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Parton Interaction Rates in the Quark-Gluon Plasma

arXiv:hep-ph/9308257v1 11 Aug 1993Parton Interaction Rates in the Quark-Gluon PlasmaM.H. ThomaInstitut f¨ur Theoretische Physik, Universit¨at Giessen,35392 Giessen, Germany(July 16, 2018)The transport interaction rates of elastic scattering processes of thermal partons in the quark-gluon plasma are calculated beyond the leading logarithm approximation using the effective per-turbation theory for QCD at finite temperatures developed by Braaten and Pisarski.

The resultsfor the ordinary and transport interaction rates obtained from the effective perturbation theory arecompared to perturbative approximations based on an infrared cut-offby the Debye screening mass.The relevance of those interaction rates for a quark-gluon plasma possibly formed in ultrarelativisticheavy ion collisions are discussed.PACS numbers: 12.38.Mh, 12.38.BxI. INTRODUCTIONInteraction or damping rates of parton scattering processes in a thermalized quark-gluon plasma (QGP) are of greatphysical significance.

For example, the inverse rates related to the elastic scattering of thermal partons (gg →gg,gq →gq, qq →qq) give the mean free paths and typical interaction times i.e., relaxation times used in the collisionterm of the Boltzmann equation.Hence they can be used for an estimate of the thermalization time of a pre-equilibrium parton gas in ultrarelativistic heavy ion collisions [1] and the maintenance of the thermal equilibriumby comparing the interaction rates versus the cooling rate. Of course thermalization times calculated in this way atfinite temperature should be valid only for situations which do not start too far from equilibrium.

On the other hand,this thermalization time does not depend on the somewhat ambiguous definition of the beginning and the end of thepre-equilibrium stage as the one deduced from numerical studies of ultrarelativistic heavy ion collisions (HIJING [2],parton cascade [3]). After all, both methods lead to similar results, namely a fast thermalization of the gluons i.e., anisotropic and exponential momentum distribution, after about 0.2 fm/c for LHC and 0.3 fm/c for RHIC [4].

The fullequilibration of the phase space density can be investigated, on the other hand, using the inelastic interaction ratese.g., gg →ggg and gg →q¯q, describing the chemical equilibration process of the QGP [4].Furthermore the elastic rates are the basic inputs for the energy loss of a parton in the QGP [5–12] and the viscosityof the QGP [13]. The first quantity is related to a possible signature of the QGP in ultrarelativistic heavy ion collisions(jet quenching) [7,14,15], while the latter provides informations about the role of dissipation in the hydrodynamicexpansion phase [16].Finally, damping rates are especially suited for studying problems of quantum field theory at finite temperaturelike gauge dependences and infrared singularities in perturbation theory [17].

In particular, the puzzle of the gaugedependence of the gluon damping rate at rest (plasmon damping) in naive perturbation theory started a recentdevelopement in finite temperature field theory, which lead to a powerful effective perturbation theory [18]. Assumingthe weak coupling limit (g ≪1) effective propagators and vertices are obtained by resumming self energies and verticesin the high temperature limit (hard thermal loops), which are used for soft external momenta of the order gT , whilebare Green’s functions are sufficient for hard momenta of the order T .

In this way gauge independent results forphysical quantities are found, which are complete to leading order in the coupling constant. (The gauge independenceof this method is still under discussion by evaluating the damping rates in a general covariant gauge.

However, usingan appropriate infrared regularization for the gauge dependent part, independence of the gauge fixing parameter canbe shown [19–22].) In addition, screening effects are included yielding an improved infrared behavior.

In summary,the Braaten-Pisarski method provides a consistent treatment for quantities which are sensitive to the momentum scalegT , meaning a crucial improvement compared to the naive perturbation theory at finite temperature.Here, we will present calculations of the interaction rates based on the Braaten-Pisarski technique and study theirphysical relevance for the QGP. Two different kinds of interaction rates are considered.

In the next section we willdiscuss the ordinary interaction rate Γ – in the following simply called interaction rate – of a parton with hardmomentum, defined by Γ ≡n σ, where n is the particle density of the QGP and σ the cross section of the scatteringprocess under consideration. In naive perturbation theory without resummation Γ ∼α2s T is found (αs = g2/4π).However, Γ turns out to be quadratically infrared divergent.

Using the Braaten-Pisarski method, Γ ∼αs T ln(1/αs)follows, which is only logarithmically infrared divergent [7,22–29]. The interaction rate is enhanced by a factor 1/αs1

compared to the one obtained from naive perturbation theory due to the infrared regularization inherent in theresummation method. The logarithmic term, reflecting the logarithmic infrared divergence, arises from the sensitivityof the rate to the momentum scale g2T , which cannot be treated within the Braaten-Pisarski method.In section 3, we will discuss the transport interaction rate defined by Γtrans = n σtrans, where the transport crosssection σtrans =Rdσ (1 −cos θ) enters the collision term of the transport equation in the case of a plasma with longrange interactions, for which the interaction rate is dominated by distant collisions [30].

