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School of Natural Sciences

arXiv:hep-ph/9308341v1 26 Aug 1993IASSNS-HEP-93/48July 1993hep-ph/9308341Remarks on Hot QCD⋆Frank Wilczek†School of Natural SciencesInstitute for Advanced StudyOlden LanePrinceton, N.J. 08540⋆Invited talk given at the Quark Matter ’93 conference, Borlange Sweden June 1993.† Research supported in part by DOE grant DE-FG02-90ER40542

ABSTRACTAfter briefly reviewing work I’ve done recently with Krishna Rajagopal on thenature of the equilibrium phase transition in QCD at finite temperature and zerobaryon number and on the possibility of forming misaligned vacuum by a quench,I discuss – also briefly – two new items. These are extension of the argumentsconcerning the order of the transition to to 2+1 dimensions, relating in this casegauge to conformal field theories, that could be explored in numerical experiments;and the energy budget at the chiral transition and its relevance to the nature ofthe transition and to quenching.2

The bulk of the material in the talk as delivered was taken from two paperswritten with Krishna Rajagopal. Here I will only give a telegraphic summary ofthose papers [1, 2] in Sections 1 and 2.

Sections 3-4 are devoted to the new itemsmentioned in the Abstract.1. Equilibrium Phase TransitionBecause of asymptotic freedom, one expects that at sufficiently high tempera-ture hadronic matter in thermal equilibrium will form a weakly interacting plasmaof quarks and gluons.

Such a plasma differs qualitatively from hadronic matter atlow temperature in two important respects. First, of course at low temperaturesone does not observe quarks and gluons but rather color neutral objects – theproperty of confinement.

Second, a large body of phenomenology connected withsoft pion theorems indicates that SU(2) × SU(2) chiral symmetry suffers a largespontaneous breakdown, in addition to being intrinsically but relatively weaklyviolated by the existence of small but u and d quark masses. It may also be usefulto consider SU(3)×SU(3) chiral symmetry, though in that case it is unclear whenit is appropriate to consider the intrinsic violation as small.

In any case, the spon-taneous portion of the breaking clearly is not contained in the standard descriptionof QCD as a weakly interacting quark-gluon plasma at high temperature.It is natural to ask whether these qualitative changes are marked by actualphase transitions, and if so what is the nature of these transitions.Is there afirst order transition with discontinuities in thermodynamic functions and releaseof latent heat, or a second order transition where the primary thermodynamicfunctions are continuous but not analytic – or perhaps no true phase transition atall? In this connection let me remind you that the passage of ordinary neutral gasinto an ionized plasma at high temperature, which in some respects is analogousto deconfinement, although it connects two very different kinds of matter, is notmarked by any singular behavior in the thermodynamic functions.3

How does one approach such questions theoretically? A powerful method isto construct order parameters, usually characterizing the state of symmetry of thesystem, that are zero in one phase but not in the other.

Classic examples of orderparameters are the magnetization or staggered magnetization which characterizeferromagnetic respectively antiferromagnetic states. These order parameters neces-sarily vanish in a state with rotational symmetry, such one expects above the Curieor Neel point.

If one has an order parameter of this kind one can be sure that thereis a singularity in passing between the states, because an analytic function cannotvanish in a finite range without vanishing everywhere. On the other hand one mayhave phase transitions with no order parameter – the thermodynamic functions onthe two sides of the transition temperature may just be different functions.

This isusually the case for liquid-gas transitions. It is not unusual to find that for a givensubstance the passage from liquid to gas is or is not marked by a true phase tran-sition depending on the value of the amibient pressure, which could not happen ifthere were an order parameter marking the difference.It would appear that the partition function Z= Tre−H/T, being a sum ofanalytic terms, must be an analytic function of the temperature – and indeed, in afinite volume, it is.

However one is interested in the limit of large volumes, wherethe sum in the trace may contain an infinite number of terms and analyticity maybe lost. If the partition function is not analytic, this must be due to some subtletyin taking the infinite volume limit.One expects that such a subtlety must beassociated with the development of correlations which extend over an infinite range.If the transition is second order, the correlation length must go continuously toinfinity at the transition point.

