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PUPT-1347, IASSNS-HEP-92/60

arXiv:hep-ph/9210253v1 20 Oct 1992PUPT-1347, IASSNS-HEP-92/60October 1992Static and Dynamic Critical Phenomenaat a Second Order QCD Phase TransitionKrishna Rajagopal⋆Department of PhysicsJoseph Henry LaboratoriesPrinceton UniversityPrinceton, N.J. 08544Frank Wilczek†School of Natural SciencesInstitute for Advanced StudyOlden LanePrinceton, N.J. 08540⋆Research supported in part by a Natural Sciences and Engineering Research Council ofCanada 1967 Fellowship.RAJAGOPAL@PUPGG.PRINCETON.EDU† Research supported in part by DOE grant DE-FG02-90ER40542. WILCZEK@IASSNS.BITNET

ABSTRACTIn QCD with two flavors of massless quarks, the chiral phase transition isplausibly in the same universality class as the classical four component Heisen-berg antiferromagnet. Therefore, renormalization group techniques developed inthe study of phase transitions can be applied to calculate the critical exponentswhich characterize the scaling behaviour of universal quantities near the criticalpoint.

This approach to the QCD phase transition has implications both for latticegauge theory and for heavy ion collisions. Future lattice simulations with longercorrelation lengths will be able to measure the various exponents and the equa-tion of state for the order parameter as a function of temperature and quark masswhich we describe.

In a heavy ion collision, the consequence of a long correlationlength would be large fluctuations in the number ratio of neutral to charged pions.Unfortunately, we show that this phenomenon will not occur if the plasma staysclose to equilibrium as it cools. If the transition is far out of equilibrium and canbe modelled as a quench, it is possible that large volumes of the plasma with thepion field correlated will develop, with dramatic phenomenological consequences.2

1. IntroductionThe QCD phase transition is of interest from several different points of view.First, there can be no doubt that it occurred in the early universe.

Second, itis reasonable to hope that in a heavy ion collision of sufficiently high energy, asmall region of the high temperature phase is created which then cools through thephase transition. Third, lattice gauge theory is well suited to calculating the equi-librium properties of QCD at high temperatures.

From all these perspectives, it isimportant to learn as much as can be learned analytically about the phase transi-tion, relying as much as possible only on fundamental symmetries and universalityarguments and as little as possible on specific assumptions and models.In a previous paper [1], one of us (F. W.) emphasized that in the chiral limitwhere there are two species of quarks with zero current algebra mass, the order pa-rameter for the chiral phase transition has the same symmetry as the magnetizationof a four component Heisenberg magnet, which has a second order phase transi-tion. In this case, and indeed for any number of quark species except zero, thereis no order parameter for a confinement/deconfinement phase transition.

Thus inthe mu = md = 0 limit the universal characteristics of the QCD phase transition(i.e. those characteristics determined by the modes which develop long correlationlengths at the phase transition) are the same as those of the N = 4 Heisenbergmagnet.In this paper we further explore the consequences of this approach to the QCDphase transition.

In the following section, we review the scenario described in [1],and establish a dictionary between QCD and the magnetic system. In order tomake the present paper self-contained, we also include in this section many of theresults from [1].In section 3, we discuss the behaviour of the pion and sigmamasses at the transition.

In section 4, we discuss the ρ and A1 mesons. In sec-tion 5, we consider how the strange quark affects the phase transition.

In section6, we go beyond the static critical phenomena of the earlier sections and discussthe dynamics of the appropriate universality class. In section 7, we discuss the3

implications of all this for cosmology and lattice gauge theory, and for heavy ioncollisions under the assumption that the plasma remains close to thermal equilib-rium through the phase transition. In section 8, we consider what phenomena wecan expect in heavy ion collisions if the system gets far out of thermal equilibriumand can be modelled as a quench.

Finally, in the last section we summarize andconclude.2. QCD and the O(4) MagnetAs discussed in [1], the physics of the QCD phase transition is qualitativelydifferent in the cases of zero, one, two, or three or more flavors of quarks.

In thissection we consider QCD with two species of quarks. (An analysis similar to theone which follows leads to the conclusion that for three or more flavors of masslessquarks, the chiral phase transition is first order.

See [1] for details.) If there aretwo flavors of massless quarks, the lagrangian is symmetric under global chiraltransformations in the group SU(2)L × SU(2)R × U(1)L+R of independent specialunitary transformations of the left and right handed quark fields, and a vectorU(1) transformation which corresponds to baryon number symmetry.

(The axialU(1) which would make the symmetry group into U(2)×U(2) is a symmetry of theclassical theory, but not of the quantum theory [2].) This chiral symmetry breaksspontaneously down to SU(2)L+R × U(1)L+R at low temperatures; and is restoredat sufficiently high temperatures.

The order parameter for this phase transition isthe expectation value of the quark bilinearMij ≡⟨¯qiLqRj⟩(2.1)which breaks the symmetry when it acquires a non-zero value below some criticaltemperature Tc.In order to describe a second-order transition quantitatively, we must find atractable model in the same universality class. For the chiral order parameter (2.1)4

the relevant symmetries are independent unitary transformations of the left- andright-handed quark fields, under whichM →U†MV . (2.2)These transformations generate an U(2)L × U(2)R symmetry, which is not quitewhat is needed, since it includes the axial baryon number symmetry which is notpresent in QCD.

This problem is solved [1] by restricting M to unitary matriceswith positive determinant, instead of general complex matrices. Matrices M inthis restricted class remain in this restricted class under the transformation (2.2)only if U and V have equal phases.

Hence, the axial U(1) has indeed been removed.The 2×2 matrices M can conveniently be parametrized using four real parameters(σ,⃗π) and the Pauli matrices asM = σ + i⃗π · ⃗τ . (2.3)In fact the order parameter can be written as a four-component vector φ ≡(σ,⃗π)and the transformations (2.2) are simply O(4) rotations in internal space.

Hence,the order parameter appropriate for the chiral phase transition in QCD with twoflavors of massless quarks has the symmetries of the standard O(4) invariant N = 4Heisenberg magnet. For smaller number of components, this sort of model is amuch-studied model for the critical behavior of magnets, with the order parameterrepresenting the magnetization of a ferromagnet or the staggered magnetization ofan antiferromagnet.If the phase transition is second order, then it will correspond to an infraredfixed point of the renormalization group.

We wish to describe those aspects ofthe critical behaviour which are universal, that is, those aspects which are deter-mined by the scaling behaviour of operators near the infrared fixed point of therenormalization group. Hence, it is sufficient to retain those degrees of freedomwhich develop large correlation lengths at the critical point.

These are just long5

wavelength fluctuations of the order parameter, which is small in magnitude nearthe critical point and therefore fluctuates at little cost in energy. Thus the mostplausible starting point for analyzing the critical behavior of a second-order phasetransition in QCD is the Landau-Ginzburg free energyF =Zd3xn12 ∂iφα∂iφα + µ22 φαφα + λ4(φαφα)2 o.

(2.4)Here µ2 is the temperature-dependent renormalized (mass)2, which is negativebelow and positive above the critical point, while λ is the strength of the quarticcouplings and is supposed to be smooth at the transition. We neglect terms withhigher powers of φ since |φ| is small near the transition.

The symmetry breakingpattern we want is M ∝1 (equivalently, ⟨σ⟩̸= 0; ⟨⃗π⟩= 0) below the transitionwhich is indeed what we find at the minimum of the potential for positive λ. Thismodel has been studied in depth for arbitrary N and spatial dimension d, and theexistence of an infrared stable fixed point of the renormalization group has beenestablished [3]. Hence, it is a model for a second order QCD chiral phase transitionfor two massless quarks.When the free energy (2.4) is written in terms of σ and ⃗π it looks much like theoriginal model of Gell-Mann and Levy [4] with two changes: there are no nucleonfields and only three (spatial) dimensions.

