17 January 2002

Physics Letters B 525 (2002) 95–100www.elsevier.com/locate/npe

Degrees of freedom and the deconfining phase transition

Adrian Dumitrua,b, Robert D. Pisarskib

a Department of Physics, Columbia University, New York, NY 10027, USAb Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA

Received 19 June 2001; received in revised form 26 November 2001; accepted 3 December 2001Editor: W. Haxton

Abstract

There is a sharp increase in the relative number of degrees of freedom at the deconfining phase transition. Characterizing thisincrease using the Polyakov Loop model, we find that for three colors, where there is a nearly second order deconfining phasetransition, the medium-induced energy loss turns on rapidly aboveTc. As expected, energy loss increases proportional to therelative number of degrees of freedom. In addition, however, there is a logarithmic dependence upon the screening mass, whichchanges markedly near the would be critical point. Thus this sensitivity to the screening mass can be used to probe non-ideal,nearly critical, behavior in the quark–gluon plasma. 2002 Elsevier Science B.V. All rights reserved.

Experiments indicate that for the central collisionsof large nuclei,A∼ 200, there are marked changes be-tween energies of

√s/A = 17 GeV, at the SPS, and

130 GeV, at RHIC [1]. Comparing centralAA col-lisions topp, the spectrum of semi-hard particles israther different. At the SPS, inAA the hardpt spec-trum, scaled by the number of binary collisions, is en-hanced overpp. At RHIC, the opposite is true: thesemi-hardpt spectrum per nucleon–nucleon collision,is suppressed in centralAA, relative either to periph-eralAA, orpp̄ [2]. This could be the result of “energyloss” [3–5], where a fast colored field loses energy asit passes through a thermal bath. In peripheralAA col-lisions, secondary hadrons are distributed anisotropi-cally in the transverse momentumpt [6]. Experimen-tally, this azimuthal anisotropy increases withpt untilpt ∼ 2 GeV, at which point it flattens [7]. This flatten-ing may also be due to energy loss [8].

E-mail addresses: dumitru@quark.phy.bnl.gov (A. Dumitru),pisarski@quark.phy.bnl.gov (R.D. Pisarski).

In the limit of infinitely large nuclei,A→ ∞, it isplausible that the initial energy density produced ina centralAA collision—at a fixed value of

√s/A—

evolves into a system in equilibrium at a temperatureT . With great optimism, assuming thatA ∼ 200 isnearA= ∞, one might imagine that the difference be-tween SPS and RHIC is because temperatures reachedat RHIC exceedTc, the critical temperature for QCD.

Thus it is of interest to know how quantities changeas one goes through the phase transition. In this Letterwe give an analysis in terms of the Polyakov Loopmodel [9–11].

In QCD, there is a large increase in the number ofdegrees of freedom at the deconfining phase transi-tion. We count degrees of freedom as appropriate forthe pressure of free, massless fields at non-zero tem-perature, so if each boson counts as one, then eachfermion counts as 7/8. In the hadronic phase, pionscontributecπ = 3 ideal degrees of freedom. By as-ymptotic freedom, at infinite temperature QCD withthree flavors of quarks is an ideal quark–gluon plasma,

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01424-1

96 A. Dumitru, R.D. Pisarski / Physics Letters B 525 (2002) 95–100

with cQGP = 4712 degrees of freedom. This is an in-

crease of more than a factor of ten.To measure the change in the number of degrees of

freedom, we introduce the relative pressure,n(T ): ata temperatureT , this is the ratio of the true pressure,p(T ), to that of an ideal quark–gluon plasma,pideal=cQGP(π

2/90)T 4

(1)n(T )≡ p(T )

pideal.

By asymptotic freedom, QCD is an ideal gas at infinitetemperature, and so

(2)n(∞)= 1.

For T < ∞, corrections to ideality are determinedby the QCD coupling constant,αs ∝ 1/ log(T ), withn(T )− 1 ∝ −αs [12].

For an exact chiral symmetry which is sponta-neously broken by the vacuum, about zero temperaturethe free energy is that of free, massless pions. Thus atzero temperature, the relative pressure is the ratio ofthe ideal gas coefficients [13]

(3)n(0)= cπ

cQGP.