In particular, sufficiently close the transition pointthe correlation length will greatly exceed any “microphysical” length associatedwith the problem, e.g.the lattice spacing (or, for QCD, Λ−1). These observationslead to the concepts of scale invariance and universality, which together form thefoundation of the modern theory of second order phase transitions.Scale invariance follows from the idea that non-trivial correlations on lengthscales much larger than the microscopic one can only depend on the ratio of the4

length in question to one overall diverging correlation length. (Generically thereshould be just one fundamental diverging length, since there is one parameter –the temperature – which must be tuned to reach the critical point.

Another wayof saying this is that all divergent quantities should be given as functions of thereduced temperature t = |T −TcTc |.) Universality follows from the idea that onlymodes with long-wavelength fluctuations in equilibrium and their low-dimensioninteractions among themselves are important for the singularities, and that the onlylong-wavelength modes that exist are likely to be those required by the symmetriesof the order parameter.

Together these principles tell us that in constructing modelsof the singularities of thermodynamic functions near second order phase transitions,it is necessary and sufficient to look for the simplest scale-invariant models in theappropriate dimension with the appropriate symmetry.How does all this apply to QCD? What are the available order parameters?For confinement, the only known order parameter is the Polyakov loop [3]P = ⟨expZiAτdτ⟩(1.1)where one is taking a thermal average, Aτ is the imaginary-time component of thegauge field written as a matrix in the fundamental representation, and the trace isover one period β = T −1 of the imaginary time.

ln P is basically the free energyassociated with adding a static color source with the quantum numbers of a quark.It vanishes when quarks are confined. However it has no reason to vanish whenquarks are not confined, and indeed it does not vanish in perturbation theory.The Polyakov loop is not expected to vanish when the color flux associatedwith the color source can be screened.

Indeed the reason why the energy cost ofinserting the color source in the confined phase is infinite is because there is a non-trivial flux extending from the source that cannot terminate, and this flux disruptsthe vacuum wherever it threads, costing a finite energy per unit length. On theother hand if we are considering a gauge theory with dynamical quarks, then thequarks that are inevitably present at non-zero temperature need only re-arrange5

themselves a bit to soak up the flux. Strictly speaking one then cannot distinguishconfinement from screening.

There is no strict order parameter for confinementin the presence of dynamical quarks at finite temperature (similar to the case ofionization of an ordinary gas).For an SU(N) gauge group, the Polyakov loop parameter transforms non-trivially under the center of the gauge group, that is ZN. When it vanishes theglobal ZN symmetry under gauge transformations which approach these centralelements at infinity is unbroken, while if it does not vanish this symmetry is broken.

(Note that local gauge invariance is never broken. )The universality class of the deconfinement transition for pure glue SU(N)is therefore described by a simple model with a discrete global symmetry group.For N = 2 it is the Ising model and for N = 3 it is the three-state Potts model.These are well studied models.

The first of them has a second-order transitionin 3 dimensions, while the second, for quite fundamental reasons, does not. Thisled Svetitzky and Yaffe [4] to predict that pure glue SU(2) would probably havea second-order, but pure glue SU(3) must certainly have a first-order, transition.After some struggles, their prediction was verified by direct numerical simulation[5].For the chiral transition with Nf massless flavors, the simplest order param-eter one can imagine is given by an Nf × Nf matrix M of scalar fields, whichparametrizes the condensate ⟨¯qiL qjR⟩.

This matrix is complex. For Nf = 2 onecan require that M is of the form σ + i⃗π · ⃗τ, with four independent real compo-nents.

It turns out that Nf = 2 is a very special case [6]. It is in the universalityclass of a four-component magnet, a model which has been extensively studied inthe condensed matter literature [7].

This model is known to have a second-ordertransition. Thus there is a scale invariant candidate theory to describe the singularbehavior near a second-order chiral symmetry restoration transition in 2 masslessflavor QCD.

For more than two flavors the situation is very different. The morecomplicated models appropriate to this case have not been studied so extensively.6

However all the evidence, both analytical [8] and numerical [9], is that they donot support a second-order transition. Thus there is no candidate theory to de-scribe the singular behavior that would occur near a second-order chiral symmetryrestoration in QCD with 3 or more massless flavors.