These two changes reflect an impor-tant distinction [5]. We are only proposing (2.4) as appropriate near the secondorder phase transition point.

This is because it is only there that we can appeal touniversality – the long-wavelength behaviour of the σ and ⃗π fields is determinedby the infrared fixed point of the renormalization group, and microscopic consid-erations are irrelevant to it. In Euclidean field theory at finite temperature, theintegral over ω of zero temperature field theory is replaced by a sum over Matsub-ara frequencies ωn given by 2nπT for bosons and (2n+1)πT for fermions with n aninteger.

Hence, one is left with a Euclidean theory in three spatial dimensions withmassless fields from the n = 0 terms in the boson sums and massive fields from therest of the boson sums and the fermion sums. Hence, to discuss the massless modes6

of interest at the critical point, (2.4) is sufficient. This means that we do not needto worry whether to introduce nucleon fields as in [4], or constituent quark fieldsas, for example, in [6].We have motivated a very definite hypothesis for the nature of the phase tran-sition for QCD with two species of massless quarks, namely that it is in the uni-versality class of the N = 4 Heisenberg magnet.

This hypothesis has numerousconsequences, which are the subject of the rest of this paper. To keep the discus-sion self-contained, in the remainder of this section we review the predictions forthe static critical exponents described in [1].2.1.

Critical exponentsFirst, we define the reduced temperature t = (T −Tc)/Tc. The exponentsα, β, γ, η, and ν describe the singular behaviour of the theory with strictly zeroquark masses as t →0.

For the specific heat one finds œ[6C(T) ∼|t|−α + less singular. (2.5)The behaviour of the order parameter defines β.⟨|φ|⟩∼|t|β for t < 0 .

(2.6)η and ν describe the behaviour of the correlation length ξ whereGαβ(x) ≡⟨φ(x)αφ(0)β⟩−⟨φα⟩⟨φβ⟩→δαβA|x| exp(−|x|/ξ) at large distances. (2.7)A is independent of |x|, but may depend on t. The correlation length exponent νis defined byξ ∼|t|−ν .

(2.8)Above Tc, where the correlation lengths are equal in the sigma and pion channels,7

the susceptibility exponent γ is defined byZd3x Gαβ(x) ∼t−γ. (2.9)We will discuss the behaviour of the susceptibility below the transition in thefollowing section.

The exponent η is defined through the behaviour of the Fouriertransform of the correlation function:Gαβ(k →0) ∼k−2+η . (2.10)The last exponent, δ is related to the behaviour of the system in a smallmagnetic field H which explicitly breaks the O(4) symmetry.

Let us first showthat in a QCD context, H is proportional to a common quark mass mu = md ≡mq. This common mass term may be represented by a 2 × 2 matrix D given bymq times the identity matrix.

We are now allowed to construct the free energyfrom invariants involving both D and M. The lowest dimension term linear inD is just trM†D = mqσ, which in magnet language is simply the coupling of themagnetization to an external field H ∝mq. In the presence of an external field,the order parameter is not zero at Tc.

In fact,⟨|φ|⟩(t = 0, H →0) ∼H1/δ . (2.11)The six critical exponents defined above are related by four scaling relations[3].

These areα = 2 −dνβ = ν2(d −2 + η)γ = (2 −η)νδ = d + 2 −ηd −2 + η . (2.12)We therefore need values for η and ν for the four component magnet in d = 3.These were obtained in the remarkable work of Baker, Meiron and Nickel[7], who8

carried the perturbation theory to seven-loop order, and used information aboutthe behaviour of asymptotically large orders, and conformal mapping and Pad´eapproximant techniques to obtainη =.03 ± .01ν =.73 ± .02 . (2.13)Using (2.12) , the remaining exponents are α = −0.19 ± .06, β = 0.38 ± .01,γ = 1.44±.04 and δ = 4.82±.05.

Since α is negative there is a cusp in the specificheat at Tc, rather than a divergence.Very different critical exponents are proposed in [8]. These authors thus implic-itly claim that a hitherto unknown fixed point theory with the symmetries of theN = 4 isotropic Heisenberg model exists, and governs the QCD phase transition.3.

The Equation of State and the Pion and Sigma MassesThe expressions which define β, γ and δ are actually special cases of a moregeneral relationship between the magnetization and the magnetic field called thecritical equation of state. The equation of state has been calculated to order ǫ2by Br´ezin, Wallace and Wilson [9].

Their result is reproduced in the appendix. Inthis section, we will use the equation of state to determine the behaviour of themasses of the pion and sigma masses near the critical point.First, we must define what we mean by the “mass” of the pion and sigma.

Wecould choose either to define the mass as an inverse correlation length or as aninverse susceptibility. We choose the latter, which is conventional in the condensedmatter literature.

Specifically, we definem−2σ=Zd3xG00(3.1)andm−2π δij =Zd3xGij(3.2)where φ0 = σ and φi = πi, i = 1, 2, 3. At any given t and H, (2.7) implies that9

whether one defines the mass as the inverse correlation length or as the inversesusceptibility is academic. However, since A in (2.7) depends on t and H, the twodifferent choices lead to different scaling behaviours for masses as functions of tand H. We shall see that with the conventional choice (3.1) and (3.2) , the massescan be extracted conveniently from the equation of state.

It is worth noting thatthe masses we have defined are related only to the behaviour of spatial correlationfunctions in the static (equilibrium) theory. They carry no dynamical information.Also, we will only be able to make universal statements about how the masses scaleat the transition.

Normalizing the magnitudes of the masses (i.e. relating themto the zero temperature masses) will require using some specific model, and hencewill not be universal.The equation of state gives the magnetization as a function of t and H. Forthe rest of this paper, we will write the order parameter as M, for magnetization,keeping in mind that M = ⟨σ⟩= ⟨|φ|⟩.

In order to define the equation of state,we first define a shifted field ˜σ = σ −⟨σ⟩= σ −M. Then the equation of state issimply the relation⟨˜σ⟩= 0.

(3.3)This relation has been expanded to order ǫ2 by Br´ezin, Wallace and Wilson [9].The result can be expressed conveniently in terms of the variables y ≡H/Mδ andx ≡t/M1/β asy = f(x)(3.4)where the function f(x) was calculated to order ǫ2 in [9], and is given in theappendix. The units in which H and M are measured are chosen so that f(0) = 1corresponds to t = 0 and f(−1) = 0 corresponds to the t < 0, H = 0 coexistencecurve.

Knowing f(x), we can calculate the value of the order parameter M for agiven H and t using (3.4) . The behaviour of the order parameter is illustratedin Figure 1.

This figure and the other ones in this section should be viewed asillustrations of qualitative behaviour rather than quantitative predictions because10

they are based on setting ǫ = 1 in the O(ǫ2) expression for f(x).The valuesfor the critical exponents themselves which we quoted in the previous section arequantitative predictions, complete with error estimates, because they are based onthe much more elaborate analysis of Baker et al. [7]From the equation of state, we can deduce the behaviour of mπ and mσ atnon-zero (but small) t and H. The masses are given bym2σ = ∂H∂M(3.5)andm2π = HM .

(3.6)The first relation follows directly from the definition (3.1) , and the second followsfrom (3.2) and from assuming that ⃗M ∥⃗H, so that a small change δH ⊥H givesa small change δM ⊥M with δM/δH = M/H. Using the equation of state, wecan rewrite (3.5) and (3.6) asm2π = Mδ−1f(x)(3.7)andm2σ = Mδ−1δf(x) −xβf′(x).

