30 November 2000

Physics Letters B 494 (2000) 215–220www.elsevier.nl/locate/npe

The effective pressure of a saturated gluon plasma

Adrian Dumitru∗, Miklos GyulassyPhysics Department, Columbia University, 538W 120th Street, New York, NY 10027, USA

Received 26 June 2000; received in revised form 5 September 2000; accepted 3 October 2000Editor: W. Haxton

Abstract

The evolution of the gluon plasma produced with saturation initial conditions is calculated via transport theory for nuclearcollisions with 0.1<

√s < 10A TeV. The effective longitudinal pressure is found to remain significantly below the lattice QCD

pressure with these initial conditions until the plasma cools to near the confinement scale. The absolute value of the transverseenergy per unit of rapidity and its dependence on beam energy is shown to provide a sensitive test of gluon saturation modelssince the fractional transverse energy loss due to final state interactions is predicted to be much smaller and exhibit a weakerenergy dependence than nondissipative hydrodynamics applied throughout the evolution. 2000 Elsevier Science B.V. Allrights reserved.

PACS:12.38.Mh; 24.85.+p; 25.75.-q; 13.85.-t

There is an ongoing experimental program to pro-duce a (transient) deconfined phase of QCD matter [1]in the laboratory via nuclear collisions at high ener-gies [2]. The production mechanism is the liberationof a large number of gluons from the nuclear structurefunctions. The plasma is produced from copious mini-jet gluons at central rapidity,y ' 0, with transversemomentumpT > p0. The rapidity density of gluonsliberated in centralA+A collisions can be estimatedfrom [3]

dN

dy(p0)=K TAA(b= 0)

∫pT >p0

d2pT

∫dxa dxb

(1)×G(xa,p2T

)G(xb,p

2T

) sπ

dσ

dtδ(s + t + u).

* Corresponding author.E-mail address:dumitru@mail-cunuke.phys.columbia.edu

(A. Dumitru).

s, t , u are the Mandelstam variables for the parton–parton scattering process, and dσ/dt denotes the hard-scattering differential cross section in lowest order ofperturbative QCD.G(x,p2

T ) denotes the LO gluondistribution function in the nucleus. The phenomeno-logical factorK = 2 accounts approximately for NLOcorrections. The nuclear overlap functionTAA(0) =A2/πR2

A determines the number of binary nucleoncollisions in head-on reactions within the Glauber ap-proach, whereRA ' 1.1A1/3 fm for massA nuclei.

For largep0, the produced gluon plasma is dilute.As p0 decreases, however, the density of gluons in-creases rapidly due to the increase ofG(x,p2

T ) asx ≈ 2pT /

√s decreases. It has been conjectured [4]

that below some transverse momentum scalep06 psatthe phase-space density of produced gluons may sat-urate sincegg → g recombination could limit fur-ther growth of the structure functions. Phenomenolog-ically, this condition may arise when gluons (per unitrapidity and transverse areaπ/p2

sat) become closely

0370-2693/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0370-2693(00)01174-6

216 A. Dumitru, M. Gyulassy / Physics Letters B 494 (2000) 215–220

packed and fill the available nuclear interaction trans-verse area. The saturation scalepsat can thus be esti-mated from

(2)dN

dy(psat)= p2

satR2A/β,

where β ∼ 1. For β = 1 the solution reported inEKRT [5] was

psat≈ 0.208A0.128√s 0.191,

(3)C1≈ 1.34A−0.007√s 0.021,

where psat and√s are in units of GeV andC1

is the average transverse energy per gluon (in unitsof psat). The focus of this Letter is to investigatewhether the final observed dEfT /dy can be used totest the predictedA and

√s dependence of the initial

dEiT /dy = C1psatdN(psat)/dy.Different gluon saturation models based on non-

linear evolution and classical Yang–Mills equations[6,7] suggest that the factorβ in (2) may varyparametrically as

(4)β(psat)= 4πα(psat)Nc

c(N2c − 1)

,

wherec∼ 1 is a nonperturbative factor proportional tothe fraction of the initial gluons in the nucleus whichare liberated. This factor was recently estimated usinglattice classical Yang–Mills methods [8] to bec≈ 1.3.