A probable interpretation [6]is that in these cases near the transition fluctuations grow so large that they inducea first-order transition.These considerations suggest the possibility of a second-order transitionuniquely in the case Nf=2. Remarkably, this pattern is consistent with ex-isting numerical evidence [10].When one has a second-order transition, it is possible to make some precisepredictions for the nature of the singularities near the transition, using the pow-erful concepts of universality and scale invariance as mentioned above.

Let meemphasize again that these are precise predictions. The most characteristic resultsare predictiions of the critical exponents, which specify the power dependence ofvarious quantities such as the magnetization (which in QCD language translatesto the magnitude of the chiral condensate) the specific heat, or many others, ont = |T −TcTc |, These predictions are similar in spirit to the famous predictions forthe behavior of moments of QCD structure functions as calculable powers of log-arithms of Q2, but the calculations involved are much more arduous.

The pointis that whereas the ultraviolet fixed point of QCD occurs at zero coupling, the in-frared fixed point governing the chiral phase transition occurs at a very large valueof the coupling, and extraordinary methods have to be used to extract quantitativeresults. Fortunately the requisite work, involving calculating hundreds of graphsin perturbation theory up to six loops, using analytic methods to estimate theasymptotic behavior of high orders of perturbation theory, and joining the two bysophisticated resummation methods, was performed in a tour de force by Baker,Meiron, and Nickel.If we accept that for two massless quarks the transition is second order whilefor three massless quarks it is first order, there is an interesting question how one7

behavior goes over into the other as the mass of the third quark varies. A verypretty possibility is that as the third (strange) quark mass is raised continuouslyfrom zero the discontinuities associated with the first-order transition get smallerand smaller, eventually vanishing at the so-called tricritical point, after which thetransition becomes second-order.

The behavior in the the immediate neighborhoodof the tricritical point is governed by a four-component asymptotically free massless(φ2)3 theory, which features logarithmic corrections to mean field theory.Thetricritical point is characterized by a particular value mcrit. of the strange quarkmass.Existing numerical evidence while very crude suggests that the physicalstrange quark mass is larger than mcrit..

However it is not impossible that thestange quark mass is quite close to the critical value, and of course in numericalexperiments one can in principle vary the masses to probe the suspected tricriticalregion. It is noteworthy that the specific heat, which generically has a cusp at thesecond-order transition, develops an actual discontinuity at the tricritical point.Another direction in which the analysis may be extended is to the consider-ation of time-dependent behavior near the critical point.

One can analyze howthe transport equations change near the critical transition. The main qualitativephenomenon is critical slowing: there is a diverging correlation time as well as adiverging correlation length, as the system finds it difficult to relax the very long-wavelength fluctuations that occur.

Thus transport coefficients generally acquiresingularities at the transition. Dynamical critical behavior probes more physicsthan the static behavior: models in the same static universality class can havedifferent dynamical critical exponents; for example, ferromagnets and antiferro-magnets are in the same static universality class but different dynamic universalityclasses, basically because the ferromagnetic order parameter, the magnetization,becomes a conserved quantity in the long-wavelength limit – unlike the staggeredmagnetization – which makes long-wavelength fluctuations relax more slowly.

QCDwith two massless quarks appears to be in the dynamic universality class of thefour-component Heisenberg antiferromaget. A major quantitative consequence ofthe analysis is that the correlation time is predicted to scale as the 3/2 power of8

the correlation length. Note that if we translate this behavior into a predictionfor the form of real-time Green functions we get funny cuts at zero frequency andwave vector, which is very different from a naive particle exchange model and ofcourse from any naive extrapolation of the behavior of weakly interacting plasma.If we take the indications for a second-order phase transition with two masslessquarks and the effective “massiveness” of the physical strange quark at face value,what are the consequences?

For numerical experiments, the we find ourselves in ahappy situation. There is a wealth of detailed, quantitative predictions waiting tobe tested.

For cosmology the situation is either dull or reassuring, depending onyour point of view. The expansion of the universe is very slow indeed on the rel-evant strong-interaction timescales, and equilibrium should be closely maintained.The non-zero masses of the up and down quarks mean that there is really nophase transition at all.