At low temperature, corrections to ideality are givenby chiral perturbation theory for massless pions,n(T )−n(0)∼ +(T /fπ)4n(0), with fπ the pion decayconstant. In QCD, pions are massive, and the relativepressure is Boltzmann suppressed at low temperature,n(T )∼ exp(−mπ/T )(mπ/T )5/2, son(0)= 0.

Given the great disparity betweencπ andcQGP, con-sider an approximation where the hadronic degreesof freedom are neglected relative to those of the de-confined phase.1 Then the relative pressure vanishesthroughout the hadronic phase,n(T ) = 0 for T < Tc.The question is then: how does the relative pressure gofrom zero atTc, when deconfinement occurs, to nearone at higherT ?

This can be answered by numerical simulations ofLattice QCD [15]. Consider first quenched QCD, withpure glue and no dynamical quarks, which is close to

1 This is valid in the limit of a large number of colors. For anSU(Nc) gauge theory coupled toNf flavors of massless quarks with

a left–right chiral symmetry,cπ =N2f

−1 andcQGP= 2(N2c −1)+

7Nf Nc/2. Gluons dominate pions in the limit of largeNc at fixed

Nf , cπ /cQGP∼ (Nf /Nc)2 [14].

the continuum limit [16]. For three colors, the Latticefinds no measurable pressure in the hadronic phase(glueballs are heavy), so our approximation ofn(T )=0 whenT < Tc is good.n(T ) increases quickly aboveTc, and is ∼ 0.8 by T ∼ 2Tc. To characterize thechange in the relative pressure, consider the ratio ofe − 3p, where e(T ) is the true energy density ofQCD, to the energy of an ideal quark–gluon plasma,eideal= 3pideal

(4)e− 3p

eideal= T

3

∂n

∂T.

Lattice simulations find that this ratio has a sharp“bump” at ∼ 1.1Tc, suggesting that the relative pres-sure changes quickly, when the reduced temperature

(5)t ≡ T

Tc− 1,

is small,t ∼ 0.1.The Lattice is more uncertain with dynamical

quarks. The pions are too heavy, and it is not near thecontinuum limit. So far, the Lattice finds thatn(T /Tc)

is about the same with dynamical quarks as without[15,17]. This suggests that the pure glue theory maybe a reasonable guide to how the relative pressureincreases aboveTc. The approximate universality ofn(T /Tc) is remarkable. At present, the Lattice findsno true phase transition in QCD, withTc smaller by∼ 0.6 than in the quenched theory [15]. Indeed, eventhe ideal gas coefficients are very different:cQGP isonly 16 in the quenched theory, versus 471

2 in QCD.The greatest change with dynamical quarks is a

small, but measurable, pressure in the hadronic phase.While in the quenched theoryn(T ) ∼ 0 for T < Tc,with dynamical quarks, althoughn(0) ∼ 0, there is anon-zero relative pressure at the critical temperature,with n(Tc) ∼ 0.1 [15]. Indeed, with no true phasetransition, an approximateTc can only be defined asthe point where the relative pressure increases sharply,reachingn∼ 0.8 by 2Tc [15].

The Polyakov Loop model [9–11] is a mean fieldtheory for the relative pressure. In a pure glue theory,the expectation value of the Polyakov Loop,�0(T ),behaves like the relative pressure: it vanishes whenT < Tc, and is non-zero aboveTc. Indeed, again byasymptotic freedom,�0 → 1 asT → ∞. The simplest

A. Dumitru, R.D. Pisarski / Physics Letters B 525 (2002) 95–100 97

guess for a potential for the Polyakov Loop is

(6)V (�)= −b2

2|�|2 + 1

4

(|�|2)2.

Defining�0 as the minimum ofV (�) for a givenb2(T ),the relative pressure is given by [9–11]

(7)n(T )= −4V (�0)= �40,

b2 > 0 above Tc (b2(T ) → 1 for t → ∞), and< 0 belowTc. Thus if the relative pressure changeswhen the reduced temperaturet ∼ 0.1, the change for�0(T )∼ n1/4 is even more rapid, within 2.5% ofTc.

