangle lattice artifacts from continuum physics.

A gauge invariant lattice action is conveniently defined in terms of link variables $U(x,\mu)\equiv U_{x}^{\mu}$, associated with a link $(x,\mu)$ leaving site $x$ in direction $\mu$, and matter variables $\psi(x)\equiv\psi_{x}$, $\bar{\psi}(x)\equiv\bar{\psi}_{x}$, associated with the sites of the lattice. We keep the continuum notation for the discrete sites $(\vec{x},x_{0})$. The link variables are elements of the gauge group $SU(N)$ and replace the continuum gauge fields $A_{\mu}$. In the continuum limit these are related as

[TABLE]

If the matter fields are fermions, as in the QCD action, $\psi$ and $\bar{\psi}$ are Grassmann variables satisfying the usual anti-commutation relations

[TABLE]

The matter fields carry spin, color and flavor indices (as indicated in Eq. (63)). The lattice discretized version of the partition function then takes the form

[TABLE]

In contrast to the symbolic notation in Eq. (89) the measure ${\cal D}U~{}{\cal D}\psi~{}{\cal D}\bar{\psi}\equiv\prod dU\prod d\psi d\bar{\psi}$ has a well defined meaning in Eq. (93). The products extend over all link- and site variables of the lattice, $dU_{x}^{\mu}$ refers to the (gauge invariant) Haar measure on the $SU(N)$ gauge group, $d\psi~{}d\bar{\psi}$ is the usual measure over Grassmann variables for each site $x$. The action depends on the gauge coupling $g^{2}$ and the quark masses $m_{i}$ ($i=1\ldots N_{f}$) as before, but has an additional dependence on the lattice spacings in spatial ($a_{\sigma}$) and temporal ($a_{\tau}$) directions. The lattice spacing, or equivalently, the bare coupling may be chosen independently in the spatial and temporal directions. This is sometimes utilized in finite temperature calculations of thermodynamical quantities.

The lattice action

Two requirements have to be satisfied by a lattice action. The first one is the reproduction of the classical continuum limit. In the limit of $a_{\sigma}=a_{\tau}\to 0$, the action must approach the continuum form of Eq. (84). The second one is local gauge invariance, i.e. an invariance under local gauge transformations $V(x)$, acting according to

[TABLE]

(The second requirement is commonly used, but actually not necessary. Actions with non-compact gauge groups recover local gauge invariance only in the continuum limit.) These requirements are not very restrictive. They leave us with a variety of possible lattice formulations. Here we state only the most popular choices, which are sufficient for what follows.

For the gauge part it is the Wilson form (Wilson, 1974), for the fermionic part two choices are Wilson fermions (Wilson, 1974) and staggered or Kogut-Susskind fermions (Kogut and Susskind, 1975; Susskind, 1977). We distinguish explicitly between spacelike and timelike quantities, to make the appearance of temperature explicit.

The Wilson form for the gauge part is defined as

$\displaystyle S_{g}={2N\cdot a_{\tau}\over g^{2}_{\sigma}\cdot a_{\sigma}}\sum_{\scriptstyle x\atop\mu<\nu<4}P_{x}^{\mu\nu}+{2N\cdot a_{\sigma}\over g^{2}_{\tau}\cdot a_{\tau}}\sum_{\scriptstyle x\atop\mu<4}P_{x}^{\mu 4},$

where

[TABLE]

contains a product of gauge field variables along the boundary of an elementary plaquette of the lattice, and $N=3$ for $SU(3)$, ${\rm Tr}$ denotes the trace in color space.

The first term of the action is a sum over plaquettes, which contain only spacelike links, the second term involves also timelike links. It can be easily checked that Wilson’s formulation (97) satisfies both requirements for a lattice action.

To find an appropriate form for the fermionic part of the action turns out to be more difficult. A naive translation of the continuum Dirac action leads to a lattice action which actually describes $2^{d}(=16)$ species of fermions in the naive continuum limit rather than the intended one species of the original continuum action. “No-Go” theorems by Nielsen and Ninomiya (1981a, 1981b) explain the failure of naive transcription and tell us that no lattice formulation of the fermionic action exists that is satisfactory in all aspects. If one insists on a local action, either continuous chiral symmetry is completely lost on the lattice or one ends up with too many flavors.

We consider two popular choices, Wilson fermions and staggered fermions. The former choice gets rid of the species doubling at the expense of breaking all continuous chiral symmetries explicitly in the limit of $M\to 0$. The latter choice keeps a $U(1)\times U(1)$ chiral symmetry for all lattice couplings, a welcome feature in view of an investigation of chiral symmetry restoration. The price is too many flavors in the continuum limit (although the number is reduced with respect to the naive formulation) and a broken flavor symmetry on the lattice. For derivations and details of lattice fermions we refer to textbooks (e.g. Rothe, 1992; Montvay and Münster, 1994) or reviews (Kogut, 1983; Creutz et al., 1983). Here we summarize only the results.

The Wilson action and hopping parameters

In the Wilson action the bare masses $m_{i}$ are hidden in the hopping parameters in spacelike ($\kappa_{\sigma}^{(i)}$) and timelike ($\kappa_{\tau}^{(i)}$) directions, one for each flavor $i$, where $i$ labels the $N_{f}$ species of flavors in the continuum limit. For the free gauge theory ($U_{x}^{\mu}=1$) and $a_{\sigma}=a_{\tau}$, $\kappa_{\sigma}^{(i)}=\kappa_{\tau}^{(i)}=\kappa^{(i)}$, the relation to the bare quark masses is

[TABLE]

with $\kappa_{c}=1/8$. For massless quarks $\kappa^{(i)}=\kappa_{c}$. In the interacting theory, the dependence of $\kappa_{c}$ on $g^{2}$ is not known. This is an undesirable feature in Monte Carlo calculations, when hopping parameter values should be translated to quark masses to check the relevance of the results for realistic mass values.

While the flavor symmetry is well defined and conserved for all lattice spacings by the Wilson action, all chiral symmetries are explicitly broken even in the massless case ($m_{i}=0$ or $\kappa^{(i)}=\kappa_{c}$). ‘All’ chiral symmetries should be contrasted with remnants of the full invariance under certain subgroups. Recall our original intention to study the phase structure within the lattice approach. A chiral symmetry restoration at finite temperature in the massless limit should be signalled by a melting of the condensate $\left\langle\bar{\psi}\psi\right\rangle$. Such a ‘melting’ is prejudiced right from the beginning, if an explicit chiral symmetry breaking is involved as it is in the Wilson formulation.

Staggered fermions and flavor symmetries

This choice of lattice fermions is frequently used in studies of the chiral phase transition. A careful inspection of the origin of the species doubling in the naive formulation suggests the possibility of eliminating the unwanted fermions by doubling the effective lattice spacing. This amounts to a distribution of the fermionic degrees of freedom over the original lattice in such a way that the effective lattice spacing for each type of Grassmann variables is twice the fundamental lattice spacing. It turns out that in $d$ dimensions $2^{d/2}$ fermion fields are necessary to place a different fermionic degree of freedom at each site of an elementary hypercube on the lattice. This indicates the origin of the integer multiple of four flavor degrees in the continuum for four dimensions, which is attainable by a description with staggered fermions.

A realization of these ideas is rather involved. The sites of the hypercube will be occupied by single-component spinors $\chi_{f}$, $\bar{\chi}_{f}$, which may be multi-component in flavor space, $f=1,\ldots,n$. (This number $n$ should be distinguished from the desired number $N_{f}$ of continuum flavors.) The $\chi$’s and $\bar{\chi}$’s are certain linear combinations of the original fields $\psi$, $\bar{\psi}$. In the end of a simulation in terms of $(\chi,\bar{\chi})$-fields, the results for the ($\psi,\bar{\psi}$)-fields have to be reconstructed.

The action for staggered fermions in terms of $(\chi,\bar{\chi})$-fields is defined by

$\displaystyle S_{F}\equiv\sum_{x,x^{\prime}}\sum_{f=1}^{n}\bar{\chi}_{f}(x)~{}{\cal Q}_{x,x^{\prime}}^{f}~{}\chi_{f}(x^{\prime})$

with

$\displaystyle{\cal Q}_{x,x^{\prime}}^{f}={1\over a_{\sigma}}\sum_{i=1}^{3}D_{x,x^{\prime}}^{(i)}+{1\over a_{\beta}}D_{x,x^{\prime}}^{(4)}+m_{f}~{}\delta_{x,x^{\prime}}$

and

[TABLE]

The only remnant of the Dirac structure is hidden in the phases $\Gamma_{\mu}$. The staggered fermion action depends on the quark masses $m_{f}$. The index $f$ labels the species of ’staggered’ flavors. Their number $n$ is often set to 1, since $n$ staggered species on the lattice correspond to $N_{f}=4\cdot n$ flavors in the continuum. The case $n=1$ comes closest to $N_{f}=3$ flavors, for which QCD is approximately chiral invariant. More precisely, for $n=1$ and $d=4$ dimensions one has 16 $\chi$’s in each elementary hypercube. It can be shown that these 16 $\chi$’s can be combined to define a quark field with four flavors in the continuum, more generally with $N_{f}=4\cdot n$ flavors. A non-zero lattice spacing breaks the flavor symmetry between the $4n$ flavors, and it breaks the chiral $U(4n)\times U(4n)$-symmetry of the continuum limit of the staggered fermion action for $m_{f}=0$ down to $U(n)\times U(n)$.

It is the $U(1)\times U(1)$ remnant of the full chiral symmetry (that is left for $n=1$ and arbitrary lattice spacing), which is the desired feature for investigating chiral symmetry restoration.

Wilson and staggered fermions appear quadratically in the action. In general, fermion fields can only occur quadratically in a renormalizable four-dimensional field theory. Thus fermions are usually integrated out. This seems to be the only tractable way in numerical and analytical calculations, since the fermion fields are Grassmann variables. The integrals are performed using the well-known formulas of Berezin integration (Berezin, 1966). The result is

[TABLE]

where ${\cal Q}$ is given by the action for Wilson fermions and by Eq. (99) for staggered fermions. The power $l$ equals to $N_{f}$, the number of continuum flavors, for Wilson fermions, and $l$ equals $n$ for staggered fermions (recall that $n=N_{f}/4$). For a positive definite ${\cal Q}$ one may use that $\det{\cal Q}=\exp({\rm Tr}\ln{\cal Q})$, and is led to

$\displaystyle Z=\int_{pbc}\prod_{x,\mu}dU_{x}^{\mu}~{}e^{-S_{\rm eff}}$

with

[TABLE]

where the trace extends over spin, color and flavor indices. Thus a simulation of Eq. (102) with $l=1$ describes $N_{f}=4-{\rm flavors}$ in the continuum limit, if staggered fermions are used for ${\cal Q}$. For Wilson fermions $\det{\cal Q}(U)$ is real and positive for values of the hopping parameter less than $1/8$ (Seiler, 1982).

Sources of errors

Also for staggered fermions (and vanishing chemical potential) $\det{\cal Q}$ is positive, but its eigenvalues are not always positive. This is an annoying feature for a representation in terms of pseudofermions. Therefore in actual simulations ($-l\cdot{\rm Tr}~{}{\cal Q}(_{x}^{\mu})$) in the effective action of Eq. (102) is replaced by ($-l\cdot{\rm Tr}\ln[\hat{\cal Q}\hat{\cal Q}^{+}(U)]$). Consider first $\hat{\cal Q}={\cal Q}$. The term ${\rm Tr}\ln{\cal Q}{\cal Q}^{+}$ is proportional to $(\det{\cal Q})^{2}$, thus the replacement induces a further doubling of flavor degrees of freedom from $n$ to $2n$. To compensate for this doubling, the number of degrees of freedom in ${\cal Q}$ is reduced by a factor of 2 by a further doubling of the effective lattice spacing. Let us call the corresponding fermion operator $\hat{\cal Q}$. Due to an even-odd symmetry of the determinant one has $\det{\cal Q}=\det\hat{\cal Q}_{\rm even}\cdot\det\hat{\cal Q}_{\rm odd}$. Thus ${\rm Tr}\ln{\cal Q}$ is actually simulated as ${\rm Tr}\ln\hat{\cal Q}\hat{\cal Q}^{+}$. This induces an error, which is not well under control. Another uncontrolled error entering the formulation for staggered fermions is due to flavor exchanging currents. The flavor symmetry is violated for a finite lattice spacing. By claiming that their contribution vanishes in the continuum limit a similar type of error may be induced as when one says that the irrelevant additive terms in the Wilson action vanish in the continuum limit. It is well-known that just the irrelevant terms of the Wilson action are essential for reproducing the right axial anomaly in the continuum limit.

Translation from lattice units to physical units

A basic step in understanding the lattice approach is the translation of lattice results into physical units. For illustration let us consider the measurement of a mass. Such a measurement on the lattice typically yields a dimensionless number of the order of 1. The dimensionless lattice mass $m_{\rm{latt}}$ is related to the physical mass $m$ via the lattice spacing $a$. From simple dimensional arguments we have

[TABLE]

in units where $\hbar\cdot c=1$. (In these units $100~{}[\rm{MeV}]\sim 3.2~{}[\rm{fm}^{-1}]$.) In other words, the lattice mass $m_{\rm{latt}}$ is measured in units of $a^{-1}$. Other variables are obtained similarly, energy densities in units of $a^{-4}$, etc.

When a lattice mass is interpreted as $m\cdot a$ and $m$ [MeV] is known from experiments, Eq. (103) gives “the” lattice spacing in physical units. More precisely it gives $a(g)~{}[\rm{MeV}^{-1}]$ at coupling $g$, if $g$ stands for the bare input parameter(s) of the lattice Lagrangian, which have been used in the measurement of $m_{\rm{latt}}$. In a pure gauge theory $g$ is the bare gauge coupling. The lattice spacing which is obtained this way is only unique, i.e. independent of the choice of the physical input $m$ [MeV], if one is in (or close to) the continuum limit.

Let us assume we have determined $a~{}[\rm{MeV}^{-1}]$ or $a[\rm{fm}]$ from a first mass measurement as indicated above and measure a second mass $\tilde{m}_{\rm{latt}}$ in lattice units. The physical value $\tilde{m}$ [MeV] is then predicted from the lattice simulation and can be compared with the experimental value. Furthermore, once $a(g)$ is known as function of $g$ in units of $\rm{MeV}^{-1}$ or fm, it makes sense to associate strong couplings ($g\gg 1$) with coarse grained lattices (say $a>1$ fm), and weak couplings with a fine grain size close to the continuum description. The very existence of such a universal mapping between bare couplings and lattice spacings in physical units is based on the renormalization group equation. In case of an $SU(N_{c})$ gauge theory with $N_{c}$ colors and $N_{f}$ massless flavors the renormalization group equation relates $g^{2}$ and $a$ in the continuum limit (for perturbatively small couplings $g^{2}\to 0$) according to

[TABLE]

If lattice calculations are performed at sufficiently small values of $g$, an observable in lattice units should scale as a function of $g$ in a way, which is determined by $a(g)$ according to Eq. (105). For a mass in lattice units this implies

[TABLE]

If such a scaling behavior is observed, the asymptotic scaling regime has been reached. The lattice spacing can then be replaced by the scale parameter $\Lambda_{\rm latt}$, which represents the only parameter in QCD with $N_{f}$ massless flavors, and has to be fixed from experiment.

The physical volume $V$ and temperature $T$ are given in terms of the lattice spacing as

[TABLE]

For $a_{\tau}=a_{\sigma}=a$, the volume and temperature may be converted in units of fm, if $a(g)$ is taken from Eq. (105).

For $a_{\tau}\neq a_{\sigma}$, the relations $a_{\tau}(g_{\sigma},g_{\tau})$ and $a_{\sigma}(g_{\sigma},g_{\tau})$ can be determined in the case of a pure gauge theory (Eq. (97)) as follows. We use the notations

[TABLE]

where $g_{\sigma}$, $g_{\tau}$ have been introduced in Eq. (97) and $\beta_{g}$ denotes the bare coupling on an isotropic Euclidean lattice. Here $\beta_{g}$ is large when $g$ is small like the physical temperature (cf. the remarks in section II B 2). Thus one may also think of $\beta_{g}$ as some kind of effective temperature (rather than effective inverse temperature as the notation suggests). The two relations which replace $g(a)$ of the isotropic case are written as

[TABLE]

The coefficients $c_{\sigma}$ and $c_{\tau}$ have been perturbatively determined by Karsch (1982). With $\xi=a/a_{\tau}$ and $g=g(a)$ from Eq. (105), Eq. (111) can be solved for $a=a_{\sigma}$ and $a_{\tau}$ in terms of $\beta_{\sigma}$ and $\beta_{\tau}$, the couplings of the pure gauge action. The relations $a_{\sigma}(\beta_{\sigma},\beta_{\tau})$ and $a_{\tau}(\beta_{\sigma},\beta_{\tau})$ replace $a(g)$ on an anisotropic lattice in thermodynamic calculations.

The critical temperature $T_{c}$

In QCD the critical temperature $T_{c}$ depends on the number of colors ($N_{c}$), the number of flavors ($N_{f}$), the current quarks masses $m_{q}$, and the volume $V$. The finite volume induces a shift of $T_{c}$ of the order of $1/V$, if $V$ denotes the $d$-dimensional volume of the system. The finite light quark masses give an effect of a few percent compared to the value of $T_{c}$ in the chiral limit. The strongest dependence comes from the number of flavors. $T_{c}$ varies about 100 MeV between $T_{c}\sim 150$ MeV for two light flavors and $T_{c}(N_{f}=0)\sim 260$ MeV for a pure gauge theory (DeTar, 1995). This is easily understood in a percolation picture. The transition is assumed to occur when a critical hadron density is reached where the hadrons start overlapping which we have briefly mentioned in the introduction. A much lower temperature is then needed for creating a critical hadron density out of light pions with a mass of the order of $T_{c}$ than out of heavy glueball states. The lightest states in a pure gauge theory are glueballs with a mass of the order of $5T_{c}$.

For a smooth crossover phenomenon $T_{c}$ is no longer defined. However, if there is a narrow temperature interval with rapid changes in thermodynamic quantities, it makes still some sense to associate a ‘pseudocritical’ or ‘crossover’ temperature $T_{c}$ with the rapid crossover region. More precisely, the crossover temperature is defined as the temperature of maximum change in an observable (e.g. the chiral condensate). Alternatively it is the temperature of the peak in some susceptibility. Both pseudocritical values for $T_{c}$ must only agree, if the susceptibility is the derivative of the particular observable whose maximum change defines $T_{c}$.

A test of asymptotic scaling

Usually one measures the critical coupling $g_{c}$ rather than the critical temperature $T_{c}$. Order parameters are plotted as functions of $g$ (or $\beta=1/(6g^{2})$) to show their behavior as a function of $T$. For simplicity let us consider an isotropic lattice, i.e. $a_{\sigma}=a_{\tau}=a$, and the limit of a pure gauge theory with bare coupling $g$. For a given number $N_{\tau}$ of lattices sites in the timelike direction the temperature may be varied by tuning $a$ via $g$ according to Eq. (108). From a physical point of view it is not surprising that the temperature may be implicitly varied via the coupling, as both are related through asymptotic freedom. Starting in the strong coupling region $g\gg 1$ and lowering $g$, the transition from the confinement to the deconfinement phase will be encountered at a certain coupling $g_{c}$. Instead $g$ can be kept fixed, but $N_{\tau}$ varied from smaller to larger values (say from $N_{\tau}=2$ to $N_{\tau}=8$). This way the transition region is passed from the deconfinement to the confinement phase, if $g$ and $N_{\tau}$ are in an appropriate range.

One would like to simulate the lattice system for large values of $N_{\tau}$ to keep the finite size effects small. For large $N_{\tau}$ a rather small coupling is necessary to reach the high temperature phase. The larger $N_{\tau}$, the smaller ist the critical coupling $g_{c}$, which corresponds to the transition temperature $T_{c}$. In practice, the computer time rises rapidly, if $g$ is small. This explains why typical extensions in time direction are limited to $N_{\tau}=$ 6, 8 or 12, when fermions are included.

The effect of fermions is a further reduction of the effective lattice spacing $a_{\tau}$ at the same coupling $g$ compared to the pure gauge case. Even larger values of $N_{\tau}$ are needed to reach the transition region. Small values of $N_{\tau}$ require stronger couplings $g$ in the presence of fermions. One should keep in mind that one is far outside the asymptotic scaling region, if dynamical fermions are simulated at small $N_{\tau}$.

A test of asymptotic scaling is the very first check, whether the measured critical coupling $g_{c}$ has some relevance for a critical temperature in the continuum limit. When the temporal size $N_{\tau}$ is increased, the transition should occur at a smaller lattice spacing $a(g_{c})$ such that $T_{c}=1/(N_{\tau}{}a(g_{c}))$ stays constant. For small couplings $g$, the relation $g_{c}(N_{\tau})$ should scale as

[TABLE]

Eq. (112) follows from Eqs. (105) and (108). Calculations for an $SU(2)$- and $SU(3)$ gauge theory have been performed up to $N_{\tau}=16$. A measurement of $g_{c}$ shows strong violations of asymptotic scaling according to Eq. (112) (see e.g. Karsch, 1992a).

Asymptotic scaling violations have been also observed in other observables. They seem to be universal. Thus it should be possible to absorb the scaling violations in a ‘renormalization’ of the bare coupling to an effective coupling which accounts for rapid fluctuations in the action in an intermediate coupling range. This idea turned out to be successful (Altmeyer et al., 1992). As a function of an effective coupling $T_{c}/\Lambda_{\overline{MS}}$ depends much weaker on $N_{\tau}$. An extrapolation to the continuum limit seems to be justified leading to (Karsch, 1992a)

[TABLE]

Translation to physical units

A second check whether the critical temperature on the lattice has some relevance for the continuum limit is a translation to physical units. The outcome should be independent of the choice of the experimental input. In the pure $SU(3)$ gauge theory $T_{c}$ is approximately independent of the physical input. It is 239(13) MeV from the string tension $\sqrt{\sigma}=420\pm 20$ MeV, 239(23) MeV from the $\rho$-mass and 225(30) MeV from the nucleon mass (Karsch, 1992a).

For QCD with two light flavors in the staggered fermion formulation the crossover temperature is estimated as 140-160 MeV (DeTar, 1995). The experimental input comes from the $\rho$-mass with 770 MeV. The critical temperature is then determined as

[TABLE]

The $\rho$-mass in lattice units $m_{\rho}\cdot a$ (depending on the bare gauge coupling $g_{c}$ and the bare quark masses $m_{q}$) is calculated as a function of $g_{c}$ and $m_{q}\cdot a$ from a fit to several zero temperature simulations (see DeTar, 1995, and references therein). Note that in Eq. (114) the $\rho$-mass [MeV] is kept fixed at its physical value of 770 MeV, while the bare quark masses $m_{q}\cdot a$ are allowed to vary to unphysical values leading to unphysical $\rho$-masses as well. As emphasized by DeTar (1995), one should further keep in mind that the nucleon to rho mass ratio is approximately 20 % above its physical value over the range of lattice parameters, where $m_{\rho}\cdot a$ is measured. Also the $m_{\pi}/m_{\rho}$ ratio comes out twice to three times the experimental value, because the bare quark masses are still too large. Thus the $\rho$-mass is not (yet) a perfect candidate for the conversion from lattice units into physical units.

In earlier lattice simulations thermodynamic quantities have been plotted as function of $6/g^{2}$. Nowadays one often finds plots vs $T$ [MeV]. The translation should be taken with care, as the mapping between $6/g^{2}$ and $T$ is not (yet) unique.

Numerical simulations

So far we have specified the measure, boundary conditions and lattice action for the path integral of Eq. (93). We have sketched, how numbers from the lattice can be translated into physical units, and what a criterion on the relevance of lattice results for continuum physics looks like. Numerical and analytical methods can be utilized to attack the functional integrations of Eq. (93). Analytical methods are usually applicable in limits where small expansion parameters are available. The hopping parameter expansion is an expansion in small values of $\kappa$, it is an expansion around the limit of infinite quark masses. The strong coupling expansion applies for large values of the bare gauge coupling $g$. As mentioned above, it is intrinsically difficult to find a small expansion parameter in the transition region, where $g$ is neither weak nor strong. Thus it is not surprising that many results on the phase structure in the vicinity of $T_{c}$ are based on numerical simulations. The most important approach is the Monte Carlo method.

Monte Carlo on the lattice. In a Monte Carlo procedure a set of field configurations $\{U^{\mu}_{x}\}^{(1)}\ldots\{U^{\mu}_{x}\}^{(N)}$ is generated such that the Boltzmann factor is absorbed in the selection of configurations. For a positive definite and real action $S[\{U^{\mu}_{x}\}]$ the expectation value of an observable $O$ defined as

[TABLE]

(a lattice version of Eq. (89) for a pure gauge theory) equals to

[TABLE]

and is approximated by the arithmetic average over a finite number of configurations

[TABLE]

The observables in Eq. (117) are evaluated on an ensemble of configurations which are representative of the coupling (temperature) $\beta^{-1}$. The configurations are selected according to the probability distribution

[TABLE]

where ${\cal D}U$ and $Z$ are short hand notations of the expressions in Eq. (115). The fundamental idea of the Monte Carlo method is stated by the law of large numbers, which tells us that under very general conditions on the probability measure $\lim_{N\to\infty}\langle O\rangle_{N}=\langle O\rangle$.

The sequence of configurations is generated with a Markov process. An easy way of constructing a Markov process, which has a particular distribution $P$ as its fixed point, is to choose the transition probabilities $Q:(\{U^{\mu}_{x}\}^{(i)}\longrightarrow\{U^{\mu}_{x}\}^{(f)})$ from an initial configuration $\{U^{\mu}_{x}\}^{i}$ to a final configuration $\{U^{\mu}_{x}\}^{f}$ to satisfy the detailed balance condition

[TABLE]

The fixed point distribution is in our application the Boltzmann equilibrium distribution. A particular way of realizing the detailed balance condition is the Metropolis algorithm (Metropolis et al., 1953).

The Metropolis algorithm is an example of a local updating procedure. One starts from an initial configuration, which may be chosen ordered ($U^{\mu}_{x}=1\hskip-2.58334pt{\rm l}$ for all links (cold start)) or disordered ($U^{\mu}_{x}\in SU(N)$ random for all links (hot start)) or mixed. A single new link variable ${U^{\mu}_{x}}^{\prime}$ is then chosen randomly and always accepted as a replacement for the old variable, if it lowers the action, i.e. $\Delta S=S(\{U\}^{\prime})-S(\{U\})<0$. Otherwise the change is accepted with a conditional probability. For $\Delta S>0$ a random number $r$ with $0\leq r\leq 1$ is selected. If $r<e^{-\Delta S}$, the new variable $U^{\prime}$ is still accepted for replacing the old one, otherwise not, one goes back to $\{U^{\mu}_{x}\}$ and repeats the steps as indicated above.

This way all links of the lattice are changed, either randomly or successively. Such a sweep through the entire lattice is counted as one Monte Carlo iteration. Usually hundreds or thousands of such iterations are necessary, before the equilibrium distribution is reached. The expectation value of the observable is then obtained according to Eq. (117), where the sum extends only over the last $N$ equilibrated (or thermalized) configurations.

When fermions are included, one comes across a technical problem due to the nonlocality of the effective action Eq. (102). The nonlocality refers to the det-term or equivalently to the Tr ln $Q$-term. For staggered fermions the matrix $Q$ has $(3N_{\sigma}^{3}{}^{.}N_{\beta})^{2}$ complex elements. In the local Metropolis algorithm this matrix should be calculated for every link in every iteration, which renders it impracticable.

The hybrid Monte Carlo algorithm. In the last decade much effort has been invested in improving algorithms for QCD with dynamical quarks (for a review see e.g. the lectures by Toussaint (1988) or Herrmann and Karsch, 1991). The integration over the fermions had led to the factor ${\det}^{l}({\cal Q})$ in Eq. (101). While ${\cal Q}$ has been a local operator, coupling only nearest neighbors on the lattice, $\det{\cal Q}$ is nonlocal. A calculation of $\det{\cal Q}$ with $\exp\{-S_{g}\}$ as Boltzmann factor should be avoided. The idea is to bosonize $\det{\cal Q}$ and to “lift” ${\cal Q}$ in the exponent. The bosonization is performed with the pseudofermion method of Petcher and Weingarten (1981).

The pseudofermion method (Petcher and Weingarten, 1981) is based on the following formula for the determinant

[TABLE]

Here $\lambda,\lambda^{*}$ are complex bosonic (pseudofermionic) variables. Note that $Q$ has to be a positive definite matrix for the Gaussian integral in Eq. (121) to converge. If we identify $Q$ with the fermionic matrix of Eq. (101), $Q$ has negative eigenvalues. This is the reason, why $Q$ is replaced by $Q^{+}Q$ at the cost of doubling the number of fermionic flavors or (for compensating this doubling) by $\hat{Q}^{+}\hat{Q}$, where $\chi$-fields live on even and $\bar{\chi}$-fields on odd sites. (Another reason for the replacement is that it is convenient for easily generating the pseudofermion fields from a heat-bath.) Via the doubling ($Q\to QQ^{+}$) configurations leading to negative eigenvalues of $Q$ are mapped on the same contribution to the effective action as those with positive eigenvalues, although $\ln\;\hbox{det }Q$ in the original effective action is not given as Tr $\ln Q$ for negative eigenvalues. Thus the error entering this way is difficult to control.

Like $\det{\cal Q}$ the action in Eq. (121) is nonlocal as well. Hence we see, when the bosonic part (${-\sum_{x,y}\lambda^{*}_{x}Q^{-1}_{xy}\lambda_{y}}$) is added to the gauge part of the QCD action to account for fermions, we need a nonlocal updating procedure. A practicable method, which is free of uncontrolled errors and nowadays often used in QCD applications, is the hybrid Monte Carlo algorithm (Duane et al., 1987). It has to be combined with the pseudofermion representation to yield a feasible simulation of QCD. The hybrid Monte Carlo algorithm is a stochastic hybrid of a Langevin and a microcanonical algorithm, where an additional Monte Carlo acceptance test has been built in to remove the systematic errors of the stochastic hybrid. The stochastic hybrid algorithm combines the virtues of the Langevin and the microcanonical (or molecular dynamics) algorithms. Both algorithms use equations of motion to select new variables.

In the Langevin algorithm (Parisi and Wu, 1981; Fukugita and Ukawa, 1985; Batrouni et al., 1985) the equation of motion is a partial differential equation, which is of first order in time. The finite step size in time entering the discretized version has to be extrapolated to zero by varying its size. This is time consuming and an unwanted feature.

The microcanonical algorithm (Callaway and Rahman, 1982 and 1983) is based on the observation that the Euclidean path integral of a quantum field theory in four dimensions can be written as a partition function for a system of classical statistical mechanics in 4 spatial dimensions with a canonical Hamiltonian that governs the dynamics in a fifth new ‘time’ variable. This time $\tau$ may be identified with the simulation time. The algorithm determines the new variables $\Phi_{x}(\tau)$ and their canonically conjugate momenta $\pi_{x}(\tau)$ in a fully deterministic way. Here one moves faster through phase space $(\phi,\pi)\to(\phi^{\prime},\pi^{\prime})\to\ldots$, but on the price that ergodicity is lost.

The stochastic hybrid algorithm (Duane, 1985) interrupts the integration of Hamilton’s equation of motion along a single trajectory with fixed energy. The interruption is made once in a while for a refreshment of momenta. The new parameter, which enters and should be optimized is the frequency of momenta refreshment. If it is low, the algorithm is as slow as the Langevin algorithm, if it is too high, ergodicity will be violated. Yet there is a systematic error in the hybrid algorithm introduced by the finite time step in the integration procedure, which leads to a violation of energy conservation.

The idea of the hybrid Monte Carlo algorithm is to absorb this energy violation in a ‘superimposed’ Monte Carlo procedure. The phase space configurations at the end of every molecular dynamics chain are subjected a Metropolis acceptance test. If the time step is too large, the energy violation too strong, the $(\Phi,\pi)$ configuration is likely to be rejected. In the case the energy is conserved, the configuration is always accepted. This way the systematic error due to finite integration steps is eliminated.

Above we have sketched a number of intermediate steps in developing the hybrid Monte Carlo algorithm. It may have indicated, how complex the problem of finding efficient algorithms is. Although the hybrid Monte Carlo algorithm moves the system fast through configuration space, it is slow due to a matrix inversion, which enters the equations of motion. The matrix inversion is usually performed with the conjugate gradient method. The number of conjugent gradient steps is proportional to $(m_{f}^{-1})$ and takes more than 90% of the CPU-time. The hybrid Monte Carlo acceptance rate is proportional to $(m_{f}^{-3/4})$ and the autocorrelation time $\propto$ $m_{f}^{-1}$ (Karsch, 1992). For example, one lattice update on a $16^{3}\times 8$ lattice with $m_{f}=0.01$ takes about 20 min on a 200 MFlop-machine. Here we see, why it is so difficult to work with small bare quark masses. Quark masses should be as small as possible to avoid an uncontrolled extrapolation to the chiral limit, but some extrapolation is unavoidable.

More generally we see, why Monte Carlo simulations of lattice QCD are so time consuming, when dynamical fermions are included. It is extremely hard to satisfy the following requirements simultaneously:

• $\bullet$

The masses are small enough to guarantee a safe extrapolation to the chiral limit.

• $\bullet$

The volume is large enough to avoid finite size effects.

• $\bullet$

The bare coupling is small enough to be in the asymptotic scaling regime.

B Pure gauge theory

1 The order of the $SU(3)$-deconfinement transition

The deconfinement transition in a pure $SU(3)$ gauge theory is nowadays believed to be of first order. Over several years there has been a controversial discussion in the literature. It was due to the subsequent use of different criteria to determine the order from a numerical analysis. In the background from statistical physics (section II) we have distinguished between ‘naive’ and ‘refined’ criteria. In this section we will illustrate their application to the deconfinement transition.

Early calculations of Kogut et al. (1983) and Celik et al. (1983) found a strong first order deconfinement transition. The conclusions were based on hysteresis effects, signals for coexisting states, and abrupt quantitative changes in bulk thermodynamic quantities like the internal energy density. One finds in fact an explicit suggestion that hydrodynamic models of the quark gluon plasma and models of hadronization should incorporate a “hard, first-order transition” (Kogut et al., 1983).

These results have been questioned by the APE-group (Bacilieri et al., 1988) claiming to see strong indications for a second order deconfinement transition. The claim has been based on finding a correlation length that grows proportional to the lattice size, as the size is increased. The APE-results led to a careful reinvestigation of the deconfinement transition in different aspects.

From section II B we recall the renormalization group analysis of Svetitsky and Yaffe (1982 a and b). There it has been argued that an effective theory for an order parameter of a (3+1)-dimensional $SU(N)$ gauge theory shares the global symmetry with 3-dimensional $Z(N)$-spin models. In particular knowledge about $Z(3)$-spin models was exploited to predict a first order phase transition for the case of an $SU(3)$ gauge theory. The knowledge refers to the absence of an IR-stable fixed point in $Z(3)$-spin systems.

A possible second order phase transition in an $SU(3)$ gauge theory would cast some doubts on the universality arguments. While it was not difficult to ‘derive’ the $Z(3)$-symmetry of the effective action in terms of the order parameter field, it was harder to argue in favor of the locality of the spin model. Thus the question of non-local interaction terms in the effective action has been addressed by Fukugita et al. (1989 and 1990a) and by Gavai et al. (1989) and Gavai and Karsch (1989) in subsequent papers.

In general the order of the transition does depend on the type of effective couplings. These couplings are generated, when the original degrees of freedom are partially integrated out. Non-local couplings could cause the $SU(3)$ transition to be of second order. This suspicion was supported by earlier results of Fucito and Vulpiani (1982) stating that already small anti-ferromagnetic next-to-nearest-neighbor (nnn-) couplings in addition to the leading ferromagnetic nearest neighbor (nn-)coupling induce a second order transition in a 3d-$Z(3)$-Potts model. It was suggested that likewise a small anti-ferromagnetic nnn-coupling could induce a second order transition in the $SU(3)$-deconfinement transition.

Both transitions in variants of the $Z(3)$-Potts model and in the original $SU(3)$ gauge theory have then been studied by applying refined criteria compared to earlier calculations. Here we report in some more detail on results by Gavai et al. (1989) on the $Z(3)$-Potts model and by Fukugita et al. (1989) on the $SU(3)$ gauge theory. In both models the first order nature of the transition has been verified with a variety of criteria, and the results of the APE-group are understood most likely as a misinterpretation of the measured data.

Let us recall from section II the distinction between naive and refined criteria. Naive criteria are signs of metastabilities, hysteresis effects, two-state-signals or jumps in thermodynamic quantities. Typically all these signatures occur in a truly first order transition. If we call them ‘naive’, it refers to their incautious use when the signatures are taken for granted from measurements for a single lattice size. Early lattice simulations were ‘naive’ in this sense. Compared to nowadays standard they were also performed on rather small lattices.

Refined criteria refer to a finite size scaling analysis of bulk and ‘special’ quantities. Examples for bulk quantities are the internal energy density $\varepsilon$, the specific heat $c$ or the order parameter susceptibility $\chi$. Finite size corrections to the average internal energy are exponentially suppressed. The shifting and rounding effects in the peaks of $c$ and $\chi$ are of the order of $1/V$ in first order transitions (see section II).

With ‘special quantities’ we mean distribution functions of the order parameter $P(O)$, and derived moments or cumulants like Binder’s cumulant, see Table II. In general they have power-law finite size corrections and are practically applicable for very large volumes, where these power-law corrections are suppressed.

Similarly correlation lengths show power-law corrections. Even their very definition in a finite volume is rather intricate compared to bulk quantities. Their practical utility for a finite size scaling analysis turns out to be questionable, as we are going to show below.

We turn now to concrete applications of both kinds of criteria. Kogut et al. (1983) have measured large gaps in the internal energies of gluons (and fermions in the quenched approximation). The lattice size was $2\times 8^{3}$ and $4\times 8^{3}$. Celik et al. (1983) have recorded two-state signals from hot and cold starts, indicating two coexisting phases with different expectation values of the Wilson line at the critical coupling. On lattices with $8^{3}$ and $10^{3}\times 2$ (or $\times 3$, 4) sites they have also found a large latent heat as difference in the gluonic internal energies ($\Delta\varepsilon/T_{c}^{4}=3.75\pm 0.25$). Lattice expressions for the latent heat and other thermodynamic quantities will be given in section III B 2.

Brown et al. (1988) have performed a concurrent study to Bacilieri et al. (1988). They still find a first order deconfining transition, but it is less strong with signs of an increasing correlation length. The lattice volumes are enlarged to $16^{3}\times 4$ and $16^{3}\times 6$ and $24^{3}\times 4$ and $24^{3}\times 6$. The values for the latent heat (obtained from the gap in ($\varepsilon+p$)) are $\Delta\varepsilon/T_{c}^{4}=2.54\pm 0.12$ for $N_{\beta}=4$ and $\Delta\varepsilon/T_{c}^{4}=2.48\pm 0.24$ for $N_{\beta}=6$. This is $60\%$ and $25\%$ smaller than the earlier values on smaller lattices, respectively.

Correlation lengths are determined from a fit of Wilson line correlation functions (cf. Eq. (136) and (137) below for the definition) with a string potential ansatz in the confinement phase and a screened Yukawa potential in the deconfinement phase. The results are increasing correlation lengths by a factor of 2 to 3 on both sides of the transition. The relatively large correlation length can be traced back to the 3d-$Z(3)$-Potts model (cf. our discussion in section II A).

Bacilieri et al. (1988) focus their study on Wilson line correlation functions. Large correlation lengths are found, increasing proportional to the lattice size. This increase is interpreted as suggestion of a second order transition. Only a careful finite size scaling analysis of correlation lengths can show that this interpretation is most likely not adequate. Such an analysis is rather involved. Therefore we devote the following section to it.

Correlation lengths, mass gaps and tunneling events

We use a notation, which is independent of the application in a spin- or gauge model. We will distinguish physical and tunneling correlation lengths $\xi^{(p)}$ and $\xi^{(t)}$ with associated mass scales $m^{(p)}$ and $m^{(t)}$ both in the finite and the infinite volume. A physical or bulk correlation length in the infinite volume limit can be defined via the decay of the connected pair-correlation $C_{c}$ as function of the distance $z$ according to

[TABLE]

Here $C$ is the unsubtracted, disconnected pair correlation and $B$ stands for the constant disconnected part, which is different from zero in the phase of spontaneously broken symmetry. The inverse correlation length $\xi^{(p)-1}=m^{(p)}$ is called the physical mass gap.

In the conventional lore it is the infinite volume correlation length, which diverges as $\beta\to\beta_{c}$ for a second order transition and stays finite in the first order case. Here we would like to stress the order of limits.

[TABLE]

In the opposite order

[TABLE]

The ‘critical’ correlation length $\xi_{L}^{(p)}(\beta_{c},L)$ diverges at $\beta_{c}$ in the infinite volume limit also in the case of a first order transition (Blöte and Nightingale, 1982; Privman and Fisher, 1983). This behavior is familiar, if we recall from section II A that singularities in second derivatives of thermodynamical potentials occur for first order transitions as well. The only difference to the second order case is the type of divergence: $\delta$- function singularities may occur in the specific heat and the magnetic susceptibility, when the infinite volume limit is taken at criticality.

Thus a diverging correlation length $\xi_{L}^{(p)}(\beta_{c},L)$ for $L\to\infty$ is not conclusive for a second order transition.

In the following discussion we consider infinite volume correlation lengths according to Eq. (126). In Table IV we compare the behavior of the physical and tunneling correlation lengths and their associated mass scales in the infinite volume limit.

In the first row of Table IV we have listed the behavior of the physical mass gap and its associated bulk correlation length, when the critical coupling $\beta_{c}$ is approached from the symmetric phase (i.e. $\beta_{c}$ from below) denoted as $\beta\to\beta_{c}^{-}$. In the first order case $\xi^{(p)}$ stays finite, while it diverges as $\beta\to\beta_{c}^{-}$ for a second order transition. In the broken phase the physical mass gap stays finite in both cases (second row).

In the lower two rows we introduce an ‘unphysical’ mass gap $m^{\rm(unp)}$ and its corresponding tunneling correlation length $\xi^{(t)}$. In the symmetric phase both quantities coincide with $m^{(p)}$ and $\xi^{(p)}$, respectively, while $\xi^{(t)}$ is always infinite in the broken phase in the infinite volume limit. Here $m^{\rm(unp)}$ may be defined via an analogous relation to Eq. (123), but this time via the decay of the disconnected correlation function

[TABLE]

The vanishing of $m^{\rm(unp)}$ defined via Eq. (129) just reflects the non-vanishing constant $B$ in the broken phase, which prevents an exponential decay of the correlation due to spontaneous symmetry breaking. The relation to the associated tunneling correlation length is not as simple as in Eq. (124).

From an analysis of the transfer matrix one expects in general for the (disconnected pair) correlation function that it is a coherent sum of its eigenvalues. If we consider only the two smallest eigenvalues above the ground state, the correlation $C(z)$ may be written as

[TABLE]

which should be compared with Eq. (129). Eq. (130) defines the tunneling correlation length $\xi^{(t)}$, or more precisely, $\xi^{(t)}$ arises in the spectrum of the transfer matrix from the level splitting associated with tunneling between different degenerate vacua. (We have anticipated this effect in section II A.)

This tunneling has a finite probability as long as the volume is finite, resulting in a finite $\xi_{L}^{(t)}$, but the potential barrier between the degenerate ground states becomes infinitely high in the infinite volume limit, leading to

[TABLE]

From Eq. (130) we recognize that as $L\to\infty$ the diverging $\xi^{(t)}$ gives rise to the constant disconnected term $B$ in the correlation function Eq. (123).

In a finite volume both length scales $\xi^{(p)}$ and $\xi^{(t)}$ are necessarily finite and (when considered at criticality, cf. Eq. (128)) both increase with increasing volume at a first order transition. Thus there is a two-fold risk of misinterpreting an increasing correlation length: the order of limits has not been properly arranged, and $\xi^{(t)}$ is intertwined with $\xi^{(p)}$.

Concrete realizations of the correlation function $C$ are spin-spin- or Wilson line-pair correlations, if we deal with applications in the $SU(3)$-deconfinement transition. We give the infinite volume definitions, written for a large, but finite volume $V=L^{3}$.

In the 3-d three-state Potts model considered by Gavai et al. (1989), the spin-spin correlations are defined as

[TABLE]

where $z=|i-j|$ and the corresponding zero-momentum projection reads

[TABLE]

where $s_{i}$ is the spin at site $i$ and $\bar{s}_{i}$ the average spin over a plane $i$, here $z$ denotes the distance between planes $i$ and $j$ along one of the principal axis of the lattice. The connected part

[TABLE]

defines the physical correlation length $\xi^{(p)}$ as

[TABLE]

in accordance with Eq. (123).

The analogous quantity $C_{0}(z)$ for an $SU(3)$-lattice gauge theory has been considered by Fukugita et al. (1989) and is the zero-momentum correlation function of Wilson lines

[TABLE]

where $i$ labels the planes, $\bar{\Omega}_{i}(z)$ is the average of Wilson lines $\Omega_{n}$ over the $i$th-plane with $n_{i}=z$, where

[TABLE]

i.e. the trace of the product over gauge field variables $U_{(n,\tau)}^{(4)}\in SU(3)$ associated to timelike links leaving the site $(n,\tau)$ in direction 4, $n$ denote the spatial coordinates. The additive constant $B$ of Eq. (123), that can be split off in $C_{0}(z)$, is given by $|<\Omega>|^{2}$.

Bacilieri et al. (1988) have calculated Wilson line pair correlations essentially with the same formulas as (136) and (137). They use a source method, where all link variables with z-coordinate 1 are fixed to the identity matrix $1\kern-3.01385pt1$. The correlation reads

[TABLE]

where $\Omega_{(x,y,z)}\equiv\Omega_{n}$ from Eq. (137).

Adaption to the finite volume. The above definitions for the various correlation lengths are applicable in the large volume limit, i.e. if $\lim_{L\to\infty}$in Eq. (126) is approximately realized. As subleading corrections from the finite volume are power-like, the large distance behavior of correlations ($z\to\infty$) is not accessible on lattices of moderate size.

For first order transitions it is rather hard to predict the finite size dependence from first principles. (The approach of Borgs and Kotecky (1990) comes closest to a rigorous derivation.) In general one has to make at some place an ansatz. The double peak Gaussian ansatz for the distribution function $P_{L}(E)$ has led to definite predictions of the finite size behavior.

Likewise it requires an ansatz for the functional form of the correlation function $C_{L}(Z)$ in a finite volume. A procedure, which has been adopted by Gavai et al. (1989) and Bacilieri et al. (1988) is to form ratios of correlation functions

[TABLE]

A finite volume ansatz for $C_{i}(z)$, which is inspired by the infinite volume decay and adapted to the choice of periodic boundary conditions on a cubic lattice is

[TABLE]

where $\xi$ stands for a generic correlation length, and $i=0$, 1. Eq. (140) includes only the first image of a source in $z$-direction, the one which is a distance $L$ apart. Multi-mirror images (accounting for images a distance $n\cdot L$ apart) should be also included, if the correlation can become large compared to the considered volume.

The careful analysis of Gavai et al. for the 3d-$Z(3)$-Potts model shows that different methods of extracting the physical correlation length $\xi^{(p)}$ from data on finite lattices do not agree in the critical region, although they do agree away from $T_{c}$. Depending on the ansatz for $R_{i}$, $\xi$ seems to grow with $L$, or to stay finite ($\xi\sim 10$ in lattice units) in the vicinity of $T_{c}$. The former behavior is compatible with $\xi=\xi^{(t)}$ as $L\to\infty$, the latter with $\xi=\xi^{(p)}$, i.e. a physical mass gap which stays finite at $T_{c}$ and indicates the first order of the transition.

$\bullet$ To summarize, it is the very definition of a correlation length in a finite volume, the order of limits ($L\to\infty$ and $\beta\to\beta_{c}$) and the occurrence of the tunneling correlation length, which impede the finite size analysis of correlation functions.

After this digression we proceed with concrete applications in the $SU(3)$-deconfinement transition. Bacilieri et al. (1988) considered ratios of finite volume correlations according to Eq. (140) with $C_{1}$ given by Eq. (138), very similar to the observables, which were later considered by Gavai et al. (1989). Rather than measuring the correlations $C_{i}(z)$, they applied a so called smearing procedure (Albanese et al., 1987) to generate operators weakly coupled to the high-energy fluctuations. We do not go into technical details, although the smearing technique is quite important for following the decay of the correlation over a large distance up to 15-$z$-slices on a lattice of cylinder geometry $12^{2}\times 48\times 4$.

Alternatively “the” correlation length $\xi$, defined via Eqs. (123) and (138) has been measured from a three-parameter fit with $C(z)=A\cdot\cosh~{}[m{1\over 2}L_{z}-z]$. The statements of the paper by Bacilieri et al. about the increasing correlation length refer to this length scale $\xi=1/m$. When $m$ coincides with the physical mass gap $m^{(p)}$, it is related to the string tension $\sigma$ via $m^{(p)}=\sigma\cdot N_{\beta}$. This relation holds in a pure gauge theory. Thus the statement about the physical correlation length $\xi^{(p)}$ tells us the behavior of the string tension in the critical region of the $SU(3)$- gauge theory.

The resolution of the contradicting statements about the order of the phase transition seems to lie in this ansatz and its interpretation

[TABLE]

In the critical region it should be probably replaced by an ansatz like Eq. (130) to account for both correlation lengths $\xi^{(p)}$ and $\xi^{(t)}$, or, if $\xi^{(t)}$ dominates, $m$ of Eq. (141) reflects the increase of $\xi^{(t)}$ rather than $\xi^{(p)}$.

A systematic finite size study of the 3d-3-state Potts model has been performed by Gavai et al. (1989). The Hamiltonian of the Potts model has been specified in section II. It is equivalent to a $Z(3)$-spin model. The volume dependence has been studied for bulk quantities like the average action $E$ and the order parameter $S$. When $S$ and $E$ are plotted as function of $\beta$, the crossover region becomes sharper with increasing volume. Also a typical manifestation of coexisting phases has been verified, the function $\beta(L,E)$ is multi-valued. One coupling (temperature) is compatible with several values of the free energy at the transition point.

Flip-flop signals in the order parameter are observed as function of the Monte Carlo time. They are reflected in a double peak structure of the order parameter distribution $P_{L}(S)$. Fig. 3 verifies what we expect from the general considerations of section II. With increasing volume the double peak structure gets more pronounced. (In a truly second order transition it fades away at larger volumes.)

In the critical region the rounding and shifting of thermodynamic quantities is governed by the thermal exponent $\lambda_{1}\equiv y_{T}=1/\nu=d/(2-\alpha)$, where $y_{T}=d$ for a first order transition (cf. section II). In particular the critical coupling should be shifted according to

[TABLE]

the width of the critical region $\sigma_{L}$ should shrink as

[TABLE]

The finite volume critical coupling $\beta_{c,L}$ and the width $\sigma_{L}$ have been extracted from a Gaussian fit for the relative population density of both phases in the critical region, and, alternatively, by the midpoint and length of the coupling interval ($0.367\leq\beta\leq 0.36703$), where flip-flops have been observed on an $48^{3}$ lattice. Results for both quantities are compatible with $y_{T}=d$, i.e. with a first order finite size scaling behavior.

The careful analysis of finite volume correlation lengths has been anticipated in the previous subsection. An estimate for the inverse physical mass gap leads to $\xi^{(p)}(\beta_{c})\sim 10$ in lattice units. Thus it stays finite at $T_{c}$ as it should for a first order transition.

The inclusion of an antiferromagnetic next-to-nearest-neighbor coupling with a relative coupling strength of $\gamma=-0.2$ does not change the qualitative result of a first order transition (Gavai and Karsch, 1989). The discontinuity in the order parameter is of the same magnitude as for $\gamma=0.0$, but the latent heat is found to be smaller by a factor of 2.

This result is interesting, as it corrects the naive expectation that a large gap in the order parameter goes along with a large latent heat. It raises the possibility that the deconfinement transition in the $SU(3)$ gauge theory is associated with a small latent heat due to slightly nonlocal effective couplings, even if the jump in the order parameter is not small. For the actual size of the latent heat in the $SU(3)$ gauge theory cf. section III B 2.

Finally we come to the $SU(3)$ gauge theory as considered by Fukugita et al. (1989). The lattice size ranges over $8^{3}\times 4$ to $36^{3}\times 4$ sites. Two-state signals have been observed for the size $N_{\sigma}=8-28$. The distinction between the two-states becomes conspicuous with increasing volume . Likewise the double peak structure in the order parameter distribution, here $P(|\Omega|)$ ($\Omega$ being the Wilson line operator of Eq. (137) is more pronounced for a larger lattice size.

The susceptibility of the Wilson line is given as

[TABLE]

Here $\Omega$ has been defined in Eq. (137), where $\hbox{Re}~{}\Omega$ is taken as the projection of $(\sum_{n}\Omega_{n}/V)$ onto the nearest $Z(3)$-axis on the complex $\Omega$-plane. The results for the maximum in the susceptibility lead to a scaling of $\chi_{\rm max}\propto V^{\gamma/(d\cdot\nu)}$ with $\gamma/(d\cdot\nu)=0.99(6)$ and a width $\Delta\beta\propto V^{-1/(d\nu)}$ with $1/(d\nu)=0.95(5)$ in reasonable agreement with the expected values of 1.

The ‘unphysical’ mass gap $m^{\rm(unp)}$ defined in Eq. (129) is obtained from a fit of $C_{0}(z)=A\cdot\cosh~{}[m^{\rm(unp)}(z-N_{\sigma}/2)]$. The expected behavior in the infinite volume limit has been shown in Table IV. For a first order transition in a finite volume $m^{\rm(unp)}$ should sharper decrease as $V\to\infty$ than for a second order transition. In fact, a crossing of the family of functions $m^{(unp)}(L,\beta)$ is observed, which is qualitatively in agreement with the expectations, although a detailed quantitative understanding is missing.

The physical mass gap $m^{(p)}$ is read off from an ansatz as Eq. (123) with $B=\left|<\Omega>\right|^{2}$, where the exponential is replaced by a $\cosh$ to account for the periodic boundary conditions. It is small, but stays finite at $\beta_{c}$ (cf. Table IV).

Fukugita et al. (1989 and 1990a) have also addressed the question of nonlocal couplings on an effective level of the $SU(3)$ gauge theory. An answer requires an explicit construction of the effective $Z(3)$ action from the original $SU(3)$ gauge theory. The $Z(3)$ variables are not introduced ad hoc (i.e. by assumption), but expressed in terms of the original field variables. An ‘effective’ $Z(3)$-spin $\omega_{n}$ is defined as the projection of $\Omega_{n}$ to the nearest $Z(3)$ axis (normalized to unit length). The generic form of an effective action in terms of $\omega_{n}$-variables is written as

[TABLE]

Here $O_{\alpha}$ are $Z(3)$-invariant monomials in $\omega_{n}$, $\beta_{\alpha}$ are the associated couplings, local as well as non-local, e.g. nnn-couplings. First correlation functions of the monomials $O_{\alpha}$ are calculated from the ensemble of $\omega_{n}$’s, generated in the original $SU(3)$-Monte Carlo runs. The number of terms labeled by $\alpha$ is truncated by hand. Next the couplings $\beta_{\alpha}$ are determined with the aid of Schwinger-Dyson equations. This method is known from Monte Carlo renormalization group calculations (Gonzalez-Arroyo and Okawa, 1987).

The results are the following. Effective two-spin couplings decrease to values less than $10^{-3}$ for distances larger than 3 between the two spins (on a $24^{3}\times 4$ lattice with $\beta=5.6925$). Three- and four-spin couplings are smaller than $(1-2)\times 10^{-3}$. The effective couplings decrease exponentially with the distance as anticipated by Svetitsky and Yaffe from the phase structure of the associated three-dimensional $SU(3)$ gauge theory.

The two-spin couplings (in particular the nnn-couplings), which are derived quantities in this case, turn out to be all positive. There is no sign of “antiferromagnetism”.

Like other numerical investigations this derivation is not rigorous. In particular it involves a truncation in the effective action, which is, however, unavoidable. In this sense none of the Monte Carlo results does actually prove the first order of the deconfinement transition, but different derivations give strong hints pointing in the same direction: the deconfinement transition in the pure $SU(3)$ gauge theory is of first order. The controversy about this topic has settled. The situation becomes considerably more involved, when fermions are included.

2 Thermodynamics on the lattice

An important contribution of lattice QCD to phenomenological applications of the QCD transition is a prediction of QCD’s equation of state. The behavior of the pressure $p$, the internal energy density $\epsilon$ and the entropy density $s$ are of most interest for seeking observable effects in heavy-ion collisions. In this section we summarize the lattice expressions of thermodynamic quantities and outline an approach due to Engels et al. (1990b) to derive the equation of state in an entirely nonperturbative framework. The method overcomes former inconsistencies observed for the pressure. Here we consider its application to the pure $SU(N)$ gauge theory followed by a discussion of finite size effects in the regions of critical and high temperature.

From section II.A we recall the continuum expressions for the energy density $\epsilon$ and the pressure $p$ as $\epsilon=(T^{2}/V)^{.}{\partial\over\partial T}\ln Z$ and $p=T{\partial\over\partial V}\ln Z$. One way of calculating these quantities on the lattice is to transcribe the derivatives which respect to $T$ and $V$ on the lattice. This has led to introducing different lattice spacings for the temporal ($a_{\beta}$) and spatial ($a_{\sigma}$) directions, and is the reason, why we have specified the lattice action directly on an asymmetric lattice. The derivatives are replaced according to

[TABLE]

After performing the appropriate lattice derivatives of ln $Z$, the lattice spacings are set equal again, $a_{\sigma}=a_{\beta}=a$.

Let us consider the first derivatives of the partition function. The order parameter for the pure gauge theory is the thermal Wilson line. Its lattice expression has been given in Eqs. (137)-(138). The order parameter for the chiral condensate will be given in Eqs.(III C 4a-b) in section III C.

The lattice expression for the internal energy density gets a contribution from the gluonic part and the fermionic part

[TABLE]

For completeness we give also the formula for the fermionic part. The continuum expression

[TABLE]

is transcribed to

[TABLE]

with $S$ given by the Wilson action or the action for staggered fermions, cf. section III A.