Here θ denotes the scatteringangle in the center of mass system. Hence the mean free path as well as the thermalization time may rather be givenby the inverse of the transport interaction rate [30].

The relevance of this rate for the QGP has been pointed out inRef. [23,31,32].

The transport interaction rate is also closely connected to the shear viscosity beyond the relaxationtime approximation [13,33].The factor 1 −cos θ cuts offsmall scattering angles and therefore small momentum transfers, too.Thus thetransport interaction rate is only logarithmically infrared divergent in naive perturbation theory (Coulomb logarithm[30]), but finite using the Braaten-Pisarski method since it is sensitive to the scale gT only.Hence dynamicalscreening of the magnetic interaction, which is included in the effective gluon propagator, is sufficient [34] leading toΓtrans ∼α2s T ln(1/αs) [13,29,31]. Note the α2s dependence as expected from naive perturbation theory since thereis no enhancement by a factor 1/αs caused by a quadratic infrared divergence.

Also the logarthmic term stems fromthe sensitivity to the scale gT instead of g2T as it is the case for Γ [29].Up to now the transport interaction rates based on the Braaten-Pisarski method have been calculated only withinthe leading logarithm approximation [13] i.e., the coefficient of 1/αs under the logarithm has not been determined.However, if we want to extrapolate the result for the transport interaction rate in the weak coupling limit to realisticvalues of αs = 0.2 – 0.5, we should not neglect this coefficient.One aim of the present paper is the calculation of the transport interaction rate beyond the leading logarithmapproximation using the Braaten-Pisarski method. Furthermore we will discuss the consequences of Γ and Γtransfor realistic values of αs.

Finally, we will compare the results of the Braaten-Pisarski method with the widely usedapproximation of the naive perturbation theory regularizing the infrared singularities simply by using the Debyescreening mass as an infrared cut-off.II. INTERACTION RATESThere are two equivalent ways of computing the interaction rates either using matrix elements or self energies [35].Considering for example elastic quark-quark scattering we may find the corresponding interaction rate Γqq from thematrix element via [8]Γqq = 12pZd3p′(2π)32p′ [1 −nF (p′)]Zd3k(2π)32k nF (k)×Zd3k′(2π)32k′ [1 −nF (k′)] (2π)4 δ4(P + K −P ′ −K′) 6Nf ⟨|M(qq →qq)|2⟩.

(1)The four momentum of the incoming particle is denoted by P = (p, p), where quark masses are neglected and p = |p|. (We consider only the case of a QGP containing up and down quarks for which the bare masses are negligible comparedto the temperature of the QGP.) The quark of the heat bath from which the quark under consideration is scattered offhas the momentum K = (k, k).

Momenta with a prime belong to the outgoing particles. The Fermi-Dirac distributionfunctions are given by nF (k) = 1/[exp(k/T ) + 1], while Nf denotes the number of thermalized flavors in the QGP.The matrix element is averaged over the spin and color degrees of freedom of the incoming particles and summed overthe ones of the final state.

The factor 6Nf comes from summing over the possible spin, color, and flavor states of thequark with momentum K. The diagrams which enter the matrix element to lowest order are shown in Fig.1a.In the case of quark-antiquark scattering we have to replace the diagrams of Fig.1a by the ones of Fig.1b. Inthe case of quark-gluon scattering we have to substitute the Fermi distributions by the Bose-Einstein distributionsnB(k) = 1/[exp(k/T ) −1], the Pauli blocking factor 1 −nF (k′) by the Bose enhancement factor 1 + nB(k′), and thefactor 6Nf by 16 (number of gluonic degrees of freedom).

If we wish to consider the scattering of an incoming gluonby the partons of the QGP we have to deal with the diagrams of Fig.1c and 1d.The second possibility to determine the quark interaction rate is given by the imaginary part of the quark selfenergy on mass shell [8]:Γq(p) = −12p [1 −nF (p)] tr [γµPµ ImΣ(p, p)] . (2)2

The equivalence of the expressions (1) and (2) can be seen from cutting the self energy of Fig.2a through the fermionlines. Eq.

(2) is the starting point for applying the Braaten-Pisarski method by considering the quark self energy ofFig.2b, where the effective gluon propagator contains the resummed one-loop gluon self energy in the high temperaturelimit [36,37]. (Note that Fig.2b contains the diagram of Fig.2a if the high temperature limit is used for the fermionloop, called hard thermal loop approximation [18], in the latter.) It is not necessary to take an effective quark-gluonvertex into account because the external quark with a momentum of the order of the temperature is hard.