The first data [9] from RHIC on Au+Au collisionsat√s = 130A GeV with dNch/dη ≈ 560 is in fact

reproduced by the EKRT saturation model [5] withβ = 1 assuming isentropic expansion (see [10]). Onthe other hand, a fit to the Phobos data using Eq. (4)requiresc ≈ 1.9. Note that the solution of Eq. (2)with β 6= 1 can be obtained from Eq. (3) by rescalingthe mass-numberA→A/β2/3, and iterating until thestationary point is reached. Extrapolating to CERN-LHC energy the minijet multiplicities are predicted tobe dN/dy = 3200 forβ = 1 versus dN/dy = 5100 forβ(psat) using Eq. (4).

While dN(A,√s )/dy systematics provide one ex-

perimental handle to test different saturation and fixedscale models of initial conditions [10], another impor-tant observable that probes collective dynamics is thetransverse energy per unit rapidity, dET /dy. In EKRTthe final value of dEfT /dy was predicted to be much

smaller than produced initially due to collective lon-gitudinal work assuming the validity of isentropic hy-drodynamics.

If the expansion proceeds in approximate localequilibrium with pressurep = c2ε and speed ofsoundc, then the energy density,ε(τ ), must decreasefaster than the expansion rateΓexp= 1/τ and leads toa bulk transverse energy loss

(5)ET (τ)

ET (τ0)= τε

τ0ε0=(τ0τ

)δ.

If local equilibrium is maintained during the evolutionδ = c2. In contrast, if the system expands too rapidly tomaintain local equilibrium, then the effective pressureis reduced (relative to that from LQCD) due to dis-sipation. The extreme asymptotically free plasma casecorresponds to free streaming withδ = 0.ET thus pro-vides an important barometric observable that probesthe (longitudinal) pressure in the plasma [11]. Therehave been of course many studies on the magnitude ofdissipative effects on this and other observables, see,e.g., [12–16]. The new twist on this old problem thatwe consider here is to extend those studies to the novelinitial conditions suggested by gluon saturation mod-els [5,7,8].

To compute the transverse energy loss due tolongitudinal work, we employ the Boltzmann equationin relaxation time approximation [12,16,17],

(6)p · ∂f (p,x)= Γrelp · u(feq(p · u)− f (p,x)

).

uµ denotes the four-velocity of the comoving frameandfeq is the chemical and thermal equilibrium phasespace distribution, towards whichf evolves at a re-laxation rateΓrel. It is important to emphasize thatthis much simplified transport equation has been ex-tensively tested against full 3+ 1D covariant partoncascade codes [14] and provides a surprisingly accu-rate equation for calculating the evolution of the trans-verse energy observable even in highly dissipative sys-tems far from equilibrium (Γrel. Γexp).

The relevant relaxation rate is given by the frac-tional energy loss per unit length,

(7)Γrel= 1

E

dE

dz,

A. Dumitru, M. Gyulassy / Physics Letters B 494 (2000) 215–220 217

which receives a contribution both from elastic andinelastic scattering,

Γrel= ρ∫ (

dσel1Eel

E+ dσin

1Ein

E

)

≈ ρQ2∫µ2

dq2 dσel

dq2

{q2

2E2 +αNc

π

q2∫µ2

dk2T

k2T

∫dx

xx

}

(8)

= ρ(

16πα2N2c

N2c − 1

)(1

slog

Q2

µ2 +αNc

πµ2 logSq2

µ2

).

In these equationsρ(τ) denotes the gluon density inthe local restframe,µ2 is the Debye screening massin the medium,Q2 ' s is the upper bound for themomentum transfer in the scattering process, andx

denotes the fraction of energy carried away by radiatedgluons. In the last step we replaced the momentumtransferq2 in the expression for the radiative energyloss by its average,Sq2 ≈ µ2 logQ2/µ2. In localthermal equilibrium the average energy per gluon andthe Debye screening scale are

(9)s

2=(ε

ρ

)2

' 9T 2,

(10)µ2= Ncg2T 2

3= 4παT 2.