The energy density contains a contribution from the vacuum, which is a $\beta$-independent infinite constant, cf. Bernard (1974). When the vacuum contributions from the gluons $\epsilon_{G,vac}$ and the fermions $\epsilon_{F,vac}$ at $T=0$ are subtracted, the resulting lattice expressions are

[TABLE]

and

$\displaystyle\epsilon_{F}=N_{f}(4N_{\sigma}^{3}N_{\tau}a^{4})^{-1}\left\langle\hbox{Tr }\left\{D^{(4)}(D+m)^{-1}\right\}\right\rangle-\epsilon_{F,vac}\;,$

where $m_{f}$ has been set to $m$ for all flavors $f$ in Eq. (153) and

[TABLE]

The notation has been introduced in section III A. Note that the energy densities scale with $(1/a^{4})$. An uncertainty in $a(g)$ in units of [MeV] is amplified to the fourth power in the results for $\epsilon$. Further we see that $\epsilon$ is expressed in certain plaquette expectation values, which are easily accessible in a Monte Carlo simulation. In principle the coefficients $c^{\prime}_{\sigma}$ and $c^{\prime}_{\tau}$ can be determined nonperturbatively from lattice simulations (for first attempts see (Blum et al., 1995a)), but in the past the perturbative values (Karsch, 1982; Trinchero, 1983) have been used in the otherwise nonperturbative expressions. The use of perturbative values is questionable from the beginning. For the $SU(3)$ gauge theory large deviations from the perturbative $\beta$-function are known to occur for $g\geq 1$, and the $\beta$-function $a^{.}dg/da$ is related to the coefficients $c_{\sigma}^{\prime}$ and $c_{\tau}^{\prime}$ via

[TABLE]

with $\xi\equiv a_{\sigma}/a_{\tau}$, cf. section III A. Therefore significant nonperturbative contributions must be expected for $c_{\sigma}^{\prime}$ and $c_{\tau}^{\prime}$ as well.

The perturbative ingredients have actually led to inconsistencies in measurements of the pressure. The pressure turned out to be negative and discontinuous across the transition point. In consequence the values for the latent heat were not consistent. For a continuous pressure the latent heat $\Delta L$ should be the same, whether it is calculated as

[TABLE]

or

$\displaystyle\Delta(\epsilon-3p)/T_{c}^{4}\equiv\Delta L_{2}\;.$

Both Eqs. (159) are simple expressions in terms of plaquette expectation values. Results of Fukugita et al. (1990a) for an $SU(3)$ gauge theory gave

[TABLE]

The discrepancy between $\Delta L_{1}$ and $\Delta L_{2}$ is clearly outside the statistical error bars. In an independent simulation Deng (1989) found

[TABLE]

for the discontinuity in the pressure.

These difficulties have been overcome by a different approach to lattice thermodynamics, which can be entirely performed within a nonperturbative framework. The additional quantity now is the $\beta$-function $a^{.}dg/da$ from QCD, which is known in a nonperturbative region of couplings $g$ from Monte Carlo renormalization group calculations. Otherwise only plaquette expectation values have to be calculated as before. This approach has been proposed by Engels et al. (1990b). The steps are as follows.

• $\bullet$

The free energy density is calculated from an integration over its derivative with respect to $\beta$, since the logarithm of the partition function is not directly accessible within the Monte Carlo approach. With $f=-{T\over V}\ln Z$ one takes the derivative with respect to the gauge coupling $\beta=2N/g^{2}$

[TABLE]

where $S_{G}$ is the gauge part of the action (e.g. given by Eq. (97)) and $P_{T}$ is a short notation of the plaquette expectation value at temperature $T$, calculated on a lattice of size $N_{\sigma}^{3}N_{\tau}$. If $P_{0}$ denotes the corresponding zero temperature expectation value, evaluated on a lattice of size $N_{\sigma}^{4}$, the difference of free energy densities $f$ at coupling $\beta$ and $\beta_{1}$ is obtained as

[TABLE]

• $\bullet$

Now an additional assumption enters, which is strictly satisfied only for an infinitely large lattice. It is the relation $\ln Z=V\partial\ln Z/\partial V$, which is valid for homogeneous systems in large volumes. From $p=T{\partial\over\partial V}\ln Z$ and Eq. (163) one obtains

[TABLE]

In Eq. (164) $\beta_{1}$ has been chosen small enough that $p(\beta_{1})\sim 0$.

• $\bullet$

A second quantity, which has appeared in Eq. (159) and is easily calculable on the lattice, is the interaction measure $\Delta$

[TABLE]

As the name suggests, $\Delta$ vanishes for an ideal gas (where $\epsilon=3p$, see section V).

• $\bullet$

The advantage of calculating first $\Delta$ rather than directly $\epsilon$ is that $\Delta$ depends no longer on $c_{\sigma}^{\prime}$ and $c_{\tau}^{\prime}$, but on the QCD-$\beta$-function $B(g)$

[TABLE]

When $\Delta$ is to be evaluated in a nonperturbative coupling regime, it requires a knowledge of the $\beta$- function for nonperturbative couplings as well. This function has been inferred from Monte Carlo renormalization group studies (Hoek, 1990; Akemi et al., 1993). The functional form suggested in (Hoek, 1990) has been used by Laermann et al. (1995) in fits to the more recent data from Akemi et al. (1993). The remaining terms in $\Delta$ and $f$ are calculated in a Monte Carlo simulation. Once we have $f$ and $\Delta$, we know $\epsilon$ and the entropy density $s$ (with $f=\epsilon-Ts$). Using this approach, the previously mentioned problems with the pressure have disappeared, $p$ stays positive and behaves continuously at $T_{c}$.

A fully nonperturbative approach alone does not guarantee that the equation of state on the lattice is relevant for continuum physics. Earlier simulations have been performed for a time extension of four lattice spacings, although $N_{\tau}\sim 10$ is needed for the continuum limit.

The $N_{\tau}$-dependence has been studied in case of the $SU(3)$-deconfinement transition by Boyd et al. (1995) (see also Karsch and Laermann, 1995). The temporal lattice size has been varied from ($N_{\tau}=4$ with $N_{\sigma}=16$) to ($N_{\tau}=6$ with $N_{\sigma}=32$), and ($N_{\tau}=8$ with $N_{\sigma}=32$). The dominant type of finite size effects depends on the temperature. At high temperatures there is an intimate relation between finite size (IR-) and finite cut-off (UV-)artifacts, although the distinction between them may naively suggest their decoupling. The relation between IR- and UV-artifacts is seen as follows. A reduction of $a_{\tau,\sigma}(g)$ [fm] as function of $g$ goes along with an increase in temperature $T$ [MeV], if $N_{\tau}$ is kept fixed. This way continuum behavior is mixed with high temperature behavior. For a small lattice spacing just the high momentum modes are cut off which give the main contribution to the energy density and pressure at high temperatures. For a larger lattice spacing at lower temperatures also lower momentum modes are cut off, but the energy density and pressure grow approximately with $T^{4}$, thus the finite cut-off effects, induced by the finite number of time slices, are largest for high temperatures.

The $N_{\tau}=1/(a\cdot T)$-dependence of the gluonic part of the energy density of an ideal gluon gas is estimated as (Karsch, 1995)

[TABLE]

For $N_{\tau}=4$, $\epsilon_{\rm{gluonic}}(4)$ is 50 % larger than the corresponding Stephan-Boltzmann value in the continuum. The leading $1/N_{\tau}^{2}$-dependence has been used by Karsch et al. to extrapolate the $N_{\tau}=6,8$-data for $\epsilon$ and $p$ to the continuum limit. Fig. 4(a-b) shows the energy density and pressure for the various lattice sizes. It prevails the expected increase of the finite size/finite cut-off effects with temperature. There are strong deviations by about 20 % from ideal gas behavior (indicated by the horizontal lines) for temperatures at least up to $3T_{c}$ in the continuum limit. It should be further noticed that the pressure $p$ increases more slowly with $T$ than the energy density $\epsilon$. This leads to a strong deviation of the velocity of sound from the ideal gas value. The velocity of sound can be an input from first principles to applications in heavy-ion collisions, cf. section V.

The latent heat is indicated by the vertical bar in Fig. 4(a), the value is taken from Iwasaki et al. (1992). It is further reduced compared to earlier results (e.g. of Brown et al., 1988). The result is

[TABLE]

The latent heat is only 30 % of the energy of an ideal gas at $T_{c}$.

The interaction measure $\Delta$ is shown in Fig. 5 for three lattices. The deviation from ideal gas behavior is largest in the transition region, it goes logarithmically to zero with increasing temperature. The interaction measure is obviously less sensitive to finite size/finite cut-off effects than $\epsilon$ and $p$ separately. The deviations from the ideal gas behavior are assumed to be due to the nonperturbative infrared structure of QCD, which is less sensitive to UV-effects. The results of Fig. 5 support this assumption and indicate that QCD’s IR-structure plays an important role even at high temperatures. As mentioned in the introduction (section I), it is too naive to argue from asymptotic freedom that QCD behaves fully perturbatively at high temperatures.

In physical units the result for the interaction measure is given by (Karsch, 1995)

[TABLE]

In Eq. (169) the string tension $\sigma$ with $\sqrt{\sigma}\sim 420$ MeV has been used as physical input. Thus the main contribution to the interaction measure comes from the zero temperature gluon condensate $\left<G^{2}\right>_{0}=2.0~{}\rm{GeV/fm}^{3}$, noting that the trace anomaly ($\epsilon-3p$) is related to the gluon condensates at zero and finite temperature according to

[TABLE]

As argued above, UV-artifacts induced by the finite temporal extension play a dominant role in the high temperature region, but a minor role in the transition region. The main contributions to the finite size effects in the transition region come from the low momentum modes, in particular, if a correlation length becomes large. In case of the pure $SU(2)$ gauge theory the standard finite size scaling analysis for second order phase transitions can be applied to the nonanalytic behavior of the free energy density. The finite size effects are controlled by the ratio $N_{\sigma}/N_{\tau}=TV^{1/3}$ rather than by $1/N_{\tau}$ as in the high temperature region. In particular the normalized critical energy density $\epsilon^{\rm{crit}}/T_{c}^{4}$ should scale as function of $T\cdot V^{1/3}$ with critical indices of the three-dimensional Ising model (cf. section II B 2). This scaling behavior has been verified by Engels et al. (1995). An extrapolation to the infinite volume limit leads to

[TABLE]

for a pure $SU(2)$ gauge theory. This result may be considered as free of lattice artifacts. It is a continuum result although not continuum physics because of two colors and the absence of dynamical fermions.

We summarize. Reliable data for QCD’s equation of state exist for the limiting cases of a pure $SU(2)$ and $SU(3)$ gauge theory. Earlier discrepancies for results on the pressure have been removed by using fully nonperturbative ingredients in the derivation. To test the relevance of the lattice results for continuum physics, a careful finite size scaling analysis has been performed. The finite size effects are under control, as the predicted dependence on $N_{\tau}$ and $N_{\sigma}$ has been confirmed in the simulations. Therefore an extrapolation to the continuum limit ($N_{\tau}\to\infty$, $a\cdot T\to 0$, $T$ [MeV] fixed) is safe. Such tests should become standard in future lattice simulations.

The fully perturbative equation of state can be also applied when dynamical fermions are included. We report on first attempts in this direction in section III C 6. However, a finite size scaling analysis gets much more involved in the presence of dynamical fermions. For sufficiently large quark masses the singularities of the first order $SU(3)$ or the second order $SU(2)$ deconfinement transitions will be rounded anyway. One then has to disentangle, which part of the rounding comes from the finite volume and from the finite masses (see section III C.2 and 3).

3 Interface tensions in QCD

In recent years the interface (or surface) tension has been frequently calculated for models of QCD. Like the latent heat the surface tension is an important measure for the strength of a first order transition. The interface occurs between phases, which coexist at the critical temperature. A large surface tension leads to a strong supercooling effect. The onset of the phase conversion is delayed, as the critical temperature is reached. One possible scenario for the phase conversion is droplet formation. Once the system has sufficiently supercooled below the critical temperature, the gain in energy from the conversion to the new phase can compensate for the energy loss in interface free energy, and the phase conversion sets in.

In applications to QCD different kinds of interfaces must be distinguished. In a first order chiral transition we have interfaces between phases with broken and restored chiral symmetry. Interfaces between the deconfinement and the confinement phases may occur in a first order deconfinement transition. They will be considered below for calculations of interface tensions. Furthermore different realizations of the plasma phase are separated by interfaces. Such interfaces do not only occur in the region around $T_{c}$, but throughout the phase of broken $Z(3)$-symmetry above $T_{c}$. One specific phase realization corresponds to a spatial domain, which is characterized by a certain expectation value of the Wilson line operator. The associated interface free energy density is called an ordered/ordered interface tension $\sigma_{oo}$, whereas an interface between the plasma and the hadronic phase leads to an ordered/disordered interface tension $\sigma_{od}$. We write ‘plasma phase’ as a common name for the deconfinement and/or chiral symmetric phase and ‘hadronic phase’ for the confinement and/or chiral symmetry broken phase. Via the relative magnitude of both quantities at $T_{c}$ one may gain some insight about the degree of wetting.

Wetting is an alternative or competing mechanism to droplet formation. One phase may spread along a 2-dimensional front into the other. Stated in a liquid/gas picture a droplet of liquid may wet an interface between a gas and a solid when these phases coexist. Whether the wetting is complete or not depends on the relative size of $\sigma_{oo}$ and $\sigma_{od}$.

In QCD the issue of wetting has been raised by Frei and Patkós (1989). It has been studied in systems which share the global Z(3)-symmetry, in the three-state Potts model and in an effective model of Polyakov loops. Later the question of wetting at the deconfinement transition has been investigated in an effective three-dimensional $Z(3)$- symmetric $\Phi^{4}$-model for the Polyakov loop (Trappenberg and Wiese, 1992). In this model the confinement phase completely wets the different realizations of the deconfinement phase at $T_{c}$.

So far these results are not applicable to QCD transitions under realistic conditions, since fermions have been excluded. Recall that the inclusion of fermion masses may completely wash out the first order nature of the chiral and the deconfinement transition. A smooth crossover phenomenon prevents the occurrence of different coexisting phases and their associated interfaces. If the hints of today will be confirmed in the future that both transitions fade away for physical fermion masses, the various interface tensions have no impact on applications in the early universe or relativistic heavy-ion collisions.

How to measure an interface tension

Consider a system, where a domain with volume $V_{H}$ in the (hadronic) phase $H$ is embedded in a second phase $Q$ (the quark-gluon-plasma phase) with volume $V_{Q}$. If we denote by $f_{h}$ and $f_{Q}$ the free energy densities of both phases, the free energy $F$ of the total system is given as

[TABLE]

$F$ differs from the sum of the domain free energies just by an amount $F_{s}$, $F_{s}$ equals the free energy which is associated with the interface separating both domains.

As long as $V_{H}$ and $V_{Q}$ are finite, and an external field $\beta$ is applied, which is homogeneous, $F_{s}$ will be actually zero. Thermal equilibrium forbids the coexistence of phases as stable configurations in a finite volume. The system will tunnel from one phase into the other. A transient coexistence can only result from metastability effects. If one wants to measure the surface tension directly as the excess free energy due to an interface in a finite system, one has to stabilize the interface by an external field gradient $\Delta\beta$. Again the order of limits is essential

[TABLE]

where $\sigma$ is the surface free energy $F_{s}$ per unit interfacial area $A$, $\Delta\beta$ stands for a generic field gradient.

For the temperature-driven transitions in QCD an appropriate ‘field’-gradient is provided by the temperature. To have a preferred direction, we consider a spacetime lattice with spatial cylinder geometry, i.e. $L_{x}\times L_{y}\times L_{z}\times N_{\tau}$ with $L_{x}=L_{y}\ll L_{z}$. Choosing half of the lattice links $(z=1,2,\dots L_{z}/2)$ at a temperature larger than $T_{c}$ (corresponding to a coupling $\beta\equiv 6/g^{2}>\beta_{c}$) and the other half $(z=L_{z}/2+1,\dots,L_{z})$ below $T_{c}$ $(\beta<\beta_{c})$, an interface between a deconfinement and a confinement phase has been enforced onto the system. It will be located somewhere ‘between’ $z=L_{z}/2$ and $L_{z}/2+1$. Early calculations of the surface tension in an $SU(3)$ gauge theory have been performed along these lines (Kajantie et al., 1990). The procedure involves the $\Delta\beta\to 0$-extrapolation in the very end. The result was $\sigma/T_{c}^{3}=0.24\pm 0.06$. It has been obtained for a time extension of $N_{\tau}$=2. The value is probably too large by an order of magnitude (cf. below). One reason are possible finite size effects due to the small extension in time direction. The other reason is that the extrapolation to $\Delta\beta\to 0$ may be not safe. An external field can lead to a suppression of fluctuations of the interface, if it is strong enough. This way the interface looks more rigid than it would be in the $\Delta\beta\to 0$ limit.

Most of the work for interface calculations in an $SU(3)$ gauge theory has been concentrating on time extensions of $N_{\tau}=2$, as the measurements are rather time consuming (Kajantie et al., 1991; Potvin and Rebbi, 1989; Huang et al. 1990 and 1991; Grossmann et al., 1992; Janke, 1992). Results for $N_{\tau}=4$ can be found in (Brower et al., 1992; Potvin and Rebbi, 1991; Grossmann and Laursen, 1993). In the following we report on work by Iwasaki et al. (1994), who extend the number of time slices up to $N_{\tau}=6$.

Iwasaki et al. apply the histogram method, which has been introduced by Binder (1981 and 1982) for Ising-type-systems, to an $SU(3)$ gauge theory. The histogram method is based on the analysis of probability distributions $P(\Omega)$ of order parameters $\Omega$ in the vicinity of the phase transition. As we have seen in section II A, the probability distribution $P(\Omega)$ develops a characteristic double peak structure close to $T_{c}$. The structure gets more pronounced as the volume is increased. This behavior had been identified as a characteristic signature for a first order transition in a finite volume, cf. Fig. 3. The valley between both peaks corresponds to configurations with interfaces, which are the more suppressed the larger the volume.

As the suppression comes from the extra costs in interface free energy, it is plausible that the value of $\sigma$ may be inferred from the position of the maxima and minimum of $P(\Omega)$ as a function of volume. In the formulas below cylinder geometry $(N_{x},N_{y}\ll N_{z})$ and periodic boundary conditions are assumed. Thus two-interfaces will be preferably be performed in the $(x,y)$-plane with total area $2A=2N_{x}N_{y}\cdot a^{2}$, $a$ being the lattice constant. The order parameter $\Omega$ will be identified with the Polyakov loop.

The ansatz for the probability distribution $P(\Omega)$ which has been chosen by Iwasaki et al. (1994) is then given as

$\displaystyle P(\Omega)=P_{1}(\Omega)+P_{2}(\Omega)+P_{m}(\Omega)$

with

[TABLE]

where the following notations are involved. The order parameters $\Omega_{1}$ and $\Omega_{2}$ denote the values in the confinement and deconfinement phases, $f_{1}$ and $f_{2}$ the corresponding free energy densities, $V_{1}$ and $V_{2}$ are the associated volumes occupied by each of the coexisting phases. The coefficients $c_{i}$ and $d_{i}$ depend on the volume. In Eq. (175) for $P(\Omega)$ we recognize the superposition of the Gaussians around the characteristic peaks for each phase. The third term $P_{m}(\Omega)$ gives the probability for finding the mixed phase. Here is the place where the interface tension enters. The probability $P_{m}(\Omega)$ depends on $\Omega$ via the volumes $V_{1}$ and $V_{2}$ which are occupied by both phases such that $\Omega V=\Omega_{1}V_{1}+\Omega_{2}V_{2}$.

The weight factors $\exp(-f_{i}V/T)(i=1,2)$ are proportional to the probabilities for the system to reside in the confinement or deconfinement phase. While $f_{1}$ and $f_{2}$ are degenerate at $T_{c}$ in the infinite volume limit, they are in general different in a finite volume. To project on $\sigma$ in a finite volume $V$, i.e. on $\sigma_{V}$ or $\hat{\sigma}_{V}\equiv\sigma_{V}/T_{c}^{3}$, one has to define $\hat{\sigma}_{V}$ in a way that the leading $V_{i}$-dependence in the exponent cancels out. The cancellation is achieved, if $\hat{\sigma}_{V}$ is defined according to

[TABLE]

Here $p_{max,1}$ and $p_{max,2}$ are the two maxima of $P(\Omega)$, while $p_{min}$ denotes the minimum between the two peaks. The powers $\gamma_{1}$ and $\gamma_{2}$ denote the weights of the contributions of $\Omega_{1}$ and $\Omega_{2}$ to $\Omega$ at the minimum, i.e. $\Omega=\gamma_{1}\Omega_{1}+\gamma_{2}\Omega_{2}$. The infinite volume limit of $\hat{\sigma}_{V}$ is the value for the interface tension in units of $T_{c}$

[TABLE]

The actual measurement of $\hat{\sigma}_{V}$ is more subtle. The subtleties concern the choice of the order parameter, the critical coupling in a finite volume and the determination of $p_{max}$ and $p_{min}$. Candidates for order parameters are the action density or the Polyakov loop. The Polyakov loop is a complex-valued observable. The distribution $P(\Omega)$ develops 4 peaks near $T_{c}$ corresponding to the confinement phase with $\Omega_{i}=0$ and 3 realizations of the deconfinement phase in the directions $\exp(i2\pi n/3),n=0,1,2$. To reduce the numerical effort for obtaining high quality data, Iwasaki et al. have projected $\Omega$ on the real axis by taking the absolute value $\Omega_{abs}$ or by rotating it with $\exp(i2\pi n/3)$ such that $-\pi/3<\;{\rm arg}\;\Omega\leq\pi/3$ and taking the real part, leading to $\Omega_{rot}$. The dependence of the results on this choice has been checked.

The choice of the critical coupling in a finite volume is a nontrivial issue. It should guarantee the correct infinite volume limit of $\beta_{c}$. Here $\beta_{c}$ has been adjusted such that the peaks of the histograms for $\Omega_{rot}$ had equal height.

The actual minimum and maxima of the histograms may be contaminated due to statistical fluctuations. Therefore the extrema were read off from 3rd order polynomial fits to the histograms in the vicinity of the extrema.

Only the leading volume dependence drops out of $\hat{\sigma}_{V}$, when it is calculated according to Eq. (177). Subleading corrections in $1/V$ arise e.g. from fluctuations of $\Omega$ in the bulk phases, capillary wave fluctuations of the interfaces and zero modes corresponding to the translation of interfaces in the direction perpendicular to the interfaces. If the geometry of the lattice deviates from an idealized cylinder in $z$- direction, interfaces are no longer restricted to the $xy$-plane, but sweep out in other directions as well. All of these finite volume effects have been taken into account by making an appropriate ansatz for the volume dependent prefactors in the formula for $p_{min}/p_{max}$. For further details we refer to Iwasaki et al. (1993).

An important point about the actual measurement of $\sigma$ concerns the generation of histograms. In case of the $SU(3)$ gauge theory high statistics histograms were available from the QCDPAX-collaboration (Iwasaki et al., 1991 and 1992a). They were used as input in the $\sigma$- measurements of Iwasaki et al. (1994). If such histograms are not available for the quantities one is interested in, one can dispose of a more advanced method to generate such histograms by means of so-called multicanonical algorithms.

Multicanonical algorithms

We briefly digress from the $\sigma$-measurement to sketch the idea of a multicanonical updating procedure. It has been developed by Berg and Neuhaus (1991) and first applied to 2-dimensional $q$-state-Potts models. For large volumes it is notoriously difficult to generate configurations corresponding to the valley between the two peaks. These configurations are suppressed by the large amount of surface free energy, which is just the quantity of interest. The reason is that usual local upgrading procedures like the Metropolis- or heat bath algorithms are designed for single-peaked distributions $P(\Omega)$. Configurations are selected with an importance sampling according to their Boltzmann weight $\exp\{-\beta S\}$. A precise calculation of the maxima and minimum of the double-peak structure of $P(\Omega)$ requires frequent tunneling through the minimum, implying exponentially large autocorrelation times $\tau\propto\exp\{2\sigma L^{d-1}\}$.

The idea of the so-called multicanonical approach is to generate an auxiliary distribution $P^{\prime}(\Omega)$ (see Fig. 6) which does not suffer from the original problems, but can be related to the canonical distribution in a controllable way. Fig. 6 gives an example for the auxiliary distribution as function of the action density $s\equiv S/V$, which is approximately flat between the two maxima. We have

[TABLE]

The index $L$ stands for the linear size $L$, $n_{L}(S)$ denotes the spectral density and $P_{B}^{\prime}(S)$ has the form of a Boltzmann weight factor. In practice one starts with a guess for the parameters in the exponent of $P_{B}^{\prime}(S)$ such that $P_{L}^{\prime}(S)$ is approximately flat. Configurations with the multicanonical probability $P_{B}^{\prime}$ are generated with a generalized Metropolis or heat bath updating procedure. The original distribution $P_{L}(S)$ (or $P_{L}(\Omega)$) is then obtained from $P_{L}^{\prime}(S)$ by reweighting (Baumann and Berg, 1985; Ferrenberg and Swendsen, 1988 and 1989).

The efficiency of the multicanonical algorithm has been well established (see e.g. Berg and Neuhaus, 1992; Janke, 1992). The algorithm has been generalized to an $SU(3)$ gauge theory by Grossmann et al. (1992), where the time extent was restricted to $N_{\tau}=2$. For $N_{\tau}=4$ Grossmann and Laursen (1993) have used the histograms of the QCDPAX collaboration (Iwasaki et al., 1991 and 1992a) and Fukugita et al. (1989a and b). Their $N_{\tau}=4$- results for $\sigma$ are compatible with the results of Iwasaki et al. (1994) which we discuss in the following.

The results of Iwasaki et al. (1994) are based on the histograms of the QCDPAX-collaboration for lattices of temporal extent $N_{\tau}=4$ and 6 and spatial sizes ranging from $N_{x}N_{y}N_{z}=12^{2}\times 24,24^{2}\times 36,20^{3},24^{3}$ and $36^{2}\times 48$. An extrapolation to the infinite volume limit leads to

[TABLE]

The results do not depend on the actual choice for the order parameter ($\Omega_{rot}$ or $\Omega_{abs}$). To assess the relevance of these numbers for the continuum limit, one has to check the scaling property, i.e. sending $N_{\tau}\to\infty(a\to 0)$ while keeping $T_{c}$ in physical units fixed. It is obvious from Eq. (180) that $\sigma/T_{c}^{3}$ violates scaling. The scaling violation is even more obvious, if the $N_{\tau}=2$ value is included ($\sigma/T_{c}^{3}=0.103(7)$). A similar violation of scaling has been observed for the latent heat $\Delta L$, when it is measured from the same data of the QCDPAX-collaboration (Iwasaki et al., 1991 and 1992a). Thus it is not surprising that the scaling violation approximately drops out, if certain ratios of $\Delta L$ and $\sigma$ are considered. Such a ratio is given by

[TABLE]

which enters the amount of supercooling and the average distance between nucleated bubbles in the early universe. Let us see, what the results for $\sigma/T_{c}^{3}$ of Iwasaki et al. (1994) imply for possible relics of the QCD transition in the early universe.

4 Phenomenological implications for the early universe

The QCD transition happened at $\sim 10^{-6}$sec after the big bang, when the universe cooled down to a temperature of $\sim(150-200)$ MeV. In the following we describe one possible scenario for the succeeding evolution which has been frequently discussed in the literature (see e.g. Kajantie and Kurki-Suonio, 1986; Fuller et al., 1987; Meyer et al., 1991; Applegate et al., 1987; Applegate, 1991; Schramm et al, 1992).

If the QCD transition is of first order and proceeds via bubble formation, the nucleation will not set in exactly at $T_{c}$, but start at some lower temperature $T_{i}$ at time $t_{i}$. The plasma phase supercools, until the gain in free energy due to the phase conversion can compensate the costs in interface free energy between the coexisting phases. The degree of supercooling cannot be too large, as the cooling of the universe goes adiabatically slowly as viewed from the scale of QCD.

In the vicinity of the QCD transition two very different time scales enter the evolution: the QCD time scale of the order of $1/T_{c}\sim 1~{}\rm{fm/c}\sim 0.33\cdot 10^{-23}~{}\rm{s}$ and the Hubble time of the order of $1/\chi=0.36\cdot 10^{-4}~{}\rm{s}\sim 10^{19}/T_{c}$ with $\chi=\sqrt{8\pi GB/3}$, where $G$ is the gravitational constant and $B$ the vacuum energy density represented by the bag constant. The time dependence of temperature $T(t)$ follows from Einstein’s equations combined with QCD’s equation of state (e.g. the bag model equation of state (Kajantie and Kurki-Suonio, 1986)). It is given by

[TABLE]

with $M_{Pl}$ denoting the Planck mass, i.e. the expansion rate is $10^{-19}$ times more slowly than typical time scales in QCD.

At time $t_{i}$ and temperature $T_{i}$ nucleation sets in, bubbles of critical radius are formed. Their growth rate is slow compared to the shock waves which they emit. The shock waves propagate with the velocity of sound $v_{sh}=1/\sqrt{3}$. They reheat the plasma to some higher temperature $T_{i}$. The release of latent heat prevents further nucleation in regions of space which have been affected by shock waves. In such regions only bubbles will grow which have been created before. The creation and growth of bubbles continues until the shock waves collide, which are emitted from various nucleation centers.

At this time ($t=t_{PT}$), when the entire universe has been affected by shock waves, nucleation shuts off. A new scale becomes important. It is the average distance R between hadronic bubbles at the time when the shock waves collide. This scale is a measure for the inhomogeneities in the baryon number density. During the coexistence of both phases most of the baryon number resides in the plasma phase. Since the hadronic bubbles are nucleated through random thermal or quantum processes, they contain on average no net baryon number. In thermal and chemical equilibrium the chemical potentials $\mu_{Q}(B)$, $\mu_{h}(B)$, associated with the elementary baryon charge, are equal in both phases (h and q). The ratio of the baryon number density in the quark ($q$) and hadron ($h$) phases $n_{B}(q)/n_{B}(h)$ is approximately given by $\exp(m_{N}c^{2}/T_{c})$, where $m_{N}$ is the nucleon mass, since the light current quark masses are almost zero.

A competing scale to the scale of baryon number density fluctuations is the neutron diffusion length. The diffusion length of neutrons depends on the age of the universe. In the period where the age of the universe is $20~{}\mu\rm{s}-1~{}\rm{s}$, the weak interactions are fast enough to ensure a uniform neutron to proton ratio. The ratio of neutrons to protons is given by $(\exp(m_{n}-m_{p})/k_{B}T)$. Protons and neutrons diffuse along with each other. Neutrons really diffuse, protons diffuse by converting into a neutron through the weak interaction, diffusing, and converting back. At the age of $\sim$1sec, when the temperature falls to $\sim 1$ MeV, the weak and electromagnetic interactions decouple. The weak interaction drops out of equilibrium. The neutrons stay neutrons, the protons stay protons. The proton diffusion length is very short, because the proton is electrically charged. The diffusion length of neutrons is larger by several orders of magnitude. If the baryon number density fluctuations survive until the decoupling of weak interactions the neutrons will diffuse out of regions which are rich in baryon number. This way baryon rich regions are transformed into neutron poor (or proton rich) regions, vice versa, baryon poor regions become rich in the number of neutrons. The ratio $\exp((m_{n}-m_{p})/k_{B}T)$ becomes spatially inhomogeneous.

Neutron rich and proton rich regions constitute different initial conditions for the primordial nucleosynthesis. An inhomogeneous universe uses neutrons less “efficiently” than a homogeneous one. (In proton rich regions 2 neutrons makes 1 alpha, while 4 neutrons are needed in a neutron rich part ($4n\to 2p+2n\to 1\alpha$). It has been shown by Applegate et al. (1987) that primordial nucleosynthesis in an inhomogeneous universe leads to a distinctive pattern in the light element abundances. In particular the production of 7Li is increased as compared to the standard scenario without baryon number inhomogeneities.

Thus the alluring prediction of this scenario is that the observation of the light element abundances nowadays provides a snapshot of the universe at its infancy (the time period of the QCD transition). Clearly it is a question of relative size of the competing scales, as to whether the inhomogeneities in the baryon number densities characterized by the mean interbubble separation $R$ transform into variations in the $n/p$-ratio determined by the neutron diffusion length. If $R$ is too large, neutron diffusion will not leave an effect. If $R$ is too small, the baryon number inhomogeneities are washed out, as both neutrons and protons diffuse over these scales. In addition back diffusion of neutrons into proton rich regions must be taken into account. They further reduce the range of length scales, over which the QCD inhomogeneities can effect the primordial nucleosynthesis.

Let us finally indicate, how the average interbubble distance $R$ depends on the surface tension $\sigma$. (Notice that in this context $\sigma$ belongs to one of the few quantities which can be calculated from first principles and do not depend on the “choice” of the scenario.) The average separation $R$ defined at the time when the shock waves collide is given by (Kajantie and Kurki-Suonio, 1986)

[TABLE]

where $v_{sh}=1/\sqrt{3}$ is the velocity of the shock waves (as introduced above), $t_{growth}$ denotes the length of the time interval between the onset of nucleation and its completion in the sense that shock waves start colliding. The duration $t_{growth}$ is known, once the fraction $f$ of the universe which has been affected by shock wave fronts goes to 1. This fraction is determined by the nucleation rate $\Gamma$ per unit time and volume according to

[TABLE]

In classical nucleation theory the rate $\Gamma$ is proportional to $T^{4}\cdot\exp(-F_{b}/T)$, where $F_{b}$ is the free energy of the bubble of critical size or the minimal work for nucleating a bubble. Before the nucleation the free energy $F_{before}=-p(V+V^{\prime})$, where $V$ and $V^{\prime}$ are the volumes of the coexisting phases. After the nucleation we have $F_{after}=-pV-pV^{\prime}+\sigma\cdot A$. For one spherical bubble it follows $F_{after}-F_{before}=-(p^{\prime}-p){4\pi\over 3}r^{3}+\sigma\cdot 4\pi r^{2}$. A bubble of critical radius satisfies $\partial(F_{after}-F_{before})/\partial r=0$, leading to

[TABLE]

It remains to express the difference in pressures of both phases via the latent heat $\Delta L$. We have $\Delta L=T_{c}{\partial\over\partial T}(p_{q}-p_{h})$ such that in the vicinity of $T_{c}$ it follows $(p_{h}-p_{q})=\Delta L\cdot(T_{c}-T)/T_{c}$ or

[TABLE]

with $\alpha^{2}=(16\pi\sigma^{3}/(3\Delta L^{2}\cdot T_{c}))$. Notice that the surface tension and the latent heat enter the nucleation rate only in the combination of $\alpha$. The degree of supercooling is determined as $(T_{c}-T_{PT})/T_{c}$, $T_{PT}$ is defined as the temperature, where the nucleation shuts off or the fraction $f(t)$ approaches 1. From $\Gamma(T)$ and $f(T)$ the degree of supercooling is finally obtained as (Iwasaki et al., 1994)

[TABLE]

and the average separation of nucleation centers as

$\displaystyle R\approx v_{sh}\pi^{1/3}e^{\tilde{\chi}/4}\cdot\tilde{\chi}^{-3/2}T_{c}^{-1}\cdot\alpha$

with

[TABLE]

For the QCD transition in the early universe $\tilde{\chi}\approx 173$, $T_{c}\approx 150$ MeV, $v_{sh}\approx 1/\sqrt{3}$. Now we are ready to insert the value for $\alpha$ which has been obtained by Iwasaki et al. (1994) from the measurement of the surface tension and the value $\Delta L$ taken from the QCDPAX collaboration (Iwasaki et al., 1991 and 1992a). The degree of supercooling turns out to be as small as

[TABLE]

and the average distance between hadronic bubbles

[TABLE]

The value of $R$ is clearly too small to lead to an observable relic in the present day light element abundances (see e.g. Meyer et al., 1991).

One should keep in mind that the calculation of $\sigma$ in the pure $SU(3)$ gauge theory does not directly apply to the QCD transition in the early universe, although the main conclusions probably would remain unchanged. Dynamical matter fields will not increase the degree of supercooling. In general matter fields weaken (or even completely wash out) a first order transition.

C Including dynamical fermions

The ambiguity in studying the order of the deconfinement transition in the $SU(3)$ gauge theory has been caused by the finite volume. A finite size scaling analysis has resolved the controversy. It is natural to attempt a finite size scaling analysis also in the presence of dynamical fermions. Other artifacts can superimpose on finite size effects. In part 1 of the following section we summarize some typical pitfalls in lattice calculations. In the second part we report on the finite size scaling analysis of the chiral transition by Fukugita et al. (1990b). The results are not yet conclusive. Possible effects of finite quark masses are disregarded, although finite masses amount to an ordering effect, which may well compete with the finite volume. Therefore we turn in part 3 to a finite mass scaling analysis. It serves to discriminate possible UV-artifacts in the chiral transition. Bulk transitions are a further manifestation of UV-effects. With the example of $N_{f}=8$- flavor QCD we illustrate bulk transitions in section 4. Results for the more realistic cases of 2 and 3 flavors are summarized in section 5. The equation of state for two-flavor QCD is the topic of section 6. In section 7 we summarize some progress which has been achieved with Wilson fermions.

1 Pitfalls on the lattice

• $\bullet$

Finite size effects. Finite size effects are specific for the Monte Carlo approach, which is frequently used in lattice calculations. They contaminate any numerical calculation which is performed in a finite volume, if the largest correlation length of the system is not small compared to the smallest linear size. In the case of first order transitions we have presented a phenomenological approach (cf. section II). The formulas of section II for the scaling of the specific heat or the susceptibility as function of the linear size $L$ hold in the zero external field limit $h=0$.

As we have seen in section III A, Monte Carlo calculations of QCD with dynamical fermions are necessarily performed at finite bare quark masses. The updating time is estimated to increase proportional to $m_{f}^{-11/4}$. The chiral limit $m_{f}\to 0$ must be extrapolated. Thus a second pitfall is a contamination of data through finite mass effects.

• $\bullet$

Finite mass effects. In section II we have stressed the similarity between the finite volume $L^{3}$ and an external field $h$, and between an external field and finite quark masses. In the vicinity of $T_{c}$ one may consider a finite mass scaling analysis at vanishing $1/L$ rather than a finite size scaling analysis at vanishing $h$. For a moment let us assume that the condition $1/L\to 0$ is satisfied to a sufficient accuracy. A mass scaling analysis then provides a tool to control the rounding of algebraic singularities in correlation functions due to finite quark masses. The precondition is a second order phase transition. (An analogous analysis may be performed for a first order transition and rounding effects due to finite masses as well). The need of extrapolation to zero masses is specific for exploring the chiral limit, when dynamical fermions are included.

A third class of pitfalls which is known from the pure $SU(N)$ gauge theory and is common to any latticized version of a continuum model are artifacts due to the UV-cut-off.

• $\bullet$

Artifacts due to the UV-cut-off. Here we distinguish three manifestations. The most obvious one occurs in the interpretation of any physical observable, when the lattice units are translated to physical units. In section III A we have already argued about the ambiguity in predicting $T_{c}$ [MeV]. Such an ambiguity occurs for the QCD transitions in the presence of dynamical quarks, when the quark mass in lattice units, e.g. $ma=0.025$, shall be translated in units of MeV, although the lattice is rather coarse grained and a splitting of a mass according to $m^{.}a$ is not unique.

A less obvious and more subtle UV-artifact is a change in the effective symmetry group of the lattice action as a function of the bare coupling. This artifact is lately a consequence of the ‘No-Go’-theorems referring to lattice regularized actions for fermions. As we have mentioned in section III A, the lattice action with massless fermions has a global $U(n)\times U(n)$ symmetry for $n$ species of staggered fermions. Only in the continuum limit the full $SU(N_{f})\times SU(N_{f})$ flavor symmetry will be restored with $N_{f}=4^{.}n$. Far outside the continuum region the lattice action has only the global $U(1)\times U(1)$ symmetry for $n=1$.

A global $U(1)\times U(1)$ symmetry may trigger a second order phase transition with $O(2)$- critical exponents, whereas the spontaneous breaking of the restored $SU(4)\times SU(4)$ symmetry in the corresponding continuum limit is supposed to induce a first order transition (Pisarski and Wilczek, 1984). Thus the order of the chiral transition may change when passing from strong to weak couplings. Such a possibility has been discussed by Boyd et al. (1991) and will be the topic of section 3.

If one expects an $SU(2)\times SU(2)$ flavor symmetry for the continuum limit, the continuum phase transition would be also of second order. In that case only the critical exponents would change in passing from strong to weak couplings.

If one wants to describe an odd number of flavors in the continuum limit within the staggered fermion formulation, one usually “corrects” for the desired number of continuum flavors on the level of determinants. The correction is performed after integrating over the fermionic degrees of freedom, whereas the fermionic operator $Q$ is that of the original lattice action. The flavor symmetry in the original action is broken and describes at least four flavors in the continuum limit. The representation of the prefactor of the determinant-term in Eq. (102) as $N_{f}/4$ cannot be derived. It should be considered as a prescription when $N_{f}$ is not an integer multiple of 4. In this case it is even more difficult to infer the symmetry that drives the phase transition and leads to a vanishing condensate above $T_{c}$.

So far the UV-effects are supposed to modify the critical indices or the order of the chiral transition. A third manifestation of the UV-cut-off are bulk transitions. Here the very transition itself is an artifact of the lattice. Bulk transitions are not restricted to QCD with dynamical fermions. They occur in pure $SU(N)$ gauge theories as well. As an illustration we consider the example of $N_{f}=8$-flavor-QCD in section III C 4. For 8 flavors the bulk transition is either dominant or superimposed on the usual finite temperature transition of QCD.

2 Finite size scaling analysis

We report on the work by Fukugita et al. (1990b) on a finite size scaling analysis of the chiral transition in the presence of fermions. The fermionic action is given by Eqs. (99). The quark mass $m_{f}$ is chosen as $0.025$ for $N_{f}=2$ flavors and as $0.025$ and $0.0125$ for $N_{f}=4$ flavors (with $l=N_{f}/4$ in Eq. (102)). The chiral condensate $\left\langle\bar{\chi}\chi\right\rangle$ and the Wilson line $\Omega$ are take as order parameters, although they strictly have this meaning only in the limiting cases $m_{f}\to 0$ and $m_{f}\to\infty$, respectively. The expectation value of the chiral condensate is calculated according to Eq. (III C 4) below. The Wilson line expectation value is defined as

[TABLE]

with $N=N_{\sigma}^{3}{}^{.}N_{\tau}$. Otherwise the notations have been introduced in section III. The expectation value $\left<\ldots\right>$ is calculated with the measure and the effective action of Eqs. (102). The temporal extent is fixed to $N_{\tau}=4$, while the spatial size is varied between $N_{\sigma}^{3}=4^{3}$ to $12^{3}$. The adopted algorithm is the R-algorithm of Gottlieb et al. (1987a) with a time step size of $\Delta\tau=0.02$. The accuracy of this algorithm has been checked against the exact hybrid Monte Carlo algorithm.

Fig. 7 shows the time history of the Wilson line Re $\Omega$ for $N_{f}=2$ and $m_{f}=0.025$ on an $N_{\sigma}^{3}\times 4$ lattice for various spatial lattice extents. It illustrates that the ‘naive’ criteria for identifying a first order chiral transition are not unique on small lattices.

The signatures of interest are signs of metastabilities and associated two-state signals.

The time evolution should be compared with Fig. 8, where the time history of Re $\Omega$ is displayed for the 4-flavor case at $m_{f}=0.025$ on an $N_{\sigma}\times 4$ lattice for several values of $N_{\sigma}$. In the $N_{f}=4$ case distinct flip-flops are seen for $N_{\sigma}=$6 and 8, and a single event for $N_{\sigma}=12$. (The tunneling rate goes to zero in the infinite volume limit.) For $N_{\sigma}=10$ no flip-flop is visible, but a metastability is observed as a two-state separation. When transitions between both phases occur via tunneling and tunneling events are rare, the phases are metastable over a long time. The system remains in the ordered phase over thousands of iterations starting from an ordered start and in the disordered phase for a random start.

A crucial ambiguity is hidden in what is called ‘long time’. A comparison of Figs. 7 and 8 shows a significant difference in time scales. The fluctuations in Fig. 7 look more irregular on a large scale. On a short time scale the interpretation may change. Consider Fig. 7 for the $8^{3}\times 4$ lattice. If the runs would have been stopped after 2000 iterations, the fluctuations (probably of statistical origin) could have been misinterpreted as a two-state signal. In the longer run of 10000 iterations they look like a statistical fluctuation.

Fig. 8 indicates for the $10^{3}\times 4$ lattice, how important a check of the dependence on the initial conditions is (whether one uses a hot or a cold start). From a single run one can never exclude that a flip-flop will occur, if one waits for a long enough time. For large volumes the metastabilities can be so pronounced that the system is in one phase over the entire simulation. The phase is interpreted as a truly stable phase and a first order transition is easily overlooked. Thus the choice of the volume has to be optimized. It may not be too small to see distinct flip-flops, but not too large to see them at all.

Further indications for the first order transition are taken from the finite size scaling behavior of various susceptibilites. Susceptibilities are ‘magnetic’ response functions which can be expressed in terms of an order parameter $O$ according to

[TABLE]

Fukugita et al. have investigated the finite size scaling of $\chi$ for different choices of the order parameter field: the Wilson line Re $\Omega$, the chiral condensate $\bar{\chi}\chi$ (not to be confused with the susceptibility itself), and the average plaquette $P\equiv\hbox{Tr }U(\partial P)$ ($U(\partial P)$ denotes the product of $U$’s along the boundary of a plaquette).

From section II and Table II we recall that the peak in the susceptibility $\chi_{max}$ should scale with the volume $V$ according to $\chi_{max}=const+a^{.}V^{p}$, where $p=1$ for a first order transition and $p<1$ for a second order transition. The constant accounts for a contribution from the regular part of the free energy, which cannot be neglected as long as the volume is relatively small.

Runs with $N_{\sigma}\geq 10$ showing no flip-flops are of no practical use for the finite size scaling test with $\chi$, since the formulas are based on a phase coexistence in the sample with a non-vanishing weight. A two-parameter fit for $a$ and $p$ of the ansatz for $\chi_{max}$ with various values of $const\;(\geq 0)$ leads to an estimate of $0.78<p<1.1$ for the Wilson line-susceptibility in the case of 4 flavors. For comparison we mention the result for the pure $SU(3)$ gauge theory with a first order transition (Fukugita et al., 1989 and 1990a), where the exponent $p$ has been estimated as $0.86<p<1.0$. The authors conclude from the estimate in the 4-flavor case that it is consistent with the first order nature of the transition.

Results for the other susceptibilites lead to even larger bounds on $p$, in particular for the 2-flavor case they are not conclusive.

For a further criterion to test on the order of the phase transition we recall the definition of Binder’s cumulant Eq. (54). The minimum of Binder’s cumulant vanishes for a second order transition, but goes to a finite value $(\geq 0)$ in the case of a first order transition. Fukugita et al. have used a slightly different normalization

[TABLE]

Here $P$ denotes the average plaquette. The minimum $V_{L,min}$ should approach 2/3 for a second order transition, as the volume increases. (For the explicit volume dependence cf. Eq. (55)). When $V_{L,min}$ is plotted as a function of $1/V$, the trend is obvious that $V_{L,min}$ does not approach 2/3 for 4-flavor QCD. This is a further indication for a 1st-order transition. For 2 flavors the deviation from 2/3 is smaller than for 4 flavors, although larger than in the pure gauge limit $(N_{f}=0)$. The difference of $(V_{L,min}-2/3)$ goes almost to zero for $N_{f}=0$. This is rather surprising, as the transition is known to be of first order for zero flavors. Thus the analysis of Binder’s cumulant turns out to be not conclusive as well.

It has been already suggested by Fukugita et al. (1990b) that the relatively large quark mass of $m_{f}=0.025$ may considerably interfere with the finite size of the system. Therefore the 2-flavor simulations have been repeated for a smaller mass value of $m_{f}=0.0125$ (Fukugita et al., 1991). The characteristic features are very similar to those for $m_{f}=0.025$, and the order of the 2-flavor transition could not be determined there either. The favored possibilities are ‘no transition for finite $m_{f}$’ or ‘a second order transition in the $m_{f}\to 0$-limit’, but a first order transition is not excluded.

This concludes our discussion of IR-artifacts. We turn to UV-artifacts in the next two sections.

3 Finite mass scaling analysis

In this section we discuss the UV-artifacts on phase transitions in some more detail. We refer to the work by Boyd et al. (1991) dealing with possible changes in critical indices and a change in the order of the chiral transition as function of the bare coupling.

In the strong coupling limit the staggered fermion action of QCD is given by Eqs. (99) with one difference. An additional coupling $\gamma$ is introduced in the action, which is related to the anisotropy in the spatial and temporal lattice spacing in the weak coupling limit. It should not be mixed up with the critical index $\gamma$ which is also considered later. The notation should be clear from the context. The fermion operator $Q$ is written as

[TABLE]

with $D$ as in Eqs. (99). The temperature is varied by varying the coupling $\gamma$ on lattices with fixed number of time slices $N_{\tau}$. The chiral symmetry restoring transition is observed at some value $\gamma(N_{\tau})$ with $\gamma(N_{\tau})\to\infty$ for $N_{\tau}\to\infty$. The role of $\gamma$ is analogous but not identical to the anisotropy parameter $\xi$ (Eq. (109)) at weak couplings which is directly related to the lattice spacings $a_{\beta}$ and $a_{\sigma}$ and this way also to the temperature $T=1/(N_{\tau}{}^{.}a_{\beta})$. The particular role of the strong cooling limit can be guessed from the special case where $\xi=1$ and $a_{\sigma}=a_{\beta}$. An attempt to reach the high temperature region in the strong coupling $(\beta=0)$-limit requires $N_{\tau}<1$.

For vanishing quark masses the action (99) for one species of staggered fermions ($n=1$) is invariant under global $U(1)\times U(1)$ transformations acting on even lattices sites according to

$\displaystyle\chi^{\prime}=e^{i\alpha}\chi\hskip 28.45274pt\bar{\chi}^{\prime}=e^{-i\beta}\bar{\chi}$

and on odd sites

[TABLE]

For finite mass $m$ in Eq. (197) this symmetry is explicitly broken to $U(1)$.

Mean field calculations of Damgaard et al. (1986) and Fäldt and Petersson (1986) and numerical calculations of Klaetke and Mütter (1990) for $SU(2)$ suggest that the chiral QCD transition is of second order in the strong coupling limit. Second order transitions are immediately washed out in the presence of an external field, as we know from statistical physics. The singularities in thermodynamical functions will be rounded, when the infinite volume limit is taken at fixed, non-vanishing quark mass.

The critical index characterizing the ‘finite mass scaling’ at criticality is $1/\delta$ (cf. Table II). It is defined as

[TABLE]

where the reduced temperature $t=(T-T_{c})/T_{c}$ is now replaced by

[TABLE]

and the order parameter

[TABLE]

is the chiral condensate of strong coupling QCD. Here we have renamed the staggered fermion fields $\chi$, $\bar{\chi}$ as $\psi$, $\bar{\psi}$ to avoid confusion with the susceptibility $\chi$ later on. It can be expressed in terms of expectation values of monomers, which are easily calculable on the lattice (see e.g. Karsch and Mütter, 1989).

Further recall the critical index $\gamma$ (cf. Table II) which specifies the singular behavior of the susceptibility $\chi$, when $T$ approaches $T_{c}$ at zero field (here zero mass)

[TABLE]

Note the order of limits. To measure $\gamma$, first the infinite volume limit has to be taken and next the zero mass limit. (In the opposite order the order parameter would vanish also in the broken phase due to tunneling events in the finite volume). Eq. (204) is still a zero mass limit.

A finite mass scaling analysis proceeds in complete analogy to a finite size scaling analysis. The rounding and shifting effects on singularities are derived from a scaling ansatz for the free energy density. The non- analytic part of the free energy density of a generic statistical ensemble is the presence of an external field (the finite quark mass) is written as

[TABLE]

where $b$ is an arbitrary scale factor as in section II, $y_{t}$ and $y_{h}$ are the thermal and magnetic critical exponents ($y_{t}\equiv\lambda_{1},y_{h}\equiv\lambda_{2}$ in our notation of section II). Choosing the scale factor

[TABLE]

the free energy density transforms to

[TABLE]

Eq. (207) implies the finite mass scaling behavior of the order parameter $\langle\bar{\psi}\psi\rangle(t,m)$ and the susceptibility $\chi(t,m)$ in the vicinity of $\gamma_{0}$ (the critical coupling)

[TABLE]

and

[TABLE]

where $F$ and $F^{\prime}$ are scaling functions. It follows that the peak of the finite mass susceptibility occurs at

[TABLE]

and scales according to

[TABLE]

Eq. (211) should be compared with Eq. (41) for the finite size scaling of the peak in the susceptibility. Eq. (210) gives the shift in the critical coupling $\gamma_{0}$ or in the reduced critical coupling $t_{0}\to t_{m}$ due to the finite quark mass.

With $\delta=y_{h}/(1-y_{h})$ it is seen that the exponent $\delta$ can be measured either from the shift in $t_{m}$ according to Eq. (210) or from Eq. (209) for the susceptibility or the mass dependence of the order parameter Eq. (208) at $t=t_{0}\equiv 0$.

In case of a second order transition the exponent $\delta$ is characteristic for the universality class of the action. Thus a measurement of $\delta$ is of much interest for verifying that the restoration of the $U(1)\times U(1)$ symmetry drives the phase transition at $\gamma_{0}$.

Boyd et al. (1991) have proposed a related quantity, the chiral cumulant, to measure $\delta$ and $t_{0}$ from finite mass calculations. It has analogous properties to Binder’s cumulant (cf. Eq. (54)) and is defined as

[TABLE]

with $x\equiv t\cdot m^{-y_{t}/y_{h}}$. It follows from Eqs. (208) and (209) that $\Delta(0,m)$ gives $1/\delta$. The slope of $\Delta$ increases with decreasing $m$

[TABLE]

in a way that the ratio $\Delta$ itself has a $\Theta$- function shape for vanishing $m$

[TABLE]

Eq. (214) follows from the definition of $\Delta$ and Eqs. (208) and (209). When the ratios of $\Delta(t,m)$ are plotted as a function of $t$ for various values of $m$, the curves cross at the $m=0$ critical point $t_{0}$. The crossing comes from the fact that $\Delta$ increases with decreasing $m$ for $\gamma>\gamma_{0}$ and decreases with decreasing $m$ for $\gamma<\gamma_{0}$. This behavior is analogous to Binder’s cumulant $(\langle O^{4}\rangle/\langle O^{2}\rangle^{2})$ where $O$ stands either for the order parameter or the internal energy $E$. This way one may extrapolate the zero mass critical point $t_{0}$ from a series of finite mass measurements, cf. also (Binder, 1981).

Such a determination of the zero mass critical coupling is in principle free of an uncontrolled extrapolation. However, corrections originate in irrelevant terms, which may not be sufficiently suppressed in the vicinity of $\gamma_{0}$, and in contributions coming from the regular part of the free energy (cf. section II). A further source which leads to interfering effects with the finite mass scaling behavior is the finite lattice volume used in the Monte Carlo simulation. As we have seen above, the formulas (208), (209), (210), (214) hold in the infinite volume limit. In general finite mass- and finite volume effects are competing ordering effects. It is a question of relative size of both effects whether one is allowed to neglect one with respect to the other.

Let us compare the mass scaling behavior of the peak in the susceptibility in the infinite volume (Eq. (209))

[TABLE]

with the finite size scaling behavior in the zero mass limit (Eq. (41))

[TABLE]

The correction coming from $L<\infty$ as $m\to 0$ is small compared to the rounding due to $m>0$ as $L\to\infty$, if

[TABLE]

following from Eqs. (215) and (216). On the other hand the quark mass has to be chosen sufficiently small in order to keep the contributions from the regular part of the free energy density small. To keep the corrections small, the quark mass must be reduced proportional to $1/\sqrt{N_{\tau}}$, when $N_{\tau}$ is increased in the contributions from the regular part.

The chiral condensate, the susceptibility and $\Delta$ have been also measured in a Monte Carlo simulation for strong coupling QCD using the monomer-dimer-algorithm (see e.g. Karsch and Mütter, 1989). The lattice size $N_{\sigma}^{3}\times N_{\tau}$ is chosen as $N_{\sigma}=$4, 8, 16 and $N_{\tau}=4$. The values for the bare fermion masses are $m=$0.005, 0.01, 0.02, 0.04 and 0.1. The results are in agreement with the expectations: the chiral condensate vanishes in the symmetric phase, when the mass approaches zero. In the critical region $\gamma=2.3$-$2.4$ the finite size effects are strong. For a smaller lattice a larger mass is necessary to keep the finite size effects negligible in accordance with the condition (218). In Eq. (218) the critical indices $\gamma$, $\delta$, $\nu$ are taken from 3-dimensional $Z(2)$ or $O(N)$-models, for which $b\sim O(2)$. Hence the finite size effects should be small for the $16^{3}\times 4$ lattice over the entire mass range.

The chiral condensate $\langle\bar{\psi}\psi\rangle$ smoothly decreases in the transition region $\gamma\sim 2.3-2.4$ as function of $\gamma$ without any signal of a first order phase transition. The distribution $P_{m}(\langle\bar{\psi}\psi\rangle)$ of the order parameter does not show any sign of a double peak structure (cf. section II). We recall that the same lattice action as Eqs. (99) with (197) in the weak coupling limit has led to a first order transition (Gavai et al., 1990), where the continuum symmetry group $SU(4)\times SU(4)$ of 4-flavor QCD seems to be sufficiently restored to induce the first order transition.

The height of the peak in the susceptibility increases with decreasing $m$, but the statistics was not sufficient to measure $\delta$ according to the formulas (208), (209). Instead $\delta$ and the critical coupling $\gamma_{0}$ are estimated from a simulation of $\Delta$ with the result

[TABLE]

The curves $\Delta(\gamma)$ for different quark masses cross in the vicinity of $\gamma=2.35$. The result for $\delta$ is in agreement with $\delta$ for the three- dimensional $O(2)$ spin model ($\delta=4.755(6)$). It justifies the hypothesis that strong coupling QCD in the staggered fermion formulation is in the same universality class as the 3-d $O(2)$ spin model. The reason is the $U(1)\times U(1)$ remnant of the $SU(N_{f})\times SU(N_{f})$ continuum symmetry for 1 species of staggered fermions.