Alsobecause the interaction rate falls offrapidly for large momentum transfers q ≡|p −p′|, Γ ∼Rdq/q3, i.e., only smallmomentum transfers contribute, a bare quark propagator is sufficient. An effective quark propagator and quark-gluonvertex would contribute to higher order in αs only.Since the final result for observables using the Braaten-Pisarski method is gauge independent we are free of choosingany gauge.

Using Coulomb gauge and the approximation p, k ≃3T ≫q ∼gT the quark interaction can be writtenas [8]Γq = CF g2T2πZ ∞0dq qZ q−qdωωρl(ω, q) +1 −ω2q2ρt(ω, q),(3)where CF = 4/3 is the Casimir invariant of the fundamental representation. The interaction rate of a hard gluonis simply obtained by replacing CF by the Casimir invariant of the adjoint representation CA = 3 [26].

The fourmomentum of the exchanged gluon is denoted by P −P ′ = K′ −K ≡Q = (ω, q), and the discontinuous parts ofthe longitudinal and transverse spectral functions, related to the effective gluon propagator ∆l,t through ρl,t(ω, q) =Im∆l,t(ω, q)/π, are given by (−q ≤ω ≤q) [38]:ρl(ω, q) = 3m2gω2q" q2 + 3m2g −3m2gω2qln q + ωq −ω!2+ 3πm2gω2q!2#−1,ρt(ω, q) = 3m2gω(q2 −ω2)4q3" q2 −ω2 + 3m2gω22q21 + q2 −ω22ωqln q + ωq −ω!2+ 3πm2gω(q2 −ω2)4q3!2#−1,(4)where m2g ≡(1 + Nf/6) g2T 2/3 may be interpreted as an effective gluon mass generated by the interaction with thethermal ensemble of the QGP. This expression was derived assuming the high temperature approximation ω, q ≪T .The integration over q can be performed analytically, whereas the one over ω has to be done numerically.

Theresult for the longitudinal part of the interaction rate corresponding to the exchange of a longitudinal gluon is givenby [7]Γlq = 1.098 CF αs T.(5)Note that the interaction rate is independent of the number of flavors Nf due to a cancellation of the factors m2gin (4) after integration. Although a larger number of thermal flavors corresponds to an increase of the number ofscattering partners enlarging the interaction rate, the screening mass is also increased cancelling this enlargement.Also the interaction rate does not depend on the external momentum p in the p ≫ω, q limit.The transverse part of the interaction rate Γtq, on the other hand, is still infrared divergent, although the infraredbehavior has been improved from a quadratic singularity in the bare two loop case (Fig.2a) to a logarithmic due todynamical screening of magnetic fields.

On the other hand, there is no static magnetic screening in the transversepart of the spectral function (or the effective propagator) i.e., the denominator of ρt vanishes in the static limitω = 0, q →0, while the denominator of ρl is given by µ2D ≡3m2g in this limit. In other words, the high temperaturelimit of the perturbatively calculated gluon self energy contains static electric screening (Debye mass µD) but nostatic magnetic screening.

In QCD, however, static magnetic screening may arise non-perturbatively from monopoleconfigurations due to the gluon self interaction. Indeed, lattice as well as semiclassical calculations show the excistenceof a magnetic screening mass, m2mag ≃15 α2s T 2 [39,40].

Thus static magnetic screening is provided on the momentumscale g2T .After all, in order to obtain an estimate of the transverse interaction rate we consider the nearly static limit (ω ≪q)of (4) [25]:3

ρl(ω ≪q) =3m2gω2q(q2 + 3m2g)2 ,ρt(ω ≪q) =3m2gωq4q6 + (3πm2gω/2)2 . (6)Inserting (6) into (3) the integrations can be done exactly, leading toΓlq = CF αs T,Γtq = CF αs T ln καs.

(7)We observe that the nearly static limit is a good approximation (within 10%) for the longitudinal rate. The logarithmicterm of the transverse rate comes from assuming a hypothetical infrared cutoffof the order g2T .

For example, themagnetic screening mass [25] or the interaction rate itself i.e., the imaginary part of the quark propagator [22,23],have been suggested as such an infrared regulator. The coefficient κ cannot be calculated using the Braaten-Pisarskitechnique but must await the developement of non-perturbative methods at finite temperature for dynamical quantities[41].For a rough estimate we proposeΓq = Γlq + Γtq ≃(2 ± 1) CF αs T,(8)assuming αs ≃0.3 under the logarithm and κ<∼2.