Assuming that the ratios/µ2 is essentially the sameeven out of equilibrium, the relaxation rate is approxi-mately given by

Γrel≈ 9πα2ρ3

ε2

(log

1

α+ 27

2π2

)(11)≡Kin9πα2ρ

3

ε2log

1

α.

We have set the double-logarithm ofQ2/µ2 ∼ 1/αequal to unity. The expression (11) in fact overesti-mates the relaxation rate at early times, because thescreening length 1/µ ∼ 1/gpsat exceeds formally thehorizon atτ0= 1/psat for longitudinal Bjorken expan-sion [18] and because we neglect the suppression of ra-diation due to formation time physics. However, sincewith psat. 2 GeV,g ≈ 2 up to the LHC energy do-main, we ignore this formal point in the discussion be-low.

The inelastic, radiative energy loss represents a sig-nificant source of uncertainty and is especially impor-tant in chemically undersaturated models of the ini-tial conditions [15,19]. Within the saturation model,gluon multiplication through 2→ 3 processes maylead to thermalization of the soft radiated gluons attimes parametrically large as compared toτ0, whilethe effect on the hard part of the gluon distribution issmall [20]. In the present Letter we do not attempt amore detailed treatment of radiative energy loss butsimply vary the factorKin ∼ 1–2 to provide a mea-sure of the theoretical uncertaintes. We note that radia-tive energy loss for gluons with modestpT < 5 GeV,is in any case significantly suppressed due to finitekinematic constraints and destructive interference ef-fects [21].

The EKRT saturation model predicts the gluondensity to be nearly chemically saturated already atthe initial time τ0 = 1/psat. This follows from theobservation that ideal-gas formulasρ ∼ T 3, ε ∼ T 4

applied with chemical potentialµg = 0 yield the same“temperature”T0 [5].

At the initial time τ0 = 1/psat, s/2 = ε20/ρ

20 =

C21p

2sat. Therefore, noting that the comoving gluon

density at timeτ0 is ρ0 = p3sat/πβ , the ratio of the

relaxation rate to the expansion rate is given by

(12)Γrel

Γexp=Kin

9α2

βC21

log1

α.

WhileΓrel∝ psat increases as a power of the energy inEq. (3), the Bjorken boundary conditions [18] forcethe system to expand longitudinally initially also atan increasing rateΓexp(τ0)= psat. The essential quan-tity that fixes the magnitude of the effective pres-sure relative to that predicted by LQCD is the ra-tio of rates in Eq. (12), which dimensionally is sim-ply a function of α(psat). The asymptotic freedomproperty [22] of QCD therefore requires that this ra-tio vanishes as

√s → ∞. In (12) the rate of how

fast it vanishes is controlled byα2Kin(psat)/β(psat).Therefore, with saturation initial conditions, asymp-totic freedom reduces theeffectivepressure actingat early timesτ ∼ τ0 and causes the initial evolu-tion to deviate from ideal hydrodynamics for a timeinterval that, as we show below, increases with en-ergy.

218 A. Dumitru, M. Gyulassy / Physics Letters B 494 (2000) 215–220

The total number of interactions during the evolu-tion up to timeτ is given by

φ(τ)≡τ∫

τ0

dτ ′Γrel(τ ′)

(13)' Kin

β

9α2 log(1/α)

C21

logτ

τ0.

The last expression holds close to the free streamingregime (“Knudsen limit”), where the number of scat-terings increases only logarithmically with time. Wefind thatφ reaches on the order of unity atτ ∗ ≈ 0.4 fmin the BNL-RHIC to CERN-LHC energy region. How-ever, the local relaxation rate atτ ∗ is still less than oron the order of the expansion rate (1/τ ∗). In a non-expandingplasmaφ ∼ 1 provides a rough equilibra-tion criterion. However, as long asΓrel is not signifi-cantly larger thanΓexp this criterion is insufficient toaddress how much collective hydrodynamic work canbe done by the plasma.

For a quantitative estimate, we must solve thekinetic equations (6). The first energy moment of thatequation together with energy conservation to replaceεeq by ε(τ ), results in the energy density evolutionequation [12,16,17]

(14)eφτε

τ0ε0= 1+

φ∫0

dφ′ eφ′ τ′(φ′)ε(φ′)τ0ε0

h

(τ ′(φ′)τ (φ)

).