(3.8)Hence, as suggested in [8], δ can be determined by measuring the ratio m2σ/m2π att = 0. In general, from f(x) we can find the pion and sigma masses for any t andH.There are two interesting limits which we will consider explicitly.

First, fort > 0 and H →0 which corresponds to x →∞, we should find the full O(4)symmetry, and hence should find that the pion and sigma masses are identical.For x →∞, the function f(x) from the appendix behaves as f(x) = cxγ. Here,11

the constant c and the exponent γ are given to O(ǫ) and O(ǫ2) respectively in (A9)and (A10) . Applying (3.7) and (3.8) , we find thatm2σ = m2π = ctγ for x →∞,(3.9)consistent with the symmetry.We can also consider the limiting case of approaching the coexistence curve.This means taking t < 0 and H →0, which implies x →−1.

In this limit, M tendsto a nonzero constant, and so from (3.6) , we obtain m2π ∝H, a familiar result forGoldstone bosons. The behaviour of the pion mass is illustrated in Figure 2a.

Theresult (3.6) may look peculiar to a particle physicist who is more familiar with thezero temperature resultm2π = 2mq⟨¯qq⟩f2π. (3.10)Before considering the sigma mass, we therefore pause here to explain how (3.10)and (3.6) are related.

We have seen that mq ∼H and that the order parameter⟨¯qq⟩∼⟨σ⟩∼M. At zero temperature, fπ is defined in terms of the axial currentby the relation⟨0 | Aαµ(0) | πβ(q)⟩= ifπqµδαβ .

(3.11)In the zero temperature linear sigma model, the axial current is given byAαµ(x) = σ(x)∂µπα(x) −πα(x)∂µσ(x) ,(3.12)which means that fπ defined in (3.10) is simplyfπ = ⟨0 | σ | 0⟩. (3.13)This result suggests that we make the identification fπ ∼M, which does indeedmake (3.6) and (3.10) equivalent.

However, it is important to remember that using12

the linear sigma model at zero temperature can not be justified by a universalityargument in the way that using it near T = Tc can. Hence the argument of thisparagraph is not a derivation of (3.6) from the zero temperature result (3.10) .

(3.6) is valid near T = Tc while (3.10) is valid at T = 0. Also, mπ in (3.10) isa mass in a 3 + 1 dimensional Lorentz invariant theory, while mπ in (3.6) is aninverse susceptibility in a 3 dimensional theory.

We have simply shown that areader familiar with one expression should not be surprised by the other.The behaviour of the sigma mass at the coexistence curve is trickier to obtainthan that of the pion mass. First, we note that in mean field theory (ǫ = 0) theequation of state is simply y = f(x) = 1+x, and mσ is easily evaluated using (3.8).

For H →0 at fixed t < 0 the result ism2σ = δ|t|βH + |t|β(δ−1)β. (3.14)Hence, in mean field theory m2σ decreases with H to a non-zero value at H = 0.However, for d < 4 when fluctuations are important, the result is quite different.In words, fluctuations of the massless pions produce new infrared singularities inthe longitudinal susceptibility, or, equivalently, make the sigma massless.

Now, letus see how this result can be obtained from the equation of state [10,11]. In thelimit H →0, f(x) ∼H while f′(x), we will see, tends to zero more slowly.

Hence,the second term in (3.8) is dominant and givesβm2σMδ−1 →f′(x) for x →−1 . (3.15)The difficulty is that from the expression (A11) for f(x) valid for x →−1, we noticethat f′(x) contains divergent terms like ǫ log(x+1), ǫ2 log2(x+1) and ǫ2 log(x+1).These terms do not exponentiate to f′(x) ∼(x + 1)p, contrary to the claims of [8].After some algebra [11], one finds the result βm2σMδ−1−1→c1 + c2y−ǫ/2 .

(3.16)Both the terms on the right side of (3.16) must be kept because they differ in their13

exponents only by order ǫ. Also for this reason, the constants c1 and c2 given inthe appendix are known only to order ǫ even though f(x) is known to order ǫ2.Qualitatively, as H is lowered at fixed t < 0, at first the c1 term dominatesand m2σ appears to be decreasing toward a non-zero value at H = 0 as in the meanfield result.

Then, the c2 term takes over and one finds that in fact the sigma massgoes to zero like m2σ ∝Hǫ/2. The behaviour of the sigma meson mass is illustratedin Figures 2b and 3.

In future lattice simulations, as mq is lowered toward zero,this behaviour should be observed. This result is an example of the power of therenormalization group techniques in obtaining universal results.

If we had chosena specific model, say that of Gocksch [6], or the Nambu-Jona-Lasinio model ofHatsuda and Kunihiro [12], we would have been able to calculate non-universalquantities far from Tc, but would basically have been limited to using mean fieldtheory, as those authors do. Then, we would have reached the incorrect conclusionthat the sigma has a non-zero mass in the chiral limit below Tc.

Here, by restrictingourselves to calculating universal quantities, we are limited to the region near thecritical point, but we can be confident in our results regardless of which specificmodel is correct and can include the effect of fluctuations.4. ρ and a1 Correlation FunctionsTo this point, we have discussed the correlation functions in the pion and sigmachannels only.

It is certainly possible to construct other spatial correlation func-tions. The next-simplest are those associated with the ρ (Lorentz vector, isospinvector) and A1 (Lorentz vector, isospin axial vector) mesons.

In Gocksch’s model[6], these have correlation lengths given simply by 2mQ(T) where mQ = πT +g⟨σ⟩is the mass of the constituent quarks in his model. Since these constituent quarksare fermions, they have a Matsubara mass of πT.

g is a coupling constant. In anymodel, there is bound to be a model dependent contribution to these correlationlengths which is smooth at Tc.

However, the pions and sigma also contribute tothe ρ and A1 correlation lengths since operators with the appropriate symmetries14

can be constructed from the pion and sigma operators. In particular,ρiα = ǫαβγπβ∇iπγ(4.1)and(A1)iα = σ∇iπα −πα∇iσ .

(4.2)Hence, the pions and sigma can make a universal non-analytic contribution to theρ and A1 correlation lengths at the critical point. In this section, we calculate thiscontribution for t > 0 and H = 0.Although we will try to define all the quantities we use, for those interested infurther details we note that we are following the notation and conventions of Amit[3].

In the region t > 0 and H = 0 the symmetry is unbroken and hence there isno distinction between the σ and π operators which we will call φα and the ρ andA1 operators which we combine into Oαβi≡ǫαβγδφγ∇iφδ. We are interested inthe scaling behaviour of the correlation function ⟨O(x)O(0)⟩≡Γ(0,0,2).

In general,by Γ(m,n,p) we mean the m-point vertex function with n insertions of φ2 and pinsertions of O. The scaling behaviour of vertex functions involving a compositeoperator like O is determined by η, ν, and the anomalous dimension of the operator.Here we are fortunate because the operators Oi are the “conserved” currents ofthe chiral symmetry, where in the 3 dimensional theory this means ∇iOi = 0.

TheWard identities arising from chiral symmetry imply that the O are not subject torenormalization [13], or, equivalently, that their anomalous dimensions are zero.For H = 0, the vertex function Γ(0,0,2) is a function of the external momentumk, the reduced temperature t, the renormalization point κ, and the φ4 couplingconstant λ. At the fixed point of the renormalization group, λ is a constant andthe vertex function satisfies the renormalization group equation [3](κ ∂∂κ −1ν −2t ∂∂t)Γ(0,0,2)(k, t, κ) = B .

(4.3)The right hand side of the equation is non-zero because although Γ(0,0,2) is not15

subject to multiplicative renormalization, it is additively renormalized. However,B does not depend on either k or t [3], and therefore does not contribute to thenon-analytic behaviour at the critical point.