A justification for the latter assumption may come from choosingµD as an upper limit and mmag as a lower limit for the integration over q. The gluon interaction rate, correspondingto the scattering processes shown in Fig.1c and 1d, is given by Γg = (CA/CF ) Γq = (9/4) Γq.

Because it is alsoindependent of the number of flavors, it holds for the QGP as well as for the pure gluon gas.Compared to the real part of the dispersion relation for thermal partons, ω ≃k ≃3T , the damping rate γ = Γ/2defined by the imaginary part of the self energy, is not really small, γg ≃3αsT for realistic values of αs.Thisanomalously large damping [22,23] indicates that interactions in the QGP are important, at least for temperaturesnot too far above the phase transition in accordance with the recently emerging picture of the QGP [42].An alternative method to the Braaten-Pisarski method based on (2) is given by inserting the t-channel diagramsof Fig.1 in the −t ≡−(P −P ′)2 ≪s ≡(P + K)2 approximation into (1), where the effective gluon propagator ∆l,t isused instead of the bare one [43]. This method has been shown to be equivalent to the self energy calculation (Eq.

(2)to (4)) in the case of the energy loss of a massive fermion in a hot plasma [8].Recently, Pisarski proposed an empirical way of including the magnetic mass and an imaginary part of the fermionself energy in the parton damping rates [44]. In the case of a hard, massless quark, using the magnetic mass of [39,40],Γq ≃0.8 CF αs T follows from his investigation assuming αs = 0.3 under the logarithm of the transverse part.

Thisresult is smaller than the estimate (8), since the transverse part of it turns out to be negative for realistic values ofthe coupling constant.Next we discuss a much simpler, widely used approximation (see, for example, Ref. [4–6]) based on the naiveperturbation theory i.e., bare propagators and vertices, where the Debye mass µ2D = 3m2g is simply introduced byhand as an infrared regulator into the gluon propagator.

It should be noted that in this case the Debye mass alsocuts offthe magnetic divergence without justification. Then the interaction rate is easily calculated fromΓ = n σ =Zd3k(2π)3 ρ(k)Zdt dσdt ,(9)where the parton momentum densities are given by ρq(k) = 12 Nf nF (k) for quarks plus antiquarks and ρg(k) =16 nB(k) for gluons, respectively.

The cross sections in the small momentum transfer limit (−t ≪s) containing theDebye mass readdσdt = ζ2πα2s(t + µ2D)2 ,(10)where the color factor ζ = 4/9 for quark-quark scattering, ζ = 1 for quark-gluon scattering, and ζ = 9/4 forgluon-gluon scattering. In contrast to the complete calculation (to leading order in the coupling constant) using the4

Braaten-Pisarski method, the result depends weakly on the number of flavors. In the case of two flavors (Nf = 2) wefindΓq(Nf = 2) ≃1.1 αs T,Γg(Nf = 2) ≃2.5 αs T(11)and in the pure gluonic caseΓg(Nf = 0) ≃2.2 αs T.(12)Comparing with (8) the simple approach leading to (11) or (12) appears to underestimate the interaction rates byabout a factor of two.

However, it agrees with Pisarski’s result [44].Finally, we discuss the consequences of the present results for typical values, T = 300 MeV and αs = 0.3, expectedat RHIC and LHC. Using (8) we arrive at relaxation timesτg ≃(0.5 ± 0.3) fm/c,τq ≃(1.0 ± 0.5) fm/c(13)indicating a rapid thermalization of the QGP and the maintenance of the local thermal equilibrium during theexpansion phase by comparing with a typical expansion time τexpan > 1 fm/c [4].

Similar results have been foundfrom Monte Carlo simulations of ultrarelativistic heavy ion collisions [2–4,45–47]. Furthermore the results (13) supportthe prediction of a two-stage equilibration i.e., there is first a thermal equilibrium of the gluonic component before acomplete thermalization is achieved later on because of the stronger interaction of the gluons compared to the quarks[1,4].III.

TRANSPORT INTERACTION RATESThe relevant physical quantities, such as mean free path and equilibration time, in a plasma with long rangeinteractions are described rather by the transport interaction rates than by the ordinary ones [23,30–32]. The transportrates are obtained from the latter by introducing a weight 1 −cosθ under the integrals of (1) or (2), defining Γtrans ≡RdΓ (1−cos θ).

Here θ denotes the scattering angle in the center of mass system: cos θ = (p·p′)/(pp′) = 1+2t/s. Thusthe transport factor 1 −cos θ = −2t/s = 2q2 (1 −ω2/q2)/s is proportional to the square of the momentum transferq2.

This additional factor q2 changes the infrared and ultraviolet behavior of the interaction rate completely. Thetransport interaction rate behaves like Γtrans ∼Rdq/q in naive perturbation theory.