A more subtle case is that of $N_{f}=2$-flavors in the continuum, where the renormalization group analysis of Pisarski and Wilczek suggests a second order transition with $O(4)$ critical exponents. This hypothesis has been recently questioned by Kocić and Kogut (1995) in an analysis of the three-dimensional Gross-Neveu model. Kocić and Kogut find mean-field scaling behavior rather than the expected universality of the two-dimensional Ising model. This sheds doubts on the conventional lore that actions for composed scalars (i.e. scalar mesons composed of quark bilinears) share the universality class with sigma models for fundamental scalars. In particular the chiral transition in two-flavor QCD might have mean-field rather than $O(4)$-critical exponents.

In Monte Carlo simulations with staggered fermions $l=n=N_{f}/4$ is set to 1/2 in the effective action Eq. (102). No sign for a first order transition has been found by Brown et al. (1990a and b) for the 2-flavor case. Thus a second order transition is favored in the massless limit. The question now arises as to whether this continuous behavior is a remnant of the strong coupling $U(1)\times U(1)$ symmetry restoration or a restoration of the desired $SU(2)\times SU(2)$ continuum symmetry with either $O(4)$- or mean-field exponents. Karsch (1994) and Karsch and Laermann (1994) performed a finite mass scaling analysis of critical exponents in 2-flavor QCD to decide which of the three possibilities is realized: $O(2)$, $O(4)$ or mean-field exponents. The exponent $1/(\beta\cdot\delta)=y_{t}/y_{h}$ was read off from Eq. (210), expressed in terms of the critical and pseudocritical couplings $g_{c}$ and $g_{pc}$, respectively

[TABLE]

The exponent $1/\delta$ was determined from the chiral cumulant Eq. (212) evaluated at $x=t=0$. The results are summarized in Table V and compared with $O(4)$, $O(2)$ and mean-field values. The QCD result for $1/(\beta\delta)$ is consistent with the $O(2)$ and mean-field results, but slightly outside the error bars of the $O(4)$ value. For $1/\delta$ the QCD result is less consistent with mean-field, but in good agreement with the $O(4)$ and $O(2)$ values.

DeTar(1995) has analyzed the scaling behavior of $\left<\bar{\psi}\psi\right>$ as function of a scaled temperature (cf. Eq. (208)) with $O(4)$ critical exponents over a wide range of available data. The agreement is good apart from $N_{\tau}=12$ data. The origin of this discrepancy has still to be clarified. Possible explanations are a shift of the crossover temperature as function of $N_{\tau}$, erroneous extrapolations of lattice mass measurements or the very scaling hypothesis with $O(4)$ exponents. A clear identification of mean-field values in two-flavor QCD would certainly initiate much effort to get a deeper understanding of critical behavior in field theories with fermions.

The caveat to see unwanted features of the strong coupling symmetries also applies to the case of three flavors in the staggered fermion formulation. In the “worst” case, a crossover phenomenon might be a result of a second order transition in the strong coupling regime, that is immediately washed out, when finite masses are included. The crossover phenomenon then would be an UV-artifact indicating that the bare coupling $g$ was yet too strong.

Once a second order phase transition has been identified, a measurement of critical indices is not a minor detail for its further characterization. As we have seen, at finite temperatures and fixed $N_{\tau}$ the continuum limit coincides with the high temperature limit. Thus one has to increase $N_{\tau}$ to shift the critical coupling towards smaller values. The large volume limit is necessary for the continuum limit at low temperatures. In Monte Carlo simulations extrapolations to zero mass, zero lattice spacing and infinite volume are unavoidable. In the vicinity of a second order phase transition the critical exponents enter the extrapolation formulas. Thus their correct identification and knowledge of their precise values are needed for taking the right limits.

If finite mass- or finite size scaling analysis are not practicable or not applicable , an alternative fermion formulation (Wilson fermions) should be explored to reestablish the type of phase transition which has been observed in the staggered formulation. In particular, when odd numbers of flavors are described in the staggered fermion formulation, it is difficult to control, how the trick of doubling and reducing the flavor degrees of freedom affects the effective symmetry of the lattice action which triggers the phase transition.

4 Bulk transitions

Bulk transitions are phase transitions at zero temperature. Their very occurrence is a lattice artifact. Latticized systems are systems of statistical mechanics with their own dynamics. They do not care about a well defined continuum limit. Generic phase transitions can occur at some critical coupling, while the temperature is zero. Their physical meaning depends on the context. They have no physical relevance for continuum QCD, if they do not “survive” the continuum limit. Thus one would like to ignore them completely, but one is not allowed to do so.

The phenomenon of bulk transitions in lattice gauge theory is known from the pure gauge sector. According to the conventional lore on four-dimensional lattice gauge theories there should be no zero-temperature phase transition for non-abelian $SU(N)$ gauge groups, which separates the strong coupling ($g^{2}\gg 1$) from the weak coupling ($g^{2}\ll 1$) region. This is a desired feature, as it should guarantee that the continuum limit of lattice gauge theory includes both the confinement properties (proven on the lattice for strong couplings) and the asymptotic freedom in the weak coupling regime. A bulk transition may in principle destroy the confinement properties of the strong coupling regime.

Monte Carlo calculations for the $SU(2)$- and $SU(3)$- gauge groups with fields in the fundamental representation have verified the conventional lore. For $N\geq 4$ first order bulk transitions have been found for the $SU(4)$ lattice gauge theory by Moriarty (1981) and for the $SU(5)$ theory by Creutz (1981). Creutz argues, however, that the confinement property is not lost at weak couplings in spite of the transition.

In $SU(2)$- and $SU(3)$ gauge theories one observes a rapid crossover phenomenon between the strong and weak coupling regime instead. This is explained by a nearby critical point in the $(\beta,\beta_{A})$ plane, where $\beta\equiv 2N/g^{2}$ as above and $\beta_{A}$ denotes the coupling of a ${\rm Tr}_{A}U(\partial P)$ term in the adjoint representation of $SU(N)$. For two colors we have a mixed $SU(2)$-$SO(3)$ lattice action. A small $\beta_{A}$ leads to a bulk transition (Bhanot and Creutz, 1981), which is absent for $\beta_{A}=0$.

The interplay of bulk and thermal transitions in $SU(2)$ and $SU(3)$ gauge theories has been recently reinvestigated by Gavai et al. (1994), Gavai (1995), and Blum et al. (1995b). A bulk transition is signalled, if the location of the phase boundary stays fixed in coupling parameter space, independently of the number of time slices. In contrast, continuum universality for the thermal transition requires a shift of the critical couplings towards smaller values, as $N_{\tau}$ is increased, cf. section III A. For the mixed fundamental/adjoint $SU(2)$-action Marthur and Gavai found a shift towards weaker couplings of both the thermal and the bulk transition boundaries at $N_{\tau}=6$. This result sheds some doubts on the interpretation of the first order phase boundary as a bulk transition.

For the $SU(3)$ mixed fundamental/adjoint action the bulk and thermal phase boundaries coalesce for $N_{\tau}=4$, but split into two lines for small enough couplings and larger values of $N_{\tau}$ ($N_{\tau}=6$, 8) (Heller et al., 1995). The shift of the thermal transition line towards weaker couplings for increasing $N_{\tau}$ supports the hypothesis of continuum universality.

The peculiar behavior, which is sometimes found for the phase structure of QCD with dynamical fermions, has been addressed to bulk transitions in mixed fundamental/adjoint actions that are induced as effective actions from the integration over fermions. It is still an open question, whether the inclusion of dynamical Wilson fermion leads to a strong enough adjoint term in the $SU(3)$-mixed action that could explain the phase structure for Wilson fermions at large $\kappa$ and $N_{\tau}=6$ (Rummukainen et al., 1995).

Similarly the integration over the eight dynamical flavors in the staggered fermion scheme has been conjectured to induce an adjoint term of the $SU(2)-SO(3)$ mixed action in the effective action. This provides a possible explanation of the bulk transition, which is seen in $N_{f}=8$-flavor QCD (for $N_{\tau}\geq 8$-timeslices) (Brown et al., 1992). We discuss the 8-flavor case now in more detail.

The action for two species of staggered fermions is given by Eqs. (99). Integration over the fermionic variables leads to

[TABLE]

with

$\displaystyle{\cal Q}=D+m\cdot a\;,$

and $D$ is the Dirac operator acting on an $SU(3)$-triplet field $\phi$ according to

[TABLE]

Here the number of continuum flavors $N_{f}$ is chosen as 8, corresponding to $l=N_{f}/4=2$ species of staggered fermions. The dynamical quark mass $m\cdot a$ is chosen flavor independently as 0.015 throughout all simulations. The gauge coupling $\beta=6/g^{2}$ is varied between 4.5 and 5.0. The lattice size is $16^{3}\times N_{\tau}$ with $N_{\tau}=4$, 8, 16 and 32.

One may wonder why the special case of 8 light flavors is of any interest at all, as only 2 (or 3) quark flavors are approximately massless in nature. Nevertheless the reason for us is a physical one. Usually the strength of the chiral transition is argued to grow with an increasing number of flavors. Earlier work on $8^{3}\times 4$ (Kogut et al., 1985; Fukugita et al., 1988), $6^{4}$ (Kogut and Sinclair, 1988a), $8^{4}$ (Kogut and Sinclair, 1988a; Fukugita et al., 1988), and $16^{3}\times 4$ and $\times 6$ (Ohta and Kim, 1991) lattices shows a strong first order transition. A comparison of the transition for 2, 3, 4 and 8 flavors reveals a strengthening of the transition as the number of flavors increases (Ohta and Kim, 1991; see e.g. also Gottlieb, 1991), where the range of time slices $N_{\tau}$ lies between 4 and 8.

Usually this tendency is interpreted as reflection of a physically plausible effect: The chiral transition gets more pronounced the higher the number of flavors is which drive the transition, whatever the ‘driving dynamics’ in detail may be. Results about the $N_{f}=8$-transition expose this tendency as a possible lattice artifact and give rise to further studies in effective models about the supposed flavor dependence of the chiral transition.

Let us first give a qualitative description of the diagram displayed in Fig. 9. It is partly conjectural and summarizes the results for the $N_{f}=8$-transition of Brown et al. (1992), see also (Christ, 1992b; Dong and Christ, 1992).

The solid line locates the bulk transition separating the parameter space in a weak coupling ($\beta\geq\beta_{c}=6/g_{c}^{2}$) and a strong coupling ($\beta\leq\beta_{c}$) phase. For a temporal extent $N_{\tau}\geq 8$ the transition becomes $N_{\tau}$-independent and occurs at the same critical coupling $\beta_{c}=4.73(1)$ for $N_{\tau}=8$ and 16 or at $\beta_{c}=4.64(1)$ for $N_{\tau}=8$ and $\beta_{c}=4.62(1)$, when the finite time step interval $\Delta\tau$ in the integration step is varied (see below). This behavior signals the bulk feature of the transition.

Identification of the lattice artifact. As mentioned above, in a usual finite temperature transition with a correspondence in continuum field theory, the critical coupling (temperature) should scale with $N_{\tau}$ such that the physical transition temperature $T_{c}=(N_{\tau}a(g))^{-1}$ remains constant (cf. section III A.). Thus it is the absence of a shift in $\beta_{c}$ when $N_{\tau}$ is increased, which suggests the possible lattice artifact. The persistence of the transition on a symmetric ($T=0$) $16^{4}$ lattice further supports the interpretation as a bulk transition.

While the strong coupling phase shares features with the usual chirally broken phase at $N_{\tau}=4$, 6, the structure of the weak coupling phase (on the right hand side of the bulk transition) is more complex and less obvious in its physical meaning. The weak coupling phase is crossed by a hypothetical finite temperature transition, which should be relevant for the continuum limit. It separates regions with chiral symmetry broken ($\left<\bar{\chi}\chi\right>~{}\neq 0$) and chiral symmetry restored ($\left<\bar{\chi}\chi\right>~{}=0$) features. The conventional finite temperature transition may be recovered on very large lattices ($N_{\tau}>32$).

The solid squares refer to parameters of actual simulations, the open squares locate the critical couplings.

In what follows we sketch the criteria which have led to the conjectures of Fig. 9.

Basic observables. Basic observables are the chiral condensate and hadron masses. Of particular interest is their dependence on the bare quark masses. The chiral condensate of staggered fermion fields, defined as

[TABLE]

with a sum over all lattice sites $x$ is estimated by

[TABLE]

For each site $x$, $h_{x}$ is an independent, complex three-vector of Gaussian random numbers. This representation of $\left<\bar{\chi}\chi\right>$ follows from the effective action in terms of pseudofermionic fields $h_{x}$, cf. section III A. Eq. (LABEL:3.112b) shows the nonlocality of the fermionic condensate. The expectation value $<<\ldots>>$ denotes an average over gauge fields and random three-vectors $h_{x}$.

The mass $m$ entering Eqs. (III C 4) should be identical with $m$ of the effective action, as it enters the determinant in the path integral. Strictly speaking the full calculations should be repeated for several quark mass values, if the quark condensate and the hadron propagators are to be checked on their quark mass dependence. The observables are then measured on new sets of equilibrated gauge field configurations depending on the quark mass via the fermionic determinant. This way of proceeding costs an enormous amount of computing time.

In the actual measurements Brown et al. have calculated the observables on the same set of configurations, generated for an action with $m\cdot a=0.015$, but with varying valence quark masses: $m_{\rm val}=0.004$, 0.01, 0.025, and 0.05. Valence quark masses enter the observables via the propagator, whereas sea quark masses are used for the determinant in simulations with dynamical fermions. The mass entering Eqs. (III C 4) should be identified with a valence quark mass.

A conclusive criterion for studying the chiral transition is the scaling behavior of the chiral condensate $\left\langle\bar{\chi}\chi\right\rangle$ and the pion mass $m_{\pi}$ as a function of the quark mass $m$ in the limit of $m\to 0$. Spontaneous chiral symmetry breaking is indicated, if $\left\langle\bar{\chi}\chi\right\rangle$ stays finite for $m\to 0$, while $m_{\pi}(m)$ should vanish in the same limit. From chiral perturbation theory and PCAC-relations it is expected that $m_{\pi}^{2}$ vanishes linearly in the quark masses. As it is argued by Brown et al., the limit $m_{val}\to 0$ for fixed $m_{sea}$ may be conclusive as well.

The simulation method is the R-algorithm of Gottlieb et al. (1987a). The R-algorithm contains finite time-step errors of the order $(\Delta\tau)^{2}$, but requires half the number of Dirac propagator inversions per unit Monte Carlo time compared to the exact hybrid Monte Carlo method of Duane et al. (1987), when it used for the 8- flavor case.

Determination of the critical coupling. An accurate determination of the critical coupling is essential for an identification of the bulk transition. Here one has to face the usual dilemma, which is encountered for strong first order transitions. If the volume is chosen too large, strong metastabilities make both phases stable within a large range of couplings. This requires long Monte Carlo runs on large lattices. Metastabilities are less pronounced when the volume is small, but then the data are contaminated with strong finite size effects.

This problem is solved in the simulations of Brown et al. by starting with a mixed phase configuration. Small changes in the coupling cause the system to rapidly evolve into one of the phases, which by itself is a typical signal for a first order transition. Starting from a mixed phase configuration, the evaluation of $\left<\bar{\chi}\chi\right>$ is followed for several values of $\beta$ to get upper and lower bounds on $\beta_{c}$.

The first order of the bulk transition is concluded from Figs. 10 and 11. Fig. 10 shows the evolution of $\left<\bar{\chi}\chi\right>$ starting from hot and cold starts for $\beta=4.65$ on a $16^{3}\times 16$ lattice. The first order is signalled by the persistence of two phases over a time scale, which is considerably larger than the equilibration time. This criterion has been mentioned in section II as one of the naive criteria which are applicable for strongly first order transitions. A further evidence comes from the jump in the order parameter seen in Fig. 11, which is interpreted as a tunneling event between coexisting phases in the transition region (cf. section II). The evolution of the order parameter for the $16^{3}\times 32$ lattice at $\beta=4.6$ is shown for a cold start. A cold start corresponds to an ‘ordered’ configuration typical for the weak coupling phase (for a finite temperature transition typical for the deconfinement phase). Since the critical coupling $\beta_{c}=4.73$, the value $\beta=4.6$ belongs to the strong coupling regime. The tunneling event that occurs at $\tau\sim 250$ time units suggests that the enforced weak coupling phase (via the initial configuration) becomes unstable at the strong coupling value $\beta=4.6$.

Indications for chiral symmetry restoration. For $N_{\tau}=4$, 6 and 8 time slices the weak coupling phase (on the right hand side of the solid line in Fig. 9) shows the typical features of chiral symmetry restoration. The condensate $\left<\bar{\chi}\chi\right>$ extrapolates linearly to zero as $m_{\rm val}\to 0$. For $N_{\tau}=16$ and/or 32 the strong coupling phase has the typical features of a chiral symmetry broken phase.

Thus one would like to observe indications of chiral symmetry restoration in the weak coupling phase. The value of $\left<\bar{\chi}\chi\right>$ becomes smaller when going from strong to weak couplings, but less small than expected from naive scaling arguments. It extrapolates to zero as $m_{\rm val}\to 0$, but in a non-linear way, while $m_{\pi}^{2}$ extrapolates to a non-vanishing value. It is speculated that for $N_{\tau}=16$ and $\beta=4.65$ the weak coupling phase is itself near a finite temperature transition region.

For a larger value of $\beta$ ($\beta=5.0$) chiral symmetric behavior is manifest: hadronic screening lengths show parity doubling, $\left<\bar{\chi}\chi\right>$ extrapolates linearly to zero as $m_{\rm val}\to 0$ and $m_{\pi}\neq 0$ for $m_{\rm val}=0$, varying little with $m_{\rm val}$. For further (technical) details we refer to the original reference (Brown et al, 1992).

Our discussion may have demonstrated that the bulk transition does not merely replace the finite temperature transition and can be ignored as a lattice artifact. It seems to be superimposed on the structure of a finite temperature transition. For a smaller number of flavors ($N_{f}=2$, 3, 4) no bulk transition has been observed. There it may be even more difficult to discriminate precursors of the bulk transition at $N_{f}=8$ from continuum behavior.

Brown et al. (1992) attempt an explanation for the bulk transition as an outgrowth of the rapid crossover region seen in the pure $SU(3)$ gauge theory for $\beta=5.6$. When passing from strong to weak couplings in this region, a strong deviation from the scaling behavior predicted by the perturbative $\beta$-function is seen (Kennedy et al., 1985). Adding light dynamical quarks to the pure gauge action, the crossover region narrows with increasing number of flavors. The sharper the crossover, the stronger the violation of perturbative scaling and the larger the increase in slope of $N_{\tau}$ vs $\beta=6/g^{2}$. A plot of $N_{\tau}$ vs $\beta_{c}$ for $N_{f}=0$, 2, 4, and 8 flavors supports this view.

In concluding we summarize the phase structure, which is expected for various temporal extents. For small values of $N_{\tau}$ ($N_{\tau}\leq 4$) one has to work with a coarse grained lattice to reach the transition region.The scale is controlled by the lattice spacing. A single finite temperature transition is observed, but its relevance to a finite temperature transition in the continuum is not obvious.

At intermediate values of $N_{\tau}$ ($4\leq N_{\tau}\leq 16$) the corresponding critical couplings fall in the crossover region. A change occurs in what is called the relevant scale. At the end of the crossover region the scale becomes loosely related to the lattice spacing, but controlled by the continuum behavior. This rapid change of relevant scales is manifest in a bulk transition preventing any smooth change of $N_{\tau}$ vs $\beta$, both quantities are apparently unrelated.

For larger values of $N_{\tau}$ ($\geq 32$) (and also a simultaneous extension of the spatial lattice size to mimic a finite temperature box) the transition region is expected to lie in a coupling regime, where the lattice is rather fine grained. The bulk transition has disappeared, a finite temperature transition recurs. This time it will be related to the chiral transition of continuum QCD (taking for granted that it does recur).

Thus the $N_{f}=8$-simulations – although far from modelling realistic QCD systems – allow an identification of lattice artifacts, which may also influence more realistic lattice simulations ($N_{f}=2$, 3) in a weakened form.

5 Results for two and three flavors

The limiting cases we have discussed in the previous sections are not close to the realistic conditions of QCD. The pure gauge theory, the limit of four and eight nearly massless flavors and the strong coupling approximation may be regarded as tools to gain some insight in the dynamical origin of the QCD transitions, when it is viewed from certain limiting cases. They can further give some hints about the stability of QCD results with respect to variations of input parameters. Of particular interest is the role of the quark masses. For example a crossover phenomenon for experimental quark masses may be understood as a result of mass values, which are too small to sustain a first order deconfinement transition and too large to sustain the chiral transition. Such a conclusion can be drawn, if unrealistic mass values have been studied before.

Next we turn to the cases of two and three flavors, which come closest to the experimental relation of two light ($m_{u}\sim m_{d}\sim 5-7$ [MeV]) and one less light ($m_{s}\sim~{}150-180$ [MeV]) flavor. Here we discuss the results by Brown et al. (1990), which are still representative for 2 light and 1 heavier flavor.

The simulations are performed in the staggered fermion formulation. As we have argued above, the staggered formulation represents intrinsically only integer multiples of 4 continuum flavors. The projection on 2 or 3 flavors is enforced by writing $l$ of the effective action in Eq. (102) as $N_{f}/4$. Although the local fermionic operator ${\cal Q}$ describes $N_{f}=4\cdot n$ ($n$ integer) flavors, one allows $N_{f}$ to take the desired continuum value and uses as prescription for the effective action

[TABLE]

with $\beta=6/g^{2}$, $U(\partial p)$ denotes the product of $U^{\prime}s\in SU(3)$ along the boundary $\partial p$ of a plaquette $p$, $N_{u,d}$ is the number of continuum up- and down flavors, that is 2, and $N_{s}$ is the number of strange flavors in the continuum, $N_{s}=1$. The Dirac operator $D$ is given by Eq. (99), it describes 4 flavors of quarks in the continuum limit. Thus the representation of the prefactor $n$ as $N_{f}/4$ cannot be derived from an integration of a local action over fermionic degrees of freedom, the local staggered fermion action leads to $N_{f}=4\cdot n$, therefore one should call Eq. (229) a ‘prescription’.

The condensate is given by Eqs. (III C 4). The algorithm evolving the gauge fields with respect to the action Eq. (229) is the R-algorithm of Gottlieb et al. (1987a) with a step size of $\Delta\tau=0.0078$ for $N_{f}=2$ and 0.01 for $N_{f}=3$. The lattice size has been fixed to $16^{3}\times 4$. Table VI lists the quark masses which have been considered together with the results for the chiral transition. The results are based on Figs. 12(a)-(d).

No transition for the 2-flavor case is concluded from the time evolution of the ordered and disordered start, cf. Fig. 12(a). No sign for metastability is seen, as the two starts mix together without clear tunneling events. In Fig. 12(b) it is the absence of a double peak structure, which is taken as indication for ‘no transition’.

Fig. 12(c) shows the case of three light degenerate flavors. Here a two-state signal is visible. Over more than 2000 time units the system stucks in the ordered (disordered) phase depending on the starting condition. For two light and one heavy flavor (Fig. 12(d)) - the case which comes closest to realistic mass relations - the order parameter evolves similarly to Fig. 12(a). Clear signs of metastability and two-phase coexistence are absent. This result has been in conflict with earlier conclusions of a clear transition on an $8^{3}\times 4$ lattice (Gavai et al., 1989; Kogut and Sinclair, 1988b) and some evidence for a first order transition (Kogut and Sinclair, 1989).

At a first glance the absence of first order signals under realistic quark mass conditions have far reaching consequences for phenomenological implications in heavy-ion collisions. Many predictions rely on the first order nature of the chiral transition for 3 flavors (see section V). From a practical point of view the alternative between a truly first order transition and a crossover phenomenon may not be distinguishable, when the volume is small. The more sensible question to ask is as to whether the crossover is rapid enough to produce a sufficiently large gap in entropy densities over a small temperature interval. The preliminary answer seems to be positive even for the $N_{f}=2$-flavor case.

The jump in entropy density occurs within a $\beta$- interval, which corresponds to a temperature interval of less than 10 MeV. In section V C we will see that an entropy jump over a finite, but small temperature range is in principle sufficient to induce multiplicity fluctuations beyond the statistical noise.

Although a translation from lattice units into physical units should be taken with care for the considered $\beta$-value of 5.171, it is of interest to estimate bounds on the critical quark masses in units of MeV. Taking $a~{}[\rm{MeV}^{-1}]$ from Born et al. (1989), the set of masses $m_{u}\cdot a=m_{d}\cdot a=0.025$, $m_{s}\cdot a=0.1$ corresponds to $m_{u}=m_{d}\sim 12$ MeV, $m_{s}\sim 50$ MeV. These mass values give an upper bound on the critical quark masses, for which the chiral transition just ceases to be of first order and happens to be of second order.

In the case of three degenerate flavors $N_{f}=3$, two state signals are observed for $m_{u,d,s}=0.025$ (cf. Table VI), while no clear signals of metastability are seen for $m_{u,d,s}=0.075$ (Gavai and Karsch, 1985; Gavai et al., 1987), leading to an estimate for the critical lattice quark masses in physical units of $12~{}\rm{MeV}\leq m_{u,d,s}^{\rm{crit}}\leq 38~{}\rm{MeV}$. These numbers should be compared with more recent results on critical quark masses obtained in Wilson’s fermion scheme, see section III C 7. Lattice quark masses in physical units are related to the current quark masses in units of MeV by an unknown multiplicative renormalization factor.

Finally we come to the reliability of the results for 2 and 3 flavors. All simulations, reported so far, have been performed on a $16^{3}\times 4$ lattice. A temporal extent of 4 time slices leads to a transition region in the coupling range, where the lattice is rather coarse grained. For a pure gauge theory $N_{\tau}\geq 10$ is necessary to reach the continuum region. The effect of fermions is to further lower the effective lattice spacing, thus an even larger temporal extent would be necessary to reach the range of asymptotic scaling here.

The “distance” from the continuum limit is visible in results of Brown et al. (1990) for hadron masses, which have been obtained in separate $T=0$-simulations on a $16^{3}\times 24$ lattice with $\beta=5.171$, $m_{u,d}a=0.025$, and $m_{s}a=0.1$. These calculations have been performed to test on the very scaling properties in the considered coupling regime. The masses of two kaons which are degenerate in their flavor contents in the continuum limit still differ by a factor of 2. The nucleon over rho mass ratio $m_{N}/m_{\rho}$ is obtained as 1.5(1) in contrast to its physical value of 1.22 . The ratio of $m_{K}/m_{\rho}=0.46(1)$ is smaller than its physical value of 0.64 . This suggests that the strange mass entering the lattice K-meson is smaller than its physical value. Hence the first order transition for 3 flavors disappears already before the strange mass adopts its physical value. The unphysical masses of the flavor partners can influence the transition dynamics in a way which is difficult to control.

Other UV-artifacts due to the coarse grained lattice may be hidden in the results. From the discussion in section III C 3 it cannot be excluded that the smooth behavior for the 2-flavor case is a remnant of the strong coupling $U(1)\times U(1)$ symmetry. Going to larger $N_{\tau}$-values, a first order transition may recur. The same warning applies to the 3-flavor case with 2 light and 1 heavy flavors, as the $U(1)\times U(1)$ symmetry at strong couplings leads to a second order transition independently of the number of fermionic flavors.

Concerning possible superimposed features from the bulk $N_{f}=8$-transition, the $N_{\tau}=4$-extent is probably below the dangerous temporal sizes, where the associated coupling values fall in the crossover region from strong to weak couplings.

From experience with other simulations it is clear that the lattice size of $16^{3}\times 4$ is not large enough to be in the asymptotic large volume region, where finite size effects can be excluded.

More recent results for the 2-flavor case of staggered fermions have been obtained for larger $\beta$-values (i.e. smaller couplings), larger temporal extent $8\times 16^{3}$, and/or smaller values of the bare quark masses. The bare quark masses are $m\cdot a=0.00625$ (HTMCGC collaboration, see Bitar et al., 1993; Gottlieb et al., 1993) and (Bernard et al., 1995a) , $m\cdot a=0.004$ (Columbia group, see Mawhinney, 1993), and $m\cdot a=0.008$ and 0.016 on $24^{3}\times 12$ lattices (Bernard et al., 1995). The qualitative conclusions are the same. There is a rapid crossover for finite quark masses. As a progress it should be mentioned that the pion mass has decreased to the order of $T_{c}$ for the lowest value of the bare quark mass. In earlier calculations (e.g. Brown et al., 1990) it has been too large for heavier mesons to decay into pions.

Another good indicator for the crossover phenomenon is the baryon number susceptibility. It is expected to be small in the low temperature phase, since the baryon number can be only changed by creating or destroying a baryon, but large in the high temperature phase, where it is sufficient to create or destroy single quarks to change the baryon number. In (Bernard et al., 1995) the coupling dependence of the baryon number susceptibility $\chi$ has been used to determine the crossover coupling $6/g_{c}^{2}$ for 2-flavor QCD at given quark masses. The baryon number susceptibility has been also plotted as function of $T$ [MeV] for various lattice sizes. The $T$-dependence is obtained from $T/m_{\rho}=1/(N_{\tau}\cdot a\cdot m_{\rho})$, if the zero temperature spectrum calculations of $m_{\rho}\cdot a$ are extrapolated to the gauge coupling and the quark mass values of the thermodynamic simulations. Once the $N_{\tau}$-dependence is under control, a plot of $\chi(T)$ is of much interest for phenomenological applications. The baryon number susceptibility enters QCD’s equation of state, hadronization processes and heavy-ion collisions.

We conclude with a summary of conjectures and results for 2 and 3 flavors (based on the work by Brown et al., 1990) in Fig. 13. Indicated are the presence or absence of the finite temperature QCD transitions as a function of the quark masses $m_{s}\cdot a$ and $m_{u,d}\cdot a$ in lattice units. The solid line indicates second order transitions, the shaded areas enclose mass values leading to first order transitions. The concave shape of the critical boundary is hypothetical. The solid circles refer to mass points, where a first order transition is seen, the solid squares, where no transition is found in the simulations of Brown et al. (1990). The dashed circle indicates the location of the physical strange- to up quark mass ratio. Error bars for the location of the various points relative to the phase boundaries have been left out. The mass point (0,0) is the chiral limit of 3 flavors with an $SU(3)\times SU(3)$ symmetry in the continuum limit and a first order transition. The pure gauge theory in the upper right corner of the diagram ($m_{u,d}=\infty$, $m_{s}=\infty$) has an exact $Z(3)$-symmetry both in the continuum and on the lattice for 3 colors. The transition is of first order.

The line ($m_{u,d}=0$, $m_{s}$) has an $SU(2)\times SU(2)$ continuum symmetry, and a conjectured second order transition above some tricritical strange quark mass, where the line of first order transitions ends. The universality class of QCD may change along the second order transition line. It has been proposed to be the universality class of an $O(4)$-model for $m_{u,d}=0$ and sufficiently large $m_{s}$, a tricritical $\phi^{6}$-theory in the vicinity of the tricritical point (Wilczek, 1992; Rajagopal and Wilczek, 1993 (cf. section IV A 1)), and Ising like along the concave part with $m_{u,d}\neq 0$ (Gavin et al., 1994). An identification of the appropriate universality class of QCD along the critical phase boundary is still in an exploratory stage. The precise location of the critical phase boundary is under debate. In particular the location of the tricritical mass point on the ($m_{u,d}=0$)-axis is yet unknown. Both fermion schemes agree about the existence of a first order transition region, but do not agree about its extension. In the staggered scheme the physical mass point lies outside, in Wilson’s scheme inside the first order region. Strong coupling artifacts act in opposite directions in both schemes. As mentioned above, the smooth behavior in the staggered formulation may be a result of the $U(1)\times U(1)$-symmetry at strong couplings, while the first order behavior in Wilson’s formulation may be caused by the light fermion doublers at strong couplings (cf. section III C 7). Results for Wilson fermions will be summarized in section III C 7.

6 The equation of state for two-flavor QCD

Blum et al. (1995a and b) have studied the energy density and pressure as a function of temperature for two light flavors in the staggered fermion formulation of QCD. The approach is based on fully nonperturbative ingredients along the same lines as in the pure gauge theory (cf. section III B 2). The generalization comes from the dependence of observables on the gauge coupling $6/g^{2}$ and the bare light quark mass $m_{q}\cdot a$. Thus the $\beta$-function has now two components

[TABLE]

For a number of points in $(a\cdot m_{q},6/g^{2})$-space the $\beta$-function has been extracted from data for $(m_{\pi}\cdot a)$ and $(m_{\rho}\cdot a)$ measured as functions of $6/g^{2}$ and $a\cdot m_{q}$. To find the change $\delta(a\cdot m_{q})$ for a given change $\delta(6/g^{2})$ such that the physics remains unchanged, the mass ratio $m_{\pi}/m_{\rho}$ has been kept fixed. An alternative way of finding two equations for the two unknown functions $(a\cdot m_{q})(a)$ and $(6/g^{2})(a)$ is to fit $m_{\pi}/m_{\rho}$ and $m_{\rho}\cdot a$ as functions of $m_{q}\cdot a$ and $6/g^{2}$, leading to $a=a(6/g^{2},m_{q}\cdot a)$. The inverse function then yields the $\beta$-function corresponding to a symmetric change of lattice spacings.

The nonperturbative $\beta$-function enters the interaction measure according to

[TABLE]

Here $P_{T,0}$ denote the average plaquette expectation values at temperature $T$ or zero, respectively. They are calculated as

[TABLE]

$\left<\bar{\psi}\psi\right>_{T,0}$ are the corresponding light quark condensates

[TABLE]

The pressure $p$ is calculated by integrating either of the two relations

[TABLE]

(cf. Eq. (163) in section III B 2) or

[TABLE]

The lower limit for the integration should be chosen so that the contribution of the integrand is negligible, ‘cold’ stands for the cold symmetric lattice at zero temperature. The energy density is then obtained from Eqs. (236), (238) combined with Eq. (232). The results are shown in the two upper curves of Fig. 14 for two light bare quark masses: The lower curves denote three times the pressure. Although there is no transition in the strict sense, there is still a rapid rise in the energy density over a temperature interval of the order of 10 MeV around a temperature of 150 MeV. Also a slower rise in the pressure is seen. A shift of the transition interval to higher temperatures is observed for lower bare quark masses. This qualitative feature seems to be at odds with observations in effective models, where finite light quark masses delay the melting compared to the chiral limit, cf. section IV A.2 and 4.

The simulations of Blum et al. (1995a) have been performed for temporal extensions of $N_{\tau}=4$ or $N_{\tau}=N_{\sigma}$ with $N_{\sigma}=8$, 12 or 16. From the discussion of finite size effects in case of the pure $SU(N)$ gauge theories (section III.B.1 and 2) we expect that the temporal extent of $N_{\tau=4}$ is certainly not large enough for interpreting the lattice results as continuum physics. An extrapolation to the high temperature region and a comparison with perturbative calculations would be a good control which is yet outstanding. Thus the equation of state for two-flavor QCD is still in an exploratory stage.

7 Simulations with Wilson fermions

To confirm the conclusions of the last sections within the lattice approach, investigations in the Wilson formulation are indispensable. Staying within the same scheme it would be difficult to control the approximations of the desired continuum symmetries.

In the staggered fermion formulation the chiral limit can be obtained by varying the bare mass $m\cdot a$ in the Lagrangian to smaller values and extrapolating $ma=0$ in the end. In contrast the chiral limit in the Wilson fermion formulation must be determined as a one-dimensional submanifold $\kappa_{c}(\beta)$ in the two-dimensional $(\beta,\kappa)$-plane, where $\beta=6/g^{2}$ denotes the inverse gauge coupling and $\kappa$ the hopping parameters. $\kappa_{c}(\beta)$ is the line of critical hopping parameters, which characterize the chiral limit. The chiral limit can be defined as the vanishing of the pion mass on zero temperature lattices. Another possibility is to determine $\kappa_{c}(\beta)$ by the location of zeros in the fermion determinant. Both definitions are in general not equivalent, as the former definition involves an average over many gauge field configurations, while the latter definition does depend on the configuration. Other definitions of $\kappa_{c}$ on finite volume lattices have been proposed by Bitar et al. (1991), Bochicchio et al. (1985) and Iwasaki et al. (1989).

In the following we discuss the QCD transitions in a space of four parameters: the number of flavors $N_{f}$, the number of time slices $N_{\tau}$, the coupling $\beta=6/g^{2}$, and the hopping parameter $\kappa$. The line of finite temperature phase transitions/crossover phenomena from the confinement/chiral symmetry broken to the deconfinement/chiral symmetric phase will be denoted by $\kappa_{T}(\beta)$, while $\kappa_{d}(\beta)$ stands for a line of bulk deconfinement transitions (further explanations follow below). It may be useful to visualize the change of parameters by a look at Fig. 15. Fig. 15 indicates the shift of the thermal transition line $\kappa_{T}(\beta)$ between the confinement and deconfinement phase as a function of $N_{\tau}$ and $N_{f}$. Larger $N_{\tau}$ shifts the transition towards the continuum limit. Note that $\kappa_{T}(\beta)$ hits $\kappa_{c}(\beta)$ only at rather strong couplings. The qualitative features hold for $N_{f}\leq 6$.

Obstacles with Wilson fermions

Fukugita et al. (1986) were the first to point out certain obstacles that may hamper the investigation of the confinement phase in the chiral limit with Wilson fermions. For $N_{f}=4$, $N_{\tau}=3$ and spatial lattice sizes of $5^{3}$ the simulations have been performed for $3.0<\beta<6.0$ and $0<\kappa<0.25$. The transition region of the deconfinement transition has been defined as the range of $\beta$ over which the Polyakov loop and the gluon energy density vary rapidly. The result was that the transition line $\kappa_{T}(\beta)$ did not meet the chiral line $\kappa_{C}(\beta)$ down to $\beta\sim 3.5$. The lines were running almost parallel to each other to the strong coupling regime without crossing. The warning of Fukugita et al. was that the line of critical hopping parameters always stays in the high temperature phase for any finite lattice size. If the conjecture were proven to be true, it would be impossible to describe the confinement phase in the chiral limit within the Wilson formulation. The warning was extended to spectroscopic calculations with Wilson fermions, which might reflect high temperature behavior at $\kappa_{c}$, if the spatial lattice size is not sufficiently large.

Results of Bitar et al. (1991) for $N_{f}=2$-Wilson fermions on $4\times 8^{3}$-lattices pointed in the same direction. The hopping parameter was varied between $0.12\leq\kappa\leq 0.19$, the coupling region was $4.5<\beta<6.0$. The phase transition line $\kappa_{T}(\beta)$ was determined in this range, e.g. the transition/crossover coupling $\beta_{T}$ lies between 5.12 an 5.13 for $\kappa=0.17$. If the chiral limit is defined as the vanishing of the pion mass on zero temperature lattices, the chiral limit is nowhere reached in the low temperature phase on the $4\times 8^{3}$ lattice in the above parameter range. The conclusion is, if the line $\kappa_{T}(\beta)$ ever reaches the chiral limit line $\kappa_{c}(\beta)$, it must be in the strong coupling regime for these small lattice sizes.

Thermodynamics with Wilson fermions has been reinvestigated by Iwasaki et al. (1992b and 1993). The number of flavors is varied between 2 and 18. The number of time slices is extended from $N_{\tau}=4$ to $N_{\tau}=6,8$ with spatial lattices of $8^{2}\times 10$ and to $N_{\tau}=18$ with $N_{x}\cdot N_{y}\cdot N_{z}=18^{2}\times 24$. We distinguish the cases of $N_{f}\geq 7$ and $N_{f}<7$.

$N_{f}<7$. In the strong coupling limit ($\beta=0$) quarks are confined in the chiral limit. For $N_{f}=6$ the critical hopping parameter is given as $\kappa_{c}=0.25$. The line $\kappa_{c}(\beta)$ is defined as the values of $\kappa$ where $m_{\pi}^{2}$ vanishes in the confinement phase with a linear extrapolation in terms of $1/\kappa$. The confining behavior is concluded from the fact that the number of iterations for the quark matrix inversion $N_{inv}$ exceeds 10 000 for $N_{f}=6$ with $N_{\tau}=4$ at $\kappa_{c}$, while it is of $O(100)$ for $N_{f}\geq 7$ under the same conditions otherwise. The large value of $N_{inv}$ is attributed to the existence of zero eigenvalues of the quark matrix.

The most interesting question concerns the existence of a crossing point of the $\kappa_{T}(\beta)$ and the $\kappa_{c}(\beta)$ lines at some $\beta>0$. In the simulations of Iwasaki et al. (1992) the answer is positive. For $N_{f}=2$ and $N_{\tau}=4$ the crossing point $\beta_{cT}$ occurs at $3.9\leq\beta_{cT}\leq 4.0$ with $\kappa_{c}\sim 0.222$. The pion mass is consistent with zero at $\beta_{cT}$. The chiral transition is of second order or a crossover phenomenon. For $N_{f}=6$ and $N_{\tau}=4$, the crossing point lies between $0.2\leq\beta_{cT}\leq 0.3$ with $\kappa_{c}\sim 0.25$, the chiral transition is of first order, the value of $m_{\pi}^{2}$ depends on the initial configuration.

For $N_{f}=2$ and $N_{\tau}=18$ the crossing point $\beta_{cT}$ is only little shifted towards the continuum region, $\beta_{cT}\sim 4.5-5.0$. The unpleasant message is that one has to go to temporal extends $N_{\tau}>18$ to obtain a confining chiral limit closer to the continuum limit (i.e. for $\beta>5.0$).

$N_{f}\geq 7$. Here we have to distinguish between a line $\kappa_{d}(\beta)$ of bulk transitions which separates the chiral limit $\kappa_{c}(\beta)$ from the confinement region, and the line $\kappa_{T}(\beta)$ of finite temperature transitions. In the strong coupling limit ($\beta=0$) the phase of deconfinement and chiral symmetry restoration is realized for $\kappa>\kappa_{d}$ ($\beta=0$). The transition is stable under an increase of $N_{\tau}$ to $N_{\tau}=18$. Therefore the transition is called a bulk transition, cf. the discussion of bulk transitions for $N_{f}=8$-Kogut-Susskind fermions in section III C 4. The $\kappa_{d}(\beta)$-line of bulk transitions extends to $\beta>0$. In addition there is the $\kappa_{T}(\beta)$-line at $\beta>0$, which reaches the $\kappa_{d}(\beta)$-line without crossing the chiral limit. For $N_{f}=7$ and 12 the quark mass is $O(1)$ in units of $a^{-1}$ at $\kappa_{d}$. If this behavior persists in the continuum limit, confinement is lost for $N_{f}\geq 7$. Recall that asymptotic freedom is lost in QCD, if $N_{f}\geq 17$.

Peculiar behavior of the phase structure has been found by the MILC-collaboration (Blum et al., 1994) for $N_{f}=2$ at $N_{\tau}=4$ or 6. At $N_{\tau}=4$ the transition is smooth for light and heavy quark masses, but sharp for intermediate masses with $\beta\sim 5.0$. This is at odds with the general expectation that the strength of the crossover should monotonically decrease with increasing masses. At $N_{\tau}=6$ a first order phase transition with clear two-state signals is seen in simulations at $\kappa=0.17$, $0.18$ and $0.19$. It has been interpreted as a bulk transition located in the vicinity of the thermal crossover (Blum et al., 1994). Fortunately the lattice artifacts seem to disappear in a more recent simulation of the MILC-collaboration (Bernard et al., 1995b) at larger $N_{\tau}$ ($N_{\tau}=8$) at $6/g^{2}=5.3$ over a range of $\kappa$-values up to $\kappa_{c}\sim 0.168$. The thermal crossover is now shifted to a slightly smaller $\kappa_{T}\sim 0.167$, in qualitative agreement with Fig. 15.

$N_{f}=2+1$ flavors and critical quark masses

The Tsukuba group (Iwasaki et al., 1995a) performed simulations on $8^{2}\times 10\times 4$ and $12^{3}\times 4$ lattices for the physically interesting case of two light and one heavier flavor. The lattice parameters are translated to physical quark masses of $m_{u,d}\sim 0$, $m_{s}\sim 400$ MeV and $m_{u,d}=m_{s}\sim 150$ MeV. The hopping parameters are chosen as $\kappa^{(u,d)}=\kappa_{c}$ for the light quarks, while $\kappa^{(s)}$ follows from measurements of $m_{\rho}\cdot a$, $2m_{q}\cdot a$ as functions of ($1/\kappa-1/\kappa_{c}$) with $\kappa_{c}(\beta)$ referring to the 2-flavor case. The results of $m_{\rho}\cdot a$ as function of $N_{f}$, $N_{\tau}$, $\beta$ are used to set the scale. If the physical $m_{\rho}$-mass of 770 MeV is identified with $(m_{\rho}(\kappa_{c})\cdot a)/a$, it follows a lattice spacing of $a\sim 0.8~{}\rm{GeV}^{-1}$ for $\beta\leq 4.7$. Once $a~{}[\rm{GeV}^{-1}]$ is known, one can find $m_{q}\cdot a$ and the associated values of ($1/\kappa^{(s)}-1/\kappa_{c}$) corresponding to $m_{q}\sim 0$, $m_{s}\sim 400$ MeV and $m_{u,d,s}\sim 150$ MeV.

One should recall that $m_{\rho}\cdot a$, i.e. the $\rho$-mass in lattice units, has been measured at rather strong couplings ($N_{\tau}=4$) on small volumes with varying quark mass input $m_{q}\cdot a$. Keeping $m_{q}$ fixed at 770 MeV assumes that $m_{q}$ [MeV] is independent of $m_{q}$ [MeV] which it is not. Furthermore the value of $a~{}[\rm{GeV}^{-1}]$ obtained from $m_{\rho}\cdot a$ and $m_{\rho}$ [GeV] at such strong couplings is certainly not universal, i.e. independent of the lattice observable. Thus the translation from lattice parameters $\beta$, $\kappa^{(u,d,s)}$ to physical parameters $a~{}[\rm{GeV}^{-1}]$ and $m_{u,d,s}$ [MeV] should be taken with care.

The time histories of the plaquette expectation value show two state signals with states depending on the initial configuration for both values of $m_{s}$ ($m_{s}=150$ and 400 MeV). Future simulations with larger temporal extensions at weaker couplings have to show, whether the first order signals survive the continuum limit or share the fate with the signals of the $N_{f}=2$-simulations at $N_{\tau}=4$, 6 (Blum et al., 1994) to fade away on larger lattices. At strong couplings the fermion doublers in the Wilson formulation are still too light to decouple. It is known that the strength of the first order transitions increases with the number of light flavors. Thus the first order at large $m_{s}$ ($\sim 400$ MeV) might be an UV-artifact of the strong couplings caused by the small number of time slices.

At present the first order behavior is in disagreement with the lattice results of the Columbia group (cf. section III C 5) and with results in effective continuum models (Meyer-Ortmanns and Schaefer, 1995) (see section IV A 4).

It is of interest to compare the estimates for the critical quark masses up to which a clear first order signal is seen with results obtained for staggered fermions and in effective models for QCD. For three degenerate flavors it is $m_{u,d,s}^{\rm{crit}}\geq 140$ MeV or $m_{u,d,s}^{\rm{crit}}\cdot a\geq 0.175(2)$ in the Wilson fermion scheme as compared to 12-38 MeV for staggered fermions (section V C 5). For nondegenerate flavors it is $m_{u,d}^{\rm{crit}}\leq 3$ MeV, $m_{s}^{\rm{crit}}\leq 54$ MeV in the $SU(3)\times SU(3)$ linear sigma model (cf. section IV A 4 below).

We summarize. Although the translation to physical mass units may suggest continuum physics, it would be prematurate to call the lattice result of a first order chiral transition for physical quark masses continuum physics of the ‘real world’.

Improved actions

The experience with Wilson fermions in the original formulation has shown that the simulations are either inconclusive due to lattice artifacts at strong couplings, or expensive due to volumes which should be even larger than for staggered fermions to avoid lattice artifacts. Thus it is natural to attempt an approach with an improved action. Improved actions are supposed to accelerate the approach to the continuum limit. One such choice has been adopted by the Tsukuba group (Iwasaki et al., 1994 and 1995b). So far the improvement concerns the gauge part of the full QCD-action, while the fermionic part remains in Wilson’s formulation. The form of the improved pure gauge action was proposed by Iwasaki (1983). Besides the usual plaquette term it contains a $1\times 2$-loop with coefficients determined by a block-spin renormalization group analysis. The preliminary results for full QCD with the renormalization group improved action look promising. The improved gauge part seems to reduce (or remove) the lattice artifacts for $N_{f}=2$ and $N_{\tau}=4$ (6) observed by the MILC-collaboration (mentioned above).

A quantitative measure for the acceleration towards the continuum limit is obtained from the difference in (inverse) couplings $\beta$, which lead to approximately the same lattice spacing $a$. The improved action needs only $\beta=2.0$, while the standard action needs $\beta\sim 5.0$ for $a^{-1}=1.01$ GeV (if the $\rho$-mass is calculated on an $8^{3}\times 16$-lattice and used as input to set the physical scale).

It is not surprising that for small lattices without improved actions the Wilson and staggered fermion schemes give different results. One manifestation of this discrepancy is given by an estimate of $T_{c}$ in physical units for 2 flavors and 4 time slices. For Wilson fermions $T_{c}$ is estimated as $221\pm 3$ MeV, whereas $T_{c}=142\pm 6$ MeV for staggered fermions (Bitar et al., 1991). A more recent estimate of $T_{c}$ from Wilson fermions leads to $T_{c}\sim 152\matrix{+18\cr-14}$ MeV (Bernard et al., 1995) as compared to $140~{}\rm{MeV}\leq T_{c}\leq 160$ MeV for staggered fermions (DeTar, 1995). Future simulations will show whether other quantities show the same tendency that both fermion formulations approach the same continuum behavior.

D QCD at finite baryon density

The most convenient way of describing quark or nuclear matter at high density is to introduce a chemical potential $\mu_{q}$ for quarks. From asymptotic freedom one would naively expect that chiral symmetry is restored and quarks are deconfined above a critical value of $\mu_{q}$, $\mu_{q}^{\rm{crit}}$, and above a critical temperature $T_{c}$. A critical line ($T_{c}$, $\mu_{q}^{\rm{crit}}$) will separate the plasma and the hadron phase in a $(T,\mu)$-diagram. The shape is indicated in Fig. 16, it is qualitatively reproduced in a bag model calculation at finite $T$ and finite $\mu_{q}$ (Cleymans et al., 1986). Note in particular that chiral symmetry is always restored above the critical temperature for $\mu_{q}=0$, $T_{c,\mu_{q}=0}$, and above the critical potential for $T=0$, $\mu_{q,T=0}^{\rm{crit}}$. In the bag model one considers an ideal gas of nucleons in the hadronic phase, and of quarks and gluons in the plasma phase. A real-valued nonnegative energy difference at $T=0$ and the critical chemical potential $\mu_{b}^{\rm{crit}}$ for baryons is then obtained, if $\mu_{b}^{\rm{crit}}$ satisfies

[TABLE]

where $M$ denotes the nucleon mass. The chemical potential for quarks and baryons are related at the transition point via

[TABLE]

Conceptually $\mu_{q,b}^{\rm{crit}}$ should be distinguished from threshold values $\mu_{q,b}^{t}$. In the bag model the threshold value for $\mu_{b}$ is given by the nucleon mass $M$

[TABLE]

because a real-valued, non-vanishing baryon number density is only obtained for $\mu_{b}>M$. Note that $\mu_{b}^{\rm{crit}}$ within the bound of Eq. (239) is always close to $\mu_{b}^{t}$ so that the finite-$\mu$ phase transition at $\mu_{b}^{\rm{crit}}$ may not be distinguishable from a treshold effect at $\mu_{b}^{t}$ in practical calculations. Threshold effects are in fact seen on the lattice (cf. the summary of results below). We turn now to the lattice approach of finite $\mu$ (and finite $T$) transitions.

The naive translation of the continuum expression on the lattice leads to quadratic divergences of the internal energy density $\epsilon$ in the continuum limit. The solution of this earlier difficulty is provided by introducing a chemical potential for quarks $\mu\cdot a$ into the fermion matrix as $e^{\mu a}$ multiplying the forward links in time direction and $e^{-\mu a}$ multiplying the backward links (Hasenfratz and Karsch, 1983).

For simplicity one may think of performing lattice simulations at finite density in the quenched approximation. It turns out that the quenched approximation leads to results which contradict the intuitive expectations. Physical observables should be independent of $\mu$ as long as $\mu$ is smaller than some threshold value $\mu_{c}$. The threshold is related to the threshold for baryon production. In the confining phase baryons made of three quarks will be produced which are bound together with an effective chemical potential $3\mu$. Once $3\mu$ exceeds the value of the baryon mass, a finite baryon density will populate the system. In the quenched approximation the threshold comes out as $\mu_{c}\cdot a=m_{\pi}^{2}/2$ even in the case of $SU(3)$, where $m_{\pi}$ is the pion mass. The threshold should depend on the baryon mass, and the baryon mass should stay finite in the chiral limit.

For a finite quark mass the chiral condensate shows some threshold behavior in the sense that $\left<\bar{\psi}\psi\right>$ stays constant for a while and then goes to zero with increasing $\mu$, but $\left<\bar{\psi}\psi\right>$ vanishes in the chiral limit as soon as $\mu>0$. Also this behavior is at variance with the expectation. A critical density should be exceeded, before chiral symmetry is restored.

Davies and Klepfish (1991) have argued that the strange threshold behavior $a\cdot\mu_{c}=m_{\pi}^{2}/2$ is really an artifact of the quenched approximation and not of the finite volume or the strong coupling, as one may have suspected as well. Their argument is based on the fact that the gauge loops contributing to the pion propagator are unaffected by the chemical potential in the quenched approximation. The failure of the quenched approximation is shown for gauge couplings $\beta=6/g^{2}$ ranging from 0.0 to 6.2 .

If the full fermion determinant $\det M$ is included in lattice Monte Carlo simulations, the problem starts for $SU(N)$ gauge theories with $N\geq 3$. In this case the determinant is a complex number for $\mu\neq 0$. Therefore the factor $\det M\exp(-S_{g})$ (where $S_{g}$ denotes the pure gauge part of the action) can no longer be used as a probability for generating configurations in a Monte Carlo simulation as it is usually possible for $\det M$ being a positive real number. Nevertheless one can generate gauge field configurations with some probability $P(U)$, which is frequently chosen as $P(U)=|\det(M)|\exp(-S_{g})$. The expectation value for an observable $O$ is then calculated as

[TABLE]

where $\left<\ldots\right>_{P}$ denotes the average over configurations with respect to the measure $DUP(U)$ and $e^{i\phi}$ is the phase of the determinant. At $\mu=0$ and $\mu=\infty$ $\left<e^{i\phi}\right>_{P}=1$. Therefore the simulation according to the above prescription is feasible for small and large values of $\mu$. For intermediate values of $\mu$ $\left<e^{i\phi}\right>$ is very small. The phase $\phi$ fluctuates violently from configuration to configuration. A report on how far one can push this method from the extreme values $\mu=0$ and $\mu=\infty$ can be found in Toussaint (1990), see also (Vladikas, 1988).

One attempt in order to minimize the large fluctuations of $\phi$ is a detailed study of the grand canonical partition function (Gibbs, 1986; Barbour and Sabeur, 1990). It can be shown that the characteristic polynomial for the fermion propagator matrix is an expansion of the grand canonical partition functions for fixed quark number. Here it is nontrivial to measure the expansion coefficients $\left<a_{n}\right>$. Monte Carlo measurements of the canonical partition functions within this expansion have been performed in an $SU(3)$ gauge theory by Barbour et al. (1988), Barbour and Sabeur (1990), Barbour and Bell (1992) and Hasenfratz and Toussaint (1992) and are reviewed by Barbour (1992).

As an analytical alternative to the Monte Carlo simulations lattice calculations have been performed at strong coupling in a $1/d$- and $1/g^{2}$-expansion (Bilić et al, 1992a and b). Here $d$ denotes the spacetime dimension and $g$ is the bare gauge coupling. The earlier pathological threshold behavior in Monte Carlo simulations (with a threshold proportional to $m_{\pi}$ and vanishing for $m_{q}\to 0$) does, fortunately, not occur in the analytical approach. Compared to the $g=\infty$-limit, at finite coupling the threshold increases with $1/g^{2}$, while the baryon mass $m_{b}$ decreases. The difference between the threshold and $m_{b}$ becomes smaller at weaker couplings. This is to be expected, as the difference is a measure for the nuclear binding energy, which is large at strong couplings. Note that the threshold value is different from the baryon mass in contrast to Eq. (241), which has been derived in an ideal gas picture for nucleons.

Above a temperature of the order of 220 MeV chiral symmetry is restored for any $\mu_{b}\geq 0$ in qualitative agreement with Fig. 16. Below this value the chiral transition is of first order, but occurs at larger critical chemical potentials for increasing $6/g^{2}$, as one would expect from the considerations above. For $\mu_{b}<\mu_{b}^{t}$ one is in the chiral symmetry broken QCD vacuum without nuclear matter (Bilić et al., 1992a).

A more detailed study of the threshold effect of zero temperature and finite $\mu$ has been performed by Bilić et al. (1992b). The threshold behavior turns out to be flavor-independent in contrast to the strong flavor dependence of the ($\mu=0$, finite $T$) transition. Apparently a threshold effect need not be a phase transition. At strong coupling, however, the threshold effect seems to coincide with the finite-$\mu$ chiral phase transition. Again the threshold value at strong coupling lies somewhat below the nucleon mass. The coincidence of the threshold effect with the finite-$\mu$ phase transition is likely an artifact of strong couplings. At the threshold value $\mu^{t}$, the baryon number density jumps from zero incidentally to the maximal occupation of lattice sites.

In the continuum limit at zero temperature one may expect to see a threshold effect at $\mu^{t}$ separately from the phase transition at $\mu^{\rm{crit}}>\mu^{t}$. In that case the threshold effect corresponds to a “transition” from the QCD-vacuum to nuclear matter with broken chiral symmetry, and the phase transition restores chiral symmetry above the critical $\mu_{q,b}^{\rm{crit}}$.

So far the lattice results confirm some qualitative features of Fig. 16, but they are seriously plagued with artifacts of strong couplings in the analytical approach, and of intermediate couplings in the Monte Carlo approach.

Mean field calculations of QCD at finite baryon density have been performed in a number of effective models for QCD in the continuum, see e.g. Asakawa and Yazaki (1989), Lutz et al. (1992) or the review by Hatsuda (1992). Some of these calculations will be also mentioned in chapter IV.