Therefore, in what follows we willdrop B. As a consequence of (4.3) , the vertex function has the formΓ(0,0,2)(k, t, κ) = Fκ1/θ , k,(4.4)where θ = 1ν −2.

By dimensional analysis, we know thatΓ(0,0,2)(k, t, κ) = bd−2Γ(0,0,2)kb , tb2, κb. (4.5)This means that (4.4) becomesΓ(0,0,2) = bd−2F κb tb21/θ , kb.

(4.6)To this point, b has been arbitrary. Therefore, we can choose it strategically.

Withthe choice b = κ(t/κ2)ν, we findΓ(0,0,2) ∼tν(d−2)g(kt−ν)(4.7)where the function g(x) satisfies g(0) = constant. If, on the other hand, we chooseb = k, the result isΓ(0,0,2) ∼kd−2˜g(kt−ν)(4.8)where the function ˜g(x) tends to a constant for x →∞.

(4.7) , which givesthe behaviour of Γ(0,0,2) as a function of t for k = 0, and (4.8) , which givesthe behaviour for t = 0 as a function of k, describe the non-analytic part of thecorrelation function. As we mentioned earlier, there will also be a model dependentbut analytic mass term for the ρ and A1.Except near the critical point, thissmooth term is presumably larger than the non-analytic term whose effects wehave calculated.

In order to observe (4.7) and (4.8) , future lattice simulations willhave to get close enough to the critical point that the non-analytic term dominatesthe analytic term.16

5. The Influence of the Strange QuarkTo this point in this paper, we have described a world with two massless quarks,and hence we have implicitly been taking the strange quark mass to be infinite.

Ifthe strange quark is massless, then Pisarski and Wilczek showed [1] that the chiralphase transition is first order. Hence, as the strange quark mass is reduced frominfinite to zero, at some point the phase transition must change from second orderto first order.

This point is called a tricritical point. There is numerical evidence[14] that when the strange quark has its physical mass, the transition is secondorder.

Hence, we devote most of this paper to analyzing the second order phasetransition. However, in a lattice simulation, the strange quark mass could be tunedto just the right value to reach the tricritical point.

In this section, we discuss thecritical exponents that would be observed in such a simulation.Let us consider the effect of adding a massive but not infinitely massive strangequark to the two flavor theory.This will not introduce any new fields whichbecome massless at Tc, and so the arguments leading to the free energy (2.4)are still valid. The only effect of the strange quark, then, is to renormalize thecouplings.

Renormalizing µ2 simply shifts Tc, as does renormalizing λ unless λbecomes negative. In that case, one can no longer truncate the Landau-Ginzburgfree energy at fourth order.After adding a sixth order term, the free energybecomesF =Zd3xn12(∇φ)2 + µ22 φ2 + λ4(φ2)2 + κ6(φ2)3 −Hσo.

(5.1)While for positive λ, φ2 increases continuously from zero as µ2 goes through zero,for negative λ, φ2 jumps discontinuously from zero to |λ|/(2κ) when µ2 goesthrough λ2/(4κ). Hence, the phase transition has become first order.

Thus atthe value of ms where λ = 0, the phase transition changes continuously fromsecond order to first order.The singularities of thermodynamic functions near tricritical points, like thesingularities near ordinary critical points, are universal. Hence, it is natural to17

propose [1] that QCD with two massless flavors of quarks and with T near Tc andms near its tricritical value is in the universality class of the φ6 Landau-Ginzburgmodel (5.1) . This model has been studied extensively [15].

Because the φ6 inter-action is strictly renormalizable in three dimensions, this model is much simplerto analyze than the φ4 model of the ordinary critical point. No ǫ expansion isnecessary, and the critical exponents all take their mean field values.

There arecalculable logarithmic corrections to the scaling behaviour of thermodynamic func-tions [15], but we will limit ourselves here to determining the mean field tricriticalexponents.In mean field theory, the correlation function in momentum space is simplyGαβ(k) = δαβ(k2 + µ2)−1. Since µ2 ∼t, this gives the exponents η = 0, γ = 1 andν = 1/2.

To calculate α and β, we minimize F for H = λ = ∇φ = 0, and findα = 1/2 and β = 1/4. To calculate δ, we minimize F for t = λ = ∇φ = 0 and findδ = 5.The result for the specific heat exponent α is particularly interesting, since itmeans that the specific heat diverges at the tricritical point, unlike at the ordinarycritical point.

This means that whereas for ms large enough that the transition issecond order the specific heat C(T) has a cusp but is finite at T = Tc, as ms islowered to the tricritical value C(Tc) should increase since at the tricritical pointit diverges. This behaviour should be seen in future lattice simulations.Finally, at a tricritical point there is one more relevant operator than at acritical point, since two physical quantities (t and ms) must be tuned to reach atricritical point.

Hence, a new exponent φt, the crossover exponent, is required.For λ ̸= 0, tricritical behaviour will be seen only for |t| > t∗, while for |t| < t∗,either ordinary critical behaviour or first order behaviour (depending on the signof λ) results. t∗depends on λ according tot∗∼λ1/φt(5.2)The mean field value of φt is obtained by minimizing the free energy F for H =18

∇φ = 0, and is φt = 1/2. These mean field tricritical exponents, α = 1/2, β = 1/4,γ = 1, δ = 5, η = 0, ν = 1/2, and φt = 1/2 would describe the real world if mswere smaller than it is, and will describe future lattice simulations with ms chosenappropriately.6.

Dynamic Critical PhenomenaTo this point, we have discussed the static critical phenomena appropriateto the equilibrium properties of the QCD plasma near its phase transition. Todiscuss dynamical phenomena, we need equations of motion.

In zero temperaturescalar field theory, Lorentz invariance requires that these equations have no firstorder time derivatives. In the finite temperature theory appropriate for discussingcritical behaviour, however, there is no Lorentz invariance, and hence we shouldexpect first order time derivatives.

In analogy with the notion of static universality,one finds that the dynamics of the long wavelength modes groups theories intodynamic universality classes containing theories described by the same equationsof motion. To specify the dynamic universality class, one needs to specify morethan the dimension of space and the number of components of the order parameter.Hence, there are usually several different dynamic universality classes which areall in the same static universality class.

In particular, it is necessary to specifywhether or not the order parameter is conserved, and which other quantities areconserved. Thus a ferromagnet will have very different dynamical behaviour froman antiferromagnet with the same number of components, even though the staticuniversality class is the same.

The ferromagnet will have a much more difficulttime thermalizing long-wavelength fluctuations, since in the k →0 limit they arerigorously stable.From this point of view, two flavor QCD behaves as an antiferromagnet. Itsorder parameter, the expectation value of a scalar or pseudoscalar quark bilinear,is not a conserved quantity.

(It is quantities of the form ¯qγ0q, not ¯qq, that are con-served). This means that a model similar to model G of Halperin and Hohenberg19

[16], is appropriate. Model G is formulated for a three component order parameter.We show below how to formulate it for the N = 4 order parameter appropriate forQCD.

One main result of the theory concerns the rate of critical slowing down nearthe transition. Just as the correlation length in space diverges at the critical point,so does the correlation time for dynamics.

Its scaling property is conventionallywritten in terms of a critical exponent z, such that the correlation time scales asξz, where ξ is the correlation length. For the model in question, we show belowthat the exponent z governing critical slowing down is predicted to be d/2 = 3/2.Let us now consider how model G of [16] must be modified to deal with thefour component order parameter of interest.

As we discussed above, it is importantto find the conserved quantities. First, there is the energy.

However, it is shownin [17] that if α, the specific heat exponent, is negative then the dynamics of theorder parameter is not affected by the presence of a conserved energy. However,the six generators of O(4) rotations are associated with six conserved quantitieswhich can be written as an antisymmetric tensorJαβ =Zd3xjαβ .