Therefore the soft as well as thehard momentum transfer regimes contribute to Γtrans. This is very similar to the energy loss of a charged particle in arelativistic plasma, where an additional factor ω2 appears compared to the interaction rate [7,8].

Quantities which arelogarithmically infrared divergent in naive perturbation theory turn out to be finite using the Braaten-Pisarski methodfor soft momentum transfers (dynamical screening). Such quantities can be calculated using the method proposedby Braaten and Yuan [48].

According to this, introducing a separation scale q⋆, the soft and hard contributions arecalculated separately. For the soft contribution (q < q⋆) resummed propagators and vertices have to be used, whereasbare Green’s functions are sufficient for the hard one (q > q⋆).

Assuming gT ≪q⋆≪T the otherwise arbitrary scaleq⋆drops out at the end by adding the soft and the hard contribution, reflecting the completeness of the effectiveperturbation theory. In the following the Braaten-Yuan method will be used for computing the transport interactionrate following the example of the energy loss [8] as close as possible.We will present the calculation of the gluon transport rate in a pure gluon plasma (Nf = 0) in detail, quoting onlythe results for the quark and gluon transport rates in a QGP of two active flavors afterwards.

The soft contributionfollows from (3) introducing the transport weight 1 −cos θ under the integral and using q⋆as an upper limit for theq-integration:Γsoftg,trans = CAg2TπsZ q⋆0dq q3Z q−qdωω1 −ω2q2 ρl(ω, q) +1 −ω2q2ρt(ω, q). (14)The integral over q can be done exactly, while the ω-integral has to be evaluated numerically.

This has been donealready in Ref. [13] yielding (m2g = 4παsT 2/3)5

Γsoftg,trans = 3CAg2T2πsm2glnq⋆2m2g−1.379= 24πα2sT 3sln q⋆2αsT 2−2.811. (15)In Ref.

[13] the hard contribution has not been computed. However, from general arguments (q⋆-cancellation) [48] weknow that the hard contribution has to be of the form Γhardg,trans = B [ln(T 2/q⋆2) + Ahard], where B = 24πα2sT 3/s andthe constant Ahard has to be determined from a detailed calculation of the hard contribution.

Thus, if we are onlyinterested in a logarithmic accuracy (leading logarithm approximation) valid in the weak coupling limit, we end upwith [13]Γg,trans = 24πα2sT 3sln 1αs. (16)However, since the strong coupling constant is not small for realistic values expected in ultrarelativistic heavy ioncollisions, the knowledge of the coefficient under the logarithm is essential.

Therefore the constant Ahard of the hardcontribution has to be determined. Before we turn to this, let us note that the ln(1/αs)-term in (16) arises from thesensitivity of Γtrans to the momentum scale gT , while the logarithmic term in the interaction rate (7) comes from asensitivity to the scale g2T [29].

Thus the transport interaction rate can be calculated completely to leading order inthe coupling constant using the Braaten-Pisarski method in contrast to Γ. These entirely different properties of thetwo different kinds of rates originate, of course, from the additional factor q2 in Γtrans.For the hard contribution (q > q⋆) it is sufficient to use the bare gluon propagator.

However, in contrast to the softcontribution the −t ≪s approximation does not hold any more and the u- and s-channel diagrams of Fig.1 cannotbe neglected. The corresponding amplitudes have been calculated a long time ago in Feynman gauge [49,50].

Sincethe on-shell matrix elements are gauge invariant, regardless of the momenta integrated over, there is no problem inadopting those results although the soft contribution has been evaluated in Coulomb gauge [8,51].However, a new difficulty arises now from the divergence of the u-channel contribution for u = (P −K′)2 →0 i.e.,−t →s. (For massless particles s+t+u = 0 holds.) The u-channel divergence can be regulated in the same way as thet-channel singularity, if we choose the transport factor (sin θ)2/2 = 2tu/s2 instead of 1 −cos θ.

This choice is justifiedbecause the transport weight 1−cosθ has been introduced only for small scattering angles [30] for which it is identicalto (sin θ)2/2. The latter also restores the t-u-channel symmetry in the transport cross sections of quark-quark andgluon-gluon scattering.

Furthermore, the shear viscosity coefficient beyond the relaxation time approximation alsocontains a factor sin2 θ [13,16,32,33]. Finally, parton collisions with scattering angles near 0◦as well as 180◦are lessimportant for achieving an isotropic momentum distribution (thermal equilibrium) indicating the physical significanceof a transport rate defined by a weight proportional to sin2 θ instead of 1 −cos θ for the equilibration process.For calculating the hard contribution of the transport interaction rate in a gluon gas we start from (1) modified togluon-gluon scattering and including the factor (sin θ)2/2:Γ g,trans(Nf = 0) = 12pZd3p′(2π)32p′ [1 + nB(p′)]Zd3k(2π)32k nB(k)×Zd3k′(2π)32k′ [1 + nB(k′)] (2π)4 δ4(P + K −P ′ −K′) 16 ⟨|M(gg →gg)|2⟩sin2 θ2,(17)where the matrix element contains the scattering diagrams of Fig.1d.