IV EFFECTIVE MODELS IN THE CONTINUUM

A Models for quark degrees of freedom

In chapters II and III we have described effective models on the lattice, where the $Z(N)$ Potts model was of particular importance. As a common feature these models could be derived from lattice QCD in some coupling or temperature limit or in a renormalization group approach. They suffer from the same shortcomings as the lattice approximation to the original model (full QCD). Their relevance for the continuum limit has to be first established. In this section we discuss effective models in the continuum up to some remarks on dual Ginzburg-Landau models on the lattice in section IV B 2. Studies of the phase structure of QCD are frequently performed in a mean-field approach. The methods which we have selected for a more detailed discussion in this section go beyond the mean-field level in various ways.

In the first part we illustrate the power of the renormalization group with the work of Wilczek (1992) and Rajagopal and Wilczek (1993), who consider a three-dimensional $SU(2)\times SU(2)$-linear sigma model as a model for the chiral phase transition region. In section IV A 2 we outline the approach of chiral perturbation theory, which is well established as description for low energy and low temperature QCD. We discuss the results of Gerber and Leutwyler (1989). The leading term in a series of actions for chiral perturbation theory is a non-linear $SU(2)\times SU(2)$ sigma model in four dimensions. Although chiral perturbation theory fails to describe the transition region (thus it cannot predict the order of the phase transition), it allows an estimate of the latent heat in case of a (hypothetical) first order transition (Leutwyler, 1992), cf. section IV A 3. In section IV A 4 we discuss the mass sensitivity of the chiral transition and the equation of state for physical meson masses (Meyer-Ortmanns and Schaefer, 1995).

1 QCD and dysprosium

In this section we report on the work by Wilczek (1992) and Rajagopal and Wilczek (1993). It concerns an effective description of QCD which is supposed to be valid just in the vicinity of the chiral transition. The approach can be summarized in three hypothesis.

In the limit of 2 massless flavors QCD is in the static universality class of an $N=4$-Heisenberg ferromagnet, when all other quark masses are sent to infinity and $T\sim T_{c}$. 2. 2.

In the limit of 2 massless flavors QCD is in the static universality class of the $\phi^{6}$-Landau-Ginzburg model, when the strange quark mass is near a (tricritical) value $m_{s}^{*}$, the remaining quark masses are sent to infinity and $T\sim T_{c}$. 3. 3.

QCD with 2 massless flavors is in the dynamical universality class of an $O(4)$-antiferromagnet.

These conjectures may be regarded as working hypothesis for lattice simulations of the chiral transition. Before we describe their predictive power, let us comment on the two basic assumptions.

• $\bullet$

QCD is well approximated by the limit of two massless flavors ($m_{u}=m_{d}=0$).

• $\bullet$

The lightest excitations in the spectrum at zero temperature maintain their dominant role up to the transition region ($T\sim T_{c}$). The transition of 2-flavor QCD is expected to be of second order.

Possible objections could be the following.

a) Even the lightest quark flavors are not massless, their finite values lead to a pion mass of $m_{\pi}\sim 135$ MeV. Unless the quark masses correspond to critical values (which is not likely), a second order transition amounts to an inadequate idealization.

  

b) From lattice calculations one knows that the chiral transition is of first order for 4 flavors, if the masses are not too large. For 2 light and 1 heavier flavor the transition is replaced by a crossover phenomenon.

  

c) It is unlikely and not plausible that a theory as intricate as QCD can be decimated to a model as simple as a scalar $O(4)$-model.

  

d) Finally the pions loose their dominant role in the transition region.

We will argue against the first three objections and postpone arguments in favor of d) to section IV A 2.

to a). The light quark masses have non-vanishing values, but their values are small compared to the energy scale of the critical temperature. Thus it makes sense to consider $m_{u}$, $m_{d}$ as perturbations around the chiral limit, where the transition should be of 2nd order, if $N_{f}=2$. The renormalization group approach can account for finite small mass values. Their “perturbing” effect on the critical behavior in the chiral limit can be parametrized with critical indices. Thus there is a predictable parametrization of the deviation from the idealized limit. We recall from chapter II that the mass plays a comparable role to the scaling field of the inverse volume in a renormalization group analysis. Power law singularities of a second order transition will be rounded due to the finite volume and due to a finite mass. As we know from chapter II, the rounding is specific for the 2nd order, if the volume is sufficiently large. The deviation from the $L=\infty$-limit can be predicted in a well-known way.

Similarly there is a good chance that the extrapolation from finite masses to the chiral limit is under control. It is under control, if the deviations can be parametrized with the critical indices of a 2nd order transition. For a check of this assumption it would be sufficient to measure certain correlations (e.g. the specific heat) at different small, but finite quark masses and compare the change in the rounding effects with the predictions of the renormalization group analysis.

to b). In view of the physical values for the current quark masses four light flavors are certainly less realistic than two. Typical signatures for a first order transition (a non-vanishing latent heat, hysteresis effects and abrupt changes in the order parameters) are absent for 2 flavors. Support for a 2nd order transition in 2-flavor QCD is taken from lattice calculations of (Bernard et al., 1992; Gottlieb, 1991; Gottlieb et al., 1989; Gottlieb et al., 1987b; Brown et al., 1990a and b). The rise in the specific heat and the critical slowing down are compatible with a 2nd order transition. As argued in chapter III, lattice results are not yet fully conclusive. Hence they do not prove the conjecture about 2-flavor QCD.

to c). Note that the hypothesis 1)-3) refer to QCD only in the critical region ($T\sim T_{c}$). There it may well be that the complicated substructure of QCD is not important for the transition dynamics. In such a case one is free to replace QCD by the most simple model belonging to the same universality class. In condensed matter physics many examples are known, where universality is used in the same spirit. An example is the critical behavior of a binary mixture like isobutyric acid plus water, which shares the universality class of an Ising model in three dimensions.

There remains the question about the right choice of the universality class. In case of 2-flavor QCD Wilczek’s proposal is the universality class of an $O(4)$-Heisenberg ferromagnet. This proposal has been recently questioned by Kocic and Kogut (1995) claiming that QCD with its quark bilinear composite scalar mesons does not belong to the same universality class as sigma models with their fundamental scalars. The warning comes from a study in a three-dimensional Gross-Neveu model that exhibits mean-field scaling rather than the expected 2-dimensional Ising universality. Let us see, how one is led to the idea of an $O(4)$-Heisenberg ferromagnet in a natural way, and where the warnings of Kocic and Kogut would change the line of arguments.

Landau’s free energy has been introduced as a first concept to discuss the phase structure, cf. section II A, Eq.(̃7). It is constructed as a power series in the order parameter field. Landau’s theory corresponds to a mean-field approximation. In general it leads to wrong predictions of characteristic singularities (e.g. the correlation length at $T_{c}$.). The physical reason are large scale fluctuations in the order parameter field, although the average magnitude of the order parameter is small.

The theoretical breakthrough came with Wilson’s renormalization group concepts (Wilson and Kogut, 1974). It leads to a systematic way of constructing an action in terms of the relevant degrees of freedom at $T_{c}$ and answers the question, why the resulting action is representative in some sense. The action is constructed as a limiting theory (fixed point theory) after a number of renormalization group transformations. Such a limiting theory will exist, if the theory is exactly scale invariant at $T_{c}$. Scale invariance implies that models at different scales ($|r|\ll\xi$, i.e. scales much smaller than the correlation length $\xi$, but larger than the microscopic scales) share the leading singularity structure.

In case of the chiral transition these renormalization group steps are not explicitly performed. The limiting theory in terms of pion fields (and their parity partners) is argued to arise out of such a procedure. More precisely only the zero modes of these fields are assumed to survive the iterated renormalization group steps. The pions are the lightest modes at zero temperature and the zero modes are the only modes that do not acquire a mass contribution $\propto(2\pi nT)$ from the Matsubara sum. Thus the renormalized mass parameter $\mu^{2}(T)$ in the resulting action should be understood as an effective mass of the zero modes which vanishes as $T$ approaches $T_{c}$ from above.

An essential outcome of the renormalization group approach is an explanation of universality. Universality defines in which sense the limiting theory is representative for a whole class of models, belonging to the same universality class. Once the order parameter field is specified, the fixed point theory in terms of these fields only depends on the dimensionality and the symmetry, which is assumed to be broken or restored at the transition. Models with the same underlying symmetry, order parameter fields and singularity structure in thermodynamic functions define a universality class.

Thus it is sufficient to find an order parameter field, construct an action in terms of this field and restrict the allowed terms by the requirement of chiral symmetry. The order parameter field for the chiral transition should at least contain the pion multiplet. For two massless flavors the QCD Lagrangian is invariant under the $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B}$-symmetry of independent $SU(2)$-rotations of left- and right-handed fields and the vector baryon number symmetry. At the phase transition, the symmetry is assumed to be broken to $SU(2)_{L+R}\times U(1)_{L+R}$. One choice for an order parameter of the chiral transition is the quark bilinear

[TABLE]

transforming under $SU(2)_{L}\times SU(2)_{R}$ according to

[TABLE]

where $U$ and $V$ represent independent unitary transformations of the left-and right-handed quark fields. In the following the quark substructure of $M^{i}_{j}$ will be disregarded. (This is the main point which is questionable according to Kocic and Kogut (1995)).

As long as $M$ are general complex $2\times 2$-matrices, the Lagrangian has too much symmetry ($U(2)\times U(2)$). A restriction of $M$ to an $SU(2)$-representation removes the additional $U(1)$-symmetry. A possible choice for the $SU(2)$-representation is $O(4)\approx SU(2)\times SU(2)$, where $M$ is parametrized in terms of four real parameters $(\sigma,\vec{\pi})=\phi^{a}$ ($a$=1,…,4)

[TABLE]

where $\vec{\tau}$ denote the Pauli matrices. Thus $M$ contains the pion multiplet along with the scalar meson $\sigma$. The action in terms of $\phi^{a}$, which is invariant under $SU(2)\times SU(2)$, is given as

[TABLE]

The action takes the same form as the Landau-Ginzburg free energy $F$ in section II A. It could be identified with $F$, if the path integral with $S$ of Eq. (247) is evaluated in the mean field approximation. Note that $S$ is an action in terms of zero modes. The Euclidean time dependence of $\phi^{a}$ has been dropped. Equivalently the $n\neq 0$-Matsubara modes of the original 4-dimensional theory are neglected. (The treatment of the $SU(2)\times SU(2)$ sigma-model in the following section IV A 2 differs in this aspect, the time dependence there is kept.)

Here the action Eq. (247) should be understood as an effective action for the chiral transition. It coincides with the familiar linear sigma model of Gell-Mann and Levy (1960) up to the absence of nucleons and the dimension 3. Nucleon and quark fields are both omitted, as they refer to “microscopic substructures”, which are claimed to be irrelevant for the transition. Moreover the action (247) agrees with the action of an $N=4$-Heisenberg ferromagnet, which is believed to model a magnetic transition in dysprosium (Malmström and Geldart, 1980). This is the reason why Wilczek calls dysprosium an “analog computer” for QCD and explains the title of this section.

The $O(4)$-model in three dimensions has been extensively studied in statistical physics (Baker et al., 1978; Lipatov, 1977; Hohenberg et al., 1976; Bervillier, 1976). It is known to have a second order transition with $\mu^{2}(T)\to 0$ as $T\to T_{c}$ from above. In the following we summarize the results for critical exponents, taken from these references and Wilczek (1992).

As usual we distinguish between the response to a stimulus in $T$ and to a stimulus in $h$, where $h$ is an external field (cf. Table II). In this context the meaning of the external field is that of non-vanishing current quark masses.

Response to a stimulus in $T$

The response of the $O(4)$-“magnet” to a deviation from $T_{c}$ is measured by the specific heat $c$ and the quark pair correlation $\xi$ in the limit, where the quark masses strictly vanish ($m_{\pi}=0$). The singular behavior of the specific heat is characterized by the index $\alpha$, the behavior of the correlation length by the index $\nu$

[TABLE]

with $\alpha=-0.19\pm 0.06$,

[TABLE]

with $\nu=0.73\pm 0.02$, where $\xi$ is determined by

[TABLE]

at large distances $|x|$. The $\pm$-signs refer to the approach of $T_{c}$ from temperatures $T>T_{c}~{}(+)$ or $T<T_{c}~{}(-)$. The exponents $\alpha_{\pm}$ and $\nu_{\pm}$ do not depend on the approach of $T_{c}$. The amplitudes $A_{\pm}$ and $B_{\pm}$ are known as well and universal.

Note that $\alpha$ is negative, thus there is only a cusp in the specific heat, no true singularity in the infinite volume limit. This is compatible with the definition of a second order transition (cf. chapter II.A), but not so common. Usually more than one of the second derivatives of the thermodynamic potential diverge with power-law singularities.

We recall from Table II that the critical index $\beta$ determines the behavior of the order parameter $\left<|\phi|\right>$ as $T$ approaches $T_{c}$ in the massless limit. We have

[TABLE]

with $\beta=0.38\pm 0.01$.

The results for the critical indices $\alpha$, $\nu$ and $\beta$ should be checked by lattice simulations of the condensate correlations Eq. (250) and the specific heat ($T\partial S/\partial T$). In principle this check can be performed, although an extrapolation to the zero mass limit and the struggle with critical slowing down are unavoidable.

Response to a “stimulus” in $m$

The simplest possibility for adding a common mass term to the action which is left-invariant by the vector $SU(2)$-symmetry and invariant under discrete $P$- and $T$-transformations is given by $m\cdot\sigma$, where $m$ denotes the common quark mass. The dictionary of Table II in chapter II tells us how we should translate the compressibility in a liquid/gas system and the susceptibility in a ferromagnet to the order parameters for the chiral transition. The corresponding quantity is the variation of the condensate as a function of $m$

[TABLE]

The approach to $T_{c}$ is characterized by the critical exponent $\gamma$ (with $\gamma_{+}=\gamma_{-}$), the amplitudes $C_{\pm}$ are universal.

The second exponent $\delta$ concerns the response of the condensate to a change in $m$ at $T_{c}$

[TABLE]

Here the results for the $O(4)\,$-model are $\gamma=1.44\pm 0.04$ and $\delta=4.82\pm 0.05$.

The critical equation of state.

Wilczek’s first hypothesis can be further exploited to predict the temperature dependence of the pion and the sigma masses in the vicinity of $T_{c}$. These functions can be calculated as a function of the underlying quark masses. The important relation is a critical equation of state, which includes the results for $\gamma$, $\beta$ and $\delta$ of the previous sections as special cases. An equation of state in a liquid/gas system can be formulated as a relation between $p$, $V$ and $T$, $p=p(V,T)$. In a ferromagnet such a relation translates to $H=H(M,T)$ with the notations of Table II, in QCD it reads $m=m(\left<\bar{q}q\right>,T)$, and $m=m(\left<\sigma\right>=\left<|\phi|\right>,T)$ in the linear sigma model. In the “magnetic” language of Brézin et al. (1973) the critical equation of state is given as

[TABLE]

where $t={(T-T_{c})\over T_{c}}$, $g$ is a universal function and $\kappa_{1}$, $\kappa_{2}$ are non-universal constants. For the $O(4)\,$-model the function $f$ has been calculated by Brézin et al. (1973) to order $\varepsilon^{2}$ in the $\varepsilon\,$-expansion. Without derivation we state the expressions for $m^{2}_{\pi}$ and $m^{2}_{\sigma}$ in terms of the function $f$ at small $t$ and $H(\equiv\left<\sigma\right>)$ (Rajagopal and Wilczek, 1992)

[TABLE]

[TABLE]

with $x\equiv{t/M^{1/\beta}}$. Note that the relations hold at finite temperature, thus we have to specify, which mass it meant by $m^{2}_{\pi}$, $m^{2}_{\sigma}$. The definitions used by Rajagopal and Wilczek (1992) are the following

[TABLE]

where $G_{\alpha\beta}(x)\equiv\left<\phi(x)_{\alpha}\phi(0)_{\beta}\right>-\left<\phi_{\alpha}\right>\left<\phi_{\beta}\right>$ and $\vec{\phi_{0}}\equiv\sigma$, $\vec{\phi}_{i}=\vec{\pi}_{i}$, $i=1,2,3\,$.

If the masses are defined according to Eq. (258), they are related to the spatial correlation functions in the equilibrium theory. They are sensitive to static screening lengths. “Static” correlation lengths should be seen in contrast to dynamic length scales, which are relevant in the context of the dynamical universality class of the $O(4)$-Heisenberg antiferromagnet.

In the chiral limit above $T_{c}$ ($t>0$ and $H\to 0$) the critical equation of state gives the expected degeneracy between $m_{\pi}^{2}$ and $m_{\sigma}^{2}$

[TABLE]

In the chiral limit below $T_{c}$ ($t<0$ and $H\to 0$) it follows from Eq. (254) that $m_{\pi}^{2}\propto H$, i.e. $m_{\pi}^{2}\propto m$, the familiar proportionality from zero temperature. On the other hand, Eq. (256) follows from $m^{2}_{\pi}={H\over M}$, which reads in a literal translation according to the dictionary of Table II

[TABLE]

This is less familiar in view of the $T=0\,$-PCAC-relation

[TABLE]

Eqs. (260) and Eq. (261) do not contradict, if $f_{\pi}$ is identified with $f_{\pi}=\left<0|\sigma|0\right>$. The last relation is compatible with a definition of $f_{\pi}$ via the axial current of the linear sigma model at zero temperature. If we had started with Eq. (261) and used $f_{\pi}=\left<0|\sigma|0\right>$, the resulting relation at $T\sim T_{c}$ may not be considered as derived from the $T=0\,$-relation (261). This point has been emphasized by Rajagopal and Wilczek (1992). The masses $m_{\pi}^{2}$ in Eqs. (260) and (261) cannot be the same as the notation may suggest. In Eq. (260) $m_{\pi}$ is the mass in the $(3+1)\,$-dim Lorentz invariant $T=0\,$-theory. In Eq. (261) it resembles a screening mass in a 3dim-theory.

The result for the sigma-mass at temperature $T<T_{c}$ in the limit of vanishing quark mass reads

[TABLE]

Here $\beta$ denotes the critical exponent of the “magnetization”, $c_{1}$ and $c_{2}$ are constants, and $\epsilon$ is the remnant of the $\epsilon\,$-expansion.

To summarize: The critical equation of state (254) may serve as a working hypothesis for lattice simulations. Its various limits are predictions for the scaling behavior of $m^{2}_{\pi}$ (Eq. (260)), $m^{2}_{\sigma}$ (Eq. (262)), or $m^{2}_{\sigma}=m^{2}_{\pi}$ (Eq. (262)) in the chiral symmetric phase. In particular the quantitative predictions for the critical indices $\beta$, $\gamma$,$\delta$ referring to the condensate as function of $m$ and $T$ are ready for numerical tests.

The role of the strange quark mass.

We come now to the second hypothesis concerning the role of the strange quark mass in the chiral transition. So far we have implicitly assumed that the remaining quark masses $m_{s}$, $m_{c}$, $m_{b}$ and $m_{t}$ are infinite. This is certainly justified for the charm, bottom and top quark masses, which are large compared to the chiral transition temperature, but the strange quark mass is just of the order of the transition temperature. Thus it can influence thermodynamic quantities in a nontrivial way. Such an influence is already visible in lattice results. If $m_{s}$ is infinite, the chiral transition seems to be of second order. We have quoted the references above. If it is zero, renormalization group arguments predict a first order transition (Pisarski and Wilczek, 1984). Numerical simulations for three light flavors verify this conjecture (see e.g. Gottlieb 1991). Hence a critical value $m_{s}^{*}$ should exist, where the second order changes into a first order transition. Usually such an endpoint, where the order changes from first into second, is called a tricritical point. The physical value of $m_{s}$ is unlikely to coincide with $m_{s}^{*}$, but it may be close by. Therefore it is tempting to describe the realistic mass parameters as a perturbation around the idealized tricritical limit. If such an ansatz is justified, the deviations from tricriticality are under control.

According to the second hypothesis a simple model which shares the universality class of QCD with two massless flavors of quarks, $T\sim T_{c}$ and $m_{s}$ near $m_{s}^{*}$ is the $\phi^{6}$-Landau-Ginzburg model. Its action reads

[TABLE]

The field $\vec{\phi}$ is the same as in Eq. (247). The explicit symmetry breaking due to $(-H\sigma)$ has been added to account for finite masses $m_{u,d}$. The $\phi^{6}$-term arises as follows. The effect of a finite strange quark mass is to renormalize the mass and coupling $\mu^{2}$ and $\lambda$ in Eq. (247). For example one contribution to the renormalization of $\lambda$ comes from a K-meson exchange between two pions. The “amount” of renormalization depends on $m_{s}$. The mere effect of a mass and coupling renormalization is a shift in $T_{c}$ as long as $\lambda$ stays positive. If $\lambda$ is negative the model gets unstable, and a $\phi^{6}\,$-term is needed for stabilization. It is easily checked that for $\lambda<0$ and fixed, the minimum of the free energy jumps discontinuously from zero to a finite value ${|\lambda|/(2\kappa)}$ when $\mu^{2}={\lambda^{2}/(4\kappa)}$. (For $\lambda>0$ and fixed, the minimum moves continuously from zero to positive values, when $\mu^{2}$ goes through zero, as one enters the broken phase.) Hence the value of $m^{*}_{s}$ can be defined as the strange quark mass for which the renormalized coupling $\lambda$ vanishes. At this point the second order of the $\lambda>0$ region changes into first order ($\lambda<0$ region).

Singularities of thermodynamic functions are universal near tricritical points. Thus one may again exploit results from statistical physics. Tricritical exponents of the $\phi^{6}$-model have been calculated by Lawrie and Sarbach (1984). Of particular interest is the result for $\alpha$, one finds $\alpha={1\over 2}$. Note that now $\alpha>0$ (in contrast to the 3-dim $O(4)$-model) indicating a true divergence of the specific heat when $T\to T_{c}$ and $m_{s}\to m_{s}^{*}$. In lattice calculations $m_{s}$ can be tuned to small values. A qualitative change in the shape of the cusp in the specific heat would be a hint on a tricritical point nearby. This illustrates the predictive power of the second hypothesis of Wilczek and Rajagopal (1992) and Wilczek (1992).

The third hypothesis concerns the chiral transition and its relation to dynamical universality classes. So far we had to deal with static properties of equilibrium QCD. The third hypothesis may be relevant for off-equilibrium situations in heavy-ion collisions. Therefore we postpone the discussion to section V D 3.

Lattice simulations in full QCD can measure critical and tricritical indices in the vicinity of $T_{c}$ (and $m_{s}\sim m_{s}^{*}$). If these results would disagree with the predictions of the above conjectures, it can have different origins. One possible reason is the size of the subleading terms in Eqs. (249), (252), (253). The subleading terms are only small as long as the parameters $T$ and $m_{s}$ are in the near vicinity of the critical values $T_{c}$ and $m_{s}^{*}$. Corrections from subleading terms do not invalidate the hypothesis. The very ansatz would fail, if the physical values for $m_{u}$, $m_{d}$, $m_{s}$ perturb the idealized situation of 2-flavor QCD too strongly to reveal the underlying substructure of the chiral limit. Furthermore $SU(2)\times SU(2)$ is a sensible choice for the symmetry group, but not unique. If $SU(2)\times SU(2)$ would be replaced by $SU(3)\times SU(3)$, it is not possible to adapt the hypothesis to the change in the symmetry group. Along with the second order of the transition the universality arguments are lost, which were so welcome for justifying a replacement of QCD by scalar $O(4)$-models.

Finally it is a question of size whether corrections due to non-zero modes and heavier mesons are negligible at $T_{c}$. In the next section we present an attempt to include heavier mesons (in a four-dimensional $SU(2)\times SU(2)$ non-linear sigma model). The results shed some doubts on the very ansatz for the 3-dimensional action in terms of $\vec{\phi}\equiv(\sigma,\vec{\pi})$.

2 The chiral transition in chiral perturbation theory

It is tempting to extrapolate results from low temperatures to the transition region. Pions as the lightest hadrons are most easily excited at low temperatures. One may expect that they remain the only relevant degrees of freedom up to $T_{c}$. Current quark masses are small compared to the scale of the transition temperature. Thus the naive conclusion would be that their effect is negligible at all. Based on these simplifying assumptions one would study the chiral limit of an $O(4)$-model as effective description for the chiral transition region ($T\sim T_{c}$). Results of Gerber and Leutwyler (1989) suggest that this way of arguing is too naive. We present their results in this section.

Chiral perturbation fails to describe the transition region, but leads to reliable predictions for the condensate and other thermodynamic quantities at small temperatures. The influence of finite quark masses on the $T$-dependence of $\left\langle\bar{q}q\right\rangle$, $\varepsilon$ and $p$ can be analyzed in the low temperature region. The effect of heavier mesonic modes may be estimated in a dilute gas approximation for somewhat higher temperatures. Non-negligible effects of finite quark masses and heavier mesons show up in the marginal validity range of chiral perturbation theory. Their extrapolations to the transition region can be summarized as follows.

• $\bullet$

Below $T=150$ MeV the nonzero quark masses reduce the temperature dependence of the condensate by roughly a factor of 2.

• $\bullet$

The effect of massive states on the energy density $\varepsilon(T)$ is even more significant. In the chiral limit the energy stored in the massive states reaches the order of the energy stored in the pions when $T\sim 130$ MeV. Nevertheless the massive states may be diluted at this temperature. The main part of the energy is rest energy of the massive states. At $T\sim 200$ MeV the mean distance between two particles is reduced to $d=0.9$ fm. This estimate is based on the density formula for a free gas. The approximation of a dilute free gas of massive states is then no longer justified. Below $T\sim 100$ MeV massive states may be neglected.

• $\bullet$

At $T\sim 200$ MeV the dilute gas approximation for the massive modes predicts a melting of the condensate, even if the pions are completely ignored. This sheds some light on the idealization of the predominant role of pions in the chiral transition (which we have anticipated in the end of the previous section).

• $\bullet$

The extrapolated value of $T_{c}$ decreases from $\sim 190$ MeV to $\sim 170$ MeV, when heavier mesons are included in the chiral limit.

• $\bullet$

$T_{c}$ decreases from $\sim 240$ MeV to $\sim 190$ MeV, when heavier mesons are included at finite quark masses. While heavier mesons accelerate the melting of the condensate, non-vanishing quark masses delay the melting by $\sim 20$ MeV.

Let us briefly recall why chiral perturbation theory is a suitable framework for describing QCD at low temperatures. Chiral perturbation theory is an expansion in small momenta. Its applicability to QCD is based on the fact that strong interactions are weak at low energies. This is a consequence of chiral symmetry. The chiral symmetry of QCD in the massless limit implies that the interaction strength is proportional to the square of the energy if the energy is small. At low temperatures the properties of the hadron gas are determined by the lightest excitations. The lightest hadrons are the pions. At low temperatures the average momenta of the pions are small. Thus the strong interactions between the pions may be treated perturbatively in the framework of chiral perturbation theory.

To be specific, we consider the $SU(2)_{R}\times SU(2)_{L}$ chiral symmetry. The particle contents is given by the three pion components $\pi^{0}$, $\pi^{-}$ and $\pi^{+}$. The fourth component corresponding to the $\sigma\,$-mode is frozen due to the non-linear realization of the symmetry. The pion field is described by a matrix field $U(x)\in SU(2)$. It transforms under global chiral rotations according to

[TABLE]

where $V_{R}$, $V_{L}\in SU(2)$ and U(x) is parametrized as $U(x)=\exp\left\{i\vec{\tau}{\vec{\varphi}(x)\over f}\right\}$. Here $\vec{\tau}$ denote the Pauli matrices, $f$ will be later identified with the pion constant, the components of $\vec{\varphi}$ represent the three components of the pion field.

If one allows only for small four-momenta of the pions, the field $U(x)$ is slowly varying. It is then convenient to expand the Lagrangian in powers of derivatives of the fields $\partial_{\mu}U$, equivalent to a power series in external momenta $p$. The ansatz is given as

[TABLE]

The upper index counts the number of derivatives $\partial_{\mu}$, thus $L^{(0)}$ does not depend on U and can be dropped. Lorentz invariance forbids odd powers in the derivatives. First we consider the chiral limit with a massless pion field. The form of $L^{(2n)}$ is then completely determined by the symmetry requirement, i.e. the invariance under $SU(2)_{R}\times SU(2)_{L}$ chiral transformations. This leads to

[TABLE]

Note that this term is just the familiar non-linear $SU(2)\times SU(2)$ sigma model. The interaction between the pion fields is described in a loop expansion with $\vec{\varphi}$-fields propagating in the loops. The only place where the finite temperature enters is in the boundary conditions, which lead to modified propagators. The power counting rules for graphs which are associated with the free energy are modified compared to the rules for chiral perturbation theory in the loop expansion at zero temperature. The only modification consists in a replacement of the role of the external momenta by the temperature. There is a one to one-correspondence between the order in the low temperature expansion and the loop expansion of chiral perturbation theory.

The expansion parameter is $T^{2}\over(8f^{2})$. Since an increasing number in these loops corresponds to a higher order in the low temperature expansion, $L^{(2n)}$, $n=1,2\dots$ is to be expanded in powers of $\vec{\varphi}$ to the order which is needed for the desired accuracy in the low T-expansion. Up to terms of order $\varphi^{6}$ we find for example for $L^{(2)}$

[TABLE]

Note that the chiral symmetry fixes $L^{(2)}$ up to one constant $f$ which may be identified with $f_{\pi}$ in the chiral limit. At the order of $p^{4}$, two further dimensionless couplings enter the effective Lagrangian. They have to be fixed from experiment.

Away from the chiral limit, the mass terms for the pion field induce an explicit breaking of the chiral symmetry in the effective Lagrangian. In the chiral limit the guiding principle for constructing $\cal L$ is an expansion in powers of external momenta. It can be maintained in the presence of mass terms, if the quark masses are small compared to the relevant physical scales of the theory. The full effective Lagrangian is Taylor expanded in powers of $m$. The derivate expansion can be superimposed on the Taylor expansion, term by term. This way additional terms involving powers of $m$ are generated, which break the chiral symmetry explicitly. The power counting in external momenta of the chiral limit goes through, if $m_{\pi}$ is taken to be of the order of $p$. Since the pion mass square $M^{2}$ to lowest order in chiral perturbation theory is given as $M^{2}=m^{2}_{\pi}=(m_{u}+m_{d})\cdot B$ (B being the $T=0$-condensate, $m_{\pi}$ being the physical pion mass), the quark masses are counted as quantities of order $p^{2}$. Inclusion of mass terms then leads to the following modifications in $L^{(2)}$

[TABLE]

$L^{(4)}$ depends now on four input parameters $l_{1},\dots l_{4}$ (Gasser and Leutwyler, 1984). Thus the main effect of finite quark masses is an increase in the number of input parameters. Once the effective action is determined and the input parameters are fixed from experiment, the thermodynamics of the pion gas can be derived in the usual way.

The partition function is given as the path integral over all pion field configurations which are periodic in the Euclidean time direction, $U(\vec{x},x_{4}+\beta)=U(\vec{x},x_{4})$.

[TABLE]

where $L^{(2)}$ is given by Eq. (269). The couplings of $L^{(2n)}$ are treated as temperature independent. The order parameter, energy density $\varepsilon$, pressure $p$ and entropy density $s$ follow from the free energy density $\cal F$ in the standard way (${\cal F}={-1\over\beta V}\ln Z$ in the large volume limit, where $Z$ is given by Eq. (270)).

To obtain a low temperature expansion of $Z$, $L_{eff}$ is expanded in powers of $\varphi$.

Let us illustrate the power counting in the expansion parameters for the case where the free energy density is to be determined including terms of order $T^{8}$. This is the accuracy, which has been achieved by Gerber and Leutwyler (1989).

To obtain $\cal{F}$ to the order of $T^{8}$, L must be expanded to order $p^{8}$, contributions come from $L^{(2)}$, $L^{(4)}$, $L^{(6)}$ and $L^{(8)}$. Tree graphs are generated from all terms $L^{(2)}\ldots L^{(8)}$, one-loop graphs contain vertices of $L^{(4)}$ and $L^{(6)}$, two-loop graphs vertices of $L^{(2)}$, $L^{(4)}$, and three-loop graphs are generated by $L^{(2)}$. Thus $L^{(2)}$ has to be specified in terms of $\vec{\varphi}$ including powers of the order $\varphi^{6}$. Divergent integrals are treated in dimensional regularization.

The leading contribution of the Lagrangian $L^{(2)}$ contains a free-field term (i.e. a term quadratic in $\vec{\varphi}$ with mass $M^{2}$, the lowest order contribution to the pion mass). Further terms of the Lagrangian are kept as perturbations depending on their importance at low energies. This way chiral perturbation theory provides a systematic expansion also for thermodynamic quantities. It goes beyond the mean-field level, where one frequently stucks in the treatment of effective models.

In the chiral limit the free energy density $\cal F$ takes the general form

[TABLE]

The correction of the order $\exp\{-M_{e}/T\}$ stands for the contribution of particles which remain massive in the chiral limit, $M_{e}$ denotes the lightest of these masses.

A similar expression can be derived in the presence of finite quark masses, when a common small expansion parameter $\lambda$ is introduced such that $T\to\lambda T,m\to\lambda^{2}m$. The formula for $\cal F$ then holds in powers of $\lambda$ and $\log\lambda$ with $\lambda$-independent coefficients $c_{mn}$ which are now nontrivial functions of $(m_{\pi}/T)$.

Free field case. Let us first ignore the interactions between the pions and consider $L^{(2)}$. It leads to the following familiar expressions for the free energy density $\cal F$, the pressure $p$ and the condensate $\left\langle\bar{q}q\right\rangle$ for an ideal gas of pions with mass $m_{\pi}$

[TABLE]

Here $\varepsilon_{0}$ is the vacuum energy density at $T=0$. In the chiral limit the result for $\cal F$ is

[TABLE]

with the $T^{4}$-law for the pressure of a free gas. The condensate is obtained as

[TABLE]

The first term on the r.h.s. gives the condensate at temperature $T=0$. Using the PCAC-relation

[TABLE]

the condensate in the chiral limit is given by

[TABLE]

where $f=f_{\pi}$ up to terms of order $m$ which are counted as order ($T^{2}$). Thus the difference shows up in $T^{4}$-terms. Formula (277) displays the melting of the condensate as the temperature increases. Simultaneously the pion density $n_{\pi}$ grows according to

[TABLE]

in the massless limit. Physically the increase in number density can be made responsible for the melting process.

For $m\neq 0$, the integral in Eq. (275) becomes a function of ${m_{\pi}\over T}$.

Including interactions up to three loop. Let us see next, how the results for $p$, $\varepsilon$ and $\left\langle\bar{q}q\right\rangle$ change, when interactions are included. It is clearly beyond the scope of this review to go into details of the three-loop calculation. Also we will not explain the way, how the various constants in the effective Lagrangian and the logarithmic scales are fixed from experimental observables (the pion mass, the pion decay constant and various scattering lengths).

Results in the chiral limit without heavier mesons. The low temperature expansion for the pressure is given as

[TABLE]

Here $\Lambda_{p}=275\pm 65$ MeV, $m_{\pi}$, $f_{\pi}$ and the I=0, D-wave scattering length have been used as experimental input.

The low temperature expansion for the energy density $\varepsilon=T({\partial p\over\partial T})-p$ follows as

[TABLE]

The general expression for the condensate can be written as

[TABLE]

with

$\displaystyle c=-f^{2}{{\left(\partial m^{2}_{\pi}\over\partial m\right)}\over{\left\langle 0|\bar{q}q|0\right\rangle}}\;,$

cf. Eqs. (275), (277) above. Hence one needs to know $p=p(m^{2}_{\pi})$ (Leutwyler, 1988). The temperature independent term $c$ has to be determined, $c=1$ in the chiral limit. The result for $\left<\bar{q}q\right>$ including terms of order $T^{6}$ takes the following form

[TABLE]

The expansion parameter $x$ shows that the temperature scale is determined by $\sqrt{8}f\sim 250$ MeV. The scale $\Lambda_{q}$ is fixed in an analogues way to $\Lambda_{p}$. Both scales $\Lambda_{p}$ and $\Lambda_{q}$ can be expressed in terms of a single scale $\Lambda_{b}$ which is fixed from $\pi\pi$-scattering. Thus the behavior of the pressure $p$ to order $T^{8}$ and of $\left<\bar{q}q\right>$ to order $T^{6}$ needs only two constants $f_{\pi}$ and $\Lambda_{b}$. This does not leave much room for phenomenological fitting and demonstrates the predictive power of chiral perturbation theory.

The results for $\left\langle\bar{q}q\right\rangle$ and the energy density $\varepsilon$ are displayed in Fig. 17.

It is tempting to read off a transition temperature of $T_{c}\sim 190$ MeV, if the three-loop curve is extrapolated to a temperature where the condensate vanishes. This extrapolation suffers from two error sources. The error bars on the three-loop curve only refer to the uncertainty in the constants $f$ and $\Lambda_{b}$, and not to the finite temperature corrections of Eq. (283). Contributions of massive states to $\cal F$ have been left out so far. They are no longer negligible, when $T$ exceeds 150 MeV. Below we will further discuss the influence of massive states.

The energy density in Fig. 18 grows with $T$ up to $T\sim 180$ MeV. The decrease of $\varepsilon/T^{4}$ for higher temperatures indicates the invalidity of the expansion scheme.

Non-vanishing quark-masses. Non-vanishing quark-masses change the general expression for the pressure, where certain constants of the chiral limit now depend on $m_{\pi}$ also via $m$. The result for the quark condensate is shown in Fig. 19. The effect of finite quark masses is that it takes longer for the condensate to melt. The extrapolated transition temperature is $\sim 240$ MeV (which lies far outside the validity range of the approximation). The error bars indicate the uncertainties in four input parameters, which have to be fixed from experiment for $m>0$.

A comparison of Figs. 17 and 19 shows that the effect of finite quark masses is substantial already inside the validity range of the chiral expansion. The reason why the tiny quark masses may cause such changes in the final results is that they are confined to pions. It is the pion mass ($\sim 140$ MeV) that enters the Boltzmann factors in the thermodynamic formulas for the hadronic phase.

Influence of heavier mesons. In an expansion around the chiral limit quark masses must be treated as perturbations when they are incorporated in an expansion of chiral perturbation theory. A perturbative treatment of $m_{u}$ and $m_{d}$ is certainly justified, the up- and down-quarks are light compared to the energy scale of $T_{c}$. Mesons containing strange quarks are accordingly heavier than the pions. Frequently they are argued to be negligible in the thermodynamics, as massive states are suppressed by their Boltzmann factors $\exp\{-M/T\}$. This argument fails for higher temperatures, in particular in the vicinity of the transition.

One way of including (part of the) heavier mesons is to extend the symmetry group from $SU(2)\times SU(2)$ to $SU(3)\times SU(3)$ (cf. section IV A 4). In the framework of chiral perturbation theory this means that also $m_{s}$ should be treated perturbatively.

A further alternative is to treat all massive mesons (others than pions) on an equal footing and consider their number density and energy density in a dilute gas approximation (Gerber and Leutwyler, 1989). In a dilute gas approximation the interactions between the mesons are neglected. This justifies an ansatz for the change in the free energy density according to

[TABLE]

The sum extends over all mesons tabulated in the particle data booklet apart from pions. It is clear that many small contributions may accumulate to a non-negligible amount. The change $\Delta\cal F$ is due to a change in the pressure. The term ${1\over 2}T^{4}h_{0}\left({T\over M_{i}}\right)$ gives the pressure of a free boson gas of particles with mass $M_{i}$. The function $h_{0}$ can be expressed in terms of Bessel functions. The induced change in the condensate (due to the mass $M_{i}$) follows from Eq. (275) with $m_{\pi}$ replaced by $M_{i}$. It is given as

[TABLE]

where

$\displaystyle T^{2}h_{1}=\int{d^{3}p\over(2\pi)^{3}}{1\over E}{1\over e^{E/T}-1}$

or equivalently

[TABLE]

Here we have used the lowest order PCAC-relation for $f^{2}_{\pi}m^{2}_{\pi}$. The derivative ${\partial M_{i}\over\partial m}$ can be estimated in the non-relativistic quark model, where it equals $N_{i}$, i.e. the number of valence quarks of type $u$ or $d$. Formula (286) explains the effect of massive states on the melting of the condensate. It accelerates the melting and reduces the transition temperature. In Fig. 20 the melting of the chiral condensate is compared for the three-loop pion contribution and the superposition of pions plus massive states. The extrapolated transition temperature is around 190 MeV, smaller than $T_{c}$ without heavier mesons, but $\sim 20$ MeV larger than in the chiral limit with heavier mesons. The shaded area reflects the uncertainty of $\left(m{\partial M_{i}\over\partial m}\right)$ entering the estimate for the contribution of massive states (Eq. (286)). The acceleration of the melting process due to the massive states is more pronounced for realistic quark masses than for vanishing ones. In the chiral limit the melting of the condensate is mainly enforced by the pions, the pions play a more distinguished role, thus the condensate is less sensitive to heavier mesons.

At temperatures $T>150$ MeV pions have lost their dominant role in melting the condensate. If lattice calculations or other approximation schemes support a value of $T_{c}>150$ MeV, it is challenging to elucidate the way of how the massive modes drive $\left\langle\bar{q}q\right\rangle$ to zero. It seems to be very worthwhile to improve on the dilute gas approximation. A suggestion due to Gerber and Leutwyler (1989) is to treat the mutual interactions between the massive states in some bootstrap model, modified and adapted to QCD (see e.g. Hagedorn, 1985 and 1983).

More recently the temperature dependence of the normalized light quark condensate has been derived by Bunatian and Wambach (1994) in a nonperturbative approach. The approach is known from phonon-phonon interactions in condensed matter physics (see e.g. Luttinger and Ward, 1960; Fulde and Wagner, 1971) and may be more appropriate to the transition region. In the chiral limit the critical temperature is identified as the point where the residue of the single-pion propagator becomes purely imaginary. The pion interactions are described by the Weinberg Lagrangian (Weinberg, 1966, 1967 and 1979). The thermodynamic potential for the interacting pion gas is determined in the Hartree approximation. After using the Gell-Mann-Oakes-Renner relation, the normalized light quark condensate follows from $\partial\Omega/\partial m_{\pi}^{2}$, where $\Omega$ denotes the difference in free energy densities at $T=0$ and $T>0$. The results confirm chiral perturbation theory up to temperatures $<100$ MeV, but are clearly different in the transition region, where chiral perturbation theory is expected to fail.

3 Clausius-Clapeyron equation for QCD

We recall the Clausius-Clapeyron equation for a liquid/vapor system in statistical physics. It has been derived in the last century by Clapeyron (1834) and Clausius (1850). The Clausius-Clapeyron equation relates the discontinuity in the order parameter to the latent heat when the phase transition is of first order. For a liquid/gas system the order parameter is the specific volume or the volume per molecule, the latent heat equals the entropy gap per molecule. The ratio of the discontinuities in the specific volume and the entropy is determined by the dependence of the critical temperature on the pressure during the transition. The relation is given as

[TABLE]

where disc stands for discontinuity.

With the dictionary of Table II Eq. (287) can be translated to QCD, where it reads

[TABLE]

The pressure is replaced by the current quark mass $m_{q}$. The conjugate variable of $m_{q}$ is the order parameter $\left<\bar{q}q\right>$ at temperature $T$. The chiral condensate is determined as response of the partition function to a change in the quark mass

[TABLE]

We sketch the derivation of Eq. (288) and show how it leads to a bound on the latent heat in a hypothetical first order chiral transition. The derivation is due to Leutwyler (1992). First note that the discontinuity in the energy density $\epsilon=(T\partial p/\partial T-p)$ can be expressed by the gap in the entropy density

[TABLE]

From Eq. (289) with $\ln Z=V(p-\epsilon_{0})$ ($\epsilon_{0}$ being the vacuum energy density) it follows that

[TABLE]

If we expand the pressure in the vicinity of $T_{c}$ according to

[TABLE]

$p$ depends on $m_{q}$ via $p_{c}$ and $T_{c}$

[TABLE]

Next we insert Eq. (293) into Eq. (291) and apply “disc” on both sides. The result is

[TABLE]

While $p$ is continuous at $T_{c}$ (also in case of a first order transition), $\partial p\over\partial T$ may jump. From Eqs. (290) and (294) we finally obtain Eq. (288).

Let us see what the Clausius-Clapeyron relation implies for $\Delta T_{c}$, i.e. the change in the critical temperature, when the physical values for the current quark masses $m_{u}$ and $m_{d}$ are replaced by their chiral limit. As $m_{u,d}$ are small compared to $T_{c}$, we may expand

[TABLE]

where $\partial T_{c}/\partial m_{u,d}$ is given by Eq. (288). It remains to replace the quark masses in Eq. (295) by means of the Gell-Mann-Oakes-Renner-relation (Gell-Mann et al., 1968) $m_{u}\left<\bar{u}u\right>_{T=0}+m_{d}\left<\bar{d}d\right>_{T=0}\sim f_{\pi}^{2}m_{\pi}^{2}\sim$ 20 MeV/fm3. The result is

[TABLE]

This relation restricts the size of the latent heat disc $\epsilon$ by the sensitivity $\Delta T_{c}/T_{c}$ of the critical temperature to the quark masses. An upper bound on disc $\epsilon$ corresponds to an upper bound on the (hypothetical) gap in the order parameter.

To get an estimate for this bound, take $T_{c}\sim 140$ MeV as a lower bound on the chiral transition temperature. The number density $n$ of the hadron gas grows with temperature. At $T_{c}\sim 140$ MeV $n$ increases by $\sim 30\%$ in passing from the physical quark masses to the chiral limit ($m_{u,d}\to 0$) (see Leutwyler, 1992). This estimate is based on a dilute gas approximation, where the change in the particle abundances is only induced by the change in their mass as function of $m_{u,d}$. If the onset of the phase transition is determined by the average volume per particle, a smaller transition temperature has to compensate for the increased density to get the same average volume. 30% increase in the density $n$ induce a decrease of 5% in $T_{c}$, i.e. $\Delta T_{c}/T_{c}\sim 5\%$ under the above conditions. For the maximal jump of $\left<\bar{q}q\right>_{T_{c}}$ one thus finds

[TABLE]

This value is in good agreement with the bound on the latent heat obtained in a large-$N$-approximation to the $SU(3)\times SU(3)$ linear sigma model (Meyer-Ortmanns and Schaefer, 1995), cf. the next section. For comparison we mention the gap in the gluonic energy density in a pure $SU(3)$ gauge theory (Laermann et al., 1995). Laermann et al. find $\hbox{disc}~{}\epsilon_{\rm{gluonic}}/T_{c}^{4}\sim 1.5$ or $\hbox{disc}~{}\epsilon_{\rm{gluonic}}\sim 0.9~{}\hbox{GeV/fm}^{3}$ for $T_{c}=260$ MeV.

For a larger value of disc $\epsilon$, say $\sim 2~{}\hbox{GeV/fm}^{3}$, the Clausius-Clapeyron relation implies ${\Delta T_{c}/T_{c}}<1\%$. Results of chiral perturbation theory indicate that such a small change in the critical temperature is rather unlikely. The condensate and the energy density are rather sensitive to the pion mass $m_{\pi}$.

Chiral perturbation theory cannot predict the order of the phase transition, as it looses its validity in the transition region, but it does provide a useful upper bound on the latent heat. This bound should be noticed in particular by phenomenologists, who like a large latent heat for visible effects in heavy-ion collisions. Scenarios based on such an assumption may be incompatible with the underlying QCD.

4 Mass sensitivity of the chiral transition

In this section we discuss two topics. The first one is the mass sensitivity of the chiral transition in the $SU(3)\times SU(3)$ linear sigma model. We give estimates for critical meson/quark masses at which the first order chiral transition becomes of second order and turns into a smooth crossover phenomenon for larger masses. The second topic is the equation of state for physical meson masses.

The aspect of mass sensitivity. The mass sensitivity of the order of the chiral transition is essential in view of realistic predictions for heavy-ion collisions. To estimate the effect of finite masses on the chiral transition we recall the analogy to a ferromagnet or a liquid/gas system.Quark masses are analogous to the pressure in a liquid/gas system or an external magnetic field in a ferromagnet. If the pressure exceeds a critical value, the first order transition from a liquid to a gas ceases to occur, it is replaced by a smooth crossover between the liquid and the gas phases. Similarly a first order transition in a $Z(3)$-Potts model becomes a second order transition for a critical value of the external field and disappears beyond this strength.

When the analogy to a statistical system is translated to an effective model for QCD, the question concerning the effect of finite masses can be posed in the following way. Are the physical meson masses too small for the chiral transition to maintain its first order? In section III we have described an attempt to answer this question within the lattice approach, cf. Fig. 13. Here we study the same question on the mesonic level. The bare quark masses $m_{s}{}^{.}a$ and $m_{u,d}{}^{.}a$ of Fig. 13 should be replaced by meson masses with and without strange quark content.

A first attempt to estimate the critical phase boundary between mass parameters of first order transitions and crossover phenomena has been made by (Meyer-Ortmanns et al., 1992) in the linear $SU(3)\times SU(3)$ sigma model. More recently Gavin, Goksch and Pisarski (1994) addressed a similar question in the same sigma model within a mean-field approach. In the remainder of this section we will focus on results of a finite mass analysis of Meyer-Ortmanns and Schaefer (1995) in a large-$N_{f}$-approximation. As a quantitative measure for the distance (in mass parameter space) between the physical and critical meson masses we consider the ratio of the associated critical to physical light quark masses $m_{u,d}^{\rm{crit}}/m_{u,d}$. If this ratio turns out to be much smaller than 1, physical masses lie deeply in the crossover region, and it is difficult to imagine any signatures specific for the phase conversion from the plasma to the hadron phase. The most attractive possibility is a ratio of the order of 1. In that case the physical masses are almost “critical”, effects due to a large correlation length should be visible. Non-universal features of the sigma model are then negligible and the decimation of QCD to an effective model in the same universality class is an allowed simplification.

Here we would like to recall that sigma models may not be the right candidates for sharing a universality class with QCD (Kocic and Kogut, 1995). As long as the deviations from the ideal second order behavior are perturbatively small, they are theoretically under control (see e.g. the finite mass scaling analysis of Boyd et al., 1992, Rajagopal and Wilczek, 1993, or Kocic and Kogut, 1993), and a comparison of critical indices in the effective model and on the lattice serves as a check of the conjectured universality class.

Choice of the model

In our application we assume that the restoration of the spontaneously broken $SU(3)\times SU(3)$ symmetry is the driving mechanism for the chiral phase transition. The deviations in the spectrum from the idealized octet of pseudoscalar Goldstone bosons are parametrized by terms which break the $SU(3)\times SU(3)$ symmetry explicitly. The assumption in our decimation is that only mesons associated with the $SU(3)\times SU(3)$ multiplets are important for the phase transition. The criterion is chiral symmetry (rather than the size of the meson masses, otherwise one should include $\rho$-mesons or others as well). The reason why we have chosen $SU(3)\times SU(3)$ rather than $SU(2)\times SU(2)$ is to account for the influence of the strange quark mass on the thermodynamics. With $SU(3)\times SU(3)$ we also include some of the heavier mesons. From section IV A 2 we know that heavier mesons are non-negligible in the transition region.

We have chosen the sigma model as an effective model for the low temperature phase of QCD ($T\leq T_{c}$). In the low temperature phase quarks are confined to hadrons, and chiral symmetry is spontaneously broken. The meson spectrum reflects some remnants of this symmetry breaking. In the transition region the use of the model becomes questionable, the model certainly fails above $T_{c}$ as a description of the plasma phase.

We use the sigma model in a similar spirit as the $O(4)$-model in section II A 1. Its action is constructed in terms of QCD’s chiral order parameter field $\phi$, where $\phi$ now is a complex $3\times 3$-matrix, parametrized as

[TABLE]

Here $\lambda_{l}$ denote the Gell-Mann matrices, $\pi_{l}$ are the pseudoscalar mesons, $\sigma_{l}$ are the scalar mesons. The mesonic order parameter field is a bilinear in the left-handed and right-handed quark fields $\phi_{ij}=\left<\bar{q}_{i}^{L}q_{j}^{R}\right>$. In the sigma model the quark structure is ignored by construction for all temperatures $T\geq 0$. In terms of $\phi$ the Lagrangian reads

[TABLE]

Note that there are 2 independent quartic terms with couplings $f_{1}$ and $f_{2}$. The determinant-terms are cubic in the components of $\phi$, $g$ is the “instanton” coupling, it takes care on the right $\eta$-$\eta^{\prime}$-mass splitting, $\mu_{0}^{2}$ is the coupling of the quadratic term. The external field $\epsilon_{0}$ gives a common mass to the (pseudo)scalar meson octet, while $\epsilon_{8}$ accounts for the right mass splitting inside the (pseudo)scalar meson octet.

Tree level parametrization at zero temperature

The parameters $\mu_{0}^{2}$, $f_{1}$, $f_{2}$, $g$, $\epsilon_{0}$, $\epsilon_{8}$ of the Lagrangian (299) should be chosen such that the model reproduces the experimental values of the (pseudo)scalar meson masses. The parametrization of the sigma model is not unique (Chan and Haymaker, 1973; Gavin et al., 1994; Meyer-Ortmanns et al., 1992). Here we are interested in a tuning of meson masses in terms of a few parameters. Suitable parameters are the external fields $\epsilon_{0}$, $\epsilon_{8}$. The induce finite quark masses according to

[TABLE]

where $\hat{m}\equiv(m_{u}+m_{d})/2$, $\alpha$ and $\beta$ are constants. Eq. (302) follows from an identification of terms in the Lagrangians for quarks and mesons, which transform identically under $SU(3)\times SU(3)$. The meson masses are determined for given $\epsilon_{0}$, $\epsilon_{8}$, once the couplings $\mu_{0}^{2}$, $f_{1}$, $f_{2}$, $g$ are specified and the condensates $\left<\sigma_{0}\right>_{T=0}$, $\left<\sigma_{8}\right>_{T=0}$ are calculated for given $\mu_{0}^{2}$, $f_{1}$, $f_{2}$, $g$. Thus we vary the quark and meson masses by varying $\epsilon_{0}$ and $\epsilon_{8}$. The chiral limit is obtained for $\epsilon_{0}=0=\epsilon_{8}$. The couplings $\mu_{0}^{2}$, $f_{1}$, $f_{2}$, $g$ are then determined from the mass input in the chiral limit, i.e. $m_{\pi}=m_{K}=m_{\eta}=0$, $m_{\eta^{\prime}}=850$, $m_{\sigma_{\eta^{\prime}}}=800$, $m_{\sigma_{\eta}}=600$, all masses in units of [MeV], $f_{\pi}=94$ MeV. Next we keep $\mu_{0}^{2}$, $f_{1}$, $f_{2}$, $g$ fixed to their values in the chiral limit and change $\epsilon_{0}$, $\epsilon_{8}$. The choice $\epsilon_{8}=0$, $\epsilon_{0}\neq 0$ leads to an $SU(3)$-symmetric case with only one order parameter field $\sigma_{0}$ for which the numerics considerably simplifies. Meson masses with almost experimental values are induced for $\epsilon_{0}=0.0265~{}\rm{GeV}^{3}$, $\epsilon_{8}=-0.0345~{}\rm{GeV}^{3}$.

The results are listed in Table VII and compared to the experimental values.

This way we have constructed a mapping

[TABLE]

between quark and meson masses.

It remains to translate the meson condensates at zero temperature to the light and strange quark condensates. In the same way as we obtained the relation (302), we find here

[TABLE]

Eqs. (305) are derived at zero temperature. We take these relations as temperature independent and use them to determine $\left<\hat{q}q\right>(T)$, $\left<\hat{s}s\right>(T)$ from the measured values for $\left<\sigma_{0}\right>(T)$, $\left<\sigma_{8}\right>(T)$, see below.

Critical meson masses in mean-field

Although the method is crude, it is instructive to get a first estimate for critical meson/quark masses. Later the results will be compared with estimates from a large-$N$-approach. In a mean-field calculation the full effective potential is replaced by the classical part in terms of two constant background fields $\sigma_{0}$, $\sigma_{8}$. For simplicity we consider here only the $SU(3)$-symmetric case, where $\epsilon_{8}=0=\sigma_{8}$, $\epsilon_{0}\neq 0$, $\sigma_{0}$ denotes a constant background field. The effect of a finite (high) temperature in a mean-field calculation is a renormalization of the quadratic term in the Lagrangian. Thus a finite temperature can be mimiced by tuning $\mu_{0}^{2}$ while keeping the other couplings $f_{1}$, $f_{2}$, $g$ fixed. For a critical field $\epsilon_{0}^{\rm{crit}}$, the first order transition just disappears, and so does the cubic term in $U_{\rm{class}}$. At $\epsilon_{0}^{\rm{crit}}$ $U_{\rm{class}}$ starts with a term proportional to $(\sigma_{0}-\sigma_{0}^{\rm{crit}})^{4}$, where $\sigma_{0}^{\rm{crit}}$ is the minimum of $U_{\rm{class}}$ for critical values $\mu_{0}^{2\rm{crit}}$, $\epsilon_{0}^{\rm{crit}}$. Thus we have

[TABLE]

[TABLE]

Here $\left.\right|_{\rm{crit}}$ means “evaluated at critical parameters”. Note that $U_{\rm{class}}$ in Eq. (308) takes the same form as a free energy functional for a liquid/gas system. It supports the analogy between a liquid/gas system and the chiral transition in QCD as mentioned in the dictionary of Table II. The vanishing of the first three derivatives in Eq. (309) determines $\sigma_{0}^{\rm{crit}}$, $\mu_{0}^{2\rm{crit}}$ and $\epsilon_{0}^{\rm{crit}}$ as functions of $f_{1}$, $f_{2}$, $g$. In the physical case of $\epsilon_{0}=0.0265~{}[\rm{GeV}^{3}]$, $\epsilon_{8}=-0.0345~{}[\hbox{GeV}^{3}]$ we obtain

[TABLE]

Such a small ratio of 3 % for the critical to physical light quark masses would mean that the chiral phase transition is easily washed out by tiny quark masses, and for physical quark masses one is left with a rather smooth crossover phenomenon.

Our interest in the mean-field result is the order of magnitude of this ratio. Recall that a first order transition can have different origins. A first one is a cubic term in the classical part of the potential. A second one is a $\phi^{6}$-term which may be needed for stabilization of the free energy when the quartic coupling picks up a negative sign due to renormalization effects. For two or more independent relevant couplings a further first order transition can be a so called fluctuation induced transition. Since the linear $SU(3)\times SU(3)$ sigma model contains two such couplings $f_{1}$ and $f_{2}$, the chiral transition may be mainly fluctuation induced. This hypothesis has been recently discussed by Gavin et al. (1994), see also Shen (1993), Paterson (1981), Jacobsen and Amit (1981).