(6.1)At the level of the unrenormalized Lagrangian, jα can be expressed in terms of theorder parameter asjαβ = ǫαβγδφα∂∂tφβ ,(6.2)but this relation need not hold upon renormalization. (Note that since we areinterested in the critical phenomena, by renormalization we mean renormalizationtowards the infrared.) In model G, the symmetry is O(3), so there are 3 conservedquantities instead of 6.

The 3 conserved quantities are the magnetization, whilethe nonconserved order parameter is the staggered magnetization of the antiferro-magnet. The equations of motion for j and φ contain two types of terms.

Thereare dissipative terms which damp the system toward the equilibrium configurationand so called mode-mode coupling terms. The latter reflect the Poisson bracket20

relations among the fields. In our case, these arehφα, Jβγi= ǫαβγδφδ(6.3)andhJαβ, Jγδi= δβγJαδ −δαγJβδ −δβδJαγ + δαδJβγ .

(6.4)The equations of motion are∂φα∂t = −Γ δFδφα+ ghφα, jβγi δFδjβγ+ θ(6.5)and∂jαβ∂t= γ∇2 δFδjαβ+ ghjαβ, φγi δFδφγ+ ghjαβ, jγδi δFδjγδ+ ζ ,(6.6)where the free energy F is given byF =Zd3xn12 ∂iφα∂iφα + µ22 φαφα + λ4(φαφα)2 +12χjαβjαβ −Hσo. (6.7)θ and ζ are Langevin noise terms.

The difference between the dissipative termsin the two equations of motion reflects the fact that j is conserved, and hencea spatially constant j can not dissipate. Note that j appears in the free energyonly as j2.

This is because any higher order terms like j2φ2 or those involving∇j are irrelevant. Because of the form of F, it turns out that the term in theequation of motion (6.6) involving (6.4) is zero.

The equations of motion givenabove determine the universal dynamics of the long wavelength modes of interestnear the critical point.With equations of motion in hand, Halperin et al. [16] go on to formulatedynamic renormalization group transformations.

These are obtained by starting21

with the theory defined with an ultraviolet cut-offΛ, integrating out modes in themomentum shell between b−1Λ and Λ and then rescaling according tox →x′ = b−1xφ →φ′ = b(d−2+η)/2φj →j′ = bcjt →t′ = b−zt . (6.8)After one such transformation, one obtains a new free energy and new equationsof motion.

If one works in the ǫ expansion, these are related to the former F andequations of motion simply by a transformation of the parameters µ, λ, Γ, γ, g,and χ. Repeated application of the renormalization group transformation thereforeleads to recursion relations for these parameters.

The next step is to find the fixedpoint of the transformation. Just as in the static case the fixed point conditionfixes η, here it fixes η, c, and z. Fortunately, to obtain c and z we only needthe particularly simple recursion relations for χ and g. Because j only appearsquadratically in the free energy, the recursion relation for χ isχ−1 →bd−2cχ−1 ,(6.9)and this is valid to all orders in ǫ.

The terms in the equations of motion proportionalto g are consequences of the O(4) symmetry, and as a result the Ward identitiesenforce the recursion relationg →bz−d+cg(6.10)to all orders in ǫ. From (6.9) and (6.10) it is clear that at a fixed point one musthave c = d/2 and, as advertised earlier, z = d/2.22

7. Implications on the Lattice and in the Real WorldFinite temperature lattice QCD simulations are ideally suited to testing manyof the predictions made in this paper.

The static correlation functions of the threedimensional theory are natural objects to consider in finite temperature (Euclidean)simulations. Also, it is much easier to vary parameters like the temperature andthe bare quark mass in a lattice simulation than in a real experiment.

Hence, itshould be possible to measure the static critical exponents of section 2, the equationof state and the scaling behaviour of the pion and sigma masses of section 3, thebehaviour of the ρ and A1 correlation functions of section 4, and the static tricriticalexponents of section 5.Present simulations [14,18] provide strong evidence that the QCD phase tran-sition is second order, and that the order parameter is M [1]. However, there areseveral reasons why present simulations can not yet be used to test the more de-tailed predictions of this paper.

The fundamental reason is that in all simulationsto date, the bare quark mass has been so large that correlation lengths do not getvery long at Tc. For example, in the work of Bernard et al.

[18], the correlationlength in the pion channel at Tc is only about 2.5 lattice lengths. Hence, in orderto study the behaviour nearer to the critical point and to measure universal prop-erties, we must wait for simulations with smaller quark masses.

The fundamentalproblem is that long correlation lengths are necessarily accompanied by numericalcritical slowing down, and this makes simulations challenging.Another hurdle to be overcome before lattice simulations can measure thecritical properties of the QCD phase transition is that any lattice implementationof fermion fields only exhibits the full chiral symmetry in the continuum limit.If Wilson fermions are used, there is no chiral symmetry at all on the lattice. IfKogut-Susskind fermions are used, four flavors of fermions are required, but there isa continuous U(1) ×U(1) chiral symmetry on the lattice.

This should give a phasetransition in the universality class of the N = 2 magnet, and has been discussedin [19]. In order to study two flavors of fermions, one takes the square root of the23

fermion determinant in the lattice action. It is not at all clear what this does tothe lattice chiral symmetries.

Hence, for Kogut-Susskind as for Wilson fermions,before we are able to test our predictions for the critical phenomena we need finerlattices so that extrapolation to the continuum limit can be done. A necessarycondition for doing this is that the results for Kogut-Susskind and Wilson fermionsagree when so extrapolated.

This condition has not yet been met [20].Hence, while we cannot test our detailed predictions against current latticesimulations, we are confident that in the future with finer lattices and longer cor-relation lengths, simulations will be able to measure critical exponents, correlationfunctions, and the equation of state, and verify our results.Let us now turn to real experiments, as opposed to those on the lattice. Here,we are not free to dial the bare quark mass.

We will see that this is unfortunate,particularly in our discussion of heavy ion collisions. First, however, let us disposeof two other possible arenas for testing our results.

The QCD phase transitioncertainly occurred in the early universe. Indeed, much work has been done onpossible observable effects of this transition, if it is first order.

Unfortunately, forphysical values of the quark masses, we have seen that the transition is second order.We can think of no observable consequences of a second order QCD phase transitionin the early universe.In [1], it was noted that since certain antiferromagnetsincluding dysprosium have order parameters in the N = 4 universality class [21],experiments on the phase transition in these materials would help us understandthe QCD transition. Alas, in these magnets there is a quartic operator like (π21 +π22)(π23 + σ2) which is allowed by the microscopic hamiltonian of the magnets [22],but not by that of QCD.

This operator makes the symmetric Heisenberg fixed pointinfrared unstable, and either makes the phase transition first order or makes itsecond order but governed by an anisotropic fixed point. Hence, neither cosmologynor dysprosium are suitable arenas for learning about the QCD phase transition.Now, we turn to relativistic heavy ion collisions.In Bjorken’s [23] pictureof such a collision, a volume of hot plasma forms and quickly reaches thermal24

equilibrium.In the center of mass frame, the incident nuclei are both Lorentzcontracted into pancake shapes. They pass through each other, and leave behinda region of hot vacuum.

In translation, this means that the baryon number of theincident nuclei ends up in that part of the plasma heading approximately down thebeam pipes, and the central rapidity region consists of plasma with approximatelyzero baryon number. In the remainder of this paper, we will attempt to use whatwe have learned about the critical phenomena associated with the chiral phasetransition to study the behaviour of the plasma in the central rapidity region as itexpands and cools through T = Tc and eventually hadronizes and becomes pionswhich fly offand are detected.The defining characteristic of a second order phase transition is the divergenceof correlation lengths.