This equation applies if the initially produced par-tons have a vanishing longitudinal momentum spreadin the comoving frame, i.e., assuming a strong cor-relation between space–time rapidity and momen-tum space rapidity [18]. The functionh(x) appear-ing in (14) is given by 2h(x)

√1− x2 = x√1− x2+

arcsin√

1− x2; it insures thatδ→ 1/3 asτ→∞.Expanding (14) to first order inφ yields

(15)τε

τ0ε0= 1−

(3

4− π

2

16

)φ +O(φ2),

giving for τ ∼ τ0

(16)δ =(

3

4− π

2

16

)9Kinα

2

βC21

log1

α.

Note thatδ→ 0 aspsat→∞ in accordance with thediscussion above.

For our numerical estimates we letα creep withtime. Since the effective temperature scales asε(τ )1/4,the effective coupling,α(Teff), increases slowly withtime approximately as

(17)α(τ)=(

12π

27

) /log

(1+ p2

sat

Λ2QCD

(ε(τ )

ε(τ0)

)1/2),

with ΛQCD = 200 MeV. Furthermore, longitudinalexpansion constrainsρ(τ)τ to remain constant. As thesystem cools, the mean center of mass energy incollisions also decreases as

(18)s(τ )= 2

(ε(τ )

ρ(τ )

)2

= s(τ0)(τε(τ )

τ0ε(τ0)

)2

.

The number of collisions betweenτ0 andτ in this caseis given by

φ(τ)= Γrel(τ0)

τ∫τ0

dτ ′(α(τ ′ )α(τ0)

)2( ε(τ0)ε(τ ′ )

)2(τ0τ ′

)3

(19)× log(1+ 1/α(τ ′ ))log(1+ 1/α(τ0))

.

We regulated the logarithmic dependence above fornumerical stability. Note that with Eq. (19), Eq. (14)is a nonlinear self-consistency equation forε(τ ).

In Fig. 1 we show the effective longitudinal pres-sure as a function of the energy density for

√s =

20,200,5400AGeV saturation initial conditions. Theratio p/ε is defined here byδ(τ ) as obtained solving

Fig. 1. The ratio of effective longitudinal pressure to energy densityas a function of the energy density along the dynamical path isshown for SPS, RHIC, and LHC saturation initial conditions [5].Solid (dashed) curves are forβ = 1 andKin = 1(2). The LQCDequation of state [1] is also shown for comparison.

A. Dumitru, M. Gyulassy / Physics Letters B 494 (2000) 215–220 219

Eq. (14) numerically. Our definition of the effectivepressure absorbesall disspative corrections to the per-fect fluid equation,

uµ∂ν[(ε + p)uµuν − pgµν]= dε

dτ+ (ε + p)

τ= 0.

For comparison the pressure of equilibrium QCDis also shown forNf = 3. This curve is obtainedfrom the Nf = 0 lattice data of [1] rescaling thenumber of relativistic degrees of freedom by 47.5/16,and assuming a transition temperatureTc(Nf = 3) =160 MeV.

Initially p/ε starts at zero in this model and re-mains small for a large time relative to 1/psat be-cause the plasma is torn apart by the initial rapid longi-tudinal expansion. The effective pressure approachesthe LQCD curve from below and reaches it at a timeτL ≈ 1–2 fm at RHIC

√s = 200A GeV, by which

time the energy density has dropped by an order ofmagnitude,εL ≡ ε(τL)= 6.5–12 GeV/fm3. For LHC√s = 5400A GeV, τL = 3–7 fm during which the

energy density falls by almost two orders of magni-tude to εL = 9.5–21.5 GeV/fm3. The quoted inter-vals correspond toKin = 1–2 using the EKRT para-metrization (3). The contrast between the dynamicalpath followed by the saturated plasma compared tothe equilibrium equation of state is striking. A qualita-tively similar behavior of the early longitudinal pres-sure has also been found from solutions of diffusionequations [23].