If the order of magnitude of the ratio (310) changes beyond the mean-field level, it sheds some doubts on the simplified description of Eq. (308) and favours the hypothesis of a fluctuation induced transition.

The large-$N_{f}$-approach

Ultimately we are interested in the temperature dependence of the meson condensates $\left<\sigma_{0}\right>(T)$, $\left<\sigma_{8}\right>(T)$. The condensates are determined as the minima of the constrained free energy density under the constraint on the average values

[TABLE]

where $\bar{\sigma}_{0,8}$ are prescribed values for the background fields, while the same average value should vanish for $\sigma_{\ell}$, $\ell=1,\ldots,7$ and $\pi_{\ell}$, $\ell=1,\ldots,8$. The relation to the full partition function is given by

[TABLE]

with ordinary integrals for the $\sigma_{0,8}$-integrations.

As a next step we quadratize the quartic part of the interaction by applying a matrix version of the Hubbard–Stratonovich transformation (Hubbard, 1959; Stratonovich, 1958; Frei and Patkós, 1990). It reads

[TABLE]

where $\phi^{\prime}(x)$ is an $N\times N$-matrix field and

[TABLE]

The path integral over the auxiliary field $\Sigma(x)$ is evaluated in a saddle point approximation. We replace $\Sigma(x)$ by $\hbox{sad}\cdot\hbox{diag(1,1,1)}$. The saddle point approximation corresponds to the leading term in a $1/N$-expansion in an $O(N)$-model (Coleman et al., 1974). The $SU(3)\times SU(3)$ linear sigma model reduces to an $O(18)$-model for $g=f_{2}=0$. The $N=18$ mesonic modes correspond to $N_{f}=3$ flavors ($N=2N_{f}^{2}$). Therefore we call our approximation “large-$N_{f}$”.

The advantage of the large-$N_{f}$-approximation is that we end up with an effectively free field theory. The only remnant of the interaction is hidden in the dispersion relation

[TABLE]

Here $Q=1,\ldots,8$ labels the particle multiplets, $m_{Q}^{2}$ are the mass squares defined by the quadratic terms in the fluctuating fields, sad is the contribution from the auxiliary field.

The final expression for $U_{\rm{eff}}(\sigma_{0},\sigma_{8},\hbox{sad})$ contains a classical part $U_{\rm{class}}$, a part coming from the quadratization $U_{\rm{saddle}}$, a zero point energy part which is dropped, and a thermal part $U_{\rm{th}}$

[TABLE]

where $g(Q)$ are the multiplicities of the multiplets, e.g. $g(1)=3$ for 3 pions etc. The full expression for $U_{\rm{eff}}$ can be found in (Meyer-Ortmanns and Schaefer, 1995).

The expression for $U_{\rm{eff}}(\sigma_{0},\sigma_{8},\hbox{sad})$ is evaluated in a high temperature expansion and—alternatively—fully numerically. The high-temperature expansion for a free field theory is standard. Since we are interested in the low temperature phase ($T\leq T_{c}$), the most we can expect from a high-temperature expansion are qualitative results. The numerical evaluation of $U_{\rm{eff}}$ looks quite straightforward, but it is hampered by imaginary parts in the effective potential. Although the final maximum in sad and the minima in $\sigma_{0}$, $\sigma_{8}$ turn out to lie in the region of real valued $U_{\rm{eff}}$, the routines encounter imaginary parts in intermediate steps. The choice of initial values for sad, $\sigma_{0}$, $\sigma_{8}$ has to be optimized to make the numerical evaluation tractable.

Results for critical quark/meson masses

In the large-$N_{f}$-approximation the chiral transition is washed out for an average pseudoscalar octet mass $\geq 203$ [MeV], if the ratio of $m_{s}/m_{u,d}$ is kept fixed at its realistic value 18.2. This gives an upper bound on the first order transition region. For the corresponding critical quark masses we find

[TABLE]

or a ratio of

[TABLE]

In the $SU(3)$-symmetric case with three degenerate flavors the common critical pseudoscalar mass is only $\leq 51$ [MeV], and $m_{u,d}^{\rm{crit}}\leq 0.9\pm 0.14$ MeV. Thus the critical mass values depend on the direction in mass parameter space. Our values are clearly below the estimates for Wilson fermions ($m_{s}^{\rm{crit}}\geq 400$ [MeV] for $m_{u,d}\sim 0$, and $m_{u,d,s}^{\rm{crit}}\geq 140$ [MeV] in the degenerate case). They are of the same order as the estimates for staggered fermions ($m_{u,d}^{\rm{crit}}\leq 12$ [MeV], $m_{s}^{\rm{crit}}\leq 50$ [MeV]) for the $N_{f}=2+1$ case, but below the bound for $N_{f}=3$, 12 [MeV] $\leq m_{u,d,s}^{\rm{crit}}\leq 38$ [MeV], cf. sections III C.5,7. Thus our ratio $m_{u,d}^{\rm{crit}}/m_{u,d}\sim 30\%$ lies between the mean-field value of $\sim 3\%$ and the lattice estimates. The intermediate value for the large-$N_{f}$-result is plausible, as “large-N” goes beyond “mean-field” due to the resummation of a certain subclass of Feynman diagrams (Jain, 1993), whereas lattice Monte Carlo simulations include all quantum fluctuations at once. It should be kept in mind, however, that a direct comparision between lattice quark masses in physical units and current quark masses (as we are using here) is questionable (cf. the argument in section III.C.5 above).

A ratio of 30% is certainly not large enough for predicting visible remnants of a nearby second order chiral transition. There is some hope that the ratio gets closer to 1, if further fluctuations are included in the effective model and the true nature of the chiral transition is fluctuation induced.

We have measured the weakening of the first order transition in the $SU(3)$-symmetric case ($\epsilon_{8}=0$). The gap in the light quark condensate above $T=177$ [MeV] slowly decreases for finite meson masses. It has disappeared for an external field strength of $\epsilon_{0}=6.6\cdot 10^{-4}~{}[\rm{GeV}^{3}]$, cf. Fig. 21. Similarly the barrier height between the coexisting minima in the effective potential decreases from $1.4\cdot 10^{-4}~{}[\rm{GeV/fm}^{3}]$ in the chiral limit to $2.1\cdot 10^{-6}~{}[\rm{GeV/fm}^{3}]$ for $\epsilon_{0}=2\cdot 10^{-4}~{}[\rm{GeV}^{3}]$ and $\epsilon_{8}=0$.

We want to conclude with an interesting speculation of Gavin et al. (1994) about the universality class of the $SU(3)\times SU(3)$ sigma model. It is based on the observation that the mass of $\sigma_{\eta^{\prime}}$ vanishes for a specific choice of tree level parameters $f_{1}$, $f_{2}$, $g$, $\mu_{0}^{2}$. If the renormalization group flow would drive the couplings towards these values in the vicinity of $T_{c}$, the light mass could lead to large correlation volumes. If the $\sigma_{\eta^{\prime}}$ were the only light mode, the universality would be Ising like (in contrast to $O(4)$).

An upper bound on the latent heat

For physical meson masses we find a sharp crossover phenomenon in the light quark condensate between $T=181.5$ and 192.5 [MeV], $\left<\bar{q}q\right>_{T}$ decreases to 50 % of $\left<\bar{q}q\right>_{T=0}$ over a temperature interval of $\Delta T=10$ [MeV]. The strange quark condensate $\left<\bar{s}s\right>_{T}$ stays almost constant up to a temperature of $\sim 200$ [MeV], cf. Fig. 22.

Similar results are found for the energy density and the entropy density (Fig. 23). Both quantities behave smoothly as a function of $T$ over the entire temperature range up to large errors in the transition region. From these errors we obtain an upper bound on a finite latent heat $\Delta L$ of

[TABLE]

which is compatible with our data. The value is in agreement with Leutwyler’s bound of $0.4~{}[\rm{GeV/fm}^{3}]$ (Leutwyler, 1992), which has been obtained from Clausius–Clapeyron–relations in the framework of chiral perturbation theory, cf. section IV A 3. Further note that it is only 10 % of the value predicted by the naive bag model equation of state. To our knowledge it is yet unclear, whether $0.2~{}\rm{GeV/fm}^{3}$ latent heat are sufficiently large to induce measurable signatures in heavy-ion experiments.

Furthermore we have calculated the difference $\epsilon(T_{2})/T_{2}^{4}-\epsilon(T_{1})/T_{1}^{4}\equiv\Delta\epsilon/\hbox{``}T_{c}^{4}\hbox{''}$. Here $T_{1}=181.5$ [MeV] and $T_{2}=192.6$ [MeV] denote the temperatures, where the rapid crossover sets in and ends, respectively. In a first order transition with a finite discontinuity $T_{1}=T_{2}=T_{c}$. Hence we can compare our value for $\Delta\epsilon/\hbox{``}T_{c}^{4}\hbox{''}=0.29$ with the ratio for the gluonic energy density $\Delta\epsilon_{\rm{gluonic}}/T_{c}^{4}$ on the lattice in a pure $SU(3)$ gauge theory. The lattice result is $\epsilon_{\rm{gluonic}}/T_{c}\sim 1.5$ or disc $\epsilon_{\rm{gluonic}}\sim 0.9~{}[\rm{GeV/fm}^{3}]$ for $T_{c}=260~{}\rm{MeV}$ as mentioned above. Our bound on the chiral contribution to the total gap in the energy density over $T_{c}^{4}$ is almost by an order of magnitude smaller than the gluonic contribution. It should be taken as a warning not to jump to conclusions from our results in view of relativistic heavy-ion experiments. So far we have only investigated the contribution from chiral symmetry to the full equation of state for a hot hadron gas. The contribution from gluonic degrees of freedom has been completely left out. It is a question of relative size, which effects in the full deconfinement/chiral transition are dominant.

It seems to us very worthwhile to elaborate alternative inclusions of heavier mesons in the chiral transition region. From the work by Gerber and Leutwyler (1989) it has become clear that heavier mesons are increasingly important when $T$ exceeds $100$ MeV. I regard our parametrization of the scalar and pseudoscalar meson masses in the $SU(3)\times SU(3)$ linear sigma model just as one ansatz for including part of the heavier mesons. This part is determined by the assumed underlying chiral symmetry. A further assumption is that the deviations from the broken $SU(3)\times SU(3)$ symmetry can be parametrized by two external fields, breaking the symmetry explicitly. The ansatz seems to be justified in the sense that the predictions of the pseudoscalar meson masses reasonably agree with experimental values.

An alternative point of view is the following. $SU(2)\times SU(2)$ is the true symmetry with only pions as idealized Goldstone bosons and one external field to account for the finite pion mass, while all other mesons are treated on an equal footing: scalar, pseudoscalar, and vector mesons. Gerber and Leutwyler (1989) take this point of view, when they describe all mesons (apart from the pion) in a dilute gas approximation to study their influence on the chiral phase transition. The quantum virial expansion of thermodynamic quantities in a hot pion gas provides a further alternative. Heavier mesons enter the second virial coefficient via their contribution to experimentally measured phase shifts of $\pi\pi$-scattering (Welke et al., 1990). A more general framework is the generalized Beth-Uhlenbeck approach (see e.g. Schmidt et al., 1990), where the scattering phase shifts of $\pi\pi$-scattering are replaced by thermodynamic $T$-matrix elements, characterizing the interacting pion gas at finite temperature and density. When the quantum virial expansion or the generalized Beth-Uhlenbeck approach are applied to a pion gas, the underlying chiral symmetry of QCD plays a less distinguished role in the description as compared to our treatment in the $SU(3)\times SU(3)$ linear sigma model.

B Models for gluonic degrees of freedom

1 A network of strings

The color flux tube models of Patel (1984a and 1984b) have been developed to describe the deconfinement phase transition. Quark degrees of freedom can be included as well. Let us first consider the case of a pure $SU(2)$- or $SU(3)$ gauge theory. We recall from section III that the spontaneous breaking of the global $Z(2)$- or $Z(3)$ symmetry is associated with the phase transition from the confinement to the deconfinement phase. The order parameter is the expectation value of the Wilson line. It vanishes, when the free energy to find an isolated test quark in the system grows to infinity. It is different from zero in the deconfinement phase.

Effective $Z(N)$-spin models for the deconfinement transition have been presented above (section III). They can be derived from the $SU(N$) lattice gauge theory in the high temperature limit and share the essential extra symmetry of QCD. In the limit of infinitely heavy masses, this is the global $Z(N)$-symmetry. For high temperatures the gauge fields are almost frozen to unity matrices. Lowering the temperature increases the disorder. One big cluster of aligned spins breaks up into several clusters. Below the transition point the system is completed disordered, if we use a “magnetic” language. The formulation in terms of spin systems is natural for $Z(N)$- models. Intuitively it is, however, much less clear, how this picture should be translated to the original $SU(N)$ degrees of freedom.

Patel’s description is complementary to the above model. It starts at $T=0$ and follows the evolution of the system as the temperature is increased. The symmetry breaking is no longer the driving force for the system to undergo a phase transition. It is replaced by entropy production.

In the first part we present a purely heuristic description. The picture is based on the flux tube model of the deconfinement transition. Flux tubes connect quarks and antiquarks in strong coupling expansions on the lattice. Thus one may not expect that they leave some remnant in the continuum limit. On the other hand, they are also the ingredients of phenomenological continuum descriptions like string models for hadrons. We add a few remarks about the yo-yo-string in the end.

Several properties of the flux tubes or strings have to be specified first. Strings in Patel’s models are characterized by three parameters: the string tension $\sigma$, the string width $w$ and the rigidity parameter $a$. We assume that there is a constant energy per unit length along the string, this is $\sigma$. The string has a constant width $w$. It has a certain resistance against bending. It has to go at least a distance $a$ apart, before it can change its orientation. The distance $a$ is of the order of 1 fm. On the lattice $a$ naturally coincides with the lattice constant, the strings bend at right angles there. Here the constant $a$ will not be tuned to zero in the end. Its role resembles more a lattice constant in models of condensed matter physics (models in continuum spacetime, where the constant $a$ is given by the physical lattice constant). In Patel’s flux tube theory the parameters $a$, $\sigma$ and $w$ must be fixed from experimental inputs.

Two further assumptions about the flux tubes must be specified to get a well defined model. Flux tubes can terminate only on quarks. Their interaction occurs at baryonic vertices. The constants $\sigma$, $w$, $a$ are treated as temperature independent. The only driving force of the phase transition is the increasing entropy of flux tubes, when the temperature is turned on.

The $SU(2)$-case

Based on these ingredients let us see, how far heuristic arguments can lead us in a pure $SU(2)$ gauge theory. The only allowed flux tube structures are closed loops differing only in size and shape, i.e. in the length of the string. Physically these loops may be interpreted as glueballs. The partition function is written as

[TABLE]

The sum runs over all loops of fixed length $l^{.}a$ (in physical units). The energy of such a loop is given as $\sigma^{.}l^{.}a$. The combinatorial prefactor $N$ gives the number of loops of length $l^{.}a$. The dependence of $N$ on the rigidity parameter $a$ is essential as it provides the possibility of a phase transition. (It depends on temperature, how often strings like to bend.)

Consider random walks of length $l$ (in lattice units) on a lattice with distance $a$ without backstepping. Backstepping should be forbidden for a physical string. This number is given as $(2d-1)^{l}$ in $d$ dimensions. The constraint that the walks should perform closed loops leads to a power law correction in $l$ to $N$. It will not be specified further, because it is irrelevant in the large-$l$-limit. The partition function is then proportional to

[TABLE]

in $d=3$ dimensions. The dimension is chosen as 3, since the string model is constructed to provide an alternative description to the 3-dim $Z(N)$-spin model for the deconfinement transition. As $T$ increases, the average length of the loops becomes larger. A phase transition is signalled, if $Z$ diverges. This happens for large $l$, if $(\sigma^{.}a/T=\ln 5)$, that is at

[TABLE]

The effective string tension $\sigma_{eff}$ defined via $\exp\{-la\sigma_{eff}/T\}$ vanishes continuously at $T_{c}$

[TABLE]

while the average length of the loops diverges. Both features suggest a second order phase transition.

To see its relation to the deconfinement transition, we have to probe the system at $T_{c}$ with a static $\bar{q}q$-pair. If it costs a finite amount of energy to isolate both quarks at an infinite distance, we have reached the deconfinement phase. Equivalently deconfinement is manifest, if the $q$ and $\bar{q}$ are a finite distance apart, but only rather loosely correlated. Color screening requires that a flux tube connects the quarks of our probe. With increasing temperature the flux reorientates itself more and more often. It oscillates between its endpoints, see Fig. 24.

Since one quark knows about its partner only via the connecting string, this information is lost, if the string gets infinitely long. This is just what happens at $T_{c}$, where it costs no extra free energy to create an infinitely long flux tube. Thus at $T\geq T_{c}$ both quarks ($q$ and $\bar{q})$ are effectively independent of each other and free. This means deconfinement.

To get a quantitative estimate of the transition temperature, we must fix the string tension and the rigidity parameter in physical units, or the product of both in lattice units. In the strong coupling approximation of the lattice theory, the lowest lying $O^{+}$-glueball at $T=0$ is given by a square loop

[TABLE]

The string parameters are assumed to be temperature independent, hence one can take this $T=0$-result to fix $T_{c}$

[TABLE]

Lattice Monte Carlo calculations give a slightly larger value.

The $SU(3)$-case

Next let us consider the pure $SU(3)$ gauge theory. Two new features must be accounted for. Quarks and antiquarks are no longer in equivalent representations of $SU(3)$. Thus a direction is associated to a string, indicating whether it terminates in a quark or an antiquark. (The quarks and antiquarks are test quarks in the pure gauge theory.) Furthermore, a flux tube representation of a baryon or antibaryon requires that a string can bifurcate at a vertex $v$, see Fig. 25. This reflects the previous assumption that flux tubes only interact at baryonic vertices. The diagrammatic rules for allowed string structures in the $SU(3)$-case follow from the allowed vertices, shown in Fig. 26. This excludes closed loops made up of an odd number of links.

The ansatz for the partition function can be chosen similarly to the $SU(2)$-case. The sum over loops has to be replaced by a sum over more complicated topological structures (of the type of nets). A non-vanishing energy $v$ is associated to each vertex to respect the additional bifurcation degree of freedom. At zero temperature the vacuum is filled with closed loops made out of nets of strings, see Fig. 27 (It is filled with “glueballs”.)

As the temperature is increased, the size and density of these structures grow. Unlike in the $SU(2)$-case, it is now not the diverging length of strings that induces a phase transition. A new qualitative feature enters, the connectivity of the network. The phase transition occurs, if an infinite network is generated. It is infinite in the sense that each string is connected with the entire volume of the lattice. Between any given pair of flux tube segments one can find a flux tube path on the lattice that connects them.

The definition of “critical temperature” can be made more precise. This type of phase transitions is well known in condensed matter physics under the name of percolation transitions. Examples for percolation transitions are gelation transitions in the context of polymer chains (Flory, 1941a and b). (We suspect that also jelly is gelatinous due to polymer networks.) To find a quantitative measure for the connectivity of the network, one has to distinguish between relevant and irrelevant links of the network. If relevant links are removed, the infinite network is destroyed. In the same sense irrelevant links are not essential for the connectivity. The fraction $f$ of relevant links depends on the temperature and the specific underlying dynamics. It can be easily estimated that ${2\over 3}<f<1$. The lower bound is realized at $T_{c}$, where the network is minimally connected. The upper bound means maximal connectivity.

Thus the challenging hypothesis is to look at the deconfinement transition as a percolation transition. Again we have to answer in a first place, what happens to a static test quark, when it is put into the network below or above the percolation transition. Below $T_{c}$ the free energy of an isolated quark is infinite, because the flux tube originating from this quark may not terminate on any closed loop which is a finite distance apart. It is excluded by the diagrammatically forbidden flux tube structures. Instead the string attached to the quark has to fuse with the string of an antiquark of the test probe. The case, where both $q$ and $\bar{q}$ can be considered as isolated or free, costs an infinite amount of free energy, since we have assumed a constant energy per unit length of the string.

Above $T_{c}$ the distance between the $q$ and $\bar{q}$ need no longer be infinite for both to loose the correlation (to behave effectively as free test probes). Already at a finite separation the correlation is weak due to the presence of an infinite network to which they are attached. Now the clusters of closed loops are connected, and the corresponding flux tube structures for a $\bar{q}q$ pair at finite distance are allowed, cf. Fig. 28. The costs in energy due to the long, dense and connected flux tubes are compensated by the gain in entropy. Thus the free energy of a configuration where the $q$ and $\bar{q}$ test quarks have lost any information about each other is finite. The $q$ and $\bar{q}$ are effectively free, we have ended in the deconfinement phase.

For a crude estimate of $T_{c}$ one has to specify the amount of extra energy due to bifurcation points.

In the heuristic way of arguing it remains to explain the first order of the deconfinement phase transition in the $SU(3)$-case. Either one refers to the percolation transition where the transition is known to be of first order, or one proves directly for the colored network of the QCD flux that it costs a finite interface free energy at $T_{c}$ when both phases coexist. Such a coexistence may be visualized as a hole in the network. Roughly speaking it is the reduction in entropy when the loose ends at the boundary of the hole have to be tied up to the network again, it costs interface energy proportional to the area.

Inclusion of matter fields

Flux tube models can also account for the inclusion of dynamical quarks. By construction the chances are good, since the flux tube picture does not rely on any symmetry argument, and the extra global $Z(N)$-symmetry is explicitly broken in the presence of dynamical quarks. Recall that the limit of a pure gauge theory may be considered as the $m\rightarrow\infty$-limit of full QCD. Thus we are interested in the effect when the quark mass is lowered or a $Z(N)$- symmetry breaking field is switched on.

Second order phase transitions in ferromagnets are known to disappear, when the magnets are exposed to an external magnetic field, independently of its strength. Similarly the second order transition of the pure $SU(2)$ gauge theory ceases to occur: there is always a finite probability for an infinitely long string to break up due to the creation of a $q\bar{q}$-pair out of the vacuum. The probability is proportional to $l^{.}e^{-2m/T}$, it is finite even if the quark mass is very large as long as $l\rightarrow\infty$. (Infinitely long strings turned out to be responsible for the second order phase transition in this picture.) The expected change is that the phase transition from the confinement- to the deconfinement region is replaced by the crossover from tightly to weakly bound $q\bar{q}$-pairs.

The $SU(3)$-case is more subtle. Again it is known from statistical physics that first order transitions are stable with respect to small perturbations of an external field. Hence we expect for large enough quark masses still a deconfinement transition, and it is still the Wilson line expectation value, which would indicate the transition by a discontinuity at $T_{c}$. However, $\langle L\rangle$ will fail to remain a good order parameter, when the quark mass is further lowered. It was the sensitivity to the $Z(N)$-symmetry which had qualified $\langle L\rangle$ as an order parameter in the $m\rightarrow\infty$ case. Now, where the symmetry is explicitly broken independently of the phase, we have to look for new criteria that tell us the phase of the system at a given temperature. It is the screening property, which is different in the high- and low temperature phase. At low temperatures, color charges are screened due to a breakage of strings. At high temperatures, flux tubes do not break, but get attached to large networks of strings, at least for very heavy masses. For lighter masses, quarks can also break up strings at high temperature. (We see that the clear distinction between low- and high T-phases may get lost depending on the value of the quark mass.)

The probability for string breakage at high temperature is determined by the energy costs for popping a $q\bar{q}$-pair out of the vacuum and the competing gain in entropy by breaking a link. Lighter masses facilitate breaking of links. Less surface energy is necessary to stabilize a hole in an infinitely connected network. Clearly there should be a critical mass, where the surface energy vanishes at $T_{c}$ and the number of broken flux tubes exceeds the critical number of irrelevant links . Once relevant links are broken, by definition, the network ceases to exist.

In Patel’s models one has a heuristic understanding of how the transition temperature should scale as a function of quark masses. As the quark mass decreases, string breakage is facilitated. It becomes more difficult to generate an infinitely connected network. Thus $T_{c}$ should increase with decreasing $m$. The opposite tendency is observed in numerical simulations.

From the above line of arguments it is suggested that the transition will always disappear in the $m\rightarrow 0$-limit. The energy costs for the creation of a $q\bar{q}$-pair are zero. Recall, however, that it is just the $m\to 0$-limit, where the chiral symmetry comes into the game and generates dynamical masses at low temperatures. In the chiral limit, the scalar mesons are believed to obtain their finite masses exclusively from chiral symmetry breaking. In the preceding sections we have investigated just the opposite effect: perturbing around the chiral limit, turning on an external field by increasing $m$ from zero towards realistic values.

In the chiral limit the strength of the first order is said to increase for an increasing number of flavors. On the other hand, in the same chiral limit, there is a gain in entropy due to easy string breakage as a remnant of the deconfinement transition. It is difficult to estimate, whether a critical number of flavors exists where one of the competing effects will win.

A crossover phenomenon at intermediate masses may result as a combination of two tendencies. The quark masses are so small that the deconfinement transition has disappeared, an infinitely connected network can no longer be formed. They are so large that effects of chiral symmetry breaking are no longer important, and the chiral transition is washed out as well. It remains to be seen, whether the realistic quark masses fall in a window where both transitions are replaced by crossover phenomena.

As long as chiral symmetry is not implemented in the massless limit, such competing effects cannot be resolved by Patel’s models, even if matter fields are included.

Our presentation so far might have suggested that the description of the deconfinement transition in terms of color flux tubes gets stuck on the heuristic level. This is not the case. A three-dimensional action can be formulated in terms of occupation number variables (Patel, 1984b). These variables live on a lattice with lattice constant $a$ and take on values in the positive integers. The flux tube variables are associated with the links, the quarks and vertex variables with the sites of the lattice. The input parameters for the action are the string tension $\sigma$, the lattice constant $a$, the quark mass $m$ and the vertex coupling $v$. If Gauss’ law is implemented, the model is equivalent to a 3-dimensional XY-model with nearest neighbor coupling in a uniform magnetic field $h$ and a magnetic field $p$. Mean-field calculations in various limiting cases confirm the conjectures which are based on the heuristic arguments (Patel, 1984b). The model can be extended to include several numbers of flavors and chemical potentials.

We summarize the virtues of the color flux tube model. It abandons symmetry breaking as driving force of the phase transition, at least at a first place. This is desirable as an alternative way of understanding the transition dynamics in mass parameter regions, where the symmetry concept is questionable. Here the deconfinement transition has been described in analogy to a gas/liquid transition at finite temperature for different values of the pressure. The pressure plays the role of the quark masses. The order parameter is the density or the volume. Its change is discontinuous, when the temperature is increased as long as the pressure is low enough. In the QCD case, $\langle L\rangle$ jumps at the transition as long as the quarks are heavy enough. As in the QCD case with non-zero masses, a symmetry in the gas/liquid system is absent that could easily distinguish between both phases. In a $(p,T)$-diagram the liquid gas phases may be smoothly connected as well. For the analogies we refer to the dictionary of Table II.

The deconfinement transition here resembles a gelation transition, the connectivity of the color flux tube network is an essential characteristic for $SU(3)$. A similar intuitive understanding of the chiral transition is still missing, when it is described in the framework of the sigma model. For the chiral transition it is less clear, what the competing effects to string breakage or network formation could be.

It seems rather worthwhile to incorporate chiral symmetry properties in color flux tube models or vice versa to supplement sigma models with flux tubes for the gluonic degrees of freedom.

We conclude with a few remarks on phenomenological string models for hadrons. Experimentalists describe the phenomenology of heavy-ion collisions with Monte Carlo codes that are based on relativistic microscopic models. String formation and fragmentation are essential ingredients. These models share some features with cellular automata. One tries to find a set of rules for string formation and breakage (fragmentation) such that observed particle multiplicities and $\left<p_{T}\right>$-spectra are reproduced for given initial conditions. An example for such a string model is the VENUS-code of Werner (1989) utilizing the concept of yo-yo-strings. Strings are formed via color exchange.

One may suspect that yo-yo-strings in VENUS share only the name with Patel’s color flux tube strings. So far it is little more than the name. The yo-yo-string is elastic, its elasticity is characterized by the string tension. Closed loops are forbidden, the flux can only terminate on colored partons (quarks, diquarks etc.). The endpoints are characterized by their flavor contents. Yo-yo-strings are allowed to stretch, break and participate in a collective motion. They cannot bend or bifurcate. There is no space for thermodynamic concepts, for a transient plasma in the early stage of the collision, no way to model a phase transition in the present formulation.

Patel’s strings are neither one-dimensional rigid–line tubes connecting the colored end-points, nor arbitrarily pliable. The rigidity constant is essential for modulating the entropy and initiating the phase transition. The range of possible applications of VENUS-type models may be extended, if the strings are not restricted to yo-yos, but allowed to close, bend and bifurcate.

2 Dual Ginzburg-Landau models

The main idea in dual Ginzburg-Landau models is to explain the confinement mechanism by a dual Meissner effect. Let us recall the Meissner effect in a superconductor. The magnetic field is expelled from the embedding superconducting medium and concentrated in a flux tube between magnetic monopoles of opposite charge. For large distances the constant energy per unit flux tube leads to a linear rise for large distances between the monopole charges. In the superconducting state of metal, the electrons generate Cooper pairs. The superconducting medium is characterized by an order parameter $\psi$, $|\psi|^{2}$ represents the density of Cooper pairs. The order parameter vanishes in the normal conducting state where Cooper pairs are absent. The transition to the superconducting phase is triggered by the condensation of Cooper pairs.

A dual Meissner effect may be an appealing explanation for confinement. In the dual Meissner effect the role of the (chromo)electric and (chromo)magnetic fields is reversed. The chromoelectric field between the chromoelectric charges of quarks and antiquarks is expelled from the embedding confinement vacuum. The chromoelectric flux tube leads to a linear rise in the energy between quarks at far distances. The transition to the confinement phase is triggered by a condensation of chromomagnetic monopoles.

The interest in models based on such a heuristic explanation has recently revived. The dual Meissner effect seems to be realized in compact lattice QED (Polyakov, 1975; Banks et al., 1977; DeGrand and Toussaint, 1980; Stack and Wensley, 1992). The main task in QCD is to identify the magnetic monopoles. Due to ’t Hooft’s idea (’t Hooft, 1981) the identification should be possible after an abelian projection. An abelian projection of QCD amounts to a partial gauge fixing. There are many ways of extracting an abelian theory out of non-abelian QCD. Popular choices are a variety of unitary gauges and the maximal abelian gauge (Kronfeld et al., 1987; Suzuki and Yotsuyanagi, 1990 and 1991; Suzuki, 1993). The maximal abelian gauge turns out to be a particularly useful way of demonstrating the dual Meissner effect. If the “non-abelian part” of the gauge freedom is fixed, the original $SU(3)$ gauge symmetry is reduced to the maximal abelian torus group $U(1)\times U(1)$. QCD becomes an effective theory of abelian “electric” charges, “photons”, and “magnetic” monopoles with respect to $U(1)\times U(1)$. The photon fields are the regular part of the abelian field, the magnetic monopoles the point-like singular part. The monopole part is isolated from the $U(1)$-field following a procedure proposed by DeGrand and Toussaint (1980). ’t Hoofts conjecture about the dominant role of monopoles seems to be realized in the maximal abelian gauge. The conjecture may be verified in several steps.

• $\bullet$

Typical indicators of confinement are the string tension, the Wilson loop, and the Polyakov loop. It is natural to test whether their expectation values from full lattice QCD can be reproduced by the abelian variables alone. This is indeed what is observed, the phenomenon is called abelian dominance (Suzuki and Yotsuyanagi, 1990 and 1991; Hioki et al, 1991; Suzuki, 1993; Matsubara et al., 1994).

• $\bullet$

More specifically one would like to know which part of the abelian residual variables is responsible for the observed abelian dominance. Here it turns out that the monopole contributions alone determine the string tension and the Polyakov loop in an $SU(2)$ and $SU(3)$ pure gauge theory (Shiba and Suzuki, 1994; Kitahara et al., 1995; Ejiri et al., 1995; Miyamura, 1995; Matsubara, 1995). The similar behavior of the abelian and the conventional Polyakov loop expectation value is not only seen in the maximal abelian gauge, but also in a number of different $U(1)$-projections.

• $\bullet$

Once the relevant degrees of freedom have been identified, the next question concerns their dynamics. Which monopoles are important in the confinement and deconfinement phases, what drives the phase transition ? Here some partial answers are known. In the confinement phase one has a long monopole in each configuration, while long monopole configurations are absent in the deconfinement region (Kitahara et al., 1995). This is easily understood from the main difference between low and high temperature QCD. The difference in the monopole dynamics comes from the bound on the maximal monopole extension due to the large or small number of time slices. The driving mechanism for the deconfinement transition is supposed to be the balance of the entropy and energy of maximal extended monopoles (cf. the preceding section for a similar explanation in the flux tube model). Further studies are, however, necessary to clarify the calculation of the entropy of monopoles (Kitahara et al., 1995 a and b). The difference in the order of the deconfinement transition for $SU(2)$ and $SU(3)$ remains to be explained.

• $\bullet$

The observation of abelian dominance and the identification of long monopoles as the most important field configurations in the confinement mechanism suggest a construction of an effective action in terms of these monopoles. Effective monopole actions have been considered in compact QED and $SU(2)$-QCD (Shiba and Suzuki, 1994 and 1995). Extended monopoles can be implemented as well by performing a block spin transformation on the dual lattice. The results obtained so far indicate an interesting scaling behavior, giving some hope that the effective action actually describes continuum physics.

The results in dual Ginzburg-Landau models look rather encouraging. The dependence on the finite volume deserves further studies on larger lattices to make sure that the dynamics of abelian $U(1)\times U(1)$ monopoles of lattice QCD survives the continuum limit. The gauge dependence of the proposed confinement mechanism is another open question. So far most of the features of the dual Meissner effect have been observed in the maximal abelian gauge. Recently Chernodub et al. (1995) have studied the pure $SU(2)$ gauge theory in different abelian projections. Their results suggest the conclusion that the topological configurations, which are relevant for confinement, do depend on the gauge. Monopole condensation as mechanism for confinement appears as specific feature of the maximal abelian gauge. So called minopoles replace monopoles in the ‘minimal’ abelian gauge, and string-like topological objects may be important as well. Such a translation of the confinement mechanism between different gauges deserves further investigations. Remarks on unitary gauges can be found in (Matsubara et al., 1995) and (Suzuki and Yotsuyanagi, 1991).

3 Some further approaches

We conclude this chapter about effective models with an outlook for further directions. Descriptions including quark degrees of freedom have been left out so far. Also the string networks of the preceding section cannot be regarded as representative for the variety of models, which have been proposed for gluonic degrees of freedom.

An important class of models for the gluonic sector are instanton descriptions of the QCD vacuum (Shuryak, 1984). The idea is that the most important (nonperturbative) contributions to the QCD-partition function come from instantons and anti-instantons at short distance scales. Dynakonov and Mirlin (1988) and Kanki (1988) have considered the instanton vacuum at finite temperature. Within a variational approach they have shown that the “evaporation” of instantons and anti-instantons out of the vacuum leads to chiral symmetry restoration as the temperature is increased.

Ilgenfritz and Shuryak (1989) and Novak et al. (1989) have extended the analysis of instanton models to the inclusion of quark degrees of freedom. The qualitative conclusion of these authors is a crossover phenomenon rather than a chiral transition, when finite quark masses are included. The results have been obtained within mean-field-type calculations.

Frequently it is the symmetry of the underlying QCD action, which is taken as a guiding construction principle for the effective action. Applied to the gluonic sector one may ask for the analogue of the chiral condensate and the role it plays in the deconfinement (and chiral) transition. A natural analogue is the gluon condensate $\langle F_{\mu\nu}F^{\mu\nu}\rangle$, if $F_{\mu\nu}$ denotes the QCD field strength tensor.

In view of the gluon condensate we recall the basic symmetry of QCD related to the condensate. At the classical level the QCD Lagrangian with massless quarks is invariant under scale transformations $x^{\mu}\rightarrow e^{\lambda}x^{\mu}$ with some real constant $\lambda$. The corresponding (classically conserved) current is the dilatation current $S_{\mu}=T_{\mu\nu}x^{\nu}$, where $T_{\mu\nu}$ is the energy-momentum tensor. On the quantum level, even massless QCD is no longer scale invariant. It is broken by the scale- or trace anomaly of QCD

[TABLE]

Here $\beta(g)$ denotes the beta-function of QCD, g is the coupling. The perturbative contribution to 1-loop order is given as $(-g^{3}/16\pi^{2})(11-{2\over 3}N_{f})$.

Schechter (1980) has proposed an effective action for both mesonic and gluonic degrees of freedom. In the limit of zero quark masses this action is invariant under the $SU(N_{f})_{L}\times SU(N_{f})_{R}$ chiral transformation ($\times U(1)_{V}$ baryon number), reproduces the axial $U(1)$ anomaly (the chiral anomaly) and in addition the trace anomaly of QCD.

For definiteness we state the result for the effective Lagrangian, as it has been used by Campbell et al. (1990)

[TABLE]

The following notations are involved. Gluonic degrees of freedom are represented by a scalar gluonium field $\chi$ with scale dimension 1. It is assumed to take a non-vanishing vacuum expectation value $\chi_{0}=\langle 0|\chi|0\rangle$, thus scale invariance is spontaneously broken on the effective level. The matter fields are represented by $U$

[TABLE]

The fields $\phi_{i},i=0,\ldots,8$ denote the nine pseudoscalar meson fields. The light quark mass matrix is given by $m_{q}=diag(m_{u},m_{d},m_{s})$. Up to prefactors depending on $\chi$ and $\chi_{0}$, the matter part of the Lagrangian is the familiar non-linear $SU(3)\times SU(3)$ sigma model (first term of $L_{m}$), plus a term proportional to the quark masses breaking chiral symmetry explicitly, plus a third term ($\propto\phi^{2}_{0}$) accounting for the ninth pseudoscalar meson. All terms are decorated with prefactors correcting for the right scale dimensions.

The specific form of the potential for the gluonic part $L_{g}$ has been proposed by Schechter (1980). It is this term which is designed to guarantee the trace anomaly according to

[TABLE]

where $B$ is the vacuum energy density of the pure gauge sector. Eq. (341) leads to the identification

[TABLE]

Campbell et al. (1990) have used the Lagrangian (339) as starting point for thermodynamic considerations.

The matter part of the Lagrangian is easily modified. Patkós (1991) has used the linear version of the $SU(3)\times SU(3)$ sigma model for $L_{m}$ to calculate the surface tension of the first order chiral transition under the inclusion of gluonic degrees of freedom.

Kusaka and Weise (1992) have chosen the Nambu-Jona-Lasinio (NJL)-Lagrangian with $SU(2)_{L}\times SU(2)_{R}$ symmetry. An interesting relation can be read off from their Lagrangian. Due to the prefactors ($\chi/\chi_{0})^{3}$ in Eq. (339), the quark condensate is not only determined by the meson condensate $\langle 0|U+U^{+}|0\rangle$, but knows about the gluon condensate as well. Such a relation reads in the effective model of Kusaka and Weise (1992)

[TABLE]

Here $\chi_{0}=\langle 0|\chi|0\rangle$ denotes the mean-field vacuum expectation value as above, $\chi_{m}$ minimizes the potential term of $L_{g}$ and $G$ is the four quark coupling.

The interesting feature about this relation arises when it is extrapolated to temperatures larger than zero. Chiral symmetry may be either restored ($\langle{\bar{u}}u\rangle_{T}\rightarrow 0$), because of the mesonic part $\langle\sigma\rangle$ has melted or because the gluon condensate $\chi_{0}$ vanishes. This leads to an unequality between the chiral $(T_{ch})$ and the deconfinement $(T_{d})$ transition temperatures

[TABLE]

The interplay between both condensates can be studied this way. A question of the type“who drives whom” in the phase transition may be answered in these models.

The chiral phase transition has been investigated both as function of temperature and/or density in a variety of NJL-models (Klimt et al., 1990; Lutz et al., 1992; Hatsuda and Kunihiro, 1985 and 1987; Kunihiro, 1989 and 1991; Bernard and Meissner, 1988). In general the calculations have been performed within a mean-field approach. The common conclusion is that under realistic conditions for coupling strengths (or quark masses) the chiral transition is replaced by a crossover phenomenon.

An extended NJL-model has been derived by Bijnens et al. (1993) as a low-energy approximation to QCD. Starting from basic QCD, gluon fields or alternatively gluon fields and quark fields are integrated out within a certain approximation in the path integral formulation. The resulting extended NJL-model includes various low-energy models in appropriate limit cases, in particular the chiral quark model of Manohar and Georgi (1984), which extends the linear sigma model of our previous sections.

Let us finally discuss the chiral quark model in little more detail. Originally this model was proposed by Manohar and Georgi as an effective model for quarks, gluons, and Goldstone bosons to explain the success of nonrelativistic quark-models. Gocksch (1991) has considered a version of the chiral quark model, where gluonic degrees of freedom are dropped, and the Goldstone part is realized as the linear $SU(2)\times SU(2)$ sigma model. The Lagrangian is given by

[TABLE]

The notation is obvious from former notations in the text. Note that the Goldstone bosons ($\vec{\pi}$) and the $\sigma$-channel are treated on an equal footing with the quark degrees of freedom. This accounts for the distinguished role the pions and the sigma-meson play among the low mass hadrons in the chiral transition. They cannot be described as states of weakly interacting quarks. (The well-known price of this treatment is a double counting. For example there is a pseudoscalar ${\bar{q}}q$ bound state as well as the ”fundamental” pion state.)

Gocksch has performed mean-field calculations at finite temperature based on the Lagrangian (346). This way it has been possible to reproduce lattice results for static hadronic screening lengths and the quark number susceptibility (Gocksch et al., 1988; Gottlieb et al., 1987c). There are different ways for chiral symmetry restoration above the transition. One possibility is a parity doubling of the known hadron spectrum (DeTar and Kunihiro, 1989; McLerran, 1986). Parity partners should be degenerate in their mass above the chiral transition, but still in the confinement phase.

The more standard symmetry restoration is realized through massless quarks. The results of Gocksch indicate that the lattice data for the $\varrho$-meson and nucleon screening masses can be understood in terms of nearby massless propagating quarks, while the $\vec{\pi}$- and $\sigma$-modes cannot be explained this way. This favors the realization of chiral symmetry restoration along the ”standard” way, i.e. via massless quarks with degenerate $m_{\pi}$ and $m_{\sigma}$ masses in the chiral symmetric phase.

The chiral quark model (346) gives a nice example for an effective model, which allows an explicit comparison with lattice results and is able to explain them. Such an agreement justifies the decimation in the very ansatz for the effective Lagrangian. In this particular case it confirms the distinguished role of the sigma and pions. It remains unclear, whether the double counting in the chiral quark model has any contaminating effect on the results.

Interesting directions for future work are combinations of “chiral” and “gluonic” effective models to study the interplay between both type of transitions. Dual Ginzburg-Landau-models may be promising candidates for describing both aspects, confinement and chiral symmetry. For first attempts see e.g. Miyamura (1995). It should be verified that the $U(1)\times U(1)$ abelian monopoles survive the continuum limit and maintain their dominant role in the presence of dynamical quarks. An interesting question then is, how the confinement mechanism in terms of abelian monopole condensation translates to different conditions for gauge fixing.

Unlike QCD there is no unique favorite action on the effective level. Therefore one should study the variation of the results under different decimations of the underlying full QCD. This applies to generic effective models.

V RELATIVISTIC HEAVY ION COLLISIONS

A Scales and observables

The experimental possibilities to test the QCD predictions of finite-temperature phase transitions from normal hadronic matter to a quark-gluon plasma are limited if we face the transition temperature of $\approx 10^{12\ \circ}K$. Most probably, the QCD transition occurred $10^{-6}$ sec after the big bang when the universe cooled down to the transition temperature. It is very questionable whether any remnants of the transition can be seen today. Promising alternatives to reproduce the QCD transition in some kind of little bang in laboratory experiments are heavy-ion collisions at ultrarelativistic energies. In the collision a large amount of the initial kinetic energy will be concentrated in a short spacetime interval and form a fireball of matter which could reach thermal equilibrium.

In the last decade, it has become technically feasible to create matter with energy densities 10-100 times that of ordinary nuclear matter (see e.g. Jacob and Satz, 1982 or Schmidt and Schukraft, 1993). The available energy for particle production is specified in terms of $\sqrt{s}$, where $\sqrt{s}$ is the total center-of-mass energy. In 1992 the AGS-machine at Brookhaven National Laboratory (BNL) has produced very heavy-ion beams of 197Au with momenta of 11.4/c GeV per nucleon (Barrette, 1994). At the CERN-SPS machine, 200 GeV/$n$ are reached for beams up to ${}^{32}S$, still ‘light’ heavy ions. Since November 1994, a Pb beam of 160$~{}GeV/c$ per nucleon is available at the SPS. Future experiments are planned starting in 2006 where a value of $\sqrt{s}=6300$ GeV/$n$-$n$ ($n$-$n$ stands for nucleon-nucleon) for lead should be reached at the CERN-LHC collider. At RHIC of BNL $\sqrt{s}=200$ GeV/$n$-$n$ will be reached in 1999.

Given the value of $\sqrt{s}$ and the mass number $A$ of colliding ions, one would like to estimate the initial temperatures that can be reached in such a collision. The basic observable is the multiplicity per unit rapidity $dN/dy$ of secondary hadrons which are emitted in the collision. If one extrapolates the known relation between $\sqrt{s}$ and $(dN/dy)_{p}$ in proton-proton collisions to central nucleus $(A)$-nucleus $(A)$ collisions, the relation is given as (Satz, 1990a)

[TABLE]

with $\alpha\geq 1.1$. This leads to multiplicity densities of about 480 at midrapidity (including neutrons and neutral pions) at the AGS (Barrette et al., 1995) for $Au+Au$, and to an estimate of 790 for the maximal $dN/dy$ (when all hadrons are included) for $Pb+Pb$ at the SPS (Margetis et al., 1995). At LHC for $\sqrt{s}=6300$ GeV/$n$-$n$ multiplicity densities of 1500-2500 are expected according to Eq.$5.1$. The multiplicity density of the final state hadrons can be related to the initial state energy density $\varepsilon$ either with (approximate) energy conservation (free flow) or entropy conservation (see section V B 1). For free flow the relation is given by

[TABLE]

Here $R_{A}$ is the nuclear radius, $m_{T}$ the transverse mass and $\tau$ the equilibration time. With $\tau\approx 1$ fm/c and $m_{T}\approx 0.5$ GeV Eq.(̃348) leads to an average initial energy density of 1.5-2.5 GeV/fm3 for central $Pb-Pb$ collisions at SPS and 4.6-7.8 GeV/fm3 at LHC. For $Si+Pb$-collisions at AGS of BNL the initial energy density has been estimated as 0.6 GeV/fm3 (Stachel and Young, 1992).

The energy density is translated to a temperature according to an ideal gas relation $\varepsilon\propto T^{4}$, where the proportionality constant depends on the number of included degrees of freedom. For three massless quark flavors, the initial temperature for $Pb-Pb$ collisions at SPS ranges from 170-190 MeV (free-flow assumption) or 160-190 MeV (isentropic expansion). If we compare these values to estimates from lattice calculations including dynamical fermions, AGS- and SPS-energies could produce temperatures slightly above $T_{c}$. Pronounced signatures of a plasma via thermal radiation can be only expected if the initial temperature $T_{i}$ is well above $T_{c}$. More recent estimates of $T_{i}$ by Shuryak (1992), Kapusta et al. (1992) and Geiger (1992a and b) give more optimistic values for $T_{i}$, $T_{i}\sim 500-600$ MeV should be reached (cf. also below).

In applying these formulas, we have implicitly made use of the fact that energy, entropy, temperature are well defined notions in describing the collision. This assumption will be discussed later on.

At this point, let us pause for a comparison to the observation of a phase transition under daily life conditions, the boiling of water. Usually one does not appreciate all the well defined experimental conditions. The fluid container has a fixed volume, the fluid is at rest and in thermal equilibrium. Calibrated thermometers are at hand, whose Hg-column grows linearly in the considered temperature range. The phase transition is easily seen as conversion from the liquid to the vapor phase while the temperature stays constant. Also heating sources are available such that tuning of the temperature does not pose a problem either. The temperature may be tuned adiabatically or as a quench as one likes.

It is not surprising that at temperatures where hadronic matter gets dissolved into its components, the experimental conditions drastically change. In collider experiments the volume is not fixed in the evolution of the plasma to the hadron gas. At high collision energies the nuclei interpenetrate each other at the collision and recede as Lorentz contracted pancakes, leaving a hot vacuum with secondaries of the collisions between them (in the central rapidity region). After hadronization this hot area is primarily a pion gas.

At LHC-energies the volume at the transition is estimated to be 5-8 times larger than the initial volume, for Pb-Pb collisions the critical volume is of the order of 800 - 1200 fm3. After the transition the system continues expanding until freeze-out, where interactions can be neglected or/and the mean free path of particles gets of the order of the size of the system. (Both definitions of ‘freeze-out’ lead to different estimates for freeze-out volumes). At LHC freeze-out volumes could be of the order of $10^{4}$ to $10^{5}$ fm3 (Satz, 1990a).

Next we come to various time scales which are involved in heavy-ion collisions. Note that the total duration of a ‘little bang’ is only of the order of $10^{-23}$ sec. (The characteristic length scale of 1 fm is a very short distance for light to pass by.) Time scales are the equilibration time $\tau_{0}$, the freeze-out time, the delay caused by a possible first order phase transition, the conversion rate of one phase into the other and the expansion rate. All of them are between say 0.1-100 fm/$c$. The largest value refers to an extreme delay of freeze-out due to strong supercooling, the smallest one corresponds to a more recent estimate for the equilibration time (Shuryak, 1992). The equilibration time is the time it takes until the system reaches a stage of local thermodynamic equilibrium, when it has passed an intermediate preequilibrium stage after the bang. ‘Local equilibrium’ means, that energy, entropy, pressure and temperature can be locally defined. In contrast to our water boiling experiment, the temperature is – after it is defined at all – a function of space and time $T(\vec{x},t)$ in a volume, whose geometry is not even fixed, but depends on the impact parameter value $b$.

Volumes of the order of thousands of fm3 and time scales of some tens of fm/$c$ can be measured with pion-interferometry, especially a time delay due to a transition (see section V C 4). The lifetime of the fireball is estimated to be 10-20fm/$c$ for Si+Pb(Au) collisions at 14.6 GeV/$n$-$n$ from pion-interferometry measurements at the AGS-machine (see e.g. Stachel, E814 collaboration, 1994).

Temperature scales which should be distinguished are the initial temperature at the onset of local equilibrium $T_{i}$, the transition temperature $T_{c}$, possible values of superheating/supercooling effects and the decoupling temperature $T_{d}$ (of the order of the pion mass).

Candidates for thermometers are thermal photon and dilepton spectra (Shuryak, 1978; see also Gyulassy, 1984). The differential cross section of dilepton production as a function of the invariant lepton pair mass $M$ and the rapidity $y$, i.e. $(d^{2}\sigma/dM^{2}dy)_{y=0}$, is predicted to scale according to exp $(-M/T)$, $T$ is the temperature of the emitting system. Unfortunately these thermometers like to hide in the background of other dileptons. Thermal dileptons are difficult to be identified in the total dileptonic yield (see section V C 2). Also pions have been proposed as thermometers for measuring the freeze-out temperature (Brown et al., 1991). Pions from the decay of the $\Delta(33)$-resonance have a characteristic $p_{T}$-distribution which is very different from that of primary pions. The ratio of $\Delta(33)$-resonances to nucleons sensitively depends on the temperature via Boltzmann factors. Thus a measurement of the $p_{T}$-distribution allows the identification of decay pions and a measurement of the temperature dependent ratio of $\Delta(33)$-resonances to nucleons.

If it is difficult to measure the temperature, let us see, how we can tune it. The initial temperature changes with the initial energy density ($\varepsilon\propto T^{4}$ for an ideal gas), and that depends on $\ln\sqrt{s}$. Thus in principle one could vary $s$ for a given nucleus $A$. Due to the $\ln\sqrt{s}$-dependence this would require a large variation in the incident beam energy, accompanied by a considerable loss in the luminosity, if $\sqrt{s}$ is reduced (Ludlam and Samios, 1988; Satz, 1990a). For asymmetric collisions between small and big nuclei, the energy can be increased by going from peripheral to central collisions, i.e. by varying the impact parameter. For symmetric collisions the realistic possibility which remains is the variation of $A$, i.e. the type of nucleus itself. Going from $S-S$ to $U-U$-collisions roughly gives a gain in energy density by a factor of $7-8$ at the price that the volume changes as well. Going from fixed target to collider experiments one expects a factor of $2-4$ in the increase of the initial temperature $T_{i}$. The estimate of $T_{i}$ at AGS and SPS is about 190-200 MeV, at RHIC $T_{i}$’s will be reached between 400-500 MeV, at LHC it may be 600-900 MeV.

Last but not least to complete our comparison to a phase transition under ’normal’ conditions, we have to find observable signatures of the QCD transition. The goal is to identify observables that could reflect almost constant pressure or temperature over an interval where energy and entropy densities ($\varepsilon$ and $s$) rapidly change, where the phase conversion takes place. Typical observables, which are at our disposal in heavy-ion collisions, are multiplicity distributions in rapidity space and average transverse momenta $\langle p_{T}\rangle$. It turns out (see section V C 1) that $\langle p_{T}\rangle$ values can be a measure for the initial pressure and temperature, while $(dN/dy)$ distributions depend on the initial values of $\varepsilon$ and $s$. Thus a $T-\varepsilon$ diagram corresponds to a $\langle p_{T}\rangle-dN/dy$ plot.

There are a variety of other signatures which are in principle sensitive to a transient plasma and to the transition dynamics as well. They are discussed in sections V.C and D.

$\bullet$ We summarize the main complications we have to face in heavy-ion experiments. The basic assumptions that the fireball is large enough and long-lived enough to reach thermal equilibrium, the application of thermodynamic concepts have to be checked and justified. Due to the expansion dynamics, different competing time and length scales are involved. The spacetime expansion from the initial thermalization until freeze-out has to be unfolded of the final observables, if one wants to compare the signatures with predictions of static, microscopic equilibrium quantities. Usually the folding is provided by a hydrodynamic description.

Hydrodynamics is a useful computational tool in estimating bulk features like leptonic or hadronic particle yields, multiplicity fluctuations, orders of lifetimes etc. Note also that hydrodynamics describes off-equilibrium situations although it is based on local equilibrium conditions. The system is expanding and cooling and out of global equilibrium. In this sense, it is also a conceptual framework to treat nonequilibrium situations in a way that all the information of equilibrium QCD is not lost, but can be built in. This information contains the equation of state, elementary cross sections, structure functions and other derived quantities of equilibrium thermodynamics which we have partly outlined above.

There are a few other concepts to treat off-equilibrium aspects in heavy-ion collisions. We will discuss the determination of the nucleation rate of hadronic bubbles in the plasma according to Langer’s approach in condensed matter physics (Langer, 1969; Csernai and Kapusta, 1992a and b) (section V D 1), an alternative process for phase conversion (large domain coarsening) in section V D 2 (Borrill and Gleiser, 1995), a calculation of transport coefficients (based on a combination of linear response theory with hydrodynamic concepts (section V D 3), and the concept of dynamical universality applied to the case far off-equilibrium (Rajagopal and Wilczek, 1992) (section V D 4).

The effect of a phase transition on the hydrodynamical flow depends on the type of phase transition dynamics. If the transition is of first order but proceeds smoothly, close to equilibrium, the effect is just to slow down the expansion. Discontinuities in thermodynamic quantities would be reflected in shock-like discontinuities of the fluid. If the transition is of first order but involves metastable states - a supercooled plasma or a superheated hadron gas - deflagrations or detonations may evolve with the possible effect of large multiplicity fluctuations (section V C 5).

Such explosive processes are one source of entropy production during the evolution. Other sources are dissipation effects and the freeze-out transition. All of them are estimated to produce little extra entropy such that the approximation of entropy conservation during the evolution seems to be justified (Blaizot and Ollitrault, 1990, and references therein).

Although dissipation effects on entropy production may be small, pure glue is rather viscous. It is amusing to estimate the viscosity of a gluon gas at tera degrees in comparison to ‘normal’ gases. For $T$ close to $T_{c}$ the shear viscosity is of the order of the $\Lambda$-QCD scale. Thus it is $10^{16}$ times the viscosity of classical gases under ’normal’ conditions (where it is $10^{-5}$ kg/$m\cdot s$) (Hosoya and Kajantie, 1985).

The main ingredients in a hydrodynamic description are the initial conditions (section V B 1) and the equation of state (section V B 2). Before we go into detail, let us finally quote some numbers from (Cleymans et al., 1986) to show that the basic condition for applying a hydrodynamic description is not violated too much: The mean free path of a particle in a medium has to be much smaller than the size $L$ of the medium. The mean free path of a quark at an initial energy density of 2.7 GeV/fm3 for $U-U$ collisions is supposed to be 0.22 fm as compared to a diameter of 15 fm of an uranium nucleus. The corresponding quark density is 9 quarks/fm3. Groups who are presently working on three-dimensional hydrodynamics for relativistic heavy-ion collisions are Venugopalan et al. (1994), Waldhauser et al. (1992) and Bravina et al. (1993).

We close this section with a warning concerning the following subsections. The reader will not find any conclusions of the type that there are strong hints for the transition to be of either first or second order. We leave this question open until the end.

B The hydrodynamic framework

The derivation of the hydrodynamic equations can be found in various textbooks, see e.g. (Landau and Lifshitz, 1959), and in their adaption to heavy-ion collisions in the work of Cooper and Frye (1974) or in reviews by (Cleymans et al., 1986) or (Blaizot and Ollitrault, 1990). For completeness we sketch the main steps in deriving an appropriate form for heavy-ion collisions. The adaption to heavy-ion collisions amounts to a suitable choice of coordinates and a set of initial conditions which is based on experimental observations.