While the hard contribution of the energy lossof a heavy fermion with mass M ≫T could be calculated exactly [8,9], this is not possible for (17). Therefore wepropose the following approximations: First we assume nB(p′) ≃nB(p) and nB(k′) ≃nB(k).

These simplificationshold as long as q = |p −p′| = |k′ −k| is not too large or −t is not of the order of s. This assumption may be justifiedbecause the transport factor (sin θ)2/2 cuts offthose momenta effectively. While we will neglect nB(p) because of⟨p⟩≃3T , we do not set nB(k) = 0.

As a matter of fact, the Bose enhancement factor 1 + nB(k) is important for theexact matching of the soft and hard parts i.e., for the cancellation of q⋆.Using the definition of the differential cross section [52] (17) may now be written asΓg,trans(Nf = 0) =Zd3k(2π)3 16 nB(k) [1 + nB(k)]Zdtdσdtgg2tus2 . (18)6

The k-integration over the Bose distribution functions gives a factor 8T 3/3, compared to 16ζ(3)T 3/π2 = 1.95 T 3neglecting the Bose enhancement factor.The differential cross section for gg →gg scattering according to thediagrams of Fig.1d are taken from Ref. [49]dσdtgg=9g464πs2−ust2 −stu2 −tus2 + 3,(19)where a factor 1/2 has been included to account for the identical particles in the final state [29].

The hard contributionfollows from (18) by restricting the t-integration from −s to −q⋆2. Assuming s ≫q⋆2, (18) together with (19) resultsinΓhardg,trans(Nf = 0) = 24πα2sT 3sln T 2q⋆2+ ln sT 2−1915.

(20)In order to proceed with the calculation we replace the Mandelstam variable s under the logarithm by its averagethermal value ⟨s⟩= 2⟨p⟩⟨k⟩= 14.59 T 2 for gluon momenta ⟨p⟩= ⟨k⟩= 2.701 T . Then we arrive atΓhardg,trans(Nf = 0) = 24πα2sT 3sln T 2q⋆2+ 1.414.

(21)Adding up the soft and hard contributions, (15) and (21), the separation scale q⋆drops out as required:Γg,trans(Nf = 0) = 24πα2sT 3sln 1αs−1.397= 24πα2sT 3sln 0.25αs≃5.2 α2s T ln 0.25αs,(22)where we have used s ≃⟨s⟩in the last equation.The result (22) may be compared to the simple approximation of using bare propagators with the Debye massas infrared regulator. Since the degree of the infrared singularity is only logarithmic this amounts approximately tousing (18), where the separation scale q⋆2 is replaced by µ2D = 4παsT 2 in the upper limit of the t-integration.

In thisway we findΓg,trans ≃24πα2sT 3sln 0.33αs. (23)Comparing to (22), one realizes that the – to leading order in αs exact – result (22) may be obtained by using aneffective infrared cut-offof 1.32 µ2D instead of µ2D.In the case of a QGP with two flavors we have to consider the diagrams of Fig.1a-c in addition.

Modifying (18)to these processes, the corresponding calculations are a little bit more involved than in the purely gluonic case. Forexample, we have to be careful about the flavors of the final state e.g., the flavors of the final state quarks of theu- and s-channel diagrams of Fig.1a and 1b have to be identical.

Furthermore there is no t-u-channel symmetry inthe quark-gluon scattering process. After all the u-channel singularity is cancelled by the transport factor (sin θ)2/2because it is only logarithmic in this case, rendering the use of an effective quark propagator unnecessary.

Also thegluon ”mass” mg depends now on Nf and, finally, in the case of quarks (or antiquarks) with momenta K and K′ wehave to replace in (18) the factor 16 by 6Nf, nB(k) by nF (k), and 1 + nB(k) by 1 −nF (k). The soft contribution forthe quark rate follows from (14) by replacing CA by CF .