How could this feature be observed here? Large volumesof space with the order parameter correlated and pointing in a direction differentthan the true vacuum (i.e.

sigma) direction will become regions in which the orderparameter oscillates coherently about the sigma direction. After hadronization,correlated volumes will turn into regions of space where the ratio of the numberof charged pions to neutral pions has some fixed value.

Since, in the standardscenario of Bjorken [23], different positions in the plasma along the beam direc-tion become different longitudinal momenta (actually, different rapidities) as theplasma expands, one would hope that a signal of a second order phase transitionwould be fluctuations in the ratio of charged to neutral pions as a function of ra-pidity. Coherent evolution of a classical pion field has been considered before byAnselm and Ryskin [24] and by Blaizot and Krzywicki [25].

However, these authorsconsidered neither an equilibrium second order phase transition as we do in thissection, nor a quench as we will in the next section.A prerequisite for the fluctuations discussed above to be observable is thatthe correlation length must get long compared, say, to Tc. To determine whetherthis does indeed happen, we must leave our universality safety net behind, sinceneither Tc nor the magnitudes of correlation lengths are universal.

From Figure 2a,it is immediately obvious that we have a problem. The longest correlation length25

is in the pion channel, and the pion mass is increasing with temperature. Thissuggests that the pion mass at Tc is larger than mπ(T = 0) = 135 MeV.

Thisis consistent with the fact that in many models the pion mass increases from itszero temperature value as the temperature is increased from zero. This result hasbeen obtained in chiral perturbation theory using the nonlinear sigma model [26]and also for the linear sigma model [27].

Hence, it seems clear that the longestcorrelation length at Tc will be shorter than (135 MeV)−1. This is to be comparedto Tc itself, which for the case of two massless quarks is around 140 MeV [18].Hence, even though the quark masses are indeed small (∼10 MeV), the magneticfield H proportional to mq is large enough to prevent any correlation lengths fromreaching interesting values.There is an appropriate quantitative criterion to determine whether a nearequilibrium second order phase transition leads to dramatic effects.

One comparesthe energy in a correlation volume just below Tc with the zero temperature pionmass to determine whether or not the correlated volume can become a large numberof pions. Using current lattice simulations [18], we can make a crude attempt atthis comparison.

The sum of the energy and pressure in a correlation volume isabout 1/4 the zero temperature ρ mass.Taken literally, this means that eachcorrelation volume becomes only one or two pions in the detector. As we havementioned, current simulations are subject to many caveats, and so this estimateshould not be taken literally.

Nevertheless, it seems clear that the physical valueof mq is large enough that in an equilibrium phase transition a correlation volumeat Tc does not evolve into a large number of correlated zero temperature pions.This is not encouraging.We pause here for an aside. The reader may be wondering why, when theseemingly small equal quark mass mu = md = mq has such deleterious effects, wehave completely neglected the difference between the up and down quark masses.It was noted in [1] that unequal quark masses allow terms of the type∆F ∝(δm)2(σ2 −π23 + π21 + π22) .

(7.1)26

If one is close enough to the critical point that this term matters, one will dis-cover an anisotropic fixed point rather than the symmetric Heisenberg fixed point.However, while the effect of a common quark mass, namely the mass of the pion,is comparable to Tc, the effect of (7.1) is much smaller. For example, the QCDcontribution to the difference in mass between the charged and neutral kaons isabout 5 MeV [28].

Therefore, we need not worry about (7.1) in real experimentssince it is much less important than the effect of the “magnetic field” proportionalto the common up and down quark mass. Of course, (7.1) could be introduced andstudied on the lattice.It seems clear that if the standard scenario for heavy ion collisions in which theQCD plasma cools through Tc while staying close to thermal equilibrium is cor-rect, then no correlation lengths will get long enough for there to be any dramaticobservable effects of the phase transition.

The chiral “transition”, like the confine-ment/deconfinement “transition,” will be a smooth crossover. If we were able todial down the quark masses and hence the pion mass, phenomena associated witha second order transition would become more prominent.

Alas, in the real world,unlike on a lattice, we have no such freedom.8. Non-equilibrium Phenomena in Relativistic Heavy Ion CollisionsWe are not done yet.

The gloomy paragraph with which we ended the pre-ceding section began with a conditional sentence. In this section we will considerthe observable effects in heavy ion collisions if the plasma does not stay close tothermal equilibrium through the transition.

There are tantalizing hints in cosmicray physics that point in this direction. Among the zoo of high energy cosmicray events known are a particularly peculiar class of events called Centauros [29].These are events with total energy of order 1000 TeV in which many (of order100) charged hadrons each with energies of a several TeV and very few photonsor electrons are seen in a cosmic ray induced shower.

In the sample of events in[29], there were 5 Centauros, representing about 1% of the events seen with energy27

of the appropriate order of magnitude. Centauros are peculiar because so manycharged pions are observed without any of the gammas that would indicate thedecay of neutral pions.

This apparent violation of isospin invariance is puzzling,unless one thinks of it in the language of a second order phase transition in whichthese events can be interpreted as the creation of a volume of QCD plasma in whichthe φ field has fluctuated throughout most of the plasma in some direction in theπ1 −π2 plane. This implies correlation throughout most of a volume of plasmalarge enough that it becomes about 100 zero temperature pions.

We convincedourselves above that this could not happen if the plasma remains close to thermalequilibrium. Hence, these Centauro events provide a tantalizing hint that it mightbe wise to consider the effects of going from the symmetric phase above Tc tothe ordered phase rapidly without maintaining thermal equilibrium.

This process,called quenching, has been much studied in condensed matter physics.We noted in the previous section that if the plasma stays close to equilibrium,one finds below Tc that the energy in a correlation volume is small compared to thezero temperature pion mass. However, since the energy density at temperatureswell above Tc is much higher than that below Tc, it is reasonable to hope that aquench, in which the energy density does not decrease in a quasi-equilibrium fashionthrough Tc, has a better chance of producing correlated volumes of plasma whichevolve into many zero temperature pions.

In this section we will begin to analyzethe observable effects if relativistic heavy ion collisions proceed via quenching. Ina real relativistic heavy ion collision, the phase transition will probably occur bysomething in between a slow equilibrium process and a quench.

Only experimentscan determine which description is more appropriate.Let us begin by describing more carefully what a quench is, in the contextof a Heisenberg magnet in greater than two dimensions with no applied magneticfield. The system starts at equilibrium at a temperature well above Tc, fluctuatingamong an ensemble of configurations with short correlation lengths.One thenimagines turning the temperature instantaneously to zero.

This means that theequilibrium configuration is now an ordered state with the field aligned throughout28

space.However, this is not the configuration in which the system finds itself.It is in one configuration from the ensemble appropriate to a high temperature.This configuration then evolves according to the zero temperature equations ofmotion (i.e.microcanonically, with no thermal fluctuations. )In a condensedmatter system, the appropriate equations of motion are those of Section 6, and inparticular they are not Lorentz invariant.

What is found [30] is that the size ofcorrelated domains grows with time in such a way that after a brief initial periodthe correlation function has the simple scaling formC(r, t) ≡⟨φ(r, t)φ(0, t)⟩= gr/L(t). (8.1)The characteristic domain size L increases with time according to L(t) ∼tp, wherethe exponent p depends only on d, N, and the dynamic universality class.

It is im-portant to note that the scaling behaviour (8.1) is obtained regardless of the initialconfiguration. Hence, it is not actually necessary that the initial configuration in asimulation (or in a heavy ion collision) be selected from a high temperature ther-mal ensemble.

Any disordered initial configuration evolves to the critical behaviour(8.1) . This phenomenon is called self-organized criticality.