Hydrodynamic equations describe the evolution of a gas (or fluid) in space and time. The gas is specified by a local temperature, pressure, energy, entropy and velocity. The equations result from constraints of energy and momentum conservation and other conserved quantities in case, such as the baryon number. If we first neglect dissipative effects (viscosity, thermal conductivity), the energy-momentum tensor of a relativistic perfect fluid in motion with velocity $u^{\mu}$ is obtained by a Lorentz boost from its rest frame as

[TABLE]

The equations for energy-momentum and baryon number conservation are

[TABLE]

and

[TABLE]

respectively, where $n_{B}(\vec{x},t)$ is the local baryon number density. Using Eq. (349), contraction of Eq. (351) with $u_{\nu}$ leads to

[TABLE]

where $\epsilon$ denotes the energy density, $p$ the pressure. An analogous equation can be derived for the entropy density $s$, which can be converted to a temperature equation in the baryon free case. Contracting Eq. (351) with $(g_{\nu\rho}-u_{\nu}u_{\rho})$ leads to the second hydrodynamic equation

[TABLE]

The next step is to choose coordinates adapted to a plasma evolution in cylinder geometry, where the $z$-axis is commonly identified with the beam axis. Now one can express Eqs. (353) and (354) in coordinates $z$ and $t$, the four-velocity $u^{\mu}$ of the matter is written as

[TABLE]

For vanishing radical velocity $v_{r}$ (which is frequently used as approximation), the remaining components of $u^{\mu}$ are parametrized according to

[TABLE]

where $\theta$ is the fluid rapidity, defined via

[TABLE]

A more convenient choice of variables are the space-time rapidity $\eta$ defined as

[TABLE]

and the proper time $\tau$

$\displaystyle\tau=\sqrt{t^{2}-z^{2}}$

with the inverse transformations

[TABLE]

Note that the fluid rapidity coincides with the space-time rapidity in the case of $v_{z}=z/t$.

In terms of these new coordinates, the hydrodynamic equations (353) and (354) for a longitudinal motion are

[TABLE]

and

[TABLE]

This set is complemented by a third equation which follows from baryon number conservation (Kajantie et al., 1983)

[TABLE]

Now we have three equations (V B)-(367) for four unknown functions: the energy density $\varepsilon$, the pressure $p$, the fluid rapidity $\theta$ and the baryon number density $n_{B}$, all of them being functions of $\eta$ and $\tau$ in a longitudinal expansion. Thus, in order to find solutions of the set (V B)-(367), we have to supply one additional equation and to specify the initial conditions. The additional equation is an equation of state relating $\varepsilon$ and $p$ or $T$ and $s$. (Other combinations are possible as well.) As an example, we will discuss the bag model equation of state, see section V B 2. Several proposals have been made for the initial conditions. Here we sketch only the Bjorken-Shuryak expansion scenario. We discuss the longitudinal solutions of Eqs. (V B)-(367) for this choice. The solutions considerably simplify in this special case, which may be one reason why they are frequently used in the hydrodynamic treatments of matter evolution in heavy-ion collisions.

Radial solutions of Eqs. (V B) are rarefaction waves propagating from the boundary into the fluid with the velocity of sound. They differ in an essential way from Bjorken’s scaling solution as they are independent of proper time $\tau$. Radial solutions enter measurements of enthalpy and pressure, cf. section V C 1.

1 Bjorken-Shuryak expansion scenario

In the Bjorken-Shuryak scenario (Bjorken, 1983; Shuryak, 1978), several experimental observations are taken into account. The phenomenon of ‘nuclear transparency’ leads to a separate treatment of the central rapidity and the fragmentation region. Nuclear transparency means the effect that a large fraction of the incoming energy is carried away by two receding nucleons in a nucleon-nucleon collision at high energy. Similarly, in a nucleus-nucleus collision the baryon contents of the colliding nuclei interpenetrates at the collision and recedes as two Lorentz contracted pancakes after the collision. The central rapidity region refers to the fluid of quanta contained in the region between the receding pancakes. In the hadronic phase it mostly consists of pions. Thus it should be a good approximation to neglect the baryon number. Setting $n_{B}=0$ leads to a first simplification of the hydrodynamic set of equations. The separate treatment of the central and the fragmentation regions is justified only if both regions are well separated in phase space. Experimental conditions should be checked to guarantee this, otherwise an analysis in this picture is not adequate (Blaizot and Ollitrault, 1990). At future colliders (RHIC and LHC) these conditions may be satisfied.

Secondly, the pronounced space-time correlations are observed in particle production in the sense that particles with large longitudinal momenta are produced at a late time, those with low momenta promptly, in the center-of-mass system. This is nothing but the twin paradox. Particles live longer in case their velocities are higher. In Bjorken and Shuryak’s ansatz the effect of time dilatation is incorporated in the boundary conditions. Consider an ensemble of particles which are produced at $z=0=t$. If it is only the proper time $\tau$ which determines the moment of disintegration, all particles which measure the same $\tau$ in their rest frame constitute an initial condition at $\tau=\tau_{0}$. That is, the initial condition refers to a hyperbola $\sqrt{t^{2}-z^{2}}=\tau_{0}$ of constant proper time $\tau_{0}$. The space-time rapidity or light cone variable $\eta={1\over 2}\ln{{t+z}\over{t-z}}$ specifies the position on this hyperbola. Two distinct positions are related via a Lorentz boost in $z$-direction. Since particles in the fluid element are supposed to move as free particles, their velocity component $v_{z}$ is given by $z/t$. The physics of a $z$-slice of a fluid element at time $t$ is equivalent to the physics of a $z^{\prime}$-slice at time $t^{\prime}=z^{\prime}/v_{z}$. This is the scaling property in Bjorken’s scaling ansatz. For $v_{z}=z/t$, the space-time rapidity $\eta$ equals the rapidity $y$

[TABLE]

if the four-momentum $\underline{p}$ is parametrized as $\underline{p}=(E,p_{z},p_{t})$, $p_{t}$ being the transverse momentum.

A third feature, which is observed in proton-proton collisions is the plateau structure of inclusive cross sections when they are plotted as functions of $y$. A plateau for central values of $y$ is also expected for nucleus-nucleus collisions. At least the particle multiplicity depends only weakly on $y$ for central rapidities (Bjorken, 1983). Accordingly, a further simplifying assumption seems to be justified. The local thermodynamic quantities like $\varepsilon,p,T,s$ depend only on $\tau_{0}$, but not on $\eta(\tau_{0})$, when the hydrodynamic expansion commences. Thus the initial condition is invariant under Lorentz boosts in $z$-direction. The dynamics preserves Lorentz covariance, which is most easily seen from the tensor equation (351). Therefore, we will look for solutions $\varepsilon(\tau),s(\tau),T(\tau)$ of Eqs. (V B) depending merely on $\tau$. Inserting $v_{z}=z/t$ in the collective four-velocity of the fluid leads to

[TABLE]

A comparison with Eq. (357) shows that the fluid rapidity can be identified with the space-time rapidity $\eta$, which furthermore coincides with the rapidity $y$. With $\eta=\theta$, Eqs. (V B) simplify to

[TABLE]

The baryon number is set to zero in the following considerations. The entropy equation for $s(\tau)$ simplifies to

[TABLE]

where the relations $u^{\eta}=0$ and $(u^{\tau}=u^{t}\cosh\eta-u^{z}\sinh\eta)$ have been used. The solution is

[TABLE]

As is seen upon integration over $d^{3}x=\tau dy\ d^{2}x$, the entropy per given rapidity interval remains constant as long as the hydrodynamic equations can be applied. This need not hold throughout all stages of the expansion, especially not close to freeze-out or in the intermediate period, where the plasma converts to the hadronic phase in one or another way. Let us assume that it approximately holds. Then the important feature of Eq. (394) is that it allows to infer the entropy density in the initial state (more precisely $s_{0}\tau_{0}$) from an observation in the final state (the pion multiplicity), see section V C 1. Under the same assumptions as above, the temperature equation simplifies in $(\tau,\eta)$-coordinates to

[TABLE]

for $\mu=0$, where by definition

[TABLE]

$c_{s}$ denotes the velocity of sound. Integration of Eq. (375) gives

[TABLE]

For a massless free gas, the speed of sound is $1/\sqrt{3}$ in units where $c=1$. Thus the temperature drops more slowly than the entropy density. In fact, the predicted decrease may be too slow, since transverse expansion has been neglected so far.

Finally, we have to solve Eq. (372) for the energy density as a function of proper time. One possibility is to use an equation of state (in principle, it should be the equation of state of QCD) to eliminate the pressure in Eq. (372). The result is that the energy density does not merely decrease because of the expanding volume in proper time, but also due to the pressure exerted by the gas of the covolume. We come back to the equation of state in the next section. Note that the velocity in transverse direction $v_{r}$ has been neglected so far. We shall sometimes abbreviate the solutions in ‘Bjorken-Shuryak’s scenario’ as Bjorken’s scaling solution.

2 The bag model equation of state

Although the bag model leads to a crude description of the equation of state for QCD, we address a short subsection to it, since it is often used in combination with the hydrodynamic equations and leads to quantitative predictions in the end. In the M.I.T.-bag model, the basic features of QCD – confinement and asymptotic freedom – are effectively incorporated via bags (Chodos et al., 1974; Johnson, 1975). In the hadronic phase, quarks and gluons are allowed to move freely or with perturbatively small interactions inside small volumes of space inside the bags. Outside the bags, the quarks are forbidden to move as free particles. The vacuum outside the bags is given a constant energy density $B$ (the bag constant), which keeps the quarks and gluons confined to the bags. During the phase transition, latent heat is necessary to liberate the color degrees of freedom. It turns out to be proportional to $B$. Its original value (0.145 GeV)4 (De Grand et al., 1975) was based on fitting the mass spectrum at $T=0$ and low density in the M.I.T.-bag model. The effective value for the ‘vacuum pressure‘ $B$ which should be used in the quark gluon plasma phase at a baryonic matter density of $n_{B}=1/$fm3 is 0.5 GeV/fm3 (Shuryak, 1988).

An additive shift $B$ in the energy density of the plasma due to the vacuum energy is obtained, if $\ln Z$ of an otherwise free gas of quarks and gluons is shifted by $-BV/T$, that is

[TABLE]

The free gas contribution follows from the usual expression for a free gas of particles and antiparticles with mass $m$, chemical potential $\mu$, and degeneracy factor $g$. In the large volume limit it is given by

[TABLE]

where the “+”-sign refers to fermions and the “-”-sign to bosons. Adding up the various contributions from bosons (gluons), fermions (quarks and antiquarks) and the vacuum, one is lead to

[TABLE]

Here $N_{c}$ is the number of colors, $N_{f}$ the number of flavors, $N_{g}$ the number of gluons and $\mu_{q}$ the chemical potential due to quarks. The standard thermodynamic relations Eqs. (II A 1) lead to the following expressions for the energy density, pressure and entropy density in the plasma phase

[TABLE]

where $a=\pi^{2}/90$. These expressions hold for $\mu=0,N_{c}=3,N_{f}=2,N_{g}=8$. The general expression for $\varepsilon_{p}$ is

[TABLE]

From Eqs. (V B 2a and b) we can easily read off the bag model equation of state in the plasma phase as

[TABLE]

which remains valid for $\mu\not=0$.

Similarly, expressions for $\varepsilon,p$ and $s$ are obtained in the hadron phase, when it is described as a free gas of the lightest mesons and baryons, i.e. pions, nucleons and antinucleons, where the baryonic contribution is sometimes omitted. For pions analytic expressions for $\varepsilon,p$ and $s$ can be derived in terms of modified Bessel functions following from Eq. (381). Contributions of the nucleon-antinucleon gas can be calculated numerically. Heavier mass particles are often omitted for moderate temperatures ($T\leq 250~{}MeV$), although the restriction to pions is rather questionable above $T_{c}\geq 150$ MeV (cf. section IV A 2). Here we state the result for the limit of a gas of massless pions

[TABLE]

such that

[TABLE]

is the bag-model equation of state in the hadron phase. The finite $T$ (and finite $\mu$) transition occurs, when the following Gibbs criteria are satisfied

[TABLE]

The indices $h,p$ stand for the hadron and plasma phases, $c$ for the critical value, $\mu$ is the chemical potential associated with nucleons. Pressure balance at $T_{c}$ relates $T_{c}$ to the bag constant

[TABLE]

The latent heat, determined as the gap in the energy densities $\varepsilon_{p}-\varepsilon_{h}$ at the transition, is $4B$ in this model, e.g. for the effective bag constant of 0.5 GeV/fm3 the latent heat is 2 GeV/fm3.

Almost by construction the bag model leads to a first-order transition at finite $T$ and vanishing $\mu$. There is a finite gap in energy and entropy densities, while the pressure is continuous. Note that in the mixed phase, the velocity of sound $c_{s}$ vanishes, since $c_{s}=dp/d\varepsilon|_{T_{c}}=0$. This is true only as long as $\mu=0$. An exception is a 2nd order transition at $\mu_{c}\not=0$, but $T_{c}=0$ for a particular value of $B$, which we will not consider further here (cf. Cleymans et al., 1986).

Eqs.(389) and (394) can be easily combined to a single equation by using $\theta$-functions as projections on the distinct phases above and below $T_{c}$. Similarly it is not difficult to formulate an equation of state for a second order transition by smoothly interpolating the step-like behavior of the bag model equation of state. Such an ansatz has been proposed by Blaizot and Ollitrault (1990) for $s(T)$. Although the interpolation is ad hoc, it provides a useful check, how sensitively phenomenological implications depend on the order of the transition. The difficulty is to derive such an equation for a second order transition within an effective model. The $O(4)$-model in three dimensions allows a second order transition, but is supposed to describe only the low-temperature phase of QCD ($T\leq T_{c}$) or the immediate vicinity above $T_{c}$, where pions and sigma mesons have not yet dissolved in their constituents.

The bag model does not provide an adequate description of the transition region. Even at the transition point, the plasma and the hadron phases are treated as non- interacting gases differing only in the degrees of freedom and the vacuum energy. An increasing number of hints warns of a naive counting of the number of modes which are taken to be the same as in the limiting cases of high and low temperatures. In particular the number of degrees of freedom of a hadron gas is a delicate problem, if the change of hadron masses as a function of temperature and density is respected. In this case a counting of pion and nucleon degrees of freedom is certainly insufficient. In the vicinity of $T_{c}$ the confinement/deconfinement properties should be implemented in a slightly more sophisticated way than with a single parameter $B$. Nevertheless the bag model is frequently used for temperatures $T\sim T_{c}$. The reason is probably its very tractable analytic form compared to (preliminary) plots of numerical simulations.

At high or low temperatures the bag model is more adequate, e.g the gluonic sector may be treated as a gas of non-interacting glueballs at low $T$ and of gluons at high $T$.

Various improvements on QCD’s equation of state have been proposed within the framework of the bag model. Finite quark masses and perturbative QCD corrections to the partons inside the bag can be taken into account (Kapusta, 1979; Shuryak, 1979) leading to corrections of $\epsilon$, $p$ and $s$. Also strange quarks, finite masses for pions, and higher mass hadrons can be included.

The ultimate goal is an equation of state from lattice QCD with dynamical quarks, which is merely based on non- perturbative ingredients (see section III B). The numerical data should then be presented in a feasible parametric form to facilitate its handling.

In the next section we will see, what we can directly learn from heavy-ion experiments about QCD’s equation of state.

C Signatures sensitive to the nature of the phase transition

Theoretical and experimental tools are described which are sensitive to the nature of the transition, in particular to its order. The sensitivity holds at least in principle. The dependence on the transition dynamics is sometimes hidden in the spacetime expansion. The signatures reveal characteristic features like time delays, nucleation of bubbles, strong correlations or large fluctuations in an indirect way.

Let us start with direct experimental tests of the equation of state. We sketch the possibilities to measure thermodynamic quantities.

1 Thermodynamic observables

The basic observables which are at our disposal in heavy-ion collisions are rapidity distributions of final state particles and their transverse momentum distributions. An extraction of the equation of state requires measurable observables which are related to $\varepsilon(T,\mu),\ s(T,\mu)$ or $p(T,\mu)$. The signals of a first-order transition in a finite volume may be qualitatively very similar to a sharp crossover phenomenon consisting in a rapid rise in the effective number of degrees of freedom over a small range of temperatures, say less than 10 MeV. These numbers are exposed in $s/T^{3}$ or $\varepsilon/T^{4}$, cf. e.g. Eqs. (V B 2) of the bag model equation of state. Thus we have to identify the observables which are related to $s,\varepsilon,p$ and $T$. Roughly speaking, temperature and pressure are measured by the average transverse momentum $\left<p_{T}\right>$ (as usual under certain restrictive conditions), energy and entropy by the particle multiplicity distribution in rapidity space.

We consider the baryon-free case, i.e. $\mu=0$. Four quantities $s,\varepsilon,p$ and $T$ have to be determined. An equation of state is a relation between any two of these four variables $(\varepsilon(p),s(T),s(\varepsilon)$). It can be obtained from experiments (and compared with theoretical predictions) if two relations are used as experimental input. For example, if $\varepsilon(T)$ and $p(T)$ were known, $s(T)$ follows from $dp/dT=s$ and $\varepsilon(p)$ is the equation of state. Alternatively, if the initial condition $s_{0}(\tau_{0})$ is fixed from a measurement of $dN/dy$, then $s(\tau),T(\tau)$, thus $s(T)$ are known from the hydrodynamic equations, and $p(T)$ follows as integral over $s$. Suppose that the second relation is provided by a measurement of $\left<p_{T}\right>$ as a function of $s^{1/3}$, where $\left<p_{T}\right>$ is related to $s/\varepsilon$ (see below) and $s^{1/3}\propto T$. The knowledge of $(s/\varepsilon)(T)$ yields $\varepsilon(T)$, thus all relations are known (Blaizot and Ollitrault, 1990).

In the following subsections we explain relations between $dN/dy$ and $s_{0}$ or $\varepsilon_{0},\ <p_{T}>$ and $\varepsilon/s$ or $T$, as well as $\left<p_{T}\right>$ and $p$ or $(\varepsilon+p)$. We will show why $\left<p_{T}\right>$ vs $\left<dN/dy\right>$ diagrams are roughly equivalent to ($T$ vs $s$) or ($T$ vs $\varepsilon$)-diagrams. In case of a phase transition a flattening of the $\left<p_{T}\right>$-distribution is expected, which has been originally proposed by van Hove (1982) as a possible signature for a phase transition.

Entropy measurements

The basic step is to identify the entropy density with the particle density in the final state. The final state particles mostly consist of pions. Pions are nearly massless bosons, thus their entropy $S$ is approximately proportional to their number $N$. For the densities we have

[TABLE]

with $\alpha=3.6$ for a free gas of massless pions. The rapidity distribution is obtained from the particle density by an appropriate integration. When the four-volume element is expressed in terms of the spacetime rapidity $\eta$ and the proper time $\tau$, it follows for $N$

[TABLE]

or for the number of particles per unit rapidity at a fixed final proper time $\tau_{f}$

[TABLE]

From Eqs. (400) and (374) we have

[TABLE]

If the initial entropy density is taken to be independent of the transverse coordinates $x$, we find that the final rapidity distribution of multiplicity is proportional to the initial entropy density

[TABLE]

where $A_{T}=\pi R^{2}$ is the transverse size of the nucleus. For example, the multiplicity per unit rapidity may be about 150 for a central collision of ${}^{16}O$ with $R\sim 3$ fm on a heavy nucleus (Blaizot and Ollitrault (1990)). If the initial (= equilibration) time is estimated as 1 fm/$c$, it implies a value of $\sim 20$ fm*-3* for the initial entropy density. We will see next what is further needed to extrapolate the initial temperature and the initial energy density.

Energy measurements

To derive a similar expression as Eq. (404) for the initial energy density $\epsilon_{0}$, we need a relation between $s_{0}$ and $\varepsilon_{0}$ and $\varepsilon_{0}$ and $T_{0}$ to eliminate $T_{0}$. From the thermodynamic relation ($\varepsilon+p=Ts$) and the bag model equation of state we find

[TABLE]

In Eq. (405) the factor 1/3 for a massless free gas has been replaced by $c_{s}^{2}$. Next we use Eq. (V B 2a) of the bag model in the plasma phase (assuming that the initial temperature is high enough for the system to be in the plasma phase)

[TABLE]

Again the exponent can be generalized to $1+1/c^{2}_{s}$. The degeneracy factor $g$ in the case of two massless $u$ and $d$ quarks, their antiquarks and gluons is given by

[TABLE]

The different factors correspond to the spin, quark, antiquark, color, flavor and gluon degrees of freedom, 7/8 arises from the Fermi-Dirac statistics. When $T_{0}$ is eliminated in Eq. (406) via (405), and (404) is solved for $s_{0}$, we obtain

[TABLE]

Eq. (408) gives the promised relation between the final state rapidity distribution and the initial energy density $\epsilon_{0}$.

Transverse momentum distributions

A second important class of observables are average transverse momentum distributions. First we consider the relation between the average transverse momentum $\left\langle p_{T}\right\rangle$ and $E/S$. For a thermalized fluid at rest the momentum distribution is isotropic. For an ultrarelativistic fluid the total momentum is equal to the energy, such that the average transverse momentum is proportional to the energy per particle (Blaizot and Ollitrault (1990)

[TABLE]

If we replace $E$ and $N$ by the corresponding densities, it remains to express the particle density $n$ in terms of the entropy density via Eq. (400)

[TABLE]

This is already a relation of the type we are looking for. Note that $\varepsilon/s$ scales approximately with $T$ and $s$ with $T^{3}$. Measuring $\left<p_{T}\right>$ as a function of $s^{1/3}_{0}$ by varying $dN/dy$ accordingly could in principle provide a check of a theoretically predicted relation between $\varepsilon/s$ and $T$.

When $\varepsilon/s$ is calculated from the bag model equation of state and plotted against $(s_{0}/s_{c})^{1/3}$, it displays a feature which turns out to be characteristic for generic $\left<p_{T}\right>$ distributions: a flattening in the region where the initial entropy density $s_{0}$ is chosen to have the critical value $s_{c}$, while a linear rise is seen for small ratios $(s_{0}/s_{c})^{1/3}$ (low temperatures) where $\varepsilon/s$ and $s^{1/3}$ are proportional to each other.

Unfortunately Eq. (410) is too simple to be true. As the QCD plasma fluid undergoes a transverse and a longitudinal expansion, the simple relation between $\left\langle p_{T}\right\rangle$ and the initial temperature via $s_{0}$ gets lost. The decoupling temperature $T_{d}$ at freeze-out and the time at which the transverse expansion sets in enter as new scales. The essential point is whether decoupling and hadronization happen before or after the transverse expansion becomes important. The results of Blaizot and Ollitrault (1990) are the following. If the system decouples before the transverse expansion sets in, $\left\langle p_{T}\right\rangle$ is not sensitive to the equation of state, but characterizes a fluid at rest at the decoupling temperature. In the opposite case (decoupling after transverse expansion), $\left\langle p_{T}\right\rangle$ is roughly that of a thermal distribution with a three-dimensional expansion at decoupling. As an effect of the longitudinal expansion, $s_{0}$ has to be replaced in the naive expression (410) in an appropriate way. The longitudinal expansion diminishes the available energy for transverse expansion.

The value of $\left\langle p_{T}\right\rangle$ is no longer proportional to $\varepsilon/s$. The modified $\left\langle p_{T}\right\rangle$-diagram as function of the initial entropy density can be compared with the diagram resulting from a bag model equation of state combined with a hydrodynamic expansion. The qualitative feature is the same: a flattening of $\left\langle p_{T}\right\rangle/T_{c}$ for initial entropy values lying in the transition region. The physical reason is a reduced pressure on individual particles, if the entropy contents is distributed among more degrees of freedom in a transient plasma. The smaller value for the pressure leads to the reduced average transverse momentum. Next we come to the relation between transverse momentum and pressure.

Enthalpy and pressure

The average transverse momentum $\left\langle p_{T}\right\rangle$ can be predicted from solutions of the hydrodynamic equations. The transverse velocity component is parametrized in terms of rapidity variables. The average transverse momentum is written in terms of the average transverse rapidity $y_{T}$ (due to thermal motion) and the fluid rapidity $\theta$ according to

[TABLE]

Enthalpy measurements can be performed via $\left\langle p_{T}\right\rangle$, if the fluid rapidity $\theta$ is determined as function of the initial enthalpy according to (Cleymans et al. 1986)

[TABLE]

where $\theta_{R}$ is a solution of the set of longitudinal hydrodynamic equations, specialized to rarefaction waves $\theta_{R}$. Solving Eq. (411) for $\theta$ with $\theta$ given by Eq. (412), we find for large $p_{T}/m_{\pi}$ that $\left<p_{T}\right>$ is proportional to $e^{\theta_{R}}$, thus

[TABLE]

It remains the express the r. h. s. of Eq. (413) in terms of measurable quantities. Write the initial enthalpy as $T\cdot s_{0}$, eliminate $T_{0}$ in favor of $\epsilon_{0}$ with an equation of state, replace $s_{0}$ and $\varepsilon_{0}$ via Eqs. (404) and (408).

Barometers are provided by $\left\langle p_{T}\right\rangle$-distributions as function of the pressure. The fluid rapidity is now considered as function of the final energy density $\epsilon_{f}$. The analogous relation to Eq. (412) reads (Gyulassy, 1984)

[TABLE]

leading to

[TABLE]

Eq. (414) is derived from a set of hydrodynamic equations, where the longitudinal expansion has been neglected for simplicity. Eq. (415) explains why $\left\langle p_{T}\right\rangle$ distributions are called barometers. If the speed of sound and the pressure $p_{f}$ in the final state are known, a measurement of $\left\langle p_{T}\right\rangle$ allows an estimate of the initial pressure $p_{0}$. Eq. (415) gives $\left\langle p_{T}\right\rangle$ for a hadron gas, which is described as a massless ideal pion gas with rarefaction waves in transverse direction, but without longitudinal expansion.

The transverse flow is overestimated as long as the longitudinal expansion is neglected. If $p_{0}/p_{f}$ or $\epsilon_{0}/\epsilon_{f}$ in Eq. (415) are replaced by the pion rapidity density $dN/dy$, the resulting $\left\langle p_{T}\right\rangle$ values are actually too large as compared to experimentally measured values. The fluid feels transverse pressure gradients when the rarefaction wave arrives at a fluid element. The arrival is somewhat delayed, since the wave moves inwards with the speed of sound. Thus some energy is already gone to the longitudinal expansion, less energy is available for the transverse motion. Formally this effect can be included by an appropriate form factor, which should be multiplied on the r. h. s.’s of Eqs. (415) and (410), cf. Gyulassy (1984), Baym (1984a and b), Blaizot and Ollitrault (1990) for further details. The terminology to call $\left<p_{T}\right>$ a “barometer” for heavy-ion collisions goes back to Gyulassy (1984). Gyulassy further identifies thermometers and seismometers in the connection of dilepton production and large multiplicity fluctuations, see sections V C 2 and V C 5 below.

We summarize. Bulk quantities like $\epsilon(T)$, $p(T)$, $s(T)$, derived quantities like the velocity of sound, the enthalpy, the energy per degree of freedom $\epsilon/s$ (Redlich and Satz, 1986) and QCD’s equation of state are calculable on the lattice. The equation of state under inclusion of dynamical fermions should finally replace the bag model equation of state which entered the derivation of various relations in the preceding section. Eqs. (404) and (413) are a remarkable example of relations between observables which are directly accessible in experiments and on the lattice. Initial energy and entropy are related to the pion multiplicity per unit rapidity. A variation of $dN/dy$ amounts to a variation of $s_{0}$ or $\epsilon_{0}$. On the other hand, $\left\langle p_{T}\right\rangle$can be a measure for the freeze-out temperature and is sensitive to the pressure. Thus a flattening of $\left\langle p_{T}\right\rangle$ in a $\left\langle p_{T}\right\rangle$ vs $dN/dy$ diagram could be nothing else but a reflection of a slow change of temperature and pressure during a rapid rise in energy and entropy densities, i.e. a rapid crossover or a first order transition in a finite volume. The typical shape of a $\left\langle p_{T}\right\rangle$ vs $dN/dy$ diagram which is schematically drawn in Fig. 29 has been actually seen at Tevatron (Alexopoulos et al., 1990), cf. Fig. 30, which gives $\left\langle p_{T}\right\rangle$ of $K^{\pm}$ and $\pi^{\pm}$ in $p-\bar{p}$ collisions as function of the charged particle multiplicity. Unfortunately a sharp crossover or a phase transition are not the only explanations for the shape. The flattening could be just a kinematic effect (lack of available energy). Kinematic constraints like energy conservation require that $\left\langle p_{T}\right\rangle$ goes to zero at the boundaries of the allowed rapidity interval (Schmidt and Schukraft, 1993). Rescattering effects or minijet production are further alternatives to explain a first increase and a subsequent flattening effect (Satz, 1990; Eskola and Lindfors, 1990). The multiplicity increases only slowly with the incident kinetic energy. The relevant variable has been identified to be $s_{0}t_{0}/s_{c}R_{0}$, where $t_{0}$ is the proper time, at which the hydrodynamical evolution starts, $s_{0}$ is the entropy density at that time, $R_{0}$ is the initial radius and $s_{c}$ the entropy density at the critical temperature (Blaizot and Ollitrault, 1990). This variable should be larger than 1 to see an effect of the transverse expansion. Thus it is not sufficient that the initial entropy density $s_{0}$ is larger than the critical value $s_{c}$, which one might naively expect. The inverse initial radius $1/R_{0}$ is proportional to $1/B$, where $B$ is the mass number of the smaller of the colliding nuclei. Therefore the use of heavier nuclei in the collision may not help, but make things worse. For a review about the available data of $\left<p_{T}\right>$-distributions we refer to Schmidt and Schukraft (1993).

2 Dileptons and real photons

Dileptons

We assume that the initial energy densities of a collision between nuclei $A$ of the projectile and nuclei $B$ of the target are high enough so that a quark gluon plasma is initially generated. Since the size of the system produced in such a collision is not too large and the electromagnetic cross section is small, all dileptons can escape, especially those from the hottest dense state of the plasma. Dileptons will be emitted from all stages of the evolution, from the initial plasma phase, an intermediate possible mixture of plasma and hadron phase till the hadron phase at freeze-out. Thus, differently from hadronic yields, dileptons are sensitive to the whole spacetime history of the evolution. The suitable kinematic variables for dileptonic cross sections are the invariant mass $M$ of the dilepton pair and its transverse momentum $p_{T}$. A typical observable is the multiplicity of dileptons per invariant mass square $M^{2}$, transverse momentum $p_{T}$ and unit rapidity interval $\Delta Y$,

[TABLE]

Here $R$ is a spacetime integral over the rate $dN/d^{4}xd^{4}p$, which is the rate at a given four-momentum $\underline{p}$ of the dilepton pair at a spacetime point $(\vec{x},t)$. The quantity $dN/d^{4}xd^{4}p$ depends on the temperature $T$ via $(\vec{x},t)$, and therefore on the phase of the system. The phase (plasma or hadron) determines the dominant production mechanisms entering the elementary cross sections and the structure functions in $dN/d^{4}xd^{4}p$. The structure functions reflect the medium where the elementary processes take place (quarks inside a pion or in a heat bath of other partons). Collective effects of the plasma may also influence the rate for a given value of $(\vec{x},t)$.

In a given phase, usually several sources are responsible for dilepton production. We are mostly interested in thermal collisions reflecting thermodynamic properties of the supposed heat bath environment. Thermal collisions are by far not the only way to produce dileptons. Other mechanisms are Drell-Yan production, preequilibrium production in the plasma phase, and dileptons from pions via $\rho$-resonance decays in the hadronic phase. Thermal dileptons result from parton collisions in a medium which can be characterized by a local temperature. They are specific for a plasma or a hadron gas which can be described with thermodynamic concepts; they are absent in hadron-hadron collisions.

Finally, we want to find out the mass range $\Delta M$, where $R$ is dominated by thermal dileptons. In this range $R$ should be sensitive to the kinetics of the phase transition and in particular to its order. Let us assume $R(\vec{x},t)$ has been calculated as a function of $T(\vec{x},t)$. The spacetime integral in Eq. (416) can be done if the missing relation $T=T(\vec{x},t)$ is known. Such a relation is provided by the hydrodynamic approach. So far it is the only framework which is detailed enough to predict particle spectra according to Eq. (416). Further simplifying assumptions are made. To these belong Bjorken’s scaling ansatz and the neglection of transverse flow, which lead to entropy conservation as function of time. As we have seen above, the hadronic multiplicity of the final state can be related to the initial conditions under the assumption of entropy conservation. Once the entropy is known as function of time, the time dependence of $T$ follows immediately if $s(T)$ is known. This is the place where the equation of state enters. Further assumptions about the kinetics of the phase transition are necessary to justify the application of hydrodynamic concepts throughout all stages of the evolution.

In the following we will outline the predictions of (Cleymans et al.,1987) for thermal dilepton spectra which are derived for different scenarios of the phase transition. The reason why we have selected this reference although it is not the most recent one in this field is that it explicitly addresses the influence of the order of the phase transition on the dilepton production rate. Although the ingredients do not represent the newest state of the art, the representation is suited for illustrating which features of dileptonic yields are sensitive at all to the dynamics of the phase transition. Later we comment on places of possible improvements and extensions.

Consider the particle rate which is differential in $x$ and $p$. The general expression for the thermal rate of lepton pairs (here written for $\mu^{+}\mu^{-}$ rather than $e^{+}e^{-}$) in the independent particle approximation of kinetic theory (Kapusta and Gale, 1987) is given by

[TABLE]

Here $M$ is the matrix element for the process of leptoproduction from particles $p_{1},\bar{p_{2}}$ with momenta $q_{1},q_{2}$, $p$ is the momentum of the lepton pair, $q_{+}$ and $q_{-}$ are the momenta of $\mu^{+}$ and $\mu^{-}$. The statistical distributions $f(q_{1})$ and $f(q_{2})$ measure the probability of finding particles $p_{1},\bar{p_{2}}$ with momenta $q_{1}$ and $q_{2}$ in the given medium. The dileptons do not receive such factors, since we assume that rescattering of the electromagnetically interacting particles can be neglected. The cross section is small, and the plasma volume is assumed not to be too large. In terms of cross sections $\sigma$, Eq. (419) reads

[TABLE]

upon integration over the lepton momenta, and $v_{q_{1},q_{2}}$ denotes the relative velocity of particles $p_{1}$and $\bar{p_{2}}$

[TABLE]

In the plasma phase the lowest order process for dilepton production is the same as in the Drell-Yan production: two quarks annihilate via a virtual photon to yield a lepton pair

[TABLE]

The momentum distribution functions $f_{1}$ and $f_{2}$ of the quarks $q$ and antiquarks $\bar{q}$ are given by the Fermi-Dirac distributions

[TABLE]

The chemical potential $\mu$ is set to zero in the end, $\underline{u}$ is the local four-velocity of the plasma fluid element in the fixed laboratory frame, $\underline{q}(\bar{\underline{q}})$ is the four-momentum of the quark (antiquark). As hydrodynamic scenario a longitudinal expansion will be used with vanishing transverse velocity of the plasma. In view of that, it is convenient to express the product $\underline{u}\cdot\underline{q}$ in terms of the transverse invariant mass $M$, the rapidity $Y$ of the lepton pair, and the plasma spacetime rapidity $\theta$ according to

[TABLE]

with the definitions of the transverse mass $M_{T}$

[TABLE]

with $v=z/t$ denoting the collective fluid velocity. If the transverse mass is large compared to the temperature, the exponentials are sharply peaked around $\theta=Y$. In this case no final $\theta$-dependence is left. Upon integration over the dilepton momentum, the result for the dilepton rate in the plasma phase at a spacetime point $\underline{x}=(\tau,x_{T},y)$ ($y$ being the local spacetime rapidity of a fluid element) and four-momentum $\underline{p}=(M_{T}\cosh Y,p_{T},M_{T}\sinh Y)$ is given by

[TABLE]

The index $P$ stands for plasma phase, $e_{i}$ are the charges of the quarks, $\alpha$ is the electromagnetic coupling, $m$ stands for the lepton mass and $K_{P}$ is a function, which depends on $\underline{x}$ via $T$, which we will not specify further now. It is characteristic for the plasma phase.

In the hadronic phase it depends on the invariant mass $M$ of the dilepton pair which process makes the leading contribution to the elementary cross section $\sigma$ in Eq. (421). For small mass pairs (small compared to the $\rho$-peak), bremsstrahlung-type emission of soft virtual photons is important. For masses $M$ well above the order of $T_{c}$, processes $h\bar{h}\to\gamma*\to\mu^{+}\mu^{-}$ play a role, where $h\bar{h}$ are hadrons others than pions. Their thermal production should be suppressed due to the relatively low temperatures, unless the in-medium masses are changed due to effects of chiral symmetry restoration, which may be dramatic! The only process that has been considered so far in (Cleymans et al., 1987) is $\pi^{+}\pi^{-}\to\rho\to\mu^{+}\mu^{-}$. The electromagnetic cross section $\sigma(M)$ is modified by the strong interactions of the pions, leading to

[TABLE]

where

[TABLE]

The pion form factor $F_{\pi}$ is treated in the vector-meson-dominance approximation, only the $\rho$-pole is kept in the sum over all $\rho$-like resonances.

Furthermore the functions $f_{1}$ and $f_{2}$ in Eq. (421) have to be replaced by Bose-Einstein distributions. The result for the dilepton rate in the hadron phase is given by

[TABLE]

where $K_{H}$ is a slightly different function from $K_{P}$ in Eq. (431).

Integration over the spacetime history

The assumption of Bjorken’s scaling solution for the hydrodynamical expansion simplifies the evaluation of the spacetime integration. The appropriate representation of the volume element is $d^{4}x=\tau d\tau dyd^{2}x_{T}$, where $\tau$ is the eigentime in a comoving reference frame with the fluid, $y$ is the spacetime rapidity (denoted as $\eta$ above), $x_{T}$ are the transverse coordinates. The transverse velocity is neglected, the transverse distributions of thermodynamic quantities are taken as step functions, $\varepsilon,p,s$ and $T$ depend then only on the eigentime $\tau$. The dileptonic yields per invariant mass, transverse momentum and unit rapidity interval $dY$ are given by

[TABLE]

The index $Ph$ indicates the dependence on the phase, $\tau_{f}-\tau_{i}$ is the eigentime duration of a certain phase or a mixture of phases, $y_{max}$ is taken to be the rapidity of the incoming beam. The equation of state enters the cooling law $T(\tau)$, the phase duration is sensitive to the dynamics of the transition.

Cleymans et al. (1987) have discussed the following scenarios

• $\bullet$

a first order transition described by a bag model equation of state proceeding in equilibrium or with supercooling and subsequent superheating

• $\bullet$

a second order transition with an equation of state taken from lattice Monte Carlo results (Redlich and Satz, 1986).

As we have seen in section V B 2, the bag model equation of state leads to the $T^{3}$ dependence of the entropy density and is typical for an ideal gas. In the baryonless plasma phase for two massless flavors, the relation is given by $s_{P}=4\cdot 37\pi^{2}T^{3}/90$. In the hadron phase described as an ideal gas of massless pions we have $s_{H}=2\pi^{2}T^{3}/15$. In the case of the second order transition $s(T)$ has been read off from numerical data of Redlich and Satz (1986), which were obtained in lattice QCD. Using the scaling solution for isentropic flow $(s(\tau)\cdot\tau=$ const., cf. section V B 1), we find the relations $T(\tau)$. When the scaled temperature $T/T_{0}$ is plotted as a function of the scaled eigentime $\tau/\tau_{0}$ (where $T_{0}$ and $\tau_{0}$ refer to the initial values chosen as $T_{0}=284$ MeV and $\tau_{0}=1$ fm/c), the characteristic difference between a first order transition proceeding in equilibrium and a second order transition is that the average temperature is higher in the quark gluon plasma phase and lower in the hadronic phase for a second order transition. Having $T(\tau)$ at hand, the integration in Eq. (437) can be done. Depending on the scenario, four rates are distinguished.

• $\bullet$

The dilepton rate in the pure plasma phase produced in the time interval between $\tau_{0}$ and $\tau_{P}$ where $T(\tau)$ is given as $T_{0}(\tau_{0}/\tau)^{1/3}$ for the bag model equation of state.

• $\bullet$

The dilepton rate in the pure hadronic phase produced during the interval between $\tau_{H}$ and freeze-out time (where the application of hydrodynamic concepts is already questionable).

• $\bullet$

If there is a coexistence of phases as it is expected in the first order case, there is a mixed phase at temperature $T_{c}$, where the plasma (hadron) phase contributes a fraction ${{dN_{P/H}\hbox{(mixed)}}\over{dM^{2}d^{2}p_{T}dY}}$, respectively. Both rates are calculated separately. Their relative weight is described by a factor $f$, which is the fraction of the entropy in the plasma phase. Its value follows from $s(\tau)\cdot\tau=$ const and Eq. (438)

[TABLE]

The entropy densities $s_{P}$ and $s_{H}$ remain constant for constant $T=T_{c}$.

The ratio of entropy densities $s_{P}/s_{H}$ gives also an estimate for the time scales $\tau_{H}$ and $\tau_{P}$, as $\tau_{H}/\tau_{P}=s_{P}/s_{H}=37/3$ if we assume for a moment that the naive counting of degrees of freedom is appropriate close to $T_{c}$. This ratio is frequently quoted in the literature, although the number 3 in the hadronic phase (resulting from 3 pions) is most likely incorrect. The large difference in entropy densities between both phases explains the long duration time of the mixed phase. It takes time to rearrange the effective degrees of freedom since they have to be reduced by an order of magnitude for the above counting when the conversion to the hadronic phase sets in.

We display the results of Cleymans et al. (1987) only for the cases of a first-order transition with Maxwell construction and a second-order transition. Upon integration over the transverse momenta, the resulting dileptonic rates $dN/dM^{2}dY$ at $Y=0$ [GeV*-2*] are shown in Figs. 31(a) and (b). The difference is the extension of the interference region where both phases contribute with dilepton production. It is considerably smaller for a second order transition. In this case, the production mechanisms change instantaneously at $T_{c}$, while those of both phases would be at work in the coexistence phase of a first order transition.

The width of the interference region does not yet provide a signal which is suitable for experiments to infer the phase transition dynamics. The sudden change in the production mechanism is reflected in the average transverse momentum $\left<p_{T}\right>$ of the dilepton pair or even more clearly in its derivative with respect to $M$. The average $\left<p_{T}\right>$ can be derived from the rate $R$ according to

[TABLE]

A plot of $d\left<p_{T}\right>/dM$ as a function of $M$ for both transition scenarios shows a peak in the $d\left<p_{T}\right>/dM$ distribution for a second-order transition. Thus a peak in the $d\left<p_{T}\right>/dM$ distribution is a possible candidate for a signal of a second-order phase transition (keeping in mind all approximations that have been applied so far). As outlined by Cleymans et al. (1987), such a sudden change in the slope of $\left<p_{T}\right>$ as a function of $M$ can be also produced by a higher mass resonance like the $\rho^{\prime}$ (1600). To disentangle the origins of bumps in the $\left<p_{T}\right>$ spectra, one should vary the initial temperature. At higher initial temperature the contribution from the plasma phase to the dilepton rate increases. A peak structure in the $\left<p_{T}\right>$ spectrum due to a 2nd order transition then shows up already for low initial temperatures, while higher initial temperatures are necessary to see a similar structure in a first order scenario.

Yet another possibility may lead to a sudden change in $\left<p_{T}\right>(M)$. Dilepton rates are calculated in a way that they do not automatically interpolate smoothly at $T_{c}$, although they should do so in the case of a second-order transition where no discontinuities should be seen in thermodynamic observables. Cleymans et al. (1987) have investigated the influence of smoothly interpolating the rates between both phases. The bump structure, however, survived the interpolation between $dN_{p}/d^{4}xd^{4}p$ and $dN_{H}/d^{4}xd^{4}p$.

In general, it is outstanding to calculate the electromagnetic current-current correlation $W(x,p)$ (whose spacetime integral is proportional to $dN/dM^{2}dp_{T}dY$) in terms of one effective model which replaces the bag model in the case of a second order transition. We have mentioned this point already in section V B 2.

So far we have been concerned with thermal rates and specific features reflecting the transition dynamics in the rate dependence of the invariant mass $M$. In heavy-ion experiments it is obviously an objective of higher priority to guarantee that thermal dileptons can be identified from variant background sources. In the order of increasing masses, we have Dalitz pairs, soft gluon bremsstrahlung, hadronic resonance decays, preequilibrium production, and Drell-Yan production in the very early stage of the collision.

Ruuskanen (1990) has derived an analytic expression for the integrated thermal rate $R$. The characteristic feature of this expression concerns the $T$ and $M_{T}$-dependence as compared to other production mechanisms ($M_{T}=(M^{2}+p_{T}^{2})^{1/2}$ denotes the transverse mass which is used instead of the transverse momentum $p_{T}$). The dependence of the thermal rates on the initial temperature $T_{i}$ is strong, i.e. proportional to $T_{i}^{6}$ as compared to the $T_{i}^{3}$-dependence for a Drell-Yan rate. Recall that $T_{i}$ at equilibration time $\tau_{i}$ can be related to the hadronic multiplicity in the final state according to

[TABLE]

if an isentropic expansion scheme is assumed. This allows an identification of thermal dileptons via their strong (quadratic) dependence on pion multiplicity fluctuations, Drell-Yan rates show only a linear dependence. At fixed multiplicity (i.e., fixed $\tau_{i}T_{i}^{3}$), the thermal rates are expected to show an approximate power law behavior proportional to $M^{-6}_{T}$ in a range of $M_{T}/T_{i}<5.5<M_{T}/T_{c}$ (Ruuskanen, 1990) with a $\left<p^{2}_{T}\right>$ dependence proportional to $M^{2}$, and an exponential decrease of the dilepton rate for large $M_{T}/T_{i}$, with $\left<p^{2}_{T}\right>$ showing a linear increase in $M$. For an initial temperature of 600 MeV, the window of the $M^{-6}_{T}$ dependence of the transverse mass spectrum lies between 1-3 GeV. Its lower end has to face a background from the hadronic phase, its upper end the Drell-Yan background.

Thus the first goal of a measurement of dilepton rates in a collision of heavy-ion nuclei is to identify a window in $M_{T}$ of the spectrum which shows the behavior of a thermal rate produced in a plasma phase. It is easier to filter out the $M^{-6}_{T}$ depending part if high initial temperatures are reached in the experiment. For detailed calculations of the different backgrounds and their competition to thermal rates we refer to e.g. Ruuskanen (1992), Weldon (1991), Kajantie and Miettinen (1981), McLerran and Toimela (1985). For more recent predictions of the dilepton spectrum at LHC energies see Redlich et al. (1992). The effect of various equations of state on the transverse expansion and the dilepton production rate is discussed in (Kataja et al., 1992). The background of dileptons from $\rho$-decays is usually considered as undesirable in view of thermal dilepton identification. Recently it has been shown by Heinz and Lee (1992) that such dileptons may contain valuable information on the lifetime of the hadronic phase; that is why they are called “fast clocks” for heavy-ion collisions. The overall judgement is that the identification of thermal rates in the different phases is difficult but not completely unrealistic. Yet it seems to be an order of magnitude more difficult to infer specific features of the transition itself from structures of $R$ or $\left<p_{T}\right>$ distributions as functions of $M$.

We summarize the approximations which enter predictions of dilepton rates along the lines we have indicated above.

• $\bullet$

Uncertainties in the background to thermal dileptons. The Drell-Yan background is the one which is understood best. Corrections due to higher orders in QCD have been assumed to be small, actually they are large in the usual deep-inelastic structure function approach. (For recent results on Drell-Yan production at collider energies including $O(\alpha_{s}^{2})$ contributions we refer to van Neerven (1995).) Deviations from the linear $A$-dependence of the structure functions on the mass number $A$ (entering the Drell-Yan cross section for colliding nuclei) are assumed to be negligible in the relevant mass range ($M=2$-6 GeV). Probably the least understood rates are those which are produced shortly after the collision but before the onset of thermal equilibrium. Effective models are needed which describe nonequilibrium properties of the evolution of the initial state. In the fluxtube model of Kajantie and Matsui (1985) and Bialas and Czyz (1984, 1985), dilepton rates depend on the strength of the color field which is formed after the color exchange between the colliding nuclei. The new production mechanism relies on plasma oscillations of quarks and antiquarks. In the mass range of a few GeV which is of special interest for thermal production, the worst perspective is that both the thermal and the Drell-Yan productions – including all their specific signatures – are buried under the yields of plasma oscillations. For further discussions, see also Bialas et al. (1988) and Bialas and Blaizot (1985).

• $\bullet$

The applicability of hydrodynamic concepts made it possible to evaluate the integral over the spacetime history of rates at a given point in phase space. It is marginal in several aspects. The volumes are not so large compared to the mean free paths, at least not close to freeze-out, where by definition the system is so diluted that interactions can be neglected. The scale of equilibration time is estimated in a purely phenomenological way. Thus it cannot really be argued, which time $\tau_{0}$ for the onset of hydrodynamic expansion is to be used. The adiabatic expansion guarantees the conservation of entropy and leads to a simple relation between entropy and time. During the phase conversion this condition may be violated.

• $\bullet$

The bag model equation of state has been used close to $T_{c}$, although strong deviations from the ideal gas behavior are expected there. Lattice Monte Carlo calculations should do their best to include the effect of dynamical fermions as realistic as possible and to derive an equation of state close to $T_{c}$. In deriving the equation of state which was used by Cleymans et al. (1987) and led to a second-order transition, the fermions are treated in the hopping parameter expansion; the involved parameters are known to be not close enough to the continuum limit. A more realistic equation of state obtained from lattice calculations with dynamical fermions can be found e.g. in (Christ, 1992), although it should not be considered as the realistic equation of state for QCD, see section III B 2.

Given the hydrodynamic approach and the equation of state, further simplifications were made in deriving the results we have shown. We have neglected effects due to

• $\bullet$

a realistic nuclear geometry and

• $\bullet$

the impact parameter dependence. In particular, the initial temperature profile depends on the impact parameter $b\not=0$.

• $\bullet$

the transverse coordinate dependence of temperature during all stages of the evolution.

• $\bullet$

the transverse flow of the hadron/plasma fluid. When transverse flow is included, hadronic rates can be reduced by an order of magnitude due to the reduced lifetime of mixed and hadronic phases, cf. Fig. 10 of Ruuskanen (1990).

• $\bullet$

last but not least the possible temperature and density dependence of masses which are involved in the dilepton production. Meson masses can be expressed in terms of the chiral condensate $\left<\bar{\psi}\psi\right>$, they should be sensitive to the order of the transition. Either $\left<\bar{\psi}\psi\right>$ vanishes smoothly when approaching $T_{c}$ from below, or two values of $\left<\bar{\psi}\psi\right>$ coexist at $T_{c}$, a large value and a small one.

Finally we refer to the work by Geiger and Kapusta (1993), Shuryak and Xiong (1993), and Ruuskanen (1992) as more recent references in this field.

When dileptons are produced from vector meson decays, the dilepton rates are sensitive to the phase transition dynamics also via vector meson mass changes in medium both as a function of $T$ and $\mu$. We have skipped this possible manifestation of the transition dynamics, as we are not aware of any specific prediction yet.

Real photons

Real photons should be understood in contrast to virtual photons in dilepton production. Like dileptons they give the electromagnetic response to the plasma evolution. They could provide clean signals in the sense that they escape from different stages of the system with rather small interaction cross sections. According to their production conditions, the three major contributions to real photons are direct photons from partonic processes in the initial state, thermal photons from both phases and decay photons produced at a late stage of the evolution. At large transverse momenta $K_{T}\geq$ 3 GeV, the lowest order ($\alpha_{s}$) contributions to direct photons are the gluon-to-photon compton scattering off (anti)quarks $q(\bar{q})g\to q(\bar{q})\gamma$, quark-antiquark annihilation to photon and gluon processes $q\bar{q}\to\gamma g$ and bremsstrahlung (with quark fragmentation into the photon). At very high collision energies, the main contribution to direct photons comes from the fragmentation of jets (Gupta, 1990). At low transverse momenta $K_{T}\leq$ 3 GeV, decays of neutral mesons $\pi^{0},\eta,\eta^{\prime}$ and $\omega$ give a large contribution. This contribution is particularly difficult to estimate, but will be measured (Stachel, 1993). The uncertainty in the estimate arises from the lack of knowledge of the $p_{T}$-distributions of these mesons in nuclear collisions. For very small transverse momenta ($K_{T}\leq 10-20$ MeV), hadronic bremsstrahlung is important (Ruuskanen, 1992).

A possible window for observing thermal photons could be in the 2-3 GeV region (Ruuskanen, 1992). Thermal photons can be produced in both phases. In earlier calculations, only the plasma contribution has been considered (see e.g. Hwa and Kajantie, 1985). Thermal photons in the plasma phase are produced via the same elementary processes as direct photons, the difference in the final thermal rate as compared to the direct production rate is a replacement of the product of structure functions by the entropy squared. The contribution from the hadronic phase has been calculated by Kapusta, Lichard and Seibert (1991, 1992) with the result that “a hadron gas shines as brightly as a quark gluon plasma” at temperatures between 150-200 MeV.

In the 1 GeV region, thermal rates are clearly below decay photons. This does not necessarily imply that thermal rates cannot be measured. An electromagnetic calorimeter with high energy resolution and fine granularity has been proposed for RHIC, which aims for measuring ratios of $(\#\hbox{thermal photons})/(\#\pi_{0})$ as function of $p_{T}\sim 3$-5% (Stachel, 1993). In view of section V D 3 we anticipate that an accurate measurement of this ratio is rather appreciable for verifying certain remnants of a hypothetical second order chiral transition. At large transverse momenta ($\geq 3$ GeV), direct photons dominate. The chances for thermal photons improve for very large multiplicities $dN_{\pi}/dy$ of pions, because thermal photons are proportional to $(dN_{\pi}/dy)^{2}$, while photons from pion decay vary with $(dN_{\pi}/dy)$. In the intermediate region, it sensitively depends on the shape of the hadronic $p_{T}$-spectra whether thermal photons have a chance of being identified or not. These $p_{T}$-spectra are not well enough understood. Real photons have certain advantages over dileptons. Prompt real photons originate from the Compton scattering of gluons and quarks. Gluons equilibrate very rapidly compared to quarks. Therefore the prompt photon measurement provides a direct (and the only direct) access to the very early initial phase of the collision which is not accessible with dilepton rates. The signal to noise ratio is expected to be larger than for the dilepton continuum.

Thus, it seems to us promising in future experiments with large $dN_{\pi}/dy$ and high resolution detectors to look for signatures of a transient plasma and even for certain substructures in thermal photon distributions which could be conclusive in view of the phase transition dynamics. The procedure to calculate the thermal rate of real photons is the same as for dileptons. Once the elementary processes are identified and emission rates are calculated in the stationary plasma or hadronic phase, the result has to be folded with the spacetime expansion. In a first approximation one can use Bjorken’s similarity flow and correct for transverse flow, nuclear geometry in further steps. For such calculations we refer to Ruuskanen (1992) and references therein. High multiplicities ($\geq 4000$) and short equilibration times favor thermal photon production. We have to wait and see whether RHIC or LHC energies do reach such multiplicities.

3 Strangeness production

In heavy-ion collisions an enhancement of certain particle ratios involving strange quarks is predicted compared to ratios in hadron hadron or hadron nucleus collisions. Ratios enhanced by a factor of 2 for $K^{+}/\pi^{+}$ have in fact been observed in experiments at the AGS at Brookhaven and SPS at CERN. A theoretical explanation in terms of a transient quark gluon plasma is not compulsory. Rescattering processes in a hot hadronic gas can also change the ratios in the same direction.

The idea to propose strangeness as a possible signature for a QGP relies on the following arguments. Strangeness production in the plasma should be facilitated by two reasons.

• $\bullet$

Independently of the assumed baryon density, the threshold energies for strangeness production in the plasma phase are much lower than the production of strange baryons or mesons in the hadron phase. In the hadron phase, strange mesons or baryons are typically made in a collision of two non-strange hadrons. The reaction with the lowest threshold energy requires already 671 MeV $(p+n\to\Lambda^{0}+K^{+}+n)$. In the plasma the threshold for $s\bar{s}$ production is equal to the rest mass of $s\bar{s}\sim$ 300 MeV. For a given temperature the density of non-interacting strange quarks is higher than the density of non-interacting kaons which will be later considered as an example. In a plasma an $s$-quark has 2 spin and 3 color degrees of freedom. If it is bound in a $K^{-}$ or a $\bar{K}^{0}$ in the hadron phase it has only 2. Also the kaon mass is $\sim$ 494 MeV compared to the current $s$-quark mass of 180 $\pm$ 30 MeV.

• $\bullet$

The second reason applies to a baryon rich environment, which can be found in collisions with high baryon stopping or in the rapidity range of fragmentation regions. The Pauli principle will prohibit the creation of $u\bar{u}$ and $d\bar{d}$ pairs compared to strange pairs. The light quarks have to supply the large Fermi energy represented by the chemical potential $\mu_{B}$, while the $s$-quarks are only suppressed by their finite mass. Moreover $\bar{u}$ and $\bar{d}$ quarks have a high probability to recombine with $u$ and $d$ quarks to form gluons. For strange quarks the recombination to gluons is less likely. Due to the volume expansion and the decreasing temperature of the fireball, the process $gg{\rightarrow\atop\leftarrow}s\bar{s}$ is soon out of thermal equilibrium, that means, it is too slow to proceed in the inverse direction. In a background which is rich in $u$ and $d$ quarks (large $\mu$), the number of (massless) $\bar{u}$ (or $\bar{d}$) quarks in a free gas is given by

[TABLE]

This number should be compared to the number of strange quarks

[TABLE]

To lowest order in perturbative QCD, $s\bar{s}$-pairs are created in collisions of two gluons and by annihilation of light quark and antiquark pairs (Rafelski and Müller, 1982). Rafelski and Müller have shown that the gluonic production is the dominating mechanism of strangeness production. For a small window of the quark mass equal to (0.5-1.) times the temperature the equilibration time is similar to the expected lifetime of the plasma phase.

For dileptons it has been sufficient (cf. the previous section V C 2) to calculate the differential rate per given spacetime volume. Due to the very small electromagnetic cross section the dileptons of high invariant mass escape from the plasma without further interaction.

Strange quarks are kept in the subsequent evolution. That is the essential difference to dilepton production. In various ways strange quarks are incorporated as hadronic constituents during the conversion of the plasma into the hadron phase. After completion of the conversion strangeness is confined to hadrons. The strange hadrons continue to react via strangeness creation, annihilation or exchange reactions until freeze-out. Thus it is not sufficient to calculate the formation rate of strangeness per unit time and volume as if the predicted rate would directly correspond to observed multiplicities (as for dilepton production).

The change $dn/dt$ in the density $n$ of strange particles as a function of time is of primary interest now. Later we shall see that in the case of a static plasma $dn/dt$ is equal to the rate $R$ if $n$ vanishes, that is at the start of the production. The change goes to zero when the equilibrium value of $n(n_{eq})$ has been reached. It is a question of time whether $n_{eq}$ can be reached at all. The two competing time scales are the equilibration time $\tau_{eq}$ defined as $n(\tau_{eq})=n_{eq}$ and the lifespan of the phase where the strangeness production is considered.