Since the longitudinal gradients atτL are muchsmaller than atτ0, the evolution beyondτL is muchmore likely to follow isentropic hydrodynamics alongthe lattice QCD equation of state. In this case onecould calculate detailed differential hadronic observ-ables along the same lines as in [24] using the condi-tions atτL as the initial conditions for 3+ 1D hydro-dynamics.

The main experimentally observable consequenceof the reduced effective pressure is shown in Fig. 2.The ratio EfT /E

iT = τf εf /τ0ε0 has been obtained

from the solution of the transport equation assum-ing εf = 2 GeV/fm3 which corresponds roughly toT ' Tc. τf is estimated assuming hydrodynamic ex-pansion from the point were the trajectories in Fig. 1reach the LQCD curve at timeτL, with the equation ofstatep/ε = a + b logε (ε in units of GeV/fm3). Theparametersa = 0.051,b = 0.092 provide a reasonable

Fig. 2. The ratios of the final to the initial transverse energy perunit rapidity are shown as a function of beam energy for centralAu + Au collisions. The initial value corresponds to the EKRTparametrization (β = 1) and the transport results are forKin = 1,2.For comparison, the final transverse energy assuming that localequilibrium was maintained throughout the evolution is also shown.

fit to the LQCD curve shown in Fig. 1. In this case,

(20)τf

τL=(

1+ a + b logεL1+ a + b logεf

)1/b

.

On the other hand, if ideal hydrodynamics wereapplicable already atτ0, the final observed transverseenergy for (1+ 1)-dimensional adiabatic expansionwould be

(21)EfT

EiT

= τf εfτ0ε0

= τf (Tf sf −pf )τ0(T0s0−p0)

= TfT0.

The last step follows both forp0,f ' 0 as well asp0,f = T0,f s0,f /4 from the condition of entropy con-servation,τs = const [5]. Strong transverse expansionleads to slightly largerEfT but we shall neglect thatsmall effect here for simplicity. Clearly, forTf ∼ Tc ≈160 MeV one would observe a much smaller trans-verse energy in the final state than in the initialstate. Moreover,EfT /E

iT would also have significantly

stronger energy dependence such thatEf

T deviatesmore and more fromEiT with increasing

√s. In this

sense isentropic hydrodynamics erases information onthe interesting initial conditions via this observable.The solutions of the transport equations clearly showa smaller decrease ofEfT and of the logarithmic slope,

κ = d logEfT /d log√s, due to final state interactions.

We find thatκ = 0.50 for the initial state (3) evolvedwith Kin = 1, κ = 0.46 withKin = 2, while κ = 0.40

220 A. Dumitru, M. Gyulassy / Physics Letters B 494 (2000) 215–220

with isentropic expansion, Eq. (21). For comparison,the initial EKRT saturatedEiT = πR2

Aτ0ε0 scales withthe higher powerκ = 0.59 according to Eq. (3). Thefractional transverse energy loss is thus less dependenton energy than for entropy conserving expansion forwhichEfT /E

iT ∝ 1/T0 ∝ 1/

√s

0.2. This is due to theincreasingly long time spent far from equilibrium inFig. 1 as the beam energy increases.

The results in Fig. 2 are encouraging from the pointof view of searching for evidence of gluon satura-tion in nuclei at high energies. Experimental data ondET /dy or dET /dη for central Au+ Au collisions atRHIC will soon provide a new test of saturation andnonsaturation models at those energies. Since we pre-dict that dissipative effects reduce considerably the ef-fective longitudinal pressure in Fig. 1, the beam energydependence of the transverse energy is expected to re-flect much more accurately the predicted power lawdependence of the initial conditions as seen in Fig. 2.We therefore conclude that the energy andA systemat-ics of the bulk calorimetric observable, dET /dy, willbe a sensitive test of saturation models of gluon plas-mas produced in the RHIC to LHC energy range.

Acknowledgements

We thank K. Eskola, K. Kajantie, L. McLerran,A.H. Mueller, D. Son, R. Venugopalan and K. Tuomi-nen for helpful criticism and discussions on thermal-ization aspects and saturation. We thank the BNL nu-clear theory group for hosting a stimulating workshopduring which this work was completed. We acknowl-edge support from the DOE Research Grant, ContractNo. DE-FG-02-93ER-40764.

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