It is also crucial forus that the domain size is not related to an equilibrium correlation length, and inparticular that in an infinite system it grows without bound.Let us now consider how the physics of quenching may be applied in relativisticheavy ion collisions. If the collision is energetic enough to create a region of plasmawell above T = Tc, the φ field will indeed be fluctuating among an ensemble ofdisordered configurations.

At the end of the process, one certainly has zero tem-perature. The question is what happens to the φ field in between.

One idealizedpossibility which has been considered by many before us and which we consideredin the previous section is that the system stays arbitrarily close to thermal equi-librium. Another idealization is that thermal fluctuation ceases instanteously.

Ina real collision, the plasma is cooling, and so cannot be exactly in thermal equilib-rium. If it cools fast enough, the configuration of the φ field will “lag,” and as in29

a quench the system will find itself in a configuration that is more disordered thanthe equilibrium configuration appropriate for the current temperature. However, itis clear that a real collision will not be an ideal quench, as thermal fluctuation willnot cease instantaneously.

Thus, a real collision fits in neither idealized category,but is somewhere in between. To the second idealization, the quench, we now turn.There are several differences between a quench in the system of interest to usand in the condensed matter system we described above.

First, at zero temperaturewe must have a Lorentz invariant field theory, with different equations of motionthan those used in the condensed matter system. While the equations of motionof section 6 describe the Lorentz non-invariant dynamics of the order parameternear T = Tc, they are inappropriate for the T = 0 dynamics of a quench.

Wepropose to use the zero temperature linear sigma model with only pion and sigmafields. Since we will be considering energies well below the rho and nucleon masses,we need not include these degrees of freedom.

The sigma could also be left out,if it were not for the fact that the appropriate initial conditions are a disorderedstate in which the pion and sigma fields are equivalent up to the effect of the barequark masses. The second difference is that unlike in the magnet, the plasma ina relativistic heavy ion collision is expanding.

This means that the descriptionin terms of configurations of the field φ will not be appropriate forever. At sometime, the energy density drops low enough that one has individual pions flying offtowards the detectors.

The third difference is that unlike in the condensed mattersystems considered in [30], we must include the effects of the bare quark masses.As we saw before, these correspond to a significant magnetic field. We proposethat quenching of a 4 component Heisenberg model with the three modificationswe have mentioned be considered an idealized model for a heavy ion collision.It is fortunate that the scenario we have just outlined, with the exception of thesignificant magnetic field, is exactly the scenario considered by Turok and Spergel[31] as a cosmological model for large scale structure formation in the early universe.They study the evolution of an O(N) sigma model in an expanding universe.

Theyfind an exact scaling solution for the non-linear sigma model in the large N limit,30

and do numerical simulations for the linear sigma model for N = 4 and N = 10.The main reason we can apply their results while we cannot use those of [30] isthat Turok and Spergel use the Lorentz invariant equations of motion appropriatefor our problem. They find that the size of correlated domains, L(t), grows at thespeed of light!

We are currently [32] extending their simulations to include theeffects of a magnetic field, and to vary the expansion rate. The magnetic field willqualitatively change the scenario.

Instead of having correlated domains with theφ field pointing in arbitrary directions on the 3-sphere, at late times the φ fieldwill be oscillating about the sigma direction. However, there will still be domainsin which the oscillations are in different directions.

If in a heavy ion collision thesize of these domains grows with the speed of light as it does in our simulations[32], the phenomenological consequences for heavy ion collisions are dramatic. Ofcourse the description we are using will only be valid for a short time.

(Bjorken [23]estimates that a hydrodynamic description will be valid for about 10 fm/c.) Sincethe plasma will be expanding slower than the speed of light, even in this shorttime domains which expand at the speed of light will grow to encompass largefractions of the total volume.

We therefore propose that if heavy ion collisions canbe modelled as a quench, this will be detected by observing clusters of pions inwhich all the pions in a region of rapidity are correlated in internal space. In someclusters, there will be only charged pions; in others, only neutral ones; and in all,charged and neutral pions will occur in some fixed ratio.We can estimate the probability distribution of the ratio R of the number ofneutral pions to the total number of pions in a correlated region.

Let us assumethat the φ field in the region is initially equally likely to be pointing in any directionon the 3-sphere. This assumption may not be strictly true because the magneticfield selects a preferred sigma direction even at high temperatures.

However, wemake the assumption in order to get a simple analytical result. Where φ startswill determine in which direction it ends up oscillating about the sigma direction.We define angles on the 3-sphere according to31

σ, π3, π1, π2=cos θ, sin θ cos φ, sin θ sin φ cos η, sin θ sin φ sin η. (8.2)Then the ratio R is given byR ≡nπ0nπ0 + nπ+π−=sin2 θ cos2 φsin2 θ(cos2 φ + sin2 φ) = cos2 φ .

(8.3)Under the assumption that all initial values on the 3-sphere are equally probable,the probability distribution P(R) is determined byR2ZR1P(R)dR = 1π22πZ0dηπZ0dθ sin2 θarccos(√R1)Zarccos(√R2)dφ sin φ(8.4)and turns out to be simplyP(R) = 12R−1/2 . (8.5)Equivalently, the probability that R < R1 is given by √R1.

If heavy ion collisionsare described by a quench, there should be large regions of the collision volumecontaining clusters of pions in which R is constant, and the values R takes indifferent such regions should be distributed according to (8.5) .As one application of (8.5) , we note that the probability that the neutral pionfraction R is less than .01 is 0.1! This is a graphic illustration of how different (8.5)is from what one would expect if individual pions were independently randomlydistributed in isospin space.

It also makes Centauro events in which less than 1% ofthe outgoing particles from a heavy ion collision are neutral pions seem much lesssurprising than they first appeared. The analysis of the Centauro data is difficultfor several reasons.

Most important of these is the limitation imposed by smallstatistics. Also, if a Centauro event occurs too high above the detector, so manysecondary photons will be produced that the event will not be recognized as aCentauro.

Third, in a Centauro event all of the particles from the collision strike asmall region of the detector and it is impossible to isolate the central rapidity region.32

This combined with the fact that the detectors do not distinguish between chargedpions and charged baryons has several unfortunate consequences. It means thatevents in which there are two or more correlated clusters of pions are not detectedas Centauros.

Only those with a single cluster are so identified. Also, the oppositetype of event in which R is close to 1, will not have a dramatic signature since therewill always be charged baryons present from the two initial nuclei.

For all thesereasons, we feel it is impossible to extract a meaningful probability distributionP(R) from the cosmic ray data.When relativistic heavy ion collisions occur at high enough energies in a labora-tory colliding beam facility, all of the difficulties of the cosmic ray experiments willbe rapidly overcome. That will be the time to look for correlated clusters of pions,and to look for a distribution like (8.5) , and hence to determine whether thesecollisions proceed by a process close to the idealized quench we have consideredhere.9.

ConclusionsThe future of the study of the QCD phase transition on the lattice looks promis-ing. As simulations improve, they will begin to investigate the plethora of staticcritical phenomena we discussed in the first sections of this paper.

Critical expo-nents, the equation of state, the critical behaviour of the pion and sigma suscepti-bilities, ρ and A1 correlation functions, and tricritical exponents are all out therewaiting to be measured.Because heavy ion collisions are dynamical processes, there are more possiblescenarios and the situation is not clear cut.We have considered two idealizedmodels. If the cooling plasma stays arbitrarily close to thermal equilibrium, ourconclusions are disappointing.

Because the pion is so heavy compared to Tc, corre-lation lengths will not become particularly long. On the other hand, in the otheridealized model we considered, in which thermal fluctuations are rapidly quenched33

and the cooling plasma is in a sense maximally out of thermal equilibrium, dra-matic phenomena are possible. Correlated volumes will form, and their size willnot be determined by any equilibrium correlation length.