Already at this point one can suspect that the appropriate “choice” of the dynamical scenario for the phase transition is even more essential here than it was in the case of dilepton production. Time scale estimates have farther reaching consequences in strangeness production. The contribution of a thermal rate is not only a question of size in the sense that it could be certainly identified if it is only large enough compared to other backgrounds. Certain rare multi-strange hadron species may completely fail as candidates for QGP signatures, if their equilibration time is just too long compared to the lifetime of the phase where they could have been produced otherwise.

Below we will describe the differential equation which determines the equilibration time for a given species in more detail.

We will not judge the chances for using strangeness enhancement as a signature for a transient plasma phase. Also we do not see a way of specifying features which could be conclusive about the underlying phase transition dynamics. We merely want to give a flavor of the complexity in calculating equilibration times. The ratio of $K^{-}/\pi^{-}$ serves only as an example. We describe the work of Kapusta and Mekjan (1986) and indicate places for further extensions or alternative treatments.

The preconditions which are taken for granted are the following: The rate constant $R$ for producing strange quarks in the plasma phase is so large that the density quickly attains its chemical equilibrium value $n_{s}^{eq}$. The system expands and cools so rapidly that the strangeness abundance - once it has been produced - is not lost to a significant extent due to annihilation into gluons. Both conditions ensure that the facilitated strangeness production in a plasma environment has a chance to become manifest in possible plasma signals.

To get numbers for particle ratios in the very end, we have to specify

• $\bullet$

the kinematical approach and the geometry. Bjorken’s ansatz of a longitudinal hydrodynamic scaling expansion is used. Possible effects due to viscosity, heat conduction and transverse expansion are neglected. Viscosity and heat conduction would prolong the decrease of temperature as a function of time, transverse expansion will accelerate it. Both effects may or may not approximately cancel.

• $\bullet$

the rapidity range. The central rapidity range has been chosen, where the baryon number approximately vanishes. Hence the calculations are performed at $\mu_{B}=0$. This may not be a favorable choice in view of the different sensitivities of $\bar{u},\bar{d}$ and $s,\bar{s}$ quarks with respect to $\mu_{B}$, but it simplifies the discussion a lot.

• $\bullet$

the equation of state. Once again the bag model equation of state is taken.

• $\bullet$

the gross features of the phase transition. Corresponding to the bag model the phase transition is assumed to be of first order. Two realizations are considered: a smooth transition via the Maxwell construction and a rapid transition with supercooling and subsequent reheating.

• $\bullet$

the measure for the strangeness contents of each phase. For the hadron phase $K^{-}$-mesons will be considered.

• $\bullet$

the elementary reactions entering the cross sections in the rate formulas.

• $\bullet$

an ansatz for the differential rate equation that determines the equilibration time.

Let us start from the end and describe the rate equation as it was used in (Kapusta and Mekjian, 1986). For an expanding plasma the change in the density of strange quarks $n_{s}$ as a function of time $t$ is given by

[TABLE]

Here $n_{s}^{eq}$ is the equilibrium density of strange quarks which depends on time via the temperature $T$. The rate $R_{p}$ is defined as the number of specific reactions per unit time and volume producing $s\bar{s}$-pairs in the plasma phase. It corresponds to $\int d^{4}p\ dN/(d^{4}xd^{4}p)$ which entered also the dilepton rate in section V C 2. Similarly to Eq. (419) the rate is calculated in the independent particle limit, that is according to the ansatz

[TABLE]

The occupation probabilities $f(\vec{k})$ are chosen as Boltzmann weights

[TABLE]

$m_{a}$ is the degenerate mass of particle $a_{1}$ and $a_{2}$. This ansatz neglects possible final state interactions due to the influence of Pauli blocking. Here it is consistent with considering a baryon free rapidity region. The prefactor $(1/1+\delta)$ prevents a double counting when gluon pairs annihilate to $s\bar{s}$-pairs, $v_{rel}$ denotes the relative velocity between the particles $a_{1}$ and $a_{2}$ and is given by the analogous formula to Eq. (422). Finally $\sigma$ denotes the elementary cross sections which contribute to $R$ in a given phase. Note also that the spacetime dependence enters $f$ via the temperature $T$.

Some further comments on the rate equation (443) are in order. The last term in Eq. (443) is a dilution term due to the volume expansion. The particle density will roughly decrease like the entropy density, that is as $1/t$ when the volume increases proportional to $t$. The first two terms of Eq. (443) characterize the rate equation for a non-expanding source. Restricting the r.h.s. of Eq. (443) to the static part, the linearized rate equation implies for small deviations from equilibrium

[TABLE]

i.e., a high rate $R$ implies a short equilibration time.

The cooling law as a function of time depends on the chosen scenario. In the plasma phase the bag model equation of state implies a decrease according to

[TABLE]

Further quantities which enter the rate equation (443) are the equilibrium density $n_{s}^{eq}$ of strange particles and the rate $R_{p}(gg\ \hbox{or}q\bar{q}\to s\bar{s})$ for producing strange quarks in the plasma phase. The density $n_{s}^{eq}$ is approximated by its Maxwell-Boltzmann limit. The rate $R_{p}$ is taken from (Müller and Rafelski, 1982), it is based on the lowest order QCD cross section, where $s\bar{s}$-pairs are created via $gg\to s\bar{s}$, $u\bar{u}\to s\bar{s}$ and $d\bar{d}\to s\bar{s}$. (It remains to be checked that suppressions due to higher orders in $\alpha_{s}$ are not compensated by large combinatorial factors in the graphical expansion.) The rate $R_{p}$ depends on $\alpha_{s}$ and $m_{s}$ as parameters, which both vary with the renormalization scheme, the energy scale, and the temperature.

Finally for solving the rate equation for $n_{s}(t)$ in the plasma phase, one has to specify the initial condition. Since the net-strangeness of two cold nuclei is zero, it is chosen as $n_{s}(t_{0})=0$, where $t_{0}$ denotes the formation time.

The dependence on this initial condition should be weak, if the final rate is close to its equilibrium value, otherwise it has to be checked carefully. The conserved charge is the difference of strange and antistrange particles. Only as long as strange and antistrange particles do not interact, their charges are conserved separately. Thus, if the $s$-quark density represented by the $K^{-}$-mesons is considered separately in (Kapusta and Mekjian, 1986), it relies on the treatment of quarks and mesons as effectively ideal gases in both phases. It is implemented by the choice for the equation of state. As lattice results have shown, the ideal gas picture is certainly not correct in the vicinity of $T_{c}$, in particular it fails if long range correlations are enforced by a second order transition. We stress this point, because in the presence of interactions between $s$ and $\bar{s}$- quarks it is not clear what a sensible measure for the strangeness contents of each phase could look like.

Next we turn to the rate equations in the intermediate phase between the onset of hadronization and its completion. The terms in the rate equation depend on the scenario. Kapusta and Mekjian (1986) have considered two alternatives, the Maxwell scenario and a phase conversion with superheating and supercooling effects.

In the first case, the transition proceeds adiabatically. The system stays in local equilibrium, Maxwell’s condition for coexisting phases is assumed to be fulfilled. The nucleation rate of hadronic bubbles is large compared to the expansion such that a fraction $(1-f(t))$ of the hadron phase is created immediately after the temperature has reached its critical value $T_{c}$. This happens at time $t_{1}$

[TABLE]

Eq. (449) follows from entropy conservation and the equation of state (cf. section V C 2 where it has been derived). Thereafter the temperature stays constant until the conversion to the hadronic phase is completed. Both phases contribute to the entropy density according to

[TABLE]

where $s_{p}$ and $s_{h}$ are the entropy densities in the plasma and the hadron phase, respectively. Combined with the time dependence of $s(t)$ it implies the time dependence of the fraction $f(t)$

[TABLE]

All plasma is converted to the hadron phase at time $t_{2}$, where $t_{2}$ follows from $f(t_{2})=0$ as

[TABLE]

The second scenario would be realized, if an interface between the hadron and the plasma phase costs much excessive energy, i.e. if the surface tension between the plasma and hadronic bubbles is large. (As we have shown in section III, this is not likely the case.) The onset of the phase transition is delayed until the temperature has dropped to a value $T_{p}<T_{c}$. At $T_{p}$ the system is assumed to transform instantaneously to the hadron phase. The hadron phase is superheated to a value $T_{h}>T_{c}$, while the plasma was supercooled to $T_{p}<T_{c}$. The entropy should not decrease (during the transition it may increase), the energy is conserved. These assumptions together with the bag model equation of state lead to a prediction of the time dependence of $T$ in the various time intervals.

In view of the final strangeness contents the interesting feature about the time evolution of temperature is that the system spends quite a long time in the mixed phase or in the overheated hadron phase, depending on the scenario and the initial temperature $T_{0}$.

Even the very ansatz for the rate equations depends on the scenario. The common part is the one which describes the approach to the equilibrium density of strange particles in a static situation. When the matter is expanding, loss terms in the density have to be included, which depend on the presence and absence of a phase mixture. In particular, a coexistence of phases as it occurs in Maxwell’s construction leads to an additional complication. For the lifespan of the mixed phase the volume occupied by the hadron phase expands while the fraction of plasma phase shrinks. This structure has to be superimposed on the common volume expansion.

In both scenarios the rate changes $dn/dt$ in $K^{-}$ mesons are followed over the period between the onset of the hadron phase until all plasma has been converted. In Maxwell’s scenario the temperature stays constant at $T_{c}$, in the supercooling/reheating scenario it drops from $T_{h}>T_{c}$ to $T_{c}$.

Finally, to solve the rate equations for the density of $K^{-}$-mesons, we have to determine the rate constants $R_{h}(T(t))$ and $R_{h}(T_{c})$ in the hadronic phase. Here the possible contributions simplify due to the restriction of a baryon free environment. The relevant processes are $\pi^{+}+\pi^{-}\to K^{+}+K^{-},\ \pi^{-}\pi^{0}\to K^{-}K^{0}$ and $\pi^{0}\pi^{0}\to K^{+}K^{-}$. The corresponding cross sections must be inserted which are inferred from experiments.

After all these specifications we are now ready to solve the rate equations for $n_{s}(t)$. In heavy-ion collisions densities are not directly observable as a function of time. Instead of densities particle ratios, in our case $K^{-}/\pi^{-}$ ratios are suitable observables. The volume drops out. Thus we have

[TABLE]

Here $n_{s}$ is the density of strange particles at freeze-out, hidden to equal fractions in $K^{-}$ and $\bar{K}^{0}$-mesons and $n_{\pi^{-}}$ is the density of $\pi^{-}$-mesons. For massless pions and under the assumption of an ideal gas, the pion number density is proportional to the pion entropy density $s\propto T^{3}$ which is almost equal to the total entropy density of the hadronic gas. This leads to

[TABLE]

This ratio should be compared to $n_{s}\hbox{(plasma)}/s_{p}$, the number of strange quarks in the plasma phase and the plasma entropy density. It has been questioned for some time what a good measure for the strangeness contents of the plasma is. Rather than $n_{s}/s$ the ratios $n_{s}/n_{d}$ ($n_{d}$ being the number of down quarks) have been earlier considered in (Rafelski, 1982). However, the quantity which should replace $n_{\pi^{-}}$ in the plasma phase is $s$ and not $n_{d}$, since all $u,d$ and gluonic degrees of freedom (“summed up” in the entropy) must hadronize into pions.

Plots of $n_{s}/s$ and $n_{s}/n_{eq}$ as a function of time can be found in (Kapusta and Mekjian, 1986) for three scenarios: the Maxwell construction, the supercooling/reheating scenario without an extra entropy production $(\Delta S=0)$ and with doubling of the entropy at $T_{c}(\Delta S=S)$. The curves display some expected features: the higher the initial temperature the faster $n_{s}$ reaches its equilibrium value. For the Maxwell scenario $n_{s}/s$ and $n_{s}/n_{eq}$ are continuous functions of time. After completion of the phase transition the final ratio of $K^{-}/\pi^{-}$ is shown in Fig. 32 as function of the initial temperature $T_{0}$ for three scenarios and three sets of input parameters corresponding to the dashed, solid and dotted curves. Input parameters are the strong coupling $\alpha_{s}$, the strange quark mass $m_{s}$, two cross sections entering the rates of $K\bar{K}$-annihilation and the proper formation time $t_{0}$ of the plasma. Obviously in the supercooling/reheating scenario with $\Delta S=0$ or $\Delta S=S$, the $K^{-}/\pi^{-}$ ratios are little sensitive to the initial temperature $T_{0}$, and the input parameters, even to the degree of supercooling and the related entropy production. The value falls in the range of 0.21-0.25. The value of chemical equilibrium is in this range, it is given by 0.217. The $K^{-}/\pi^{-}$ values, predicted here for heavy-ion collisions should be compared to the same ratios in $\bar{p}p$-collisions where $K^{-}/\pi^{-}$ is 0.07 $\pm$ 0.02.

In the Maxwell scenario such an enhancement by a factor about 3 is only seen for an “optimistic” choice of parameters (dashed curves in Fig. 32) for all values of $T_{0}\geq$200 MeV. For a “realistic” choice (solid lines in Fig. 32) it is only seen for $T_{0}>250$ MeV, and for a “pessimistic” set (dotted lines in Fig. 32) only for large values of $T_{0}$. The physical reason for the higher sensitivity of the Maxwell scenario to the input parameters is likely due to the continuous time evolution throughout all stages of the evolution. No jumps in thermodynamic quantities at $T_{c}$ cut off the history of the plasma evolution.

The enhancement of $K^{-}/\pi^{-}$ by a factor of 3 compared to $\bar{p}p$-collisions at the same beam energy is to be interpreted as an indirect sign for a transient plasma phase. Without the first order phase transition the lifetime of the hadron phase would be shorter, so short that the time would not suffice to saturate the $K^{-}/\pi^{-}$ ratio nearly to its equilibrium value. Thus, in contrast to dileptons, $K^{-}$-mesons are not a direct probe of the high temperature plasma phase, their abundance is influenced by the phase transition and the subsequent evolution in the hadron phase.

This example for an estimate of the strangeness enhancement may have illustrated the urgency of calculating the equation of state, the latent heat, the surface tension, the temperature dependence of couplings and masses and the order of the phase transition from first principles. Theoretical calculations may help to find the most realistic scenario. In particular we have seen the simplifying power of the bag model equation of state which does not justify its use. For strong correlations between particles close to $T_{c}$, the very indicator itself may fail as a measure for the strangeness contents of both phases. Also the gain and loss terms in the rate equations which take care on the dynamical expansion, are sensitive to details of the phase transition scenario.

The approach of Mekjian and Kapusta (1986) can be generalized in various aspects, which we mention rather briefly below. For our limited aspect of specific signatures for the phase transition a detailed discussion of a more general framework would only serve to further support the overall impression: it is extremely difficult to unfold the underlying phase transition dynamics from strangeness production.

Koch et al. (1986b) have considered the strangeness production in the plasma phase within a framework, where the baryochemical potential is different from zero. The rate equation is considered in an alternative form including a linear loss term in $n_{s}/n_{s}^{eq}$. The volume expansion is investigated for different time dependences of $V$ and $T$, one following from the bag model equation of state and one chosen by hand to simulate a more rapid expansion. In addition, Koch et al. have checked the influence of Pauli blocking on final states by replacing the momentum distributions of strange quarks in a suitable way.

A more general set of rate equations in the hadronic phase has been considered in (Koch et al., 1986a). The underlying physical picture is similar to the Maxwell scenario of Kapusta and Mekjian (1986). It is more general in including baryons and antibaryons in addition to pions and kaons. A wider spectrum of possible reactions in the hadron phase has to be taken into account: strangeness production and exchange reactions, annihilation processes. The abundances not only of $\pi$ and $K$-mesons, but also of $N,Y,\Xi,\Omega$ and their antiparticles have to be followed in their evolution. An additional term appears in the rate equation for the mixed phase which is specific for the chosen hadronization process (here the so called fragmentation-recombination scenario). Needless to say that the way in which the strangeness abundance of the plasma is modified during the phase conversion depends on the choice of the hadronization process. Unfortunately there is yet a whole variety of fragmentation models at one’s disposal.

We conclude with a short and incomplete confrontation to the experimental status. The ratios of $K^{+}/\pi^{+}$ and $K^{-}/\pi^{-}$ have been measured in the E 802 experiment (Abbott et al., 1990; Abbott et al., 1991) at the AGS (Brookhaven) and the NA34 experiment at SPS (CERN) (Van Hecke, 1991). Both experiments are measuring in a baryon rich region of phase space, thus we cannot compare the results with the theoretical predictions of (Kapusta and Mekjian, 1986) for a baryon free environment. In both experiments the same trends are seen: an enhancement of $K^{+}/K^{-}$ ratios by roughly a factor of 2 when going from hadron hadron to hadron nucleus and nucleus nucleus collisions, whereas the $K^{-}/\pi^{-}$ ratio increases less strongly or stays approximately constant. It should be mentioned that the ratios $K^{+}/\pi^{+}$ and $K^{-}/\pi^{-}$ are not the only measure for strangeness enhancement. Another quantification is the strangeness suppression factor $\lambda=2s\bar{s}/(u\bar{u}+d\bar{d})$ as evaluated by Wroblewski (1985) for pp-collisions and applied to S+S-collisions at 200 GeV/n-n (Seyboth, 1992; Foley, 1992). With this criterion one finds a factor 2 of strangeness enhancement as compared to pp- and pA-collisions.

This does not mean that a plasma has been observed. For many strange particles the equilibrium values in a hot extended hadron phase are similar to those in a quark gluon plasma. Therefore the enhanced particle ratios do not uniquely signal a transient plasma. If there is another way to get strangeness into equilibrium by adjusting parameters in a hadronic cascade model, one will never be able to distinguish this explanation from a transient plasma. A system in equilibrium has lost the memory how it got there. There may be exceptions for finding a unique signal of a plasma (Rafelski, 1991). Enhanced multistrange (anti)baryon production (antihyperons $\bar{\Lambda},\bar{\Xi}^{-}$) have been proposed as special tests for a plasma environment (Eggers and Rafelski, 1991). A hot hadron gas cannot overcome their large mass barriers.

In Table VIII we recapitulate the ingredients, which have led to the prediction of $K^{-}/\pi^{-}$ enhancement in the baryon free case. Some freedom of choice is given for the geometry, the rapidity range, and the initial conditions. This freedom corresponds to the possibility of experimental tuning of certain parameters like the initial energy density. In contrast to that one is truly not allowed to choose between various versions for the kinematical approach, the equation of state, the gross features of the transition, the rate formula, the contributing elementary processes and the ansatz for the rate formula or the differential rate equation. These ambiguities are due to uncertainties in the theoretical description and shed some light on the reliability of predicted numbers in the very end. For example the choice of a strong supercooling scenario must be regarded as ad hoc.

4 Pion interferometry

Interferometry is nowadays a well-known technique. It has been developed by Hanbury-Brown and Twiss (1954) almost 40 years ago in astrophysics, where it is a tool to measure the size of various stellar objects in the visible and radio frequency ranges. Fig. 33 shows a source which is assumed to emit identical particles from positions $P_{1}$ and $P_{2}$. The particles are later observed at positions $P_{3}$ and $P_{4}$. Both emission points may contribute to both observation points, even if the particles are non-interacting, but have small relative momenta. The reason is the symmetrization (antisymmetrization) of the quantum-mechanical wave function in the case of bosons (fermions). A correlation function is constructed from the number of counts at $P_{3}$ and $P_{4}$. The particles which are detected in astrophysical interferometry experiments are photons.

The overwhelming part of particles which are radiated in heavy-ion collisions are pions. Thus we describe pion interferometry as a particularly useful tool to estimate the lifetime and the final-state size of the source which radiates the pions. From the discussion in the previous sections it is evident that we need some measurements delivering information about the equation of state. Production rates of dileptons and strange particles strongly depend on the duration of a certain phase. Three extensive quantities must be measured to obtain an equation of state for a static gas with no conserved charge. In heavy-ion collisions the total energy and entropy can always be estimated from experiment (cf. section V C 1). If we know the volume and the fraction of collective energy at a given time of the collision, we have a single value in an equation of state corresponding to the conditions at that time. The correlation function measured in pion interferometry could give an estimate for the size at freeze-out. The main reason why we are reporting on this experimental tool is the possibility to measure the prolongation of lifetime due to a phase transition.

Let us see how we can infer the spacetime structure of the source from correlations in momentum space. The correlations refer to two-particle correlations in the pion’s momenta distributions. They are defined as

[TABLE]

where $P(\vec{p},\vec{q})$ denotes the probability to find two pions with three-momenta $\vec{q}$ and $\vec{p}$ in the same event. In terms of rates, $C(\vec{p},\vec{q})$ is given as

[TABLE]

Thus $C$ is measurable as a ratio of two-pion to one-pion inclusive yields. Its width is a measure for the inverse source size. The weaker the correlation the smaller its width, the larger is the source which emits the pions.

Two-pion correlations can arise from different origins. The only one we will discuss in some more detail are Bose-Einstein correlations due to the quantum mechanical symmetrization of the outgoing wave functions of identical bosons. In the same way electrons in a metal are anticorrelated in their spatial distribution due to the fact that they occupy only a finite volume in momentum space.

Other causes of correlations due to final-state interactions are hadronic and Coulomb interactions. Coulomb interactions lead to positive correlations between particles of opposite charge if the relative momentum is small. It has to be checked how far their contribution is averaged out due to a comparable number of pairs with equal and opposite charges.

A third source for correlations comes from resonance decays. Pions are detected that are not directly emitted from the hadronic gas at freeze-out, but result from decays of heavier mesons. Such pions may contaminate the information about the source size.

Bose-Einstein correlations are the only origin for correlated emission from the radiating source if the source is completely chaotic. If we follow the evolution of the plasma through the phase transition to the final state of a hadron gas, it consists to $\sim$90% of pions in the baryon free region at freeze-out, where the pion gas is already rather dilute. (Present experiments deal with equal nucleon and pion numbers (AGS) and a ratio of 1:7 (SPS) for S+S, 1:5 (SPS) for S+Ag, the baryon free region in an idealization up to now). The assumption of an incoherent source for a dilute pion gas seems to be a good approximation to start with.

Therefore we focus on the case of a chaotic pion source. The probability $P(\vec{p},\vec{q})$ of measuring two pions with momenta $\vec{p}$ and $\vec{q}$ in the same event is given as (Pratt, 1984)

[TABLE]

where

[TABLE]

furthermore

[TABLE]

is the average momentum of the pion pair and

[TABLE]

denotes the relative four-momentum. The function $g(\vec{K},x)$ gives the probability of emitting a pion of momentum $\vec{K}$ from a spacetime point $x,\ g(\vec{K},x)$ is called the emission function. Thus, in the simplest case, where the source is incoherent and only Bose-Einstein correlations relate the wave functions of emitted particles, the correlation function essentially measures the expectation value of $\cos(kr)$ weighted by the product of emission functions, $r=x-y$ is the relative distance of the pions at their source). Stated differently, the measurable correlation in momentum space is the Fourier transform of the emission functions. (A general ansatz for the two-particle correlation function due to final state interactions in rapidity space can be written in a similar form.) The probability $P(\vec{p},\vec{q})$ depends on six degrees of freedom. Under a further constraint on the spacetime evolution of the source or on its symmetry properties, it may be possible to determine the emission function $g$ from a measurement of $P$, $g$ depends originally on seven degrees of freedom. The final spacetime distribution described by the product of $g()$’s is a result of the spacetime evolution of the plasma passing a possible phase transition to the hadronic phase. The spatial distributions of the pions at the moment of freeze-out constitute the radiating pion source we are interested in. Hence the ansatz for $g(\vec{K},x)$ depends on two ingredients: the equation of state and the dynamical concept relating the spacetime dependence to thermodynamic quantities.

Let us recall why a phase transition is expected to delay the break-up. The moment of freeze-out or break-up $\tau_{B}$ is determined by the degree of dilution. Equivalently, it is given by the value of its entropy density $s(\tau_{B})$. For fixed initial energy density and equilibration time $\tau_{0}$, the initial entropy density for a plasma is considerably higher than for a hot pion gas to start with. The total entropy is assumed to be approximately conserved during the expansion. Thus it takes longer to reach the same low entropy density at freeze-out if one starts in the plasma phase. A phase transition must have occurred at an intermediate stage.

In Fig. 34 (from Bertsch, 1989) we visualize how the prolongated lifetime is expected to influence the effective spatial dimension of the source. The beam axis is chosen in $z$-direction and labelled $x_{L}$. The two transverse directions are $x_{sidewards}(x_{s})$ and $x_{outwards}(x_{o})$, $\vec{x}_{o}$ points in the direction where the detector is placed. This coincides only with the direction of the main pion stream if the pions are viewed in a frame in which they are emitted perpendicular to the cylinder. Fig. 34 shows a snapshot of the pion sources at freeze-out for both scenarios: a plasma at the beginning or a hot and dense pion gas. In the latter case, the high density of pions in the formation zone leads to a rescattering. Thus the transverse dimensions at freeze-out would clearly exceed the size of the formation zone if the formation zone were at rest. The longitudinal expansion of the source modifies the picture in the following way. The collective flow of the pions singles out a preferred direction. Recall that in interferometry measurements pions are only detected which have almost the same momentum. For a moving source the probability to detect such pairs in a certain direction is reduced. Only pions which come from the same side of the cylinder as the detector have a chance to arrive in the detector with nearly the same momenta. This reduces the transverse dimensions of the pion cloud considerably in spite of the rescattering effects (Pratt, 1984; Bertsch, 1989).

A first-order phase transition which proceeds via nucleation of the plasma to the hadronic phase will alter the picture. The plasma lives only a very short time, thus its transverse expansion may be neglected. In the intermediate phase, a possible scenario is that the plasma breaks up into droplets of hadrons. The conversion to the hadronic phase is slow, because the rearrangement of degrees of freedom takes time. The pions are emitted over a longer period at a smaller rate, hence they have less chance to rescatter. The result is a small sideward and a large outward-dimension of the cloud at the moment of last interactions. This is illustrated in Fig. 34. The hope to identify a (strong) first order phase transition via pion interferometry is essentially based on this picture: to measure a large source size in the outward direction at freeze-out.

Pratt (1986) has derived a functional dependence of the emission function $g()$ for two equations of state (the bag model equation of state to simulate a phase transition and an ultrarelativistic pion gas without a phase transition). As dynamical concept he has used the hydrodynamical equations for two geometries, a spherically symmetric exploding plasma and Bjorken’s scaling solution. For Bjorken’s scenario, an analytic expression can be obtained, for the spherically exploding plasma the hydrodynamic equations have been solved for different initial conditions. The change in the dynamics which is caused by a phase transition is more pronounced the longer the mixed phase lasts where the pressure stays constant while the energy is absorbed in latent heat. Qualitatively the following features can be seen in a plot of $C(K,k)$ as a function of $k$, the relative momentum. Higher average momentum $K$ leads to a broader correlation corresponding to a smaller effective source size. This reflects the collective expansion of the system. One effect of a phase transition is a reduction of the explosive velocity. The other effect is only apparent in $C$ when the relative momentum is chosen parallel to the average momentum. For larger values of $K$, the correlations drop to smaller values indicating a larger size or a longer lifetime of the source. The strength of the signals depend, however, on the initial conditions which have been chosen for the spherically exploding fireball. They may fade away under realistic conditions. In order to get a hint on a transient phase transition in a cylinder geometry, the orientation of the relative momentum $k$ with regard to the average momentum $K$ has to be chosen in an appropriate way. The large outward dimension shows up only in correlations with $\vec{k}$ parallel to $\vec{K}_{per}$ ($\vec{K}_{per}$ being the transverse component of the average momentum).

The difference between sideward and outward dimensions in a cylinder geometry has been numerically investigated by Bertsch et al. (1988). This calculation is based on a cascade description of the hadronic phase starting at a time where the plasma begins to convert to hadronic matter. At a first view such a treatment seems to be more appropriate for the hadronic phase, especially in the period close to freeze-out, where the system is too diluted to justify a hydrodynamic approach. On the other hand it turns out that a hydrodynamic expansion in the sense of a collective transverse and longitudinal expansion velocity may be reobtained from a more microscopic starting point, if realistic cascade descriptions are used (Stachel, 1993). In the cascade approach of Bertsch et al. individual pions are followed along their classical trajectories. The plasma is assumed to break up into “blobs” which emit and reabsorb pions at a certain rate. Entropy conservation is no longer an ingredient, its production during the phase conversion comes out as a result. Also the freeze-out time is predicted in this approach. The break-up is described to occur instantaneously, i.e. without supercooling delay.

Again one has to specify an ansatz for the emission function. The classical treatment of pion motions leads to an emission function which is too singular to display interference effects. To implement interference effects, the momentum dependence of the source function is smoothed. Bertsch et al. (1988) have derived the following parametrization

[TABLE]

The spacetime coordinates are $x$, the proper time $\tau=\sqrt{t^{2}-z^{2}},r_{per}$ denotes the transverse position, $y_{r}\equiv\tanh^{-1}(z/t)$ is the spacetime rapidity, the average three-momentum $\vec{K}$ is written in terms of the transverse components $\vec{K}_{per}$ and the momentum rapidity $y_{\vec{K}}\equiv\tanh^{-1}(\vec{K}_{z}/E)$. The values for $\tau_{0},R$ and $y_{0}$ are chosen as $\tau_{0}=9$ fm/c, $R=3.3$ fm and $y_{0}=0.76$. Arguments for this choice can be found in the original reference. The result for the rate $C(\vec{p},\vec{q})$ to find a pion pair with momenta $\vec{p}$ and $\vec{q}$ is obtained from Eq. (463) as an integral over the transverse positions. The integral which is left is given as

[TABLE]

Here $k_{per}\equiv|(\vec{p}-\vec{q})_{per}|$ has a sideward and an outward component. The result is a Gaussian in sideward direction, $C$ measures the transverse size of the source which is the same in sideward and outward direction as long as the time dependence of the distribution is neglected. If the transverse relative momentum is chosen in the outward direction (i.e. $\vec{k}_{perp\ s}=0,\vec{k}_{\perp\ 0}\not=0$), the Gaussian in $\vec{k}_{perp\ 0}$ is modified by the $\vec{k}_{perp\ 0}$-dependence of $E_{\vec{p}}$ and $E_{\vec{q}}$, an effective size in outward direction is sensitive to the time correlation as it is seen in Eq. (467). The numerical results for sideward and outward correlations are displayed in Figs. 35(a) and (b).

The reduced correlation for outward (compared to sideward) relative momenta reflects the effectively larger size of the source in this direction, as has been anticipated above (cf. Fig. 34).

Expression (467) for the pion correlation partly motivates an ansatz which is sometimes used in real interferometry experiments (see e.g. Schmidt and Schukraft, 1993). That is a threefold Gaussian in $\vec{k}_{perp\;0}$, $\vec{k}_{perp\;s}$ and $\vec{k}_{L}$ (the longitudinal component of $\vec{k}$ instead of the rapidity $y_{\vec{K}}$). The possibility is implemented to have different length scales $R_{T_{out}},R_{T_{side}}$ and $R_{T_{L}}$ in all three directions, cf. Eq. (469) below.

So far we have reported on pion interferometry as a tool to estimate the lifetime of a pion radiating source. Naturally the question arises as to whether it is also possible to resolve spatial inhomogeneities of bubble structures in the mixed phase. This question has been recently addressed by Wieand et al. (1992). The answer is in principle positive. Pion interferometry may be a tool to resolve the grain structure of the hadronic phase. Clearly the bubble structure should not be too fine-grained, otherwise it is not resolvable for the detectors. In practice it does not seem to be a realistic possibility. The theoretical predictions have not yet reached a stage where it is clear what should be verified experimentally.

The status of experiments.

A few aspects of the experimental status of interferometry measurements are described which are restricted to applications in heavy-ion experiments. The primary observables are the double-pion and single-pion inclusive distribution functions $N(p_{1},p_{2})$ and $N(p)$ which determine the correlation $C$ according to Eq. (455). In Bjorken’s geometry, $C$ depends on five independent variables, the transverse and longitudinal components of the average momenta $K_{transv}$, $K_{L}$ or the rapidity $Y$, and the sideward, outward and longitudinal components of the relative momenta ($k_{per\;s}$,$k_{per\;0}$,$k_{L}$, respectively). An ansatz which is frequently used by experimentalists is a simple Gaussian in the relative momentum of the pion pair. A comparison to the analytic expression Eq. (467) may indicate the gap between theory and experiment. The parametrization which has been used in the NA35 experiment at CERN-SPS (Baechler et al., 1992) is a Gaussian in all three components of the relative momentum

[TABLE]

where $k_{per\;s}$ is orthogonal to $K_{transv}$ and $K_{L}$, $k_{per\;0}$ is parallel to $K_{transv}$ and orthogonal to $K_{L}$, and $k_{L}$ is orthogonal to $K_{transv}$ and parallel to $K_{L}$. Furthermore, $\lambda$ is the coherence or chaoticity parameter which is assumed to absorb the Coulomb corrections of final state interactions. It is 1 for an incoherent boson source. The parameters $R_{per\ s},\ R_{per\ 0}$ and $R_{L}$ are the observables which are determined from a fit. Note that the simple Gaussian ansatz of Eq. (469) describes only spatial correlations. The time correlation is more complicated if a phase transition occurs. The parameters $R$ include some information on the spacetime evolution of the system (Ferenc et al., 1992). The data for $R_{L}$ [fm] as a function of rapidity $Y$ as seen by an observer at fixed $Y_{obs}=2.5$ are consistent with a longitudinally expanding source. This result is not trivial. Predictions for a static source are quite different (Makhlin and Sinyukov, 1988). The longitudinal radius $R_{L}$ can be related to the freeze-out or decoupling time $\tau_{f}$. The relation between $R_{L}$ and $\tau_{f}$ for a scaling expansion is given as (Makhlin and Sinyukov, 1988)

[TABLE]

in a comoving frame, $T_{f}$ is the freeze-out temperature, $\left\langle m_{T}\right\rangle$ the average transverse pion mass. From the measured values of $R_{L},\ \tau_{f}$ is estimated to be $4.5<\tau_{f}<6.3$ fm/c for $100<T_{f}<200$ MeV (Ferenc et al., 1992). The difference between the outward and sideward extensions $R_{per\ 0}-R_{per\ s}$ should be a measure for the duration of pion emission as was argued above. The source dimension in the outward direction has two contributions (cf. Fig. 34): the effective geometrical depth (which is comparable to the sideward extension and can be larger than the geometrical size of the formation zone due to rescattering effects) and the effective prolongation due to a reduced particle emission rate in case of a phase transition. In Fig. 36 from Ferenc (1992) it is seen that the difference in the transverse dimensions is negligible.

For a specification of the rapidity intervals, see Ferenc (1992). A more careful estimate for the duration of particle emission is $\Delta\tau_{f}\leq 2$ fm/c. This rules out a long-lived source ($\geq 10$ fm/c) as a result of a first-order phase transition with supercooling delay. Finally, the transverse size at decoupling time $R_{\tau_{f}}$ turns out to be 1.5-2 times larger than the geometrical size if the geometrical size is estimated as a transverse size of the projected transverse density distribution. This is compatible with a high density formation zone, from which the pions first rescatter before they freeze out. The freeze-out volume becomes independent of the formation volume in this case, it merely depends on the particle density.

The statistics of the data has still to be improved to check the reliability of the projections of $R_{per}$ on $R_{per\ s}$ and $R_{per\ 0}$. One should also keep in mind that the results for the different length scales could be an artifact of the simplified ansatz (469) for $C$. It is unlikely that just one parameter $\lambda$ can absorb the full complexity of different origins of pion correlations. Another complication may arise from rescattering with the surrounding target spectator matter (Schmidt and Schukraft, 1993), a “dirt effect” which we have not mentioned so far. The relative weights of the contributions to pion correlations of different origins depend on the impact parameter. One would like to ignore these effects, unfortunately one is not allowed to do so.

For comparison we also mention results of pion-interferometry measurements with the AGS-machine at BNL (Stachel, E814 collaboration, 1994; Xu, E814 collaboration, 1994). The data have been measured for central 14.6 GeV/$n-n$ Si+Al- and Si+Pb-collisions. At freeze-out the result is the following. In the center-of-mass frame the pions are emitted from a source with transverse radius $R_{per}=6.7$ fm and longitudinal radius $R_{L}=5.0$ fm (on average). The source is nearly spherically symmetric. If here the transverse radius is compared to the initial transverse size of the system $R_{per}(Si)=2.9$ fm, one finds a transverse expansion of the system by a factor of 2.3. Assuming an expansion velocity of $v/c=0.3$ $(0.2)$, the timescale of the expansion follows as 10 (15) fm/$c$. The uncertainty in the expansion velocity is due to resonance decays. The pion-interferometry results lead to a freeze-out volume of $V=2400$fm3. A kind of consistency check is provided by the estimate for the volume, if freeze-out is defined to occur when the average distance from a given nucleon to the nearest pion is $d=\sqrt{\sigma_{\pi n}/\pi}$, where $\sigma_{\pi n}$ is the pion-nucleon cross section. For a freeze-out temperature of 140 MeV and $\sigma_{\pi n}=62$ mb one finds $V=2750$fm3 at freeze-out in reasonable agreement with the interferometry estimate.

In these measurements the estimates for the radii are not obtained from a direct fit of a certain functional form to the data, but by use of a dynamical model (realized in the event generator RQMD (Sorge et al., 1989)) with known spacetime characteristics of the source. Two-particle correlations are constructed from the RQMD-generated single-particle distributions in a way that the Bose-Einstein effect is imposed (Xu, E814 collaboration, 1994). The realistic experimental parameters are imposed as input data in the RQMD-code.

Although there is no indication for a first order transition in these experiments, the important conclusion is that the hadron gas does have time enough to be in thermal (and chemical) equilibrium at freeze-out. This supports one of the basic underlying assumptions for applying thermodynamic concepts.

Signals of a strong first order transition should not be missed by pion-interferometry. A prolongation of lifetime by an order of magnitude should be visible for a certain projection of the relative pair’s momentum. Non-appearance of such signals is less conclusive. Specific signals for a second order transition with a large correlation length have not been invented so far. Experimental groups which are at present dealing with pion, kaon and rho-interferometry are the E802 collaboration (Abbot et al., 1992), (Akiba et al., 1993), the E814 collaboration (see Xu, 1994, and Stachel, 1994) at Brookhaven, and the NA35 collaboration (Seyboth, 1992; Ferenc, 1992), the NA44 collaboration (Sarabura et al., 1992), and the WA80 collaboration (Peitzmann, 1993; Peitzmann et al., 1992 and 1993a and b) at CERN. A chance for future high statistics experiments may lie in kaon interferometry. The yield of kaons is considerably lower than that of pions, but kaons are less plagued by contamination due to resonance decays (Ferenc, 1992b).

5 Multiplicity fluctuations

The possible signatures for a first-order transition which we have discussed in the preceding sections are based on a prolongated duration of the matter evolution. Multiplicity fluctuations rely on additional entropy production which might be generated in explosive processes during the plasma conversion to hadronic matter. Several candidates for hadronization processes have been under discussion: nucleation in the bulk, “boiling by cooling” (van Hove, 1983), or an adiabatic procedure, where the system stays in local thermal equilibrium, pressure and temperature remain constant during the conversion. These scenarios do not lead to strong multiplicity fluctuations.

In this section, we describe alternatives to the adiabatic scenarios. These are hadronization processes via deflagrations or detonations. They are compatible with a strong first order phase transition and a liberation of a large amount of latent heat, supercooling and/or superheating effects during the transition. One of the proposed physical pictures is the following (van Hove, 1985). Energy and entropy densities change drastically over a small temperature and pressure interval in the vicinity of $T_{c}$. Therefore the proposed hadronization mechanism has to provide a fast liberation of the large entropy and energy amount of the plasma, a large energy has to flow into the hadron phase in an efficient way. The main part of hadronization occurs at the phase transition. Hadronization via evaporation of hadronic bubbles before the transition is negligible in this picture. Close to the transition temperature, the color fields (induced by the color charges of the plasma) collapse to flux tubes creating something like a network of strings (cf. Patel, 1984 a and b, and section IV B 1). Due to string formation, stretching and breaking, plasma droplets will be stopped in their longitudinal expansion or break up in a few more droplets. In the next stage, the main part of the hadronization is assumed to proceed via ejection of hadrons through the surface of the plasma blobs with the kinematics and thermodynamics of a deflagration. Deflagration is said to occur only at the outer surface, but not in the interior (in the “bulk”) of the plasma droplets. The plasma blobs themselves are small in size (a few fermi at most).

Let us see which constraints from energy momentum and positive entropy production are obtained for deflagration or detonation scenarios. Again we use a hydrodynamic description and follow Cleymans et al. (1986) to sketch the derivation of kinematic constraints. Deflagrations start at the surface of the plasma, detonations in its interior and form hadronic bubbles. In the case of deflagrations the front separating both phases moves more slowly inwards than the hadrons escape outwards. In the rest frame of quark matter the hadrons and the front move in opposite directions. In detonations the velocities of the front and the hadrons are both directed outwards, but at a different speed. Usually the speed of the hadrons is assumed to be lower than the speed of the front. In the case that the speed of the hadrons is higher, the relative velocity of the hadrons with respect to the front is negative. This explosion is called an eruption and is very unlikely as we will see.

Let us go to the rest frame of the front. Pictorially, in this frame the front “eats up” the plasma which wants to pass the front. In the case of deflagrations the hadron velocity exceeds the velocity of quark matter leading to

[TABLE]

Here $\theta_{h,q}$ are the fluid rapidities in the hadron and quark phases, respectively. In usual detonations the quark matter is “eaten up” faster than the hadronic bubbles are formed ($\theta_{h}<\theta_{q}$), in eruptions quark and hadronic matter velocities have opposite sign in the rest frame of the front, cf. Fig. 37(a)-(c).

Thus the difference between deflagrations and (usual) detonations lies in the relative magnitude of their velocities in the rest frame of the separating front. To derive constraints for pressure, energy and entropy densities, we go to the rest frame of the front, where the flow of quark and hadronic matter is steady. Derivations $\partial/\partial\tau$ vanish. Further, we specialize the interface separating both phases to a (1+1)-dimensional planar interface. The interface is chosen to be infinitesimally thin. Energy momentum conservation $\partial_{\mu}T^{\mu\nu}=0$ across the front leads to

[TABLE]

in $(t,z)$ coordinates. The energy-momentum tensor is taken as that of a perfect fluid (apart from the place of discontinuity). The velocity is parametrized in terms of the fluid rapidity $\theta$ as usual according to $u_{0}=\cosh\theta$ and $u_{z}=\sinh\theta$. Continuity of $T^{zz}$ and $T^{z0}$ across the front leads to

[TABLE]

for a perfect fluid. Solving for $\tanh^{2}\theta_{h,q}$ and implementing the unequalities for $\theta_{h}$ and $\theta_{q}$ Eq. (475) gives

[TABLE]

and

$\displaystyle\varepsilon_{q}-p_{q}<\varepsilon_{h}-p_{h}\ \qquad\hbox{for detonations}\;.$

Imposing the bag model equation of state (section V B 2) with $c_{s}=1/\sqrt{3}$, this means

[TABLE]

The unequality in Eq. (478) for detonations shows that a strong supercooling is necessary for $\varepsilon_{q}$ being small enough to satisfy Eq. (478). (The gap between the energy densities of both phases may be unrealistically large in the bag model, but it is obvious that some supercooling is necessary.) The next constraint on the allowed values of $\varepsilon$ and $p$ comes from the second law of thermodynamics. The entropy flux into the front may not be larger than out of the front. The condition of entropy increase across the discontinuity leads to ${{s^{2}_{h}}/{s^{2}_{q}}}\geq{{(\varepsilon_{h}+p_{h})(\varepsilon_{h}+p_{q})}/{((\varepsilon_{q}+p_{q})(\varepsilon_{q}+p_{h}))}}$. For a perfect fluid, where dissipation effects are absent, entropy is strictly conserved inside each phase $(\partial_{\mu}(su^{\mu}))=0)$. It is also approximately conserved across the front (equality sign above), if the discontinuity in $\varepsilon$ is infinitesimally small (van Hove, 1983).

In general, however, deflagrations and detonations are irreversible processes and do produce entropy. The equation for $s_{h}^{2}/s_{q}^{2}$ can be exclusively expressed in terms of $\varepsilon_{h}$ and $\varepsilon_{p}$ by using the thermodynamic relation $T^{.}s=\varepsilon+p$ and the equation of state. Finally the velocities must be smaller than the velocity of light. This means

[TABLE]

The combined constraints for $s_{h}^{2}/s_{q}^{2}$ and (479) limitate the values for $\varepsilon_{q}$ and $\varepsilon_{h}$ which are compatible with detonations or deflagrations to a small window. Thus reliable results for $\varepsilon_{q}$ and $\varepsilon_{h}$ and the equation of state could lead to an exclusion of these scenarios.

Deflagration und detonation solutions of the hydrodynamic equations have been constructed in a numerical analysis by Gyulassy et al. (1984). The plasma expansion is neglected in the initial state, the baryon number is set to zero. The constants from energy conservation and entropy increase are evaluated as indicated above, the equation of state is taken from the bag model. The results are the following.

Detonations and deflagrations use the latent heat in a different way. Detonations need strong supercooling and subsequent strong superheating of the hadron phase. Deflagrations are possible under less extreme conditions. They are compatible with small supercooling. The calculations have been extended from discontinuities along a single surface to several ones, that means bubble deflagration rather than surface deflagration. A surface deflagration shock had been considered by van Hove (1985). It seems to move too slowly into the plasma for providing a realistic possibility of plasma conversion to the hadron phase.

Entropy producing processes can lead to enhanced rapidity density fluctuations, peaks in $dN/dy$ of the final state hadrons. These peaks may be isolated or overlapping, depending on the resolution $\delta y$ and the size of a typical scale $\Delta y$ over which $N$ fluctuates. Large $\left\langle p_{T}\right\rangle$-momenta are another possible effect. Multiplicity fluctuations are sometimes called seismometers (Gyulassy, 1984). Experimental hints for “crests” of plasma bubbles are taken from cosmic ray events (Iwai et al., 1976 and 1982). The observations are at least compatible with an explanation of deflagration processes in QCD-plasma conversions (Gyulassy et al, 1984; van Hove, 1985).

Let us assume that the QCD transitions are truly sharp crossover phenomena. The question arises as to whether there is still a chance for observing deflagrations. The answer seems to be positive, if we look at the derivation of deflagrations e.g. in van Hove (1985). The essential ingredients are a large variation of $\varepsilon$ and $s$ over a small interval of $T$ and $p$. An ingredient is the small value of $p$ as compared to $T^{.}s=\varepsilon+p$. In case of a crossover phenomenon the transition interval is defined as a region of rapid variation of $dp/dT$. Van Hove (1985) has studied the implications for deflagrations in both cases, a first and a second order transition. The results are quite similar in both cases. Deflagrations may be realized in heavy-ion collisions in spite of a smooth crossover if it is only sharp enough. The result is not surprising. Nucleus-nucleus collisions are performed in a finite space-time volume. From numerical simulations we know that a strictly first order transition and a sharp crossover phenomenon are practically indistinguishable in a finite volume, if the measurement is performed for a single size. Similarly deflagrations may be practically the same for first order transitions and a crossover phenomenon in a finite volume. The finite volume constitutes part of the physical boundary conditions in heavy-ion collisions. Experimental results will not expose finite volume roundings as artifacts of approximations to a hypothetical infinite volume world.

6 Intermittency analysis

Experimental manifestations of the QCD transitions are frequently discussed under the assumption that they are of first order. As we have seen in sections III and IV, there are indications that the finite temperature transition resembles more a crossover phenomenon. The realistic set of quark masses may be close to a “critical” set in mass parameter space, which could lead to a second order transition. A standard physical picture of critical phenomena in spin systems is a diverging correlation between spins leading to clusters of aligned spins of arbitrary length scales. The $3-d-Z(2)$-spin system is supposed to share the universality class of an $SU(2)$ pure gauge theory, an $O(4)$-ferromagnet is assumed to share the class of 2-flavor QCD.

Thus the question arises how a picture of clusters of aligned spins in color or isospin space can be transferred to typical observables in heavy-ion collisions, e.g. to particle multiplicities. One of the possible answers is intermittent behavior in rapidity distributions. (For a particular realization of a second order transition in case of a hot pion gas see also section V D 3 below).

The concept of intermittency was originally introduced in studies of turbulent behavior of fluids (Zel’dovich et al., 1987). In general, intermittency can be defined as the appearance of structure in random media. In the context of particle physics, it has to deal with large fluctuations of charged particle density in small regions of phase space. In heavy-ion collisions intermittency refers to certain moments of rapidity distributions. Other distributions like energy, pseudo-rapidity and azimuthal angles can be considered as well (Ochs and Wosiek, 1988).

Let us first consider a toy model that is universal enough for applications in different areas. Consider a set of $N$ balls distributed in a box of total size $R$ into $M$ cells of size $L(M=R/L)$. In the case of rapidity distributions, they correspond to $N$ particles per given unit of some kinematic range. A cell in a box is a rapidity interval in a certain available rapidity range. For an Ising model, the balls are the spins on a spacetime lattice $R$ subdivided into cells of size $L$. We are interested in the ball distributions if the resolution of the lattice is made finer and finer $(M\to\infty)$. There are many ways to realize this limit. One possibility is to keep $R$ constant and let $L$ going to zero. This limit is of interest for rapidity distributions, where $L$ stands for the bin size $\delta y$ of the rapidity interval. Another realization is the thermodynamic limit with $R\to\infty$ for fixed $L$. This limit is usually taken when critical phenomena are discussed in the infinite volume limit.

The same type of limit is of interest in a second order finite temperature transition in spacetime continuum, which is described by a model of lattice gauge theory. One may expect that intermittency in the limit of ($R\to\infty$, $L$ fixed) has a correspondence in the limit of ($R$ fixed, $L\to 0$).

We define the $\ell$th normalized moment $f_{\ell}$ for a given distribution of $N$ balls as

[TABLE]

Here $K_{m}$ denotes the number of balls in the $m$th cell. First, we keep $N$ fixed and vary $M$. In the extreme case of an equidistribution of $N/M$ balls in each box, it is easily seen that $f_{\ell}(M)$ is independent of the grain size, i.e.

[TABLE]

In the other extreme, a strong fluctuation (where all balls are concentrated in one box), we find a logarithmic dependence of $\ln f_{\ell}$ on the “resolution size” $L$

[TABLE]

Such a behavior of a given distribution with $N$ balls is called intermittent.

More generally, intermittent behavior is attributed to average values of $f_{\ell}$. The weighted average $\left\langle\dots\right\rangle$ of the $\ell$th moments runs over an ensemble of configurations in a $d$-dimensional volume $R^{d}$ divided into $M=(R/L)^{d}$ cells of equal size $L^{d}$. In a multiplicity measurement it could be an average over all events, in the Ising model it is the thermodynamic average with Boltzmann weights.

Intermittency is called the property of fluctuations around some average distribution which lead to a power law behavior of $\left<f_{\ell}\right>$ in the number of cells $M$, or equivalently if

[TABLE]

Here $\lambda_{\ell}>0$ are constant and $g_{\ell}(R)$ is independent of $L$, $L$ being the linear cell size. The constants $\lambda_{\ell}$ are called intermittency indices, they are a measure of the strength of intermittent behavior.

Results for $\lambda_{\ell}$ and $g_{\ell}$ in case of a d-dimensional Ising model can be found in (Satz, 1989). The essential point where criticality enters the derivation of intermittency in an Ising model is scale invariance at $T_{c}$. While the balls of the toy model are the same independently of the partition of the lattice, one has first to argue, why one may choose the same type of variables on all length scales in an Ising model. This is justified by selfsimilarity as $T\to T_{c}$, if $1\ll L\ll\xi$ is satisfied. For a given $\ell$ the intermittency indices can be expressed in terms of the more familiar critical exponents. Like critical exponents they are universal for all models belonging to the same universality class.

The mere divergence of a correlation length $\xi$ is not sufficient for introducing intermittent behavior. In a 1-$d$-Ising model, where $\xi$ has an essential singularity rather than a power-law singularity as in the 2- and 3-dimensional cases, the normalized momenta $f_{\ell}$ are bounded from above for all $\ell$ (Hajduković and Satz, 1992).

We turn now to intermittent behavior in heavy-ion experiments. To identify this behavior in rapidity distributions, we consider moments $\cal C_{\ell}$ which have been introduced by Bialas and Peschanski (1986) according to

[TABLE]

Here $M$ is the number of intervals of size $\delta y$ in a given rapidity interval $\Delta y=M\cdot\delta y$, $p_{m}(m=1,\dots,M$) denotes the probability for finding particles in any of these rapidity intervals $dp_{M}$. The total distribution $P(p_{1},\dots,p_{M})dp_{1}\dots dp_{M}$ of probabilities for finding particles in the intervals $dp_{1}\dots dp_{M}$ is normalized such that

[TABLE]

The variable $t$ stands for a collection of kinematic cuts like the energy of the collision. The average moment $\left\langle\cal C_{\ell}\right\rangle$ is obtained as the sum over all configurations in rapidity space weighted by the probability distribution $P(\;)dp$

[TABLE]

In terms of rapidity variables genuine intermittent behavior is signalled, if

[TABLE]

in the limit of $\delta y\to 0$, i.e. a logarithmic dependence on the resolution size in the limit of increasing resolution, which is equivalent to the definition in Eq. (483). Small values for $\delta y$ correspond to a fine resolution in rapidity space, $\Delta y$ is the full considered rapidity interval. The lower bound on $\delta y$ is given by the experimental resolution. The intermittency indices $\phi_{\ell}$ vary between $O<\phi_{\ell}\leq\ell-1$. They are a measure for the strength of the intermittent behavior. Intermittency occurs if selfsimilar fluctuations exist on all scales $\delta y$. In the case of rapidity distributions intermittency should be seen in contrast to dynamical fluctuations.

In real experiments we have to deal with finite size systems. Statistical fluctuations around the probability distribution $dp_{1}\dots dp_{M}P(p_{1},\dots,p_{M};t)$ will always provide a noisy background for fluctuations of dynamical origin that we are trying to identify. Unless the multiplicity in the events is very high, the probability $p_{m}$ for finding particles in the rapidity interval $m$ is different from the measured fraction $K_{m}/N$ of the total multiplicity $N$. As filter of dynamical fluctuations Bialas and Peschanski (1986) have proposed to consider scaled factorial moments $\left\langle F_{\ell}\right\rangle$ of the distribution $Q$ which is actually measured in the experiment

[TABLE]

where

[TABLE]

and $B$ is the Bernoulli distribution.The statement is that the scaled moments $\left\langle{\cal C}_{\ell}\right\rangle$ of a probability distribution $P$ are equal to the scaled factorial moments $\left\langle F_{\ell}\right\rangle$ of the experimental distribution $Q$. It can be shown that $\left\langle F_{\ell}\right\rangle=\left\langle{\cal C}_{\ell}\right\rangle$. Notice the advantage, if one uses $\left\langle F_{\ell}\right\rangle$ instead of $\left\langle{\cal C}_{\ell}\right\rangle$. In Eq. (491) $\left\langle F_{\ell}\right\rangle$ is exclusively expressed in quantities which are directly measured. The average in Eq. (491) can be evaluated by absorbing the weights $Q(...)$ in the selection of events.

Further information about the origin of intermittency is contained in the $\ell$-dependence of $\phi_{\ell}$. Equivalently, one can use the $\ell$-dependence of anomalous fractal dimensions $d_{\ell}$ which are related to $\phi_{\ell}$ according to

[TABLE]

At least two physical mechanisms are known which may lead to real intermittent behavior in heavy-ion collisions (real in contrast to fluctuations which are induced by two-particle correlations). One of these mechanisms is the QCD transition if its correlation length diverges at $T_{c}$, or, adapted to the finite volume, if its correlation length extends the typical volume in the collision. In this case, $d_{\ell}$ should be approximately independent of $\ell$. Other origins are selfsimilar cascades which are attempts to model the evolution of the plasma to the hadronic phase. Here $d_{\ell}$ will depend on $\ell$ in general. Unfortunately, the $\ell$-dependence of $d_{\ell}$ does not allow a unique identification of the origin of intermittency. Neither does the independence on $\ell$ guarantee a phase transition nor does its dependence exclude a transition. According to the work by Bialas and Peschanski (1986), it may be possible to tune the parameters of a selfsimilar cascade in such a way that $d_{\ell}$ does not depend on $\ell$. On the other hand, multifractal phase transitions are known which lead to $\ell$-dependent $d_{\ell}$ (although the latter possibility is not predicted by QCD).

We add a remark on dynamical fluctuations which are due to two-particle correlations. In this case the scaled moments first rise with decreasing bin size, then they saturate to some constant value below a certain resolution size $\delta y_{0},\,\delta y_{0}$ gives also the typical size of the dynamical fluctuations. Two-particle correlations can result from resonance decays or Bose-Einstein correlations. For resolutions $\delta y>\delta y_{0}$ two-particle correlations are mainly responsible for apparent intermittency effects. At least in $e^{+}e^{-}$-reactions the observed intermittent behavior can be fully explained by two-particle correlations within a certain particle production scheme. For heavy-ion collisions the particle production processes in the matter evolution are less well understood, at least the contribution from Bose-Einstein correlations can be tested experimentally, see e.g. Schmidt and Schukraft (1993).

We summarize what a measurement of the anomalous fractal dimension $d_{\ell}$ can tell us. Assuming that the contribution from two-particle correlations can be subtracted, the following alternatives can be distinguished.

• $\bullet$

If $d_{\ell}\sim 0$, the fluctuations are of purely statistical origin. A first order transition with a correlation length smaller than the size of the system would be compatible with such a scenario.

• $\bullet$

If $d_{\ell}$ strongly depends on $\ell$, it seems rather unlikely that the QCD transition itself is responsible for inducing such a multifractal behavior. More likely is a cascade mechanism at work whose vertices should be chosen to reproduce the observed $\ell$-dependence of $d_{\ell}$. For a cascading process one expects $d_{\ell}$ to depend linearly on $\ell$ (Bialas and Peschanski, 1988).

• $\bullet$

If $d_{\ell}$ depends weakly on $\ell$, but is clearly different from zero, a second order transition provides an explanation for intermittency, but not the only one.