Putting everything together we end up withΓg,trans(Nf = 2) =1 + Nf6 24παsT 3s′ln 0.17αs≃6.6 α2s T ln 0.17αs,Γq,trans(Nf = 2) =1 + Nf6 32παsT 33s′′ln 0.14αs≃2.5 α2s T ln 0.14αs,(24)7

where s′ = (1+Nf/6)/(1/sgg+Nf/6sgq) and s′′ = (1+Nf/6)/(1/sgq+Nf/6sqq). Here sgg is the Mandelstam variablefor two gluons, sgq for one gluon and one quark, and sqq for two quarks in the initial state, leading to ⟨s′⟩= 15.13 T 2and ⟨s′′⟩= 17.65 T 2.Applying the transport interaction rates (22) and (24) to ultrarelativistic heavy ion collisions we encounter a seriousproblem: The results (22) and (24), obtained to lowest order perturbation theory, become negative for realistic valuesof αs > 0.2.

Hence we cannot draw any conclusions regarding thermalization times at RHIC and LHC from (24).However, the validity of the Landau collision integral containing the transport cross section beyond the logarithmicapproximation depends on the condition that the characteristic length l over which the distribution function variessignificantly must be large compared with the screening length 1/µD i.e., l µD ≫1 [30], which is not fulfilled for1/l ≃T and µD ≃4√αsT . Thus the physical significance of the transport interaction rate is somewhat obscure inthe QGP.The unphysical, negative results for the transport rates for realistic values of αs arise from the separate calculation ofthe soft and hard contributions, which works only in the weak coupling limit g ≪1.

In the soft and hard contributionsthe assumption T ≫q⋆≫gT is essential for achieving the cancellation of q⋆. Of course, this assumption cannotbe fulfilled for g>∼1 rendering (15) and (21) negative, although each contribution is positive by itself without anyrestriction on q⋆, since they are equivalent to integrals over squares of magnitudes of matrix elements.

A similarproblem occured in (23) by using µD as an upper limit for the t-integration instead of a regulator in the gluonpropagator and assuming s ≫µD, which does not hold for g>∼1 any more.The problem of an unphysical, negative result already appeared in the computation of the energy loss of a heavyquark in the QGP [9]. There it was argued that by using the effective perturbation theory for the entire momentumrange this problem may be circumvented, increasing, however, the complexity of the calculation drastically.

It is notpossible simply to use the effective gluon propagator for the whole integration range, since it is well defined only forω, q ≪T . Gauge invariance and completeness in the order of the coupling constant then demand the use of effectivevertices and quark propagators in addition, corresponding to including higher orders in g [9,48].

Thus, in order toguarantee positive results and gauge invariance at the same time for values of the coupling constant g>∼1, one has togo beyond the lowest order in g. In the weak coupling limit, on the other hand, the lowest order contribution, hereα2s ln(1/αs), is sufficient. Furthermore such a unphysical behavior for realistic values of the coupling constant hasalso been observed in the calculation of the gluon plasma frequency beyond leading order [53].

In all of these cases,a negative result shows up, if g exceeds a critical value of about 1.Finally, we will discuss the implications of the transport interaction rates for the viscosity of the QGP[13,16,32,43,54–56]. In Ref.

[13,16] the shear viscosity coefficient η has been calculated fromηi ≃415ǫiΓηi,(25)where i = q, g, ǫi is the energy density of the quarks and gluons in the QGP, and Γηi = 2 Γi,trans [13]. (The factor oftwo comes from using the weight sin2 θ instead of (sin θ)2/2 in the definition of the shear viscosity coefficient [33].

)Inserting (24) into (25) we findη = ηg + ηq = T 3α2s0.11ln(0.17/αs) +0.37ln(0.14/αs). (26)This should be compared to the most elaborate calculation of η (albeit in the leading logarithm approximation) usinga variational method for the Boltzmann equation resulting in η = 1.16 T 3/[α2s ln(1/αs)] [43].

For αs < 0.1 both resultsare comparable.IV. CONCLUSIONSBefore presenting our conclusions, we discuss the validity of the approximations used here.

The distinction betweensoft and hard momenta and the cancellation of the separation scale q⋆, which are the foundations of the Braaten-Pisarski and the Braaten-Yuan method [18,48], rely on the weak coupling limit assumption g ≪1. In contrast, realisticvalues of αs > 0.2 imply g > 1.5.

Since αs is expected to decrease only logarithmically with increasing temperature,g ≪1 is not even fulfilled at the extreme temperature of the Planck scale. On the other hand, the Braaten-Pisarskimethod is nothing but an improvement of the usual perturbation theory at finite temperature which should work attemperatures above twice the critical according to comparisions with lattice QCD [57,58].

Furthermore, comparingthe effective gluon ”mass” mg, calculated using the non-perturbative Hartree approximation, with perturbative results8

shows a difference between the both results by less than 30% at g = 1.5 [41]. Also in the case of the energy loss of anenergetic quark in the QGP it has been shown that the result depends only weakly on the assumption gT ≪q⋆≪T[11].