Indeed they may grow atthe speed of light until hadronization occurs. This will have the consequence thatclusters of large numbers of pions will be detected in which the ratio of neutral tocharged pions will be constant.

This ratio will be different in different clusters, andwill follow a probability distribution which is skewed towards having few neutralpions. We eagerly await the verdict of experiment as to which scenario is moreappropriate.Acknowledgements: We are grateful for the fruitful discussions we had with BertHalperin.

We both also acknowledge the hospitality of Harvard University, wherepart of this work was completed.APPENDIXIn this appendix, we reproduce the equation of state to order ǫ2 [9], and givevarious other results used in section 3.The equation of state is found by doing an ǫ expansion of the relation⟨˜σ⟩= 0 ,(A.1)where ˜σ = σ −M.The resulting expansion can be expressed in terms of thevariables y ≡H/Mδ and x ≡t/M1/β, where1β = 2 +6N + 8ǫ + 4(N + 5)(7 −N)(N + 8)3ǫ2 + O(ǫ3)(A.2)andδ = 3 + ǫ + N2 + 14N + 602(N + 8)2ǫ2 + O(ǫ3) . (A.3)We are interested in ǫ = 1 and N = 4.

We choose to measure fields in units suchthat y = 1 at t = 0, and x = −1 at the coexistence curve (H = 0, t < 0). The34

equation of state isy = f(x) . (A.4)The function f(x) is given byf(x) = 1 + x+ǫ2(N + 8)1 +ǫ2(N + 8)N −1 + 6 ln 2 −9 ln 3 + (N −1) ln(x + 1)×3(x + 3) ln(x + 3) + (N −1)(x + 1) ln(x + 1) + 6x ln 2 −9(x + 1) ln 3+ǫ2(N + 8)2n 12(10 −N)(x + 1)ln2(x + 3) −ln2 3+ 36ln2(x + 3) −(x + 1) ln2 3 + x ln2 2−54 ln 2ln(x + 3) + x ln 2 −(x + 1) ln 3+ 3(N −1)ln 274(x + 1) ln(x + 1)+ 212 + 17N −4N2N + 8(x + 3) ln(x + 3) + 2x ln 2 −3(x + 1) ln 3+ (N −1)(x + 1) ln(x + 1) ln(x + 3) −N2 (N −1)(x + 1) ln2(x + 1)+ N −1N + 8(19N + 92)(x + 1) ln(x + 1) −2(N −1)(x + 6)I1(ρ) −6(x + 1)I1(3/4)−6(N −1)I2(ρ) −(x + 1)I2(3/4)+ 4(N −1)I3(ρ) −(x + 1)I3(3/4)o+ O(ǫ3)(A.5)whereρ ≡x + 34(x + 1)(A.6)andI1(ρ) ≡ρZ0du ln uu(1 −u)p1 −u/ρ −1−∞Zρdu lnuu(1 −u)I2(ρ) ≡ρdI1dρI3(ρ) ≡I1(ρ) + 2I2(ρ) .

(A.7)The behaviour of I1(ρ) near the coexistence curve is given byI1(ρ) ∼14ρln2 4ρ + 2 ln ρ,ρ →∞. (A.8)35

The leading terms of f(x) for large x aref(x) ∼1 + 3 ln(4/27)2(N + 8) ǫ + O(ǫ2)xγ ,x →∞(A.9)whereγ = 1 +N + 22(N + 8)ǫ + (N + 2)(N2 + 22N + 52)4(N + 8)3ǫ2 + O(ǫ3) . (A.10)Of course, this result for γ and the results for β and δ in (A2) and (A3) areconsistent with the scaling relations (2.12) .

We saw in section 3 that the behaviour(A.9) determines the sigma and pion masses for H = 0 and t > 0.In section 3, in order to determine m2σ near the coexistence curve we neededthe behaviour of f′(x) for x →−1. In this region, f(x) is given by [11]f(x) ∼(x + 1)(1 + ǫ"N −12(N + 8) ln(x + 1) + 3(1 + 3 ln(2/3))2(N + 8)#+ ǫ2"(N −10)(N −1)8(N + 8)2ln2(x + 1)+N −14(N + 8)2hN + 27 + 18 ln 2 −9 ln 3 −60N + 8iln(x + 1)#+ O(ǫ3)).

(A.11)One next inverts (A.11) to obtain x + 1 in terms of y = f(x),(x + 1) = c1y + ˜c2y1−ǫ/2 + Oy2−O(ǫ),(A.12)differentiates the result with respect to x, and obtainsβm2σMδ−1 →f′(x) =1c1 + c2y−ǫ/2 for x →−1 ,(A.13)withc1 =9N + 8 1 −ǫ2(N + 8)h(N + 8) ln 2 −9 ln 3 + 25N2 + 142N + 769(N + 8)i!+ O(ǫ2)(A.14)36

andc2 = ˜c21 −ǫ2= N −1N + 8(1 −ǫ21 −1(N + 8)29(N + 8) ln 3 + 22N + 116)+ O(ǫ2) . (A.15)As H is lowered to zero at fixed t < 0, m2σ at first tends toward a constant, andthen goes to zero according to m2σ ∝Hǫ/2 when the c2 term takes over from thec1 term.

For N = 4, this occurs whenHMδ <∼c2c12/ǫ∼131 + 0.96ǫ2/ǫ∼0.4 for ǫ = 1 . (A.16)Of course, the numerical value for ǫ = 1 should not be taken too seriously.

Thequalitative result is clear nevertheless.We end this appendix by describing how the figures in section 3 were obtained.When evaluated at ǫ = 1, the expression (A.5) for f(x) has several problems. First,at large x it does not grow as xγ.

Rather, it increases like x ln2 x. Also, for x →−1,f(x) given by (A.5) does not satisfy (A.12) .

In fact, for x <∼−0.95, f(x) < 0which is unphysical. Both of these problems arise because we are setting ǫ = 1 inan expansion of f(x) to order ǫ2 which is valid for ǫ →0.

In order to illustratethe correct qualitative behaviour, we constructed a function f(x) which smoothlyinterpolates between (A.9) at large x, (A.12) near x = −1, and (A.5) in between.Using this function f(x), we obtained Figure 1 by solving H/Mδ = f(t/M1/β)for M at various values of t and H. We then calculated the results for m2π andm2σ shown in Figures 2 and 3 using (3.6) and (3.8) . Because of the limitationsimposed by working at ǫ = 1, all three figures should be viewed as illustrations ofqualitative behaviour.37

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FIGURE CAPTIONS1) The order parameter M as a function of reduced temperature t ≡(T −Tc)/Tcfor magnetic fields H = 0, 0.002, 0.005, and 0.02. Like t, both M and H aredimensionless.

They are obtained from their dimensionful counterparts bydividing by non-universal dimensionful constants defined in such a way thatfor t < 0 and H = 0 the order parameter is given by M = (−t)β, and fort = 0 it satisfies M = H1/δ.2) m2π (Figure 2a) and m2σ (Figure 2b) as functions of t for H = 0, 0.002, 0.005,and 0.02. Since M and H are in dimensionless scaled units, so are m2π andm2σ.

For t = 0, m2π = H(δ−1)/δ and m2σ = δm2π. For H = 0 and t > 0 (and forlarge enough t for any H) m2σ = m2π ∼tγ.

For t < 0 and H →0, m2π ∼Hand the sigma mass decreases to zero as shown in Figure 3.3) m2σ as a function of H for t = −0.2. For large H it behaves as if it will benon-zero at H = 0, but in fact for H →0 it decreases like m2σ ∼Hp whereto lowest order in ǫ, p = ǫ/2 = 1/2.40

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