Interesting relics that might have occurred for a stronglyfirst-order transition, including gravity waves, inhomogeneous nucleosynthesis, andpossible production of exotic matter, do not occur. For heavy ion collisions, if ther-mal equilibrium is produced and maintained, the conclusion is similar.

Even theinteresting long-range correlations that might have been expected to arise near asecond order transitions, and to give rise to interesting phenomenological signa-tures as I shall discuss in a moment, are not quantitatively significant due to thefinite pion masses (that is, inverse correlation lengths) that are not much smaller,and maybe not at all smaller, than the temperature at the transition. Howeverthat’s a very big if as you know, and it is interesting to consider another quitedifferent idealization of the physics.9

2. Quenching and Misaligned PatchesIf instead of cooling a bar of iron slowly through its Curie point one suddenlyplunges it into ice water, one is said to have quenched the magnet.

It may be thatit is not inappropriate to model what occurs following a heavy ion collision, as thehot plasma comes into contact with cold empty space outside, as a quench. Insofaras it is plausible to map the important degrees of freedom for QCD into a magnetmodel, i.e.

if the pions and the chiral condensate are mainly what we must keeptrack of, the analogy is quite close.The dynamical evolution following a quench is quite different from that forcooling through equilibrium.One expects that broadly speaking whereas nearequilibrium the question is the typical size of correlated regions, which individualgrow, shrink, come into being and pass away, following a quench the system isracing to the ordered state, the dynamics is unidirectional, and the question is howthe domains evolve (grow) in time.There is a substantial condensed matter literature on problems of this type,including very recent work [11]. Also related problems arise in the cosmologicalmodels (“texture models”) where the fluctuations responsible for triggering theformation of structure in the universe are ascribed to the formation, growth, andeventual collapse of domains during a cosmic phase transition [12].

The QCD prob-lem has its own special features, however, and requires a fresh analysis. The mainnew features that are central to the QCD problem are the fact that the symmetry isintrinsically broken, and that one considers hyperbolic rather than diffusive equa-tions.

In many condensed matter problems diffusive equations for the dynamicalvariables of interest are appropriate, because these variables are in contact withother degrees of freedom (e.g. phonons ) whose effect is modelled by appropriatediffusive transport equations.

However in the QCD problem, and probably also insome condensed matter problems, one should use the true microscopic equationswhich of course are hyperbolic.The fact that the symmetry is intrinsically broken means that every region of10

space is heading toward the same ground configuration, and interest focuses onthe nature of the approach. Do different regions of space relax independently tothe ground configuration (small oscillations around the σ direction), or do largealigned regions form and then relax coherently?

The latter possibility could havedramatic consequences, because it would mean that large regions of space have amisaligned condensate [13], a classical field configuration that might be expectedto emit classical pion radiation – to “lase” pions – as it relaxes.To model a QCD quench we take a representative configuration of the σ and πfields from the ensemble weighted by the free energy at the pre-quench temperatureT, and then evolve it according to the zero temperature equations of motion. Wefind that for broad ranges of the parameters large coherent structures do formfollowing a quench.

We believe that the fundamental mechanism underlying thisbehavior is the following. The (mass)2 of a Nambu-Goldstone field is zero in thetrue ground state, due to a cancellation between an intrinsically negative bare(mass)2 and a positive contribution from the interaction with the condensate:m2 = −µ2 + 2λv2 = 0 (groundstate)(2.1)However in a quench the vacuum expectation value v2=⟨σ⟩2 starts centeredaround zero, not its ground state value µ2/2λ.

Thus m2 can easily be negative,and according the the dispersion relationω2 = k2 + µ2(2.2)modes with sufficiently small spatial frequencies will grow – and the longer thewavelength, the more the growth. Though the real situation (that is, our idealizedmodel of a quench) is more complicated in various ways, including the non-linearityof the equations, the phenomenon suggested by this simple picture does occurroughly as expected.Taking a realistic intrinsic symmetry breaking into account does not destroythe phenomenon, because the intrinsic (mass)2 of the pion is substantially less than11

µ2. (The ratio of the two is basically the square of the ratio of pion and sigmamasses; see section 4 below).Thus there seems to be a good chance that large regions of misaligned vacuummight form.