We conclude with a short glance at the experimental status. Intermittency has been found in $e^{+}e^{-}$-reactions, hadron hadron- and heavy-ion collisions, see e.g. (Abreu et al., 1989; Buschbeck, 1989; Åkesson et al., 1990, respectively). Qualitatively, different experiments agree about the result that intermittency has been observed and different mechanisms are at work, as the observed behavior is not uniform. Quantitatively, they disagree. In Fig. 38 we display the dependence of the anomalous fractal dimension on $\ell\equiv p$ at $\sqrt{s}=20$ GeV taken from Bialas and Hwa (1991). Obviously, the $p$-dependence of $d_{p}$ weakens when passing from lepton-hadron to hadron-hadron and nucleus-nucleus collisions. In (Bialas and Hwa, 1991) this is interpreted as a result of selfsimilar cascade mechanisms in the first two cases, but as a phase transition in the $S-Ag,Br$ central collisions. The data for $Ag,Br$ collisions which are consistent with an almost constant nonzero value of $d_{\ell}$ suffer from low statistics (Schmidt and Schukraft, 1993). Another tendency which is seen in Fig. 38 is a decrease of $d_{p}$ when the process becomes more complex, but this decrease is less strong for small values of $p$. Thus it may be more than an artifact of generally small values of $d_{p}$ that $d_{p}$ is almost independent of $p$ for heavy-ion collisions.

Less encouraging are the high statistics data taken at CERN SPS (Bloomer et al., 1992) for ${}^{32}S+S$ and ${}^{32}S+Au$ collisions. The slope of $\ln\left\langle F_{2}\right\rangle$ vs $(-\ln\delta\eta)$ ($\eta$ being the pseudorapidity) is consistent with zero or less than zero. Thus one of the conclusions of the authors is that there is “no need for new physics” in the sense of fluctuations of unknown dynamical origin.

Another systematic trend seems to be well established by the data. As we have mentioned above, two-particle correlations may look like intermittency signals as long as the resolution is not high enough. In fact, for lighter projectiles like oxygen or silicon the data for intermittency indices $\phi_{2}$ can be well fitted as a function of the average particle density $\left\langle\rho\right\rangle$. Such a scaling law has been proposed by Seibert (1990) and is based on the assumption that only two-particle correlations are responsible for the observed intermittent effects. The optimistic news are that for heavier projectiles like sulfur-induced reactions such an explanation is not sufficient. For heavier projectiles, there seems to be a chance for seeing collective effects. These could be due to a nearly second-order transition or a cascade mechanism during the matter evolution. Last but not least detector effects should be kept in mind as possible source for contaminating the data. For further details on the experimental status we refer to the review of Schmidt and Schukraft (1993) and references therein.

Thus the main task in an intermittency analysis is to disentangle fluctuations of different origins: geometry (impact parameter dependence), statistics (finite number of particles), resonance decays or Bose-Einstein correlations, hadronization along cascade mechanisms, and a phase transition with a large correlation length. Recently work by Campi and Krivine (1995) questions any connection between the observation of intermittent behavior in mass distributions of finite nuclei and a second order phase transition. In any case, one has to face a superposition of several backgrounds on the type of multiplicity fluctuations one is primarily interested in. Once an interesting structure is filtered from the backgrounds, the $\ell$-dependence of intermittency indices may indicate the specific dynamical origin of this structure. The situation resembles that for dilepton rates, where the main practical problem was an identification of thermal rates. Once the thermal rates are identified, the remaining task is to look for conclusive structures of the transition scenario (cf. section V C 2).

D Theoretical concepts for off-equilibrium situations

1 Nucleation rate of hadronic bubbles

Many phenomenological implications assume a scenario of a first-order transition which is often described by a bag model equation of state. The scenarios differ in assumptions about the entropy production during the transition, the amount of supercooling, the production of latent heat.

In this section, we describe an approach of Csernai and Kapusta (1992a and b) for calculating the nucleation rate of hadronic bubbles in the plasma phase. In principle, such a calculation can select a realistic scenario. For example, it should tell us whether a Maxwell-construction of two-phase equilibrium is realistic in view of the total duration of the plasma evolution to the final hadronized state, which is of the order of $10-20$ fm/c. It requires a nucleation rate large enough that the latent heat is conducted fast enough from the bubble surfaces for the bubbles to grow. Naturally transport properties of the latent heat are determined by properties of the “conducting” medium surrounding the bubble. Such properties are the thermal conductivity, the shear and bulk viscosity, if a hydrodynamical description is used. In our case of an expanding plasma on its way to hadronization, the plasma viscosity will turn out to determine the nucleation rate of the hadronic bubbles.

The prediction of a definite value for the nucleation rate and the time delay due to a first order transition is rather ambitious and involves unavoidably a number of assumptions and approximations. Deviations from equilibrium have to be admitted during the phase transition, which need not be perturbatively small. One has to deal with an expanding system of many particles interacting according to the excitation spectrum of QCD.

Quite generally, the nucleation rate of one phase out of another (here the hadronic phase out of the plasma phase) has the form

[TABLE]

This is the usual expression for the rate of a thermally activated process. More precisely, it is the probability to form a (hadronic) bubble of critical size per unit time and unit volume. The difference $\Delta F$ is the activation energy to form a bubble of critical size. More generally, it is the difference in the free energies between two configurations with and without a condensed part of the new phase (it need not have the form of a droplet). The more delicate part is the prefactor, which describes some sort of quantum and/or thermal fluctuations. It sensitively depends on the context and the approximation scheme.

A specific realization of Eq. (495) is the famous formula of Callan and Coleman (1977) for the decay rate of the ‘false’ vacuum. The $\Delta F$ in the exponent is given by the Euclidean action $S_{4}$ in four dimensions of the considered field theory, evaluated at a solution $\phi$ of the Euclidean equation of motion satisfying $\phi\to 0$ for $x^{2}+t^{2}\to\infty$ as boundary condition. The prefactor depends on $S_{4}$ and a ratio of determinants depending on some zero-temperature (effective) potential $V$. The occurrence of determinants in the prefactor is a common feature of formulas for decay rates when fluctuations are taken into account in the Gaussian approximation. Although Callan and Coleman’s formula is sometimes mentioned in the context of phase transitions in the early universe, it is not appropriate for calculating a decay rate at finite temperature in a situation out of equilibrium, which is encountered in heavy-ion collisions.

A suitable framework is provided by Langer’s approach (Langer, 1969), which has been generalized to relativistic field theory and applied to QCD by Csernai and Kapusta (1992a). We briefly summarize the main ingredients in Langer’s formalism. The zero-temperature infinite volume potential in Callan and Coleman’s formula is replaced by a so called coarse-grained free energy, which is the appropriate quantity for describing a phase coexistence in a finite volume. Langer considers an effectively classical system of $N$ degrees of freedom, represented by a set of $N$ coordinates and their conjugate momenta $\eta_{i}$, $i=1,\ldots,2N$. These variables need not describe the system on a microscopic level, but enter the coarse-grained free energy as some kind of collective coordinates. Whereas the Euclidean action $S_{4}$ (mentioned above) depends on microscopic variables, the coarse-grained free energy is formulated in terms of ‘mesoscopic’ variables, which are collective coordinates.

The $\Delta F$-term in Eq. (495) keeps its meaning as the change in (coarse-grained) free energy due to a formation of one droplet of critical radius $R_{*}$.

The prefactor $I_{0}$ of the general formula accounts for (thermal) fluctuations in a specific way. It is written as a product of two terms

[TABLE]

a statistic factor $\Omega_{0}$, which is a measure of phase space volume around the saddle point, and a dynamical factor $\kappa$. While $\Omega_{0}$ will be seen to depend on equilibrium quantities, $\kappa$ knows about off-equilibrium features of the dynamical expansion. The dynamics may be described by hydrodynamical equations of motion. Thus it is no longer restricted to the regime of $O(4)$-invariant Euclidean equations of motion in equilibrium situations. A direct manifestation of off-equilibrium features is $\kappa$’s dependence on the thermal conductivity in a liquid-gas system or on the shear and bulk viscosities ($\rho$ and $\eta$ respectively) in the case of QCD, cf. Eq. (517) below. This is clearly a generalization as compared to the approach of Callan and Coleman, but also Langer’s formulation does not apply to generic off-equilibrium systems. The basic assumption is that the conversion proceeds via nucleation. The nucleation rate comes out as time-independent by construction. The phase transition is described as motion of a probability current $\rho$ in phase space. It starts at $\{\eta_{0}\}$, the metastable phase, passes most likely a saddle point $\{\bar{\eta}\}$, where the current is assumed to be stationary

[TABLE]

and ends in the stable phase configuration $\{\eta_{f}\}$. The configuration $\{\bar{\eta}\}$ differs from $\{\eta_{0}\}$ just by one critical droplet of new phase. To allow a stationary current across the saddle in phase space, there has to be a heat bath in the background, which replenishes the metastable phase at the same rate as it is lost in the phase conversion process. The assumption in Langer’s derivation is that such a heat bath exists, it is in equilibrium prior to each interaction with the converting phase. The probability for a transition between two configurations $\{\eta\}$ and $\{\eta^{\prime}\}$, induced by thermal fluctuations of the heat bath, is determined by the same Hamiltonian $F$ as it enters the dynamical part of the system which transforms into the new phase.

The dynamical part is assumed to be governed by equations of motion of the form

[TABLE]

In general, $F$ plays the role of a Hamiltonian, in an application to a hydrodynamical system $F$ has the specific form of a coarse grained free energy. Later $F$ will contain the information which is specific for the underlying QCD. The matrix $M$ is the mobility matrix, in our application the coefficients $M_{ij}$ can be read off from a comparison with the equations of relativistic fluid dynamics. Eq. (498) enters the time development of the distribution function $\rho(\{\eta\},t)$ which is associated with the variables $\eta_{i}$ and gives the probability density over configurations $\{\eta\}$. For equilibrium configurations $\rho$ is proportional to the Boltzmann weight $\exp[-F\{\eta\}/T]$, where $F$ are the costs in (coarse grained) free energy to generate such a configuration. Skipping some steps in the derivation, the time evolution of $\rho$ can be finally written in the form of a continuity equation

$\displaystyle{{\partial\rho\{\eta\}}\over{\partial t}}=-\sum_{i}{{\partial J_{i}}\over{\partial\eta_{i}}}$

with

[TABLE]

where $F$ is the Hamiltonian and $T$ is the temperature. The probability current across the saddle is identified with the decay rate. The prefactor $\kappa$ gives the initial exponential growth rate of a bubble which has just exceeded its critical size $R_{*}$. It is defined as

[TABLE]

To obtain $\kappa$ from the time evolution of $\rho$, Eq. (500) is solved for small deviations $\nu_{i}=\eta_{i}-\bar{\eta}_{i}$ from the saddle $\{\bar{\eta}\}$. Note that this is a further assumption entering the derivation of $\kappa$. The deviations from the saddle must be small for the saddle point approximation to make sense. For details about the statistical prefactor $\Omega_{0}$ we refer to the original literature (Langer, 1969; Langer and Turski, 1973 and 1980).

Let us turn next to the application in QCD (Csernai and Kapusta, 1992a and b). We have to specify the collective variables $\eta_{i}$, the coarse grained free energy $F$, the hydrodynamic equations of motion and their energy momentum tensor $T_{\mu\nu}$. The mobility matrix $M$ and the stationary configuration $\{\bar{\eta}\}$ must be identified.

The collective variables $\eta_{i}$ are chosen as energy density $e(\vec{r})$ and flow momentum $\vec{P}(\vec{r})$ at positions $\vec{r}$ in the system.

The choice of a coarse grained free energy $F$ for QCD is more subtle. The hydrodynamic approach is suitable for describing nucleation rates in a fluid, since droplet formation occurs on a semi-macroscopic level where a large number of particles contributes to a single droplet. The concept of the coarse grained free energy is introduced when hydrodynamics is derived from microscopic kinetic theory (Van Kampen, 1964; Jackson 1960). The macroscopic system is divided into cells of a given volume. Specific densities ($e$ and $\vec{\cal P}$ in our case) are assigned to each cell. If the partition function of the microscopic system is evaluated under these cellular constraints on the microscopic variables, one obtains an effective action (in a more field theoretical language), which is called the coarse grained free energy (over $k_{B}T$). (Note that this concept is very similar to the constrained effective potential used in section IV A 4. In the constrained effective potential the entire system is considered as one cell subjected to a constraint on the vacuum expectation value of the order parameter field.)

The underlying QCD dynamics enters the interaction part $f(e)$ of the coarse grained free energy $F$ for a small volume of nonuniform energy density $e$. The grain size of $F$ is obviously essential for the approximation to make sense. If the grain size is too small, the use of hydrodynamic variables is not meaningful, if it is too large, the coarse grained free energy cannot resolve the substructure of the fluid. This happens if phase separation occurs within single cells, $F$ would be a convex function of the local average energy density $e$ in this case. Similarly the constrained effective potential becomes convex in the large volume limit. Thus an ansatz for $f(e)$ which leads to a non-convex shape of $F$ as function of $e$, will indirectly implement the right grain size which is sensitive to nonequilibrium properties like droplet growth and phase separation. Note that $F$ is non-convex, but real, the non-convex shape is not an artifact of a perturbative approximation scheme, in which a thermodynamic potential is evaluated in the infinite volume limit. The very grain structure is implemented in the choice of hydrodynamic variables, implicitly involving averages over cells.

As ansatz for $f(e)$ Csernai and Kapusta have chosen a fourth order polynomial in $(e-e_{0})$, where $e_{0}$ is the location of the top of the potential barrier. The coefficients depend on the energy densities in the hadron and plasma phase, and on the curvature $f^{\prime\prime}_{0}$ of $f(e)$ at $e_{0}$. In the full ansatz for $F$ one is left with two free parameters, $f^{\prime\prime}_{0}$, and the coefficient $K$ of the derivative term $(\nabla e)^{2}$. Both parameters can be expressed in terms of the interface tension $\sigma$ and the interface thickness $2\xi_{0}$. Here $\xi_{0}$ denotes the correlation length defined at the top of the barrier by $\xi_{0}^{2}\equiv-K/f^{\prime\prime}_{0}$. Note that both quantities ($\sigma$ and $\xi_{0}$) can be calculated from first principles, i.e. from the underlying QCD Lagrangian.

The equations of motion are the equations of relativistic fluid dynamics, $\partial_{\mu}T^{\mu\nu}=0$. In the baryon-free region of the quark gluon plasma we have to deal with a relativistic system. Without a net baryon number there is no distinguished reference frame to define the flow four velocity $u_{\mu}$ of the matter. Also the pressure is not small compared to the energy. Therefore $u_{\mu}$ is defined as velocity of the collective energy flow. The stress-energy-momentum tensor for a relativistic dissipative fluid depends on the shear and bulk viscosities $\eta$ and $\zeta$, respectively. It should be noticed that for a vanishing dissipation part the growth rate of bubbles would vanish. Similarly it has been noticed by Langer and Turski (1973) that the growth rate in a liquid-gas transition vanishes near the critical point, if the thermal conductivity is set to zero. The bubbles can only grow, if the latent heat is conducted from the bubble surface.

We skip further details of the derivation and come to the results. The exponential suppression factor in Eq. (495) depends on the change in free energy $\Delta F$, if a hadronic bubble has formed in the QCD plasma at $T<T_{c}$ due to a thermal fluctuation. It is given as a sum of a volume and a surface term according to

[TABLE]

where the indices $h$ and $p$ indicate the corresponding phases. The critical radius $R_{*}$ is given by

[TABLE]

where $p_{h}$ and $p_{p}$ denote the pressure in the hadron and the plasma phase, respectively. (Bubbles with $R<R_{*}$ collapse, those with $R>R_{*}$ expand.)

The statistical prefactor $\Omega_{0}$ can be expressed in terms of the surface tension $\sigma$, the bubble radius $R$, the correlation length $\xi_{p}$ in the plasma phase and the volume $V$ according to

[TABLE]

The dynamical prefactor $\kappa$ which determines the exponential growth of a fluctuation $\nu$ of a droplet of critical size is derived as

[TABLE]

where $\Delta w$ is the enthalpy difference between both phases. Note that $\kappa$ is determined in terms of equilibrium quantities $\sigma$, $R_{*}$ and $w$, which are in principle accessible in lattice calculations, and nonequilibrium quantities, which are the transport coefficients $\eta$ and $\zeta$. For an ultrarelativistic gas the bulk viscosity $\zeta$ is much smaller than the shear viscosity $\eta$, which has been estimated to leading order in perturbative QCD by Baym et al. (1990). For two flavors the result is

[TABLE]

where $\alpha_{s}$ is the strong coupling constant at finite temperature.

Eqs. (496), (503) and (504) with $\Delta F$ from Eq. (501) imply for the nucleation rate

[TABLE]

The input parameters are chosen as follows: $\sigma=50~{}\hbox{MeV/fm}^{2}$, $\alpha_{s}=0.23$, $\xi_{P}=0.7$ fm, $\eta_{P}=14.4T^{3}$ and $B^{1/4}=235$ MeV for the bag constant. The pressure and the enthalpy difference $\Delta w$ between the hadronic and the plasma phase are calculated from the bag model equation of state. We would like to add a comment on the input. More recent results on the surface tension in pure $SU(3)$ gauge theory show that the estimate of $50~{}\hbox{MeV/fm}^{2}$ which has been used by Csernai and Kapusta is too large by an order of magnitude, see e.g. section III B 3 or (Kajantie, 1992). It would be also quite interesting to update the equation of state, replacing the bag model values by more recent lattice results, since the bag model equation of state fails in particular in the transition region.

The nucleation rate $I$ naturally defines a nucleation time if $I$ is multiplied by the volume of a bubble of critical size

[TABLE]

This time scale is a measure for the delay due to supercooling. It should be distinguished from the total time duration of the phase transition. It just refers to bubbles of critical size, but neglects bubble growth. When $\tau$ is plotted as a function of ($T/T_{c}$), one sees that 5% of supercooling are necessary for the nucleation to set in, when the above input parameters are used. A use of updated input parameters may considerably reduce the degree of supercooling.

A hint on the self-consistency of the approximation scheme is provided by the temperature dependence of the critical bubble radius $R_{*}$, which is determined by Eq. (502). The radius diverges at $T_{c}$, but rapidly reduces to typical hadronic scales and even below. At $T/T_{c}=90\%$, $R_{*}\sim 0.73$ fm. This is comparable to the correlation length in the plasma phase $\xi_{P}$ and of the order of the surface thickness. For such values of $R_{*}$ the approximation scheme breaks down. Therefore the question arises as to whether the phase transition has been completed before or after these low temperatures are reached. The quantity of interest is the time dependence of temperature. In the early universe $T$ drops so slowly with time that the phase transition is most likely over before the approximations loose their validity.

For heavy-ion collisions the time dependence of $T$ has been estimated by Csernai and Kapusta (1992b). The $T(t)$-dependence in the bulk phase is known, once an equation of state is combined with an ansatz for the expansion scenario, e.g. Bjorken’s scaling ansatz. The time dependence of $T$ during the phase conversions is a more subtle problem. In the Maxwell construction $T$ is independent of $t$ right from the beginning. To find $T(t)$ for a given nucleation rate $I$, one starts from a dynamical equation which couples $T(t)$ to the fraction of space which has been converted to the hadronic phase.

The result for $T(t)$ clearly shows supercooling, superheating effects and a time delay in the temperature decrease as compared to a Maxwell scenario. When the temperature has decreased to $\sim 0.8T_{c}$, the nucleation is sufficient for reheating the system up to $\sim 0.95T_{c}$, where further nucleation is stopped. From that temperature on the previously created bubbles grow further until the conversion to the hadronic phase is complete. Compared to the Maxwell scenario the time of completion is delayed by 11 fm/c leading to an increase in volume at that completion time. The increase is linear in $t$ in Bjorken’s ansatz. If the entropy density at completion is only determined by the equation of state in the pure hadronic phase, but independent of the conversion procedure, a delay of 11 fm/c implies 30% of extra entropy generation to keep the entropy density fixed. This is non-negligible in an extrapolation from the final pion multiplicity to the initial temperature. It has to be checked, whether the amount of extra entropy generation is an artifact of the bag model equation of state.

Finally we note that the critical bubble radius in the late period of the phase conversion is about 1 fm (not too large in view of the finite size of the system and not too small for the approximation to hold).

Coming back to the question posed in the beginning of this section: Is a Maxwell construction a realistic scenario ? The time dependence of temperature during the transition based on the derived nucleation rate $I$ (Eq. (506)) comes out not so different from a Maxwell construction, which is usually taken as an ad hoc assumption. It remains to be checked whether the similarity can be maintained, if one improves on various approximations, and most recent input data are used for the surface tension, the surface thickness, and the enthalpy difference.

Time dependent nucleation rates

In Langer’s formalism for deriving the nucleation rate an embedding heat bath has been assumed to guarantee a stationary flow in probability space or a time independent nucleation rate. Langer’s description does not apply to a situation far out of equilibrium. The remnants of off-equilibrium are contained in the viscosity coefficients of the nucleation rate. It may well be that such a heat bath is not realized during all stages of phase conversion in heavy-ion collisions. Some time dependence of the nucleation rate is natural, at least at the onset and the completion of nucleation. A calculation of time dependent rates requires a different formalism.

Boyanovsky and Aragao de Carvalho (1993 and 1994) have analyzed the thermal activation via sphaleron transitions in a (1+1)-dimensional field theory, where they use a real time formalism. They obtain a time-dependent nucleation rate from the real time evolution of the initial density matrix along the unstable direction in configuration space. The rate is rather sensitive to the initial state, which is typical for an off-equilibrium process. The initial state is a metastable state that has been formed after a period of rapid supercooling.

One may think of the phase transition in heavy-ion collisions in a similar way. The initial state for the conversion process is a metastable state arising from a rapid cooling rather than being in local equilibrium with a heat reservoir. The time evolution of the conversion may come out quite differently, if the heat bath is not replenished during the conversion process. A comparison of nucleation rates calculated in one or the other way would be quite useful for assessing the estimates of time delay and extra entropy production, which should be visible in heavy-ion collisions.

Recent results from the lattice and studies in effective models suggest that the transition is weakly first order – if it is of first order at all. In this case nucleation may completely fail as conversion mechanism from the plasma to the hadron phase. Large domain coarsening might replace nucleations as $T$ approaches $T_{c}$ from above. Nonequilibrium aspects in weak first order transitions are discussed in the next section.

2 Large domain coarsening

Different nonequilibrium aspects may be relevant for the transition dynamics. Here we distinguish only between nucleation and large domain coarsening or spinodal decomposition. Nucleation of hadronic bubbles in a first order QCD transition has been the topic of the preceding section. The physical assumption there was that the system is still in a homogeneous plasma phase, as $T_{c}$ is reached from above. Some supercooling below $T_{c}$ is necessary, until bubbles of critical size can be formed and the phase conversion proceeds via nucleation. This scenario is likely a realistic description for a strong first order transition. The formal manifestation for the underlying physical assumption is the saddle point evaluation of the path integral in Langer’s formulation, as we have seen above. The saddle point approximation is justified as long as the fluctuations about the homogeneous metastable ground state are small. The system should be well localized in the metastable minimum as the temperature drops below $T_{c}$.

The nonequilibrium dynamics of the phase conversion may drastically change if the transition is weakly first order (Borrill and Gleiser, 1995). In spite of a potential barrier the system may no longer be well localized in the metastable minimum, before the temperature drops below $T_{c}$. Large amplitude fluctuations are allowed, subcritical bubbles can form above $T_{c}$. At $T_{c}$ the system is found in a completely mixed phase of high and low temperature “components”. The phase mixing is assumed not to be an artifact of the finite volume. (Recall that spontaneous symmetry breaking strictly occurs only in the infinite volume limit. Here the volume is chosen large enough that the tunneling rate between the different phases is negligible.) If the system starts at $T_{c}$ from a mixed phase and then cools below $T_{c}$, the phase separation may evolve by domain coarsening in a way that resembles spinodal decomposition rather than nucleation.

Borrill and Gleiser (1995) have studied the phase mixing at $T_{c}$ as a function of the strength of the first order transition. The framework are discretized scalar field theories in $3+1$ dimensions. The homogeneous part of the free energy density contains a quadratic, cubic, and quartic term in a real scalar field. The couplings in the quadratic and cubic part are assumed as temperature dependent. The precise form is motivated by an effective potential for the electroweak phase transition, but the results are applicable to QCD as well. The strength of the first order transition has been tuned by the quartic coupling $\lambda$, keeping the cubic coupling fixed. Small values of $\lambda$ correspond to a high barrier in the effective potential and a strong first order transition, large values of $\lambda$ to a weak first order transition. The system is always considered at fixed temperature $T_{c}$. It is prepared to be initially well localized in the high temperature phase. It is then evolved by a discretized Markovian Langevin equation with white stochastic noise to mimic the coupling with a thermal bath.

Let us denote the fractions of the total volume in the high (low) temperature phase divided by the total volume $f_{0}$ ($f_{+}$), respectively. The fraction of the total volume in each phase, $f_{0,+}(t)$, is then measured as a function of time, until the final ensemble-averaged equilibrium values $f_{0,+}^{\rm{eq}}$ are reached. The qualitative results of Borrill and Gleiser are the following. For small values of $\lambda$ (strong first order transitions) the system remains localized in the initial metastable minimum, $f_{0}^{\rm{eq}}\sim 1$. For large enough $\lambda$ (sufficiently weak first order transitions), $f_{0}^{\rm{eq}}$ approaches 0.5, corresponding to a complete phase mixing at $T_{c}$.

A further interesting observation is made, if the equilibrium fractional population difference at $T_{c}$

[TABLE]

is plotted as a function of $\lambda$. The curve suggests a second order phase transition as function of $\lambda$ (the control parameter of the strength of the first order finite-$T$ transition). There seems to exist a critical strength of the first order transition (obtained for $\lambda=\lambda_{c}$), so that for larger values of $\lambda$ (weaker transitions) $\Delta F$ soon approaches zero, corresponding to a perfect thermal mixing $f_{0}^{\rm{eq}}=0.5=f_{+}^{\rm{eq}}$. For smaller values of $\lambda$ (stronger transitions), $\Delta F_{\rm{eq}}$ remains close to 1, the system is trapped in its homogeneous initial state. Therefore Borrill and Gleiser proposed, $\Delta F_{\rm{eq}}$ to be used as order parameter for the ‘transition’ from strong to weak first order transitions, $\Delta F_{\rm{eq}}$ is a quantitative measure for how strong the finite-$T$ transition is.

The prediction of complete thermal mixing at $T_{c}$ relies on a cooling process, which is slow compared to other equilibration scales in the system. Keeping $T$ fixed at $T_{c}$, this assumption was implicitly built in the derivation. The narrow transition region between the two areas of strong and weak first order transitions may be related to the fact that the order parameter for the finite $T$ transition $\left<\phi(t)\right>$ is not conserved. For conserved order parameters in binary mixtures and long internal equilibration times the transition between the two regimes is smooth (Borrill and Gleiser, 1995, and references therein).

To conclude we summarize some possible implications for QCD. The results of Borrill and Gleiser have shown that the nonequilibrium transition dynamics may change from nucleation to large domain coarsening in passing from strong to weak first order transitions. The interpolation between both schemes need not be smooth, but may be a phase transition itself. In QCD it is likely that the finite temperature transitions are rather weakly first order (if they are of first order at all), or rapid crossover phenomena, which may be indistinguishable from a practical point of view. The role of the control parameter of the strength of the transition will be played by the current quark masses. If the cooling of the plasma is slow compared to equilibration time scales of amplitude fluctuations about the metastable ground state, the QCD transition may proceed via large domain coarsening rather than supercooling with subsequent nucleation.

Observable signatures in heavy-ion collisions will in general depend on the dominant process during the phase conversion. Large domain coarsening may result in similar effects as they are expected for second order transitions, while strong supercooling would lead to explosive effects in the other extreme case.

In view of the results of Borrill and Gleiser, a good quantitative estimate of the strength of the first order/crossover phenomenon does no longer appear as a minor detail, once the order of the transition is determined. It is particularly important, if the regimes of nucleation and spinodal decomposition are separated by a narrow range in the control parameter of the potential barrier. Thorough estimates of time scales for the cooling process and relaxation phenomena during the phase conversion are needed to turn the speculations of today into predictions for observable signatures.

3 Transport coefficients

As we have seen in section V D 1, dissipation effects like shear and bulk viscosities of the plasma will influence the nucleation rate of hadronic bubbles, if the transition proceeds via nucleation. Viscosity and thermal conductivity are further sources of entropy production during the evolution from the initial to the final state. Although it is sometimes claimed that the amount of extra entropy is negligible compared to the entropy generated during a hypothetical supercooling scenario, it is worthwhile being calculated; it may well be that there is no supercooling at all.

Corresponding to this extra entropy, dissipation leads to an increase in the final rapidity density of observed particles. The entropy as a function of time enters various production rates, thus an extra entropy will influence predictions of final rates as well.

Transport processes result from gradients in thermodynamical parameters. Gradients in the temperature lead to a thermal current, in the velocity field to a friction force, in color distributions to a color current. They are irreversible and typical nonequilibrium phenomena which have to be treated with nonequilibrium methods. At length we have discussed results from equilibrium thermodynamics. Typically they refer to the existence of phase transitions, critical parameters and ingredients for the equation of state. In order to incorporate all this knowledge in nonequilibrium effects, the equilibrium has to play the role of an expansion point, at which small perturbations are allowed. Thus it is not surprising that transport coefficients are calculated within some linear response approach. The linear response refers to the response of the energy momentum tensor to the thermodynamical forces (friction forces, temperature gradients, etc.).

Two approaches are known which we call the kinetic and the phase space approach. The kinetic approach emphasizes particle aspects. Transport processes arise as the result of collisions between particles (partons or hadrons, depending on the phase). In the vicinity of $T_{c}$, kinetic theory is expected to fail. A hydrodynamic treatment seems to be more adequate, which is incorporated in the phase-space approach. In principle, the phase-space approach applies to all temperatures. The transport coefficients are expressed in terms of expectation values of retarded Green’s functions. Following Zubarev (1974), Hosoya et al. (1984) elaborated the phase-space approach for the hydrodynamic regime (although it is not necessarily restricted to that). The hydrodynamic description is apparent in the choice of variables, locally defined temperature and velocity fields. As usual this assumes that the mean free time of constituent particles is much shorter than the typical relaxation time of the system.

Only linear terms in gradients of inhomogeneous distributions are kept. The system has to be in a late stage of preequilibrium expansion rather close to equilibrium to satisfy this condition. Under these restrictions one obtains as an expression for the shear viscosity

[TABLE]

in the limit of $\varepsilon\to 0$, $T_{\mu\nu}$ denote the components of the energy momentum tensor. The formulas for the heat conductivity $\chi$ and the bulk viscosity $\zeta$ are very similar, they mainly differ in the components of $T_{\mu\nu}$. The expectation value $\left<\ldots\right>_{ret}$ stands for a retarded Green’s function in Minkowski time. It is related to the thermal expectation value $\left<\ldots\right>_{0}$ with respect to the thermal equilibrium distribution according to

[TABLE]

Eq. (512) is the expression we are finally interested in. On the other hand, expectation values which are easily accessible on the lattice are thermal Green’s functions in Euclidean time $\tau$. They are expressed via thermal expectation values

[TABLE]

The procedure of analytic continuation from Euclidean to Minkowski time is standard and can be found in textbooks, see e.g. (Kadanoff and Baym, 1962). Here we only sketch the logics.

In order to relate the retarded Green’s function $G_{R}(\vec{x},t)$ to the thermal function $G_{\beta}(\vec{x},\tau)$, one proceeds via their Fourier transforms $\tilde{G}_{R}(\vec{p},p_{0})$ and $\tilde{G}_{\beta}(\vec{p},\omega_{n})$, respectively. Both have almost identical representations in terms of the spectral functions $\rho(\vec{p},\omega)$. We have

[TABLE]

Here $\tilde{G}_{\beta}(\vec{p},\omega_{n})$ is only defined on a discrete (but infinite) set of Matsubara frequencies $\omega^{\prime}=\omega_{n}$ (due to the periodicity in Euclidean time). Next one performs an analytic continuation of the Fourier coefficients $\tilde{G}_{\beta}(\vec{p},\omega_{n})$ for all (non real) $\omega^{\prime}$. If there is no essential singularity at $\omega^{\prime}=\infty$, the unique continuation is

[TABLE]

The spectral function is then given by the discontinuity of $\tilde{G}_{\beta}(\vec{p},\omega^{\prime})$ across the real axis. Once we know $\rho$, the Fourier transform of the retarded Green’s function can be calculated according to

[TABLE]

with the same $\rho$ as in Eq. (514). The retarded Green’s function $G_{R}(\vec{x},t)$ then follows as a Fourier transform of Eq. (516). Thus in the ideal case we are done if we are able to extract the spectral function from certain correlators in Euclidean space. The spectral function is not only essential for analytic continuation. It also allows an identification of the physical degrees of freedom. In a given channel $C$, $\rho_{c}$ has a pole for each physical degree of freedom with quantum numbers $C$, and a brunch cut for every allowed multiparticle state with the same $C$.

Practically, it is very difficult to determine $\rho$ in lattice calculations. $G_{\beta}(\vec{x},\tau)$ is only known for a finite (and even small) number of points $\tau_{i}$. Therefore $\tilde{G}_{\beta}(\vec{p},\omega_{n})$ cannot be calculated as integral $\int^{\beta}d\tau e^{i\omega_{n}\tau}$ (…), but the spectral representation of $\tilde{G}_{\beta}(\vec{p},\omega_{n})$ is just the quantity of interest.

What can be done in spite of these difficulties has been shown by Karsch and Wyld (1987). Karsch and Wyld have pointed out the possibilities and limitations of calculating transport coefficients in lattice gauge theory. The approximation for extracting the spectral function is to make an ansatz for $\rho$ involving certain unknown parameters, to calculate $\tilde{G}_{\beta}(\vec{p},\omega_{n})$ and $G_{\beta}(\vec{p},\tau)$, and to fix the unknown parameters by fitting $G_{\beta}(\vec{p},\tau)$ with Monte Carlo data.

Kubo-type formulas (cf. Eq. (510)) have been used with an energy momentum tensor of pure $SU(3)$ lattice gauge theory. The ansatz for the zero-momentum spectral function is given as

[TABLE]

It involves three parameters $A$, $\gamma$ and $m$. The spectral function of a free field theory contains just two, $A$ and $m$, thus $\gamma$ represents the interactions. Let us label the different transport coefficients $\alpha_{i}$ with index $i$. Correspondingly the spectral functions $\rho_{i}(\omega)$ needed for the retarded Green’s functions $G_{R_{i}}(t)$ in the associated Kubo formulas depend on parameters $A_{i}$, $\gamma_{i}$ and $m_{i}$. The transport coefficients read in terms of $(A_{i},\gamma_{i},m_{i})$ (Karsch and Wyld (1987))

[TABLE]

This expression can be easily reformulated in dimensionless quantities which are directly measurable on the lattice: $m_{\ell}=m\cdot a,\gamma/m,1/T=N_{\tau}a$ and $A_{\ell}=Aa^{5}$, $a$ being the lattice constant. The goal is to determine $A_{\ell},m_{\ell}$ and $\gamma/m$ from fits of $G_{\beta}^{(i)}(\tau)$. For example $G_{\beta}^{(i)}(\tau)$ for the shear viscosity is given in terms of space-space-off-diagonal correlations according to

[TABLE]

This quantity has been evaluated on an $8^{3}\times 4$ lattice. Thus the time distance $\tau$ in lattice units is restricted to three values 0, 1, 2, it cannot be larger than 2. In fact, only $G^{(\eta)}(\tau)$ could be measured up to distance 2. For the other correlators $G^{\zeta}$ and $G^{\chi}$, the signal is already lost in the noise at $\tau=2$.

The measurements have been performed above and below the critical coupling of the deconfinement transition, which is at $6/g^{2}=5.68$ for the given lattice size. The mass parameters $m_{i}$ are easier accessible than $A_{i}$ and $\gamma_{i}$. They follow from the exponential decay of the correlations. The interesting result is that the correlations drop rapidly above and below $T_{c}$. Especially above $T_{c}$, there still seem to be massive modes in the system with a large effective mass. Thus there is no indication for an abrupt change from a free glueball gas below $T_{c}$ to a free gluon gas above $T_{c}$. Only at low temperatures, the effective masses approach the lowest glueball masses obtained from independent calculations (Berg and Billoire, 1983), in the vicinity of $T_{c}$ they differ. Yet one should keep in mind the typical caveats for any lattice calculation. In particular, the extraction of masses from short distance properties could lead to artificially large values. For the interaction parameter $\gamma$, Karsch and Wyld obtain an upper bound, resulting in an upper bound for the shear viscosity $\eta$, $\eta/T^{3}<9.5$. The bound is in agreement with analytic estimates in the collision time approximation of the kinetic approach, cf. Hosoya and Kajantie (1985).

We summarize. The original objection to derive nonequilibrium properties in Minkowski space from Euclidean field theory in thermal equilibrium is divided into two steps. Based on a linear response approach, first the nonequilibrium properties are expressed in terms of equilibrium quantities in Minkowski space. The second step is the analytic continuation between Minkowski and Euclidean spacetime. The essential ingredient is the spectral function $\rho$. A basic obstacle to derive it from Euclidean time correlators measured on the lattice is the rather limited set of time points. At best, one can only guess an ansatz for the spectral function that is compatible with the time correlator. In addition, the correlators $G_{\beta}$ of the matrix elements of $T_{\mu\nu}$ turned out to drop so rapidly that the small values for $G_{\beta}$ are lost in the noise for large distances. Large distances are needed to project on lowest excitation states, to justify the very ansatz for $G_{\beta}$. Thus the values for $\tau$, the time distance in lattice units, are too small both in number and size. From a realistic point of view, it seems to be impossible to unfold the rich analytic structure of the spectral function from a lattice approach. The main part of it will be hidden in the Monte Carlo noise. Further discussions of transport coefficients in the lattice approximation and comparisons to the results of kinetic theory can be found in (Horsley and Schoenmaker, 1986) and (Schoenmaker and Horsley, 1988).

Heat conductivity or viscosities are relaxation phenomena of the energy-momentum with respect to temperature gradients, friction forces etc. Their impact on the phase transition concerns the phase conversion (cf. the preceding section). A direct manifestation of the phase transition should be visible in a different relaxation phenomenon associated with color degrees of freedom. It is noteworthy that one of the most challenging properties of a quark gluon plasma - its color conductivity - poses itself as a transport phenomenon. In analogy to the Mott transition from an electric insulator to a conductor, the transition from the hadronic color insulator to the plasma color conductor should be manifest in the transport coefficient of color conductivity, which drops to zero either smoothly or suddenly, when $T_{c}$ is approached from above.

Selikhov and Gyulassy (1993) have found a surprisingly small value for the color conductivity, which is determined by the ratio of the static color electric and magnetic screening masses, $m_{e}$ and $m_{m}$, respectively. (Their assumption is $m_{m}\ll m_{e}$. This assumption is not borne out by lattice simulations in an $SU(2)$ and $SU(3)$ gauge theory, where $m_{m}\sim m_{e}$ is found from $T_{c}$ up to 2-3 $T_{c}$ (Boyd, 1995).) The color conductivity coefficient is calculated from the linear response of the system to a weak external field. The reason for the rapid damping of collective color modes is rapid color diffusion. Color diffusion is characterized by a specific relaxation time scale $t_{c}$, which measures the precession frequency of the color of a parton in a fluctuating background field. This time scale turns out to be much smaller than a typical momentum relaxation scale $t_{p}$, which has been used in (classical) non-abelian transport theory (see e.g. Heinz, 1986; Baym et al., 1990; Eskola and Gyulassy, 1993). Its small value explains the poor conductivity.

Color transport phenomena are still in their infancy and deserve further attention in future work. One may think about constructing a device for a ‘color ampèremeter’.

We conclude with some remarks on a promising new direction in transport theory, which is suited for applications in heavy-ion collisions. It has been studied by Zhang and Wilets (1992) and is based on a closed time-path Green’s function technique (Schwinger, 1961; Keldysh, 1964 and 1965) combined with the loop expansion scheme of Cornwall, Jackiw and Tomboulis (1974). The framework leads to transport equations which include the generalized Boltzmann equations in the special case of the quasiclassical limit. Its virtue is a unified description of equilibrium and off-equilibrium systems. The deviations from equilibrium in the considered off-equilibrium situations are not restricted to a small size, whereas small deviations are necessary for justifying a linear response approach.

In particular the framework is well suited for incorporating the phase transition. The order parameter (here chosen as the chiral condensate of the Nambu-Jona-Lasinio model) is a dynamical variable that enters the generalized Boltzmann equations. The Boltzmann equations reduce to the Vlasov equation in the Hartree approximation. The Vlasov dynamics controls the collision dynamics, in particular medium effects of chiral symmetry breaking in transverse flow. The transverse flow is directly sensitive to the chiral condensate.

A further numerical elaboration of these microscopic transport equations could lead to predictions of observable signatures for chiral symmetry restoration. Such predictions should be more reliable (in the sense of more realistic) than any predictions based on equilibrium thermodynamics. Ultimately full QCD should be substituted for the NJL model used in the analysis of Zhang and Wilets. A less ambitious extension would incorporate three rather than two flavors, or vary the effective microscopic model for QCD to include gluonic degrees of freedom as well.

4 Dynamical universality and disoriented chiral condensates

Most of the proposed experimental consequences of a QCD transition at finite temperature are based on the assumption of a first order transition scenario. As manifestation of a second order transition we have discussed intermittent behavior in section V C 6. In this section we describe further consequences of a second order transition under an additional assumption of rapid cooling from the high to the low temperature phase. The underlying theory is the 3-dimensional $O(4)$-model of section V A, which is supposed to describe 2-flavor QCD in the vicinity of $T_{c}$ and predicts a second order transition in the chiral limit. At criticality long range order in an $O(4)$-model leads to large clusters of aligned spins, or translated to the particle contents of the sigma model, large clusters in three-dimensional space, where the ratio of charged to neutral pions is fixed. The question arises as to whether the cluster size is sufficiently large to have observable effects. Due to the finite quark masses the second order of the transition is an idealization anyway, (unless the masses take tricritical values). Although the up- and down masses are only a few MeV, the induced zero temperature pion mass is $\sim 135$ MeV and further increases with temperature (Rajagopal and Wilczek, 1993). As lightest particles in the spectrum the pions lead to the largest correlation length. Thus the largest correlation length is smaller than $(135~{}\hbox{MeV})^{-1}$ which is not large as compared to $T_{c}^{-1}$ (if we use for $T_{c}\sim 140$ MeV for two flavors (Bernard et al., 1992)). The number of pions arising from a typical cluster of “aligned $O(4)$-spins” in isospin space can be estimated from the energy which is stored in a typical correlation volume in the vicinity of $T_{c}$ (Rajagopal and Wilczek, 1992). The estimate is based on lattice results and says that not more than one or two pions belong to the same correlation volume. Thus the hope of seeing clusters of hundreds of pions with a fixed ratio of neutral to charged pions seems to be gone. The upper bound on the maximal size of a correlation volume and the energy stored in this volume are based on equilibrium thermodynamics. They apply to a phase transition in heavy-ion collisions, if the system remains always close to local equilibrium, in particular during the phase transition.

The situation may drastically change, if the transition proceeds far out of equilibrium via a quench. Let us recall what a quench is in spin systems. The system starts at high temperature in a completely disordered state. Now the temperature is instantaneously turned to zero. An example is a bar of iron which is plunged into ice water rather than slowly cooled through the transition point. The high temperature spin configuration is far out of equilibrium, when the thermal fluctuations are suddenly switched off. The equilibrium configuration at zero temperature would be a well ordered ground state. In condensed matter physics it is well known what happens to high temperature spin configuration after a quench (Hohenberg and Halperin, 1977; Halperin et al., 1974). The system evolves according to the zero temperature equations of motion, which determine the universal dynamics of the long wavelength modes near $T_{c}$. Theories which are described by the same equations of motion for the long wavelength modes define a dynamical universality class. In contrast to static universality classes dynamical universality classes are not only specified by the space dimension and the number of order parameter components, but also by those quantities which are conserved. The important result of condensed matter physics (Bray, 1990; Newman et al., 1990) is that the linear size of a correlation volume $L(t)$ grows with time according to $L(t)\sim t^{p}$, where $p$ depends on the characteristic parameters of the dynamical universality class. It is crucial in view of the application to QCD that $L(t)$ does not depend on equilibrium correlation lengths. From the digression to condensed matter physics let us return to QCD.

The interesting suggestion of Rajagopal and Wilczek (1993) is that QCD belongs to the dynamical universality class of an $O(4)$-anti-ferromagnet in the vicinity of $T_{c}$. The implications of the hypothesis are discussed in the following.

The question of interest is the specific way, how different regions of space relax to the ground state at low temperature. In case of an $O(4)$-model the ground state at low $T$ is characterized by small oscillations around the $\sigma$-direction in isospin space. The vacuum expectation value of the $O(4)$-field is $\left<\phi\right>=\left<\sigma,\vec{\pi}\right>=(f_{\pi},\vec{0})$. If the cooling is adiabatically slow, it is likely that different clusters of aligned spins relax independently to the ground state configuration. In contrast, if the cooling proceeds via a quench, the long-wavelength modes relax much more slowly to the equilibrium configuration. This allows the formation of large clusters of misaligned spins characterized by a ‘misaligned’ condensate value $\left<\vec{\pi}\right>\neq\vec{0}$. Such regions of misaligned condensates have been called Disoriented Chiral Condensates (DCC) by Bjorken et al. (1993). Bjorken et al. had speculated about events, in which the pion yield comes in clusters of pions aligned in a single direction in isospin space over a large fraction of the collision volume. When these regions relax coherently to the true ground state with $\left<\sigma\right>\neq 0$, a specific radiation of pions is emitted. This radiation is proposed as observable signature in heavy-ion collisions. The ‘misaligned’ high temperature vacuum is determined by oscillations around some fixed direction $(\sigma,\vec{\pi})$ in classical field space. The radiated pions correspond to a coherent configuration in that direction. Pions originating from one such cluster of misaligned condensate have a fixed ratio of charged to neutral pions. The probability distribution for a given charge ratio $R$ is predicted as (Rajagopal and Wilczek, 1993)

[TABLE]

with $R=\sharp\pi^{0}/(\sharp(\pi^{0}+\pi^{+}+\pi^{-}))$.

Correlations in space are not directly accessible in heavy-ion collisions. Thus a further relation is needed to translate correlations in space to measurable correlations in rapidity space. Such a relation is provided by Bjorken’s scaling ansatz. In Bjorken’s description different positions in space become different positions in rapidity space. Given $z$ and $t$ we know the rapidity $y={1\over 2}\ln((1+v)/(1-v))$, cf. section V B 1).

Thus an observable effect of a quenched QCD transition could be constant ratios of neutral to total number of pions over large volumes in rapidity space. Fluctuations in pion multiplicities of this type should be distinguished from fluctuations originating from exploding plasma blobs, where the criterion to form a blob is a genuine nucleation process rather than a fixed direction in isospin-space, cf. section V.C.5.

Generic spin systems and an $O(4)$-model as effective description for QCD share similarities in view of dynamical universality, but also differ in several aspects, which we list in the following.

• $\bullet$

At high temperature the $O(4)$-vector will not really fluctuate in arbitrary directions from cluster to cluster due to the ‘external field’. The non-vanishing pion mass singles out a preferred direction even at high temperatures assuming that the model has not lost its validity at high temperatures. The analysis of Bray(1990) and Newman et al. (1990) has to be extended to account for the effect of finite quark masses (see Rajagopal and Wilczek, 1993).

• $\bullet$

The succeeding quench as instantaneous cooling is certainly an idealization of the realistic cooling process. In Bjorken’s scaling ansatz, $T$ drops according to $T=T_{0}(\tau_{0}/\tau)^{c_{s}^{2}}$, more slowly than the corresponding entropy density if transverse expansion is neglected.

• $\bullet$

Strictly speaking it is not possible to talk about a well-ordered ground state at $T=0$. The plasma is expanding until freeze-out, where the temperature is larger than zero. After freeze-out it does no longer make sense to call some parameter ‘temperature’.

• $\bullet$

Furthermore a description in terms of field configurations is not convenient when individual particle aspects become more relevant. In the very end detectors register single particles. Even before freeze-out the language of kinetic theory is more adequate.

• $\bullet$

The equations of motion at zero temperature have to be Lorentz invariant in the QCD case. The proposal is to use the $T=0$-$O(4)$ linear sigma model to guarantee Lorentz invariant equations of motion.

• $\bullet$

The plasma is expanding, whereas the ferromagnet is static.

In spite of the complications in heavy-ion collisions there is at least a chance that a fast cooling amplifies the role of long-wavelength pionic modes during the phase conversion of the quark gluon plasma. However, for realistic coupling strengths in the sigma model, their relaxation to the low $T$-equilibrium configuration goes too fast to allow the formation of large regions of ‘misaligned’ vacuum (Gavin et al., 1994b). Gavin et al. performed a numerical simulation of the zero-temperature $O(4)$-linear sigma model in the presence of an external field $H$ with couplings in the Lagrangian that induce physical values for $f_{\pi}=92.5$ [MeV], $m_{\pi}=135$ [MeV] and $m_{\sigma}=600$ [MeV]. In particular the quartic coupling is strong ($\lambda=20$). In contrast, simulations at weak couplings corresponding to unrealistic light masses ($m_{\pi}=0.3$ [MeV] and $m_{\sigma}=1.8$ [MeV]) reproduce the formation of domains with a field slowly varying about some nonzero value $\left<\vec{\pi}\right>$. The ‘collapse’ of these domains to the true ground state goes along with a characteristic pion radiation, where the number of pions in a typical clump could be large enough for detection.

Under realistic conditions the cooling in heavy-ion collisions may not be fast enough to sufficiently amplify the long-wavelength modes, and the wavelengths of the lightest modes may not be long enough for the formation of disoriented chiral condensates. For further references to disoriented chiral condensates see (Krzywicki, 1994; Huang, 1995), (Rajagopal, 1995, and references therein).

A useful tool which helps in assessing the chance for observing the predicted pion radiation, is an energy budget for the chiral and the deconfinement transition (Wilczek, 1994). The energy density which is stored in the chiral vacuum (associated with chiral condensation) should be compared with the energy density of the quark gluon plasma for temperatures $T\geq T_{c}$. The decimation of QCD’s degrees of freedom to pions and the description of the cooling process as a quench are certainly idealizations. Within the idealized limits we have seen the predictive power of the renormalization group approach. As we have argued in sections IV A.1 and 2, the basic open question is, how important the heavier modes are in the relative to the light modes associated with spontaneous chiral symmetry breaking. One measure for their importance is their contribution to the total energy density. The accumulating effects of heavier modes can bury the interesting (predictable) structures arising from the singularities in the chiral 2-flavor transition. A quench does not guarantee, but improves the chance for seeing relics of the chiral transition.

Support from ‘experiment’ is taken from a particular class of cosmic ray events called Centauros (Rajagopal and Wilczek, 1993). In Centauros isospin invariance seems indeed to be violated. The number of charged pions strongly exceeds the number of neutral ones. For Centauros see e.g. Lattes et al., 1980.

VI SUMMARY AND CONCLUSIONS

The ultimate question of the order of the finite temperature QCD transition for three colors, two light and one heavier flavor with physical values for the quark masses is open. Closest to this physical case come lattice simulations with two light and one heavier flavor in the fermion schemes with staggered and Wilson fermions. At present their results are in disagreement, predicting a crossover phenomenon in the staggered formulation (section III C 5) and a first order transition in the Wilson formulation (section III C 7). Both results may be still artifacts of the strong couplings inherent in the simulations.

Partial answers about the order of the QCD transitions are known in limiting cases. The first (second) order of the deconfinement transition in the $SU(3)$ ($SU(2)$) Yang Mills theory seems to be well established (section III B 1 and II B 2). General agreement is also found about the second order of the chiral transition in the limit of two massless flavors when dynamical fermions are included. An open question is the universality class of two-flavor QCD (whether it is $O(2)$, $O(4)$ or mean-field). Consistent results in different fermion formulations are found for the first order of the chiral transition in case of $N_{f}\geq 3$ flavors with sufficiently light masses (section III C 5 and III C 7).

An interesting topic for future studies is a determination of the critical strange quark mass $m_{s}^{*}$ at which the first order of the chiral transition changes into second order. For larger values of $m_{s}$ it stays of second order for $m_{u,d}=0$ and disappears otherwise. The location of the ‘critical’ phase boundary between mass regions of first order transitions and crossover phenomena could tell us, how far the physical quark masses are from ‘critical’ quark masses. It would be appealing both from a theoretical and an experimental point of view, if they are close by. The powerful renormalization group approach becomes then applicable for finite mass scaling predictions. Remnants of a nearby second order transition should lead to large correlation lengths and pronounced effects in RHIC-experiments (section III C 5, III C 7 and IV A 4).

Lattice calculations start from first principles as they simulate QCD in its discretized version. They are the nonperturbative tool for studying the phase transition region, where many expansion schemes break down. But they are plagued with artifacts from the finite volume, the finite lattice constant and unphysical mass parameters. IR-artifacts may amount to a harmless rounding of an infinite volume singularity or a misleading double-peak structure, suggesting a first order transition though it is truly of second order (section II A 2). UV-artifacts may result in a ‘harmless’ change of critical exponents ($O(2)$ rather than $O(4)$) (section III C 3), but also in bulk transitions being a pure lattice artifact without any impact on continuum physics (section III C 4).

Progress on the lattice is not only a question of powerful machines (like special purpose computers). Progress in the past came also from a better control of UV- and IR-artifacts with finite size and finite mass scaling analysis, using refined criteria which have been developed in statistical physics. Extrapolations to the infinite volume and zero mass limits are unavoidable in numerical simulations. Therefore a precise measurement of critical exponents is worth the effort assuring that the right extrapolation formulas are used (section III C 5).

The earlier discrepancies between predictions from staggered and Wilson fermions shrink. Wilson fermions seem to have a chiral limit in the confinement phase with broken chiral symmetry at low temperatures, if the lattice is large enough. The lattice size must be even larger than for staggered fermions to reach the continuum limit, too large for practicable simulations, unless improved actions accelerate the approach to the continuum limit. Work in this direction is in progress (section III C 7).

The main contributions from the Euclidean lattice formulation of QCD to physical applications in RHIC-experiments should be reliable predictions of in-equilibrium properties. To these belong the equation of state, the tension of interfaces between coexisting phases, and the velocity of sound. A derivation of the equation of state under the inclusion of dynamical fermions is still in an exploratory stage. The dependence on the lattice size has not yet been analyzed, and the simulations are performed for 2 rather the 2+1 flavors, but the ingredients are fully nonperturbative (section III C 6). This is an important progress compared to earlier calculations using perturbative and nonperturbative input data in an inconsistent way (cf. also section III B 2).

From results in the pure gauge sector it has become clear that strong deviations from the phenomenological bag model equation of state must be expected in the transition region. Deviations from the ideal gas behavior are pronounced up to temperatures of the order of 2-3 $T_{c}$ (section III B 2). So far many phenomenological applications are based on the bag model equation of state. An update is called for, once the lattice results are well established.

With a reliable calculation of interface tensions one could exclude certain scenarios, which are under discussion for RHIC-experiments, right from the beginning. Thus a small value for the interface tension is incompatible with a strong supercooling scenario (section III B 3).

Studies of the phase structure of QCD in effective models in the spacetime continuum may give useful hints about the underlying physical mechanisms. Results in dual Ginzburg-Landau models suggest that abelian $U(1)\times U(1)$ monopoles are the most important field configurations in the confinement mechanism (section IV B 2) in the maximal abelian gauge. Confinement would then have an explanation as a dual Meissner effect. The type of most relevant topological field configurations does depend on the gauge.

Nonequilibrium aspects enter the evolution of the initially created hot and dense ‘fireballs’ in RHIC-experiments and the transition dynamics during the phase conversion. Most of the nonequilibrium approaches in the past have studied small deviations from in-equilibrium properties. Field theoretical descriptions of nonequilibrium phenomena are still in its infancy. Recently they became increasingly important for the electroweak transition in the early universe. These tools may have fruitful applications to QCD as well. For example it is yet unclear, whether the conversion from the plasma to the hadron phase proceeds via nucleation or large domain coarsening (section V D 1 and V D 2).

For phenomenological discussions of QCD transitions in lab-experiments theoretical concepts are sometimes missing. Thus it seems to be rather difficult to derive the equilibration time of the hot initial state from first principles. Formulas are missing for particle production rates in media with long-range correlations. Such correlations exist at $T_{c}$, if the transition is almost of second order. An interpolation between nuclear structure functions and thermal Bose-Einstein or Fermi-Dirac distributions would be desirable to close the gap between the initial off-equilibrium and the final in-equilibrium medium.

Altogether the judgement of the experimental situation is the following. Even from a very conservative point of view, extended and dense systems have been observed which display some collective features. Some of them can be explained with a transient plasma, but alternative mechanisms explain the observed features as well. To date the question of the order of the transition has been of secondary importance in heavy-ion experiments. Once the creation of a plasma phase may be taken for granted, one can focus on substructures exhibiting details of the transition dynamics (section V).

Our aim was to list the most important signatures which are at least in principle sensitive to the transition, and in case it is hopeless to see an effect to argue why it is hopeless (see e.g. section V C 3).

In Table IX we summarize signatures in various experimental tools which may be indicative for a first or second order transition and display alternative explanations in a separated column. Crosses (x) stand for an affirmative answer which generally should be taken with care, “/” stands for “not known”. Explanations of further notations can be found in the corresponding sections.

We conclude with the following conjecture. For physical quark masses the deconfinement and the chiral transitions are replaced by rapid crossover phenomena. The rapid change is seen in the energy and entropy density over a temperature interval of the order of 10 MeV, and in variations of the order parameter expectations values like the Polyakov loop and the light quark condensate. It would be interesting to investigate on the phenomenological side when a change in one of these quantities is considered as rapid enough for producing multiplicity fluctuations in rapidity space beyond the statistical noise or clear deviations in dilepton production rates from the standard rates without a crossover. Is a temperature interval of 10 MeV resolvable from an experimental point of view ? Finding a quantitative answer to these questions is a challenge for further studies.

Acknowledgements.

The author wishes to thank J. Kapusta and E. Shuryak for an invitation to the workshop on Strong Interactions at Finite Temperature at the ITP in Santa Barbara in August 1993 and for discussions with the participants of this workshop. I would also like to thank J. Cleymans, A. Patkós, H.-J. Pirner, K. Redlich and H. Satz for discussions. In particular I am indebted to G. Boyd, F. Karsch, and J. Stachel for critical comments on parts of the manuscript.