For two colors, (6) is a mean field theory for asecond order deconfining transition [18]. The� field isreal, and so the potential defines a mass:(m�/T )

2 =(1/Zs)∂2V/∂�2, with

(8)m�(T )

T∝ �0 ∼ n1/4,

whereZs is the wave function normalization con-stant for �, Zs = 3/g2, up to corrections of orderg0 [19]. This is measured from the two point func-tion of Polyakov loops in coordinate space,∝ (1/r)×exp(−m�r) asr → ∞.

For three colors,� is a complex valued field, and aterm cubic in� appears inV (�), −b3(�

3+ l∗3)/6. Thisproduces a first order deconfining transition, where�0jumps from 0 atT −

c to �c = 2b3/3 atT +c [10]. The�

field has two masses, from its real(m�) and imaginary(m̃�) parts. AtT +

c ,√Zs m�/T = �c; from the Lattice,√

Zs m�/T ∼ 0.3 [15], which givesb3 ∼ 0.45. Thissmall value of b3 reflects the weakly first orderdeconfining transition for three colors [15,16]. Themass for the imaginary part of� is

√Zs m̃�(T )/T ∝√

b3� ∼ n1/8; at T +c , m̃�/m� = 3. With dynamical

quarks, in principle a term linear in�, −b1(�+ �∗)/2,can also appear inV (�) [20]. If the pion pressure isincluded belowTc, however,b1 is very small,� 0.03.

Thinking of �0 provides a useful way of viewingthe deconfining phase transition. For a strongly firstorder transition—as appears to occur for four or morecolors [21]—�0, jumps from zero belowTc, to avalue near one just aboveTc. As �0 is near one, thedeconfined phase is presumably well described as anearly ideal quark–gluon plasma [22]. In this case,there is a hadronic phase belowTc, and a quark–gluonplasma fromTc immediately on up.

In contrast, for three colors the deconfining transi-tion is weakly first order. As the energy density is dis-continuous atTc, for small t the relative pressure islinear in the reduced temperature

(9)n(T )∼ 3rt,

herer ≡ e(T +c )/eideal(Tc) is the ratio of the energies

at Tc, in the deconfined phase versus an ideal quark–gluon plasma. For quenched QCD,r ∼ 1/3 [16],which givesn(T ) ∼ t , and so�0(T ) ∝ t1/4. Exceptvery nearTc, this simple estimate agrees with morecomplicated analysis usingb3 �= 0 [10,11]. For exam-ple, at only 5% aboveTc, this estimate gives�0 ∼0.051/4 ∼ 0.5. For three colors, then, there is a (non)-ideal quark–gluon plasma only at temperatures above∼ 2Tc; betweenTc and ∼ 2Tc, the Polyakov Loopdominates the free energy, going from∼ 0.5 at 1.05Tcto ∼ 1 by 2Tc.

The difference between these two scenarios: astrongly first order transition, where�0(T ) is approxi-mately constant aboveTc, and nearly second order be-havior, where�0(T ) changes significantly, is in prin-ciple observable. As an example, consider energy lossfor a fast parton, with a high energyE. We first givea general discussion of energy loss in a medium [4,5],and then discuss the differences between a strong firstorder transition, and one which is nearly second order.

We introduce the energy scale [5]

(10)Ecr = m2�

λL2,

whereλ is the mean free path andL is the thicknessof the medium. The high-energy jet loses energy byradiating gluons with energyω <E. There are severalcontributions to the total energy loss of the jet,!E,depending on the energy of the radiation. For thecontribution fromω > Ecr, which exists ifE > Ecr,effectively only one single scattering occurs (this is theso-called factorization regime) and so that contributionis medium independent [5]. In what follows we ratherfocus on the medium-induced energy loss, from theregion whereω is less thanEcr.