The radiation from such a region as it relaxes will be quite structuredand unusual.The 4-component classical field (σ,⃗π) will oscillate in the planedetermined by sigma and some definite direction of the vector ⃗π. The radiationis a coherent configuration of pions in that direction.

These radiated pion clumpsshould have small relative momentum, and each one will have a fixed ratio ofcharged to neutral pions. The probability distribution for a given charge ratio isProb.

(R) = 12R−12(2.3)whereR ≡π0π0 + π+ + π−. (2.4)This distribution is, of course, highly non-Gaussian.

For example the probabilityof having less than 1% neutral pions is 10% !3. Equilibrium Phase Transition One Dimension DownIt would be interesting to test the logic of the argument leading to the expec-tation of a possible second order deconfinement phase transition for color SU(2)pure glue theory and chiral symmetry restoration phase transition for two flavorsof massless quarks, but definitely first order for deconfinement in color SU(3) pureglue theory or chiral symmetry restoration with three or more flavors, in otherexamples.

A simple possibility lies near at hand: one can study the correspond-ing questions in one fewer spatial dimension. While the direct relevance of suchstudies to any achievable laboratory situation is doubtful, they have the advantageof being relatively easy to access by numerical experiments.

Also as we shall seesome rather striking situations might arise.12

Following the same arguments as before, we relate the possible second ordertransitions of 2+1 dimensional gauge theories to the existence of 2 dimensionalscale invariant theories with the same symmetries.The study of 2 dimensional scale invariant theories has been a large thrivingactivity in recent years [14]. In two dimensions scale invariance leads to a muchlarger and more powerful symmetry group of conformal symmetry transformations,and use this symmetry to carry the analysis of scale or conformal invariant theoriesa long way.

In favorable cases critical exponents can be calculated exactly andthere are tractable algorithms for calculating just about any correlation functionof interest. Thus a question of interest becomes, which conformal field theoriescorrespond to which gauge theories at their critical points.It seems quite plausible that the SU(N) pure glue theories have deconfinementphase transitions associated with the ZN central symmetries discussed above.

ForN = 2 one has the symmetry class of the Ising model, for N = 3 the three-statePotts model, and so forth.Whereas in 3 dimensions the phase transition wassecond order for N = 2 but necessarily first order for N = 3, here they can bothbe second order. As I said there is a quite a full undertanding of the correlationfunctions of these models, though it may not be entirely trivial to set up thedictionary translating these results into gauge theory language.The issue of chiral symmetry breaking in 2+1 dimensions is complicated bytwo factors, which make the situation very different from what one has in 3+1dimensions, where one has breakdown of a continuous chiral symmetry.First,there is no chirality in 2+1 dimensions; and second, there can be no spontaneousbreaking of a continuous symmetry.

Let me recall these facts for you.In 2+1 dimensions we can use (essentially) the ordinary two by two Paulimatrices as gamma matrices, say (γ0, γ1, γ2) = (σ2, iσ1, iσ3). These are chosenin such a way that they are all imaginary, so that the Dirac equation can besolved with real (Majorana) spinors.However we shall focus on fermions withthe quantum numbers of quarks, which are necessarily complex anyway.More13

important for present purposes is the fat that the product γ0γ1γ2 reduces to a purenumber. This implies that it is impossible to make a chiral projection.

Anotherfact that will be important to us is that the parity operation of reflection in oneaxis acts as (say) γ0γ2 on the spinors, so that the operator ¯ψψ is odd under parity(and time reversal).There is no spontaneous breaking of continuous symmetries in 2 dimensionalquantum field theory at zero temperature (Coleman theorem, [15]) – or for 2+1dimensional theories at non-zero temperature (Mermin-Wagner theorem, [16]).These are very nearly the same theorem, since the zero-frequency componentsof the putative Nambu-Goldstone fields have the infrared singularities (long wave-length fluctuations) of the 2 dimensional theory. These fluctuations are actuallydivergent at long distances; their divergence undoes the hypothesized symmetrybreaking or long-range order.

This shows the internal inconsistency of that hy-pothesis that the continuous symmetry breaks, leading to the theorems.Still there is something to investigate. The theory with f flavors of masslessquarks has a U(f) flavor symmetry and also a parity symmetry.