Those observations indicate that the assumption g ≪1 should be merely regarded as a mathematical trick andnot as a physical restriction. Therefore we believe that the Braaten-Pisarski method not only provides a consistenttreatment of QCD at high temperatures taking into account at the same time important physics as collective effects ofthe non-ideal relativistic plasma e.g., screening, but also gives results for realistic situations which are correct withinabout a factor of 2 [59], as long as logarithmic factors ln(const/αs) are not too close to zero or negative as it is thecase for the transport interaction rate.We have estimated the ordinary interaction rate of thermal quarks and gluons by using the effective perturbationtheory of Braaten and Pisarski, for which the use of an effective gluon propagator is sufficient in the case of thermalpartons.

Due to the missing static magnetic screening in the transverse part of the effective gluon propagator and theabsence of an imaginary part of the quark propagator we still encounter a logarithmic infrared singularity. Assuminga reasonable cut-off, a rough estimate has been obtained, Γg = (6.0 ± 3.0) αs T for gluons and Γq = (2.7 ± 1.3) αs Tfor quarks, which corresponds to relaxation times of the order τ = (0.5 ± 0.3) fm/c for gluons and τ = (1.0 ± 0.5)fm/c for quarks.

This indicates a rapid thermalization of the gluon component (two-stage equilibration [1]) and amaintenance of the local thermal equilibrium during the expansion phase of the possibly formed QGP at RHIC andLHC in accordance with computer simulations of ultrarelativistic heavy ion collisions [2,3,45].On the other hand, in a plasma with long range interactions as in QCD the physically relevant quantity shouldbe the transport rather than the ordinary interaction rate. The transport rate follows from the ordinary one byintroducing a transport weight containing the scattering angle in the center of mass system.

Due to this factor theinfrared behavior of the interaction rate is completely changed. The transport rate turns out to be finite using theBraaten-Pisarski method because dynamical screening suffices now.

We have calculated the transport interaction ratefor thermal quarks and gluons beyond the leading logarithm approximation by decomposing it into soft and hardparts according to the prescription of Braaten and Yuan [48]. The soft contribution has been computed by using theeffective gluon propagator of the Braaten-Pisarski method, while the hard contribution has been treated using barepropagators and vertices.Compared to the ordinary interaction rate the transport rate is reduced by a factor of αs caused by the improvedinfrared behavior due to the transport weight.

For a QGP of two active flavors Γg,trans ≃6.6 α2s T ln(0.17/αs) forgluons and Γq,trans ≃2.5 α2s T ln(0.14/αs) for quarks have been found. The surprisingly small values of the coefficientsunder the logarithm show that Γtrans is only meaningful for αs<∼0.1.

Improving the calculation by using the Braaten-Pisarski method over the entire momentum range increases the complexity of the calculation enormously, includinghigher orders of g. Hence no statement about the consequences (e.g., thermalization times, mean free paths) forrealistic values of the coupling constant can be given here. For this purpose, at least a calculation beyond the lowestorder perturbation theory is required.

However, the transport interaction rate obtained here suggests that it maybe much smaller than the ordinary rate. Thus the realization of a local thermal equilibrium in relativistic heavy ioncollisions seems to be questionable assuming the transport rates to be responsible for thermalization, in contrast tocomputer simulations.

However, neither in HIJING [2] nor in the parton cascade [3] transport cross sections for thefundamental parton interactions are used.Furthermore the shear viscosity, which is proportional to the inverse of the transport interaction rate [13], has beenobtained for the first time beyond the leading logarithm approximation. For values of αs < 0.1, for which the result iswell defined, it is large and comparable to the ones obtained by using the leading logarithm approximation [13,16,43].This supports the idea that dissipation cannot be neglected in hydrodynamic descriptions of the expansion phase ofthe QGP in ultrarelativistic heavy ion collisions [16].Finally, we have compared the rates obtained from the Braaten-Pisarski method, which are complete to leadingorder in the coupling constant, with the widely used approach of using bare propagators including the Debye mass asan infrared regulator.

While the latter approximation works well for transport rates and energy losses [8], it seemsto underestimate the ordinary rates. This observation suggests that the use of the Debye regulator is justified forquantities which are logarithmically infrared divergent in naive perturbation theory as the transport rates or theenergy loss, but might be questionable for quadratically infrared divergent quantities as the ordinary interaction rate.ACKNOWLEDGMENTSThe author would like to thank T.S.

Bir´o for valuable discussions. The work was supported by the BMFT and GSIDarmstadt.9

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FIG. 1.

Lowest order Feynman diagrams for qq →qq (a), q¯q →q¯q (b), qg →qg (c), and gg →gg (d) scattering.FIG. 2.

Lowest order quark self energy contributions to the quark interaction rate using naive perturbation theory (a) andusing the Braaten-Pisarski method (b).11

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