For very small frequency,ω < ELPM ≡ λm2� , the

formation time [3–5]tf ∼ ω/m2� of the radiation from

the hard jet is short, and so incoherent radiation takesplace. This is the so-called Bethe–Heitler regime;the contribution to!E is just a sum from single

98 A. Dumitru, R.D. Pisarski / Physics Letters B 525 (2002) 95–100

scatterings onL/λ scattering centers. In the high-energy limitE,Ecr �ELPM the region of phase spacewith ω < ELPM contributes little to!E and will beneglected.

The largest contribution is rather from the Landau–Pomeranchuk–Migdal (LPM) regime, where succes-sive scatterings coherently interfere [3–5]. Integratingthe radiation intensity distribution overω from zero tosome energyE∗ yields a total energy loss of [4,5]

(11)−!E ∼ 3αsπ

√EcrE∗ log

2E

Lm2�

.

There is a logarithmic sensitivity to the infrared scalem� [4,23]. When the jet energyE is less than thefactorization scaleEcr, we can integrateω all the wayup toE∗ =E, so

(12)−!E ∼ 3αsπ

√EEcr log

2E

Lm2�

, E < Ecr.

Note that−!E should not exceedE. This requiresthatm� is not so small that the logarithm overwhelmsthe ∼ αs . However,Ecr is small nearTc, so minijetswith energies of at least a few GeV are aboveEcranyways, and Eq. (12) does not apply.

Rather, for jet energies greater thanEcr, the totalmedium-induced energy loss is given by integratingoverω up to the factorization scaleEcr; settingE∗ =Ecr in (11)

(13)−!E ∼ 3αsπEcr log

2E

Lm2�

, E > Ecr.

To compute the critical energyEcr, we need theinverse mean free path,λ−1. This is approximatelythe product of the density,ρ, times the elastic crosssection,σel. The elastic cross section is quadraticallydivergent in the infrared. This divergence is naturallycut off bym�, so the elastic cross sectionσel ∝ α2

s /m2�,

and

(14)m2�

λ∝ ρ.

The scaleEcr is then proportional toρ; this followsautomatically from our assumption thatλ−1 ∼ ρσel.Thus in the high-energy regime aboveEcr, ignoringthe logarithmic dependence uponm�, energy loss isproportional toρ; below that scale, to

√ρ. This has

been emphasized by Baier et al. [5].

Before giving estimates ofm� and λ, we canunderstand how energy loss changes, depending uponthe order of the deconfining phase transition. For anearly second order transition,m�/T is small nearTc, Eq. (8), and then increases rapidly. As the energyloss!E depends logarithmically onm�, for smallm�energy loss is enhanced. This is directly analogousto critical opalescence. In contrast, for a stronglyfirst order transition,m� is large atT +

c , with m�/T

approximately constant with increasing temperature.What are reasonable values form� and λ? In the

extreme perturbative regime,T � Tc, (static) electricfields are heavy, with a mass∝ √

αs T , while the staticmagnetic fields are light,mmag ∝ αsT . The inversemean free path,λ−1, equals the damping rate for agluon with momentum∼ T , and is∝ αsT .

At temperatures∼ 2Tc, this ordering is reversed, asstatic electric fields are significantly lighter than staticmagnetic fields:m� ∼ 2.5T , while the static mass formagnetic glueballs ismmag ∼ 6T [24]. There are noestimates of the damping rate; we guess thatγ ∼λ−1 ∼ T . This seems reasonable for a quasiparticlewith such a mass, asγ /m� ∼ 1/2.5 = 0.4 is less thanone. If the width were much larger, then it would notmake sense to speak of quasiparticles. Conversely, ina strongly coupled system, it is unreasonable to thinkthat the width could be much smaller than the mass.

In the derivation of energy loss, implicitly it isassumed that multiple scatterings of the hard jet areindependent of each other. This requires that therange of the potential is smaller than the mean freepath,m−1

� < λ [4,5], which is equivalent to havingquasiparticles with relatively narrow width.

At 2Tc, then,m2�/λ∼ (2.5T )2T . Below 2Tc, by our

mean field analysis, then,m2�/λ ∼ 6.25T 3n(T ). No-

tice that asm2� ∼ √

nT 2, andγ = λ−1 ∼ √nT , that

γ /m� ∼ n1/4: the � quasiparticles become narrowerasT → T +

c . Consequently, for temperatures nearTc

(15)Ecr ∼ 20 GeV× n(T )

(T

Tc

)3

.