As we have seeneven a common mass terms, or a quark-antiquark condensate without any preferreddirection in flavor space, breaks parity. Thus if we assume that a flavor singletcondensate develops at zero temperature, but goes away at high temperature, thenthere is the possibility of a phase transition with Z2 symmetry, conrresponding toparity restoration.

If this phase transition is second order, then it could be mappedonto the Ising transition.I think it would be quite interesting to carry out appropriate numerical exper-iments on lower-dimensional QCD at finite temperature, to check whether second-order transitions with the predicted exponenets in fact occur. Hansson and Zahed[17] have also emphasized that the high temperature behavior of the 2+1 dimen-sional gauge theory appears to be tractable theoretically and to form a good testingground for analytical methods which can also be applied to the more difficult 3+1case.14

4. Energy Budget for Chirality and DeconfinementIt is an interesting exercise to estimate the energy locked up in the chiralvacuum, and to compare it with the energy of the glue near the phase transition.To estimate of the energy density in the chiral vacuum, begin with the sigmamodel potentialV (σ) = −µ2σ2 + λσ4 .

(4.1)Elementary calculations lead to the vacuum expectation value ⟨σ⟩=pµ2/2λ,the sigma mass mσ = 2µ and the energy density E = −µ4/4λ at the minimum,where the symmetric vacuum energy density is normalized to zero. Identifying⟨σ⟩= fπ we arrive at the estimateE = −fπ2m2σ8(4.2)of the condensation energy in terms of observables.

Inserting experimental values,one finds|E| ≈5 × (100 Mev)4 . (4.3)This estimate, based on classical reasoning and on taking the broad observed σresonance as the embodiment of the field in the sigma model, is certainly crude.However it may not be inappropriate to use it to draw one tentative qualitativeconclusion, as follows.The energy density for an ideal gas of SU(N) color gluons at temperature T isEglue = (N2 −1)π215T 4 ≈4.8 × T 4 (for N = 3) .

(4.4)There is also a contribution from Nf species of massless quarks, with the prefactor74NNf replacing N2 −1 in (4.4). For N = 3 and two light flavors, this quarkcontribution is just a bit larger than (4.4).

The simple qualitative point I want15

to stress is that the sorts of energy densities associated with the chiral conden-sation are, for T∼100 Mev and for the stated values of N and Nf, roughlycomparable to the energy densities of the entire quark-gluon plasma. This com-parison is important in thinking about the question whether the singular parts ofthe thermodynamic functions, which are the universal quantities we can addresstheoretically, are quantitatively important in the relative to the thermodynamicfunctions themselves.

For example is the predicted cusp in specific heat a majorfeature or just a tiny dimple on an otherwise smooth curve? This is a non-universalquestion, whose answer will depend on details of the microscopic theory includingthe absolute value of the transition temperature (which is very much conditionedon such details).

The estimates above suggest that if T ≈100 Mev, as is perhapssuggested by the numerical work, then the energy controlled by the value of theorder parameter is not much less than the total energy, so that the singular partsshould may not be overly difficult to discern. Existing evidence suggests that Tis a little larger than this, but on the other hand probably one should not expectthe gluon plasma to acquire its full (non-interacting) energy immediately followingthe transition.

In any case, it is important to emphasize that the foolproof wayto look for manifestations of critical singularities is to study the behavior of theorder parameters and closely related quantities, which is completely dictated bythe universal theory.The issue of the energy difference is also important in assessing the appro-priateness of the quench model as an idealization.The essence of the quenchphenomenon is that the modes associated with spontaneous symmetry breakingare slower to relax to equilibrium than the generality of modes. Thus when energyis suddenly drained from the quenched system these modes are trapped in the pre-existing configuration, but at a low temperature.

Clearly the most favorable caseis when the energy associated with the symmetry-breaking modes is commensuratewith the total energy, so that we are not making absurd demands on the coolingprocess. Expansion of the plasma provides an excellent cooling mechanism, but weshould not require miracles of it.16

Acknowledgements: As mentioned before, the first two sections report joint workwith Krishna Rajagopal, who also made some helpful remarks on the other sections.I am also grateful to Chetan Nayak and to Bert Halperin for important insightsregarding the 2+1 dimensional theory.17

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