This number is avery crude estimate, obtained bytaking L ∼ 5 fm and Tc ∼ 175 MeV. For QCD,n(Tc)∼ 0.1, so thenEcr ∼ 2 GeV.

It is interesting to note that while the system is be-coming increasingly dilute nearTc for a nearly secondorder transition, typical minijets withE on the order

A. Dumitru, R.D. Pisarski / Physics Letters B 525 (2002) 95–100 99

of several GeV automatically have energies larger thanEcr. This implies that (i) some part of their energy lossis medium independent, from the factorization regimeω > Ecr; and that (ii) their medium-induced energyloss is given by Eq. (13), not (12). Physically, this isbecause the formation timetf ∼ ω/m2

� of the radiationfrom the hard jet grows liken−1/2 asT → T +

c . In all,Ecr is small nearTc, and the regime where!E scaleswith

√ρ shrinks. That regime only emerges asEcr in-

creases: by 2Tc, with n(2Tc) ∼ 1, Ecr has risen dra-matically, to∼ 160 GeV! Thus, the energy loss (13)turns on very rapidly asT increases fromTc. In partthis is because of the factor ofT 3 in Eq. (15), which ispresent whatever the order of the phase transition. Forgauge theories with three colors, such as QCD, how-ever,additionally there is an increase∝ n(T ) [15–17].

With these numbers, the logarithmic sensitivity tothe changes in the screening mass can be significant.For E = 25 GeV,L = 5 fm, T = Tc, then if m� ∼2.5Tc, log(2E/(Lm2

�))∼ 2.3. Including the change inthe screening mass nearTc, with m� ∼ 2.5Tcn

1/4, thelogarithm changes to log(2E/(Lm2

�))∼ 3.5. This is anincrease by about 50%. At smallerE, the sensitivity tom� is even stronger.

Even with the limitations of our approximations, itis clear that since the density vanishes asT → T +

c ,any contribution from the deconfined phase vanisheslike some power ofn(T ). 2 For example, estimates in-dicate that dilepton production from the (nearly ideal)deconfined phase [26] is about as large as that fromthe hadronic phase [27]. NearTc, dilepton productionfrom the deconfined phase should be strongly sup-pressed, quadratically in the density

(16)dNe+e−

d4x∝ n2T 4.

This assumes that the density scales as the relativepressure,n(T ). Whatever the exact form, it is clearthat dileptons from the deconfined phase will turn onlater (in T , or

√s/A) than energy loss, which close

to Tc is linear in the density, and thus inn(T ). If true,then there is a sequential series of effects: increasingT

2 Quasiparticle models also provide a fit to the pressure down toTc. In these models, the pressure decreases asT → T+

c becausethermal masses increase. Conversely, asT increases aboveTc,dilepton production from increasingly light quarks goes up: at lowdensity∼ n2, as in (16), modulo factors of log(n) [25].

from Tc, as the relative pressure increases, first energyloss turns on, then dileptons.

Admittedly, our approximations are very crude.Nevertheless, they should not obscure the basic point,which is that for three colors, when the quark–gluon plasma first appears—at temperatures fromTcto 2Tc—it will not look like a perturbative quark–gluon plasma. Instead, it may well be dominated bythe change in a single mass scale,m�. All scatter-ing processes are affected by the drop in the screen-ing mass asT → T +

c . Processes involving small an-gle scattering, such as the elastic cross section, arestrongly affected, and vary as∼ 1/m2

�. For others,such as energy loss, or viscosity [28], the dependenceuponm� is only logarithmic.

Acknowledgements

We thank R. Baier, K. Bugaev, F. Gelis, M. Gyu-lassy, L. McLerran, A. Peshier, D.T. Son, I. Vitev, andJ. Wirstam for helpful discussions. A.D. acknowledgessupport from DOE Grant DE-FG-02-93ER-40764;R.D.P. from DOE Grant DE-AC-02-98CH-10886.

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