Transport coefficients of the quark-gluon plasma in ultrarelativistic limit
aa r X i v : . [ h e p - ph ] O c t Transport coefficients of the quark-gluon plasma inultrarelativistic limit
Stefano Mattiello
Institut f¨ur Theoretische Physik, Universit¨at Gießen, D-35390 Gießen, Germany
Abstract
The calculation of the transport coefficients of the strong quark-gluon plasma(sQGP), i.e. sheer viscosity η , heat conductivity κ and bulk viscosity ζ , inthe first Chapman-Enskog approximation is presented. Their formulationin terms of two-fold integrals depending on the particle interaction, knownas relativistic omega integrals, is derived and their evaluation in the ultra-relativistic limit is worked out assuming a cross section independent of therelative and total momentum of two colliding particles. We find a suppressionof the bulk viscosity and a pronounced temperature dependence for the sheerviscosity and heat conductivity. However, at high temperatures, they scalewith the third and second power of the temperature respectively as expected.Furthermore, we find that all results in this ultrarelativistic expansion aredominated by the leading order contribution. Key words:
Quark-gluon plasma, Viscosity, diffusion, and thermalconductivity, Kinetic and transport theory of gases
PACS:
1. Introduction
Transport properties of the strongly interacting quark-gluon plasma (sQGP),in particular the bulk and shear viscosities, which describe the hydrodynamicresponse of the system to energy and momentum density fluctuations, haveattracted remarkable attention in the last years. The experimental findings ofthe heavy-ion collisions at the Relativistic Ion Collider (RHIC) have pointedto a small specific sheer viscosity, i.e. the ratio sheer viscosity to entropydensity η/s , and led to the announcement of the discovery of the nearly per-fect fluidity of the sQGP [1, 2]. The key role of the viscous corrections has
Preprint submitted to Nuclear Physics A October 24, 2018 een also addressed already by the very first v data for Pb-Pb collisions atthe LHC [3]. Indeed, the shear viscosity plays a crucial role in the estimationof the initial condition of the heavy ion collisions [4].Additionally, the hybrid model VISHNU that couples viscous hydrody-namics for the macroscopic expansion of the QGP to the hadron cascademodel for the microscopic evolution of the late hadronic stage is a verypromising tool for a more precise extraction of the QGP shear viscosity [5].Therefore, several studies on shear and bulk viscosity have been performedin the last few years, for the confined [6, 7, 8] as well as for the deconfinedphase [9, 10, 11, 12, 13]. Additionally, also for the thermal conductivity mod-els have been developed [14, 15]. Therefore a dynamic consistent calculationof all transport coefficients, i.e. the shear viscosity η , the bulk viscosity ζ and the heat conductivity κ within a model describing the strong couplingproperties of the QGP is mandatory. Recently, we have performed an inves-tigation of the viscosity η of the sQGP [9] in a dynamical way within kinetictheory in the ultrarelativistic limit using the virial expansion approach in-troduced in Ref. [16]. In this approach the choice of the interaction betweenthe partons in the deconfined phase plays a crucial role to reproduce thethermodynamic properties of the QGP in comparison to lattice QCD. By us-ing a generalized classical virial expansion we have calculated the correctionsto a single-particle partition function starting from an interaction potentialwhose parameters are fixed by thermodynamical quantities. From the samepotential one can evaluate the transport cross section and then the sheerviscosity [9].We have found η/s ≈ .
097 which is very close to the theoretical lowerbound. Furthermore, for T ≤ . T c the ratio is in the range of the experi-mental estimates 0 . − . α V to be employed for the determination ofthe transport cross section which enters the transport coefficients. In thisway, we systematically calculate all three transport coefficients η , ζ and κ .This paper is organized as follows. In Section 2, following Refs. [12, 20]we review the formal expression for the transport coefficients in the firstorder of the first Chapman-Enskog approximation. In particular, we give aformulation in terms of relativistic omega integrals. In the ultrarelativisticlimit these integrals are evaluated for cross sections which are independenton the relative and total momentum of two colliding particles.In Section 3 we calculate and discuss the transport coefficients for ourmodel within the ultrarelativistic approximation. The conclusions in Sec-tion 4 finalize this work.
2. Theory
A elegant and clear derivation of the general expression for the trans-port coefficient in the first Chapman-Enskog approximation can be found inRef. [19]. In the following we directly recall the results for the heat conduc-tivity and the sheer and bulk viscosity in the first order of this approximationof a gas of free particles with mass m at temperature Tη = 110 T γ c , (1) κ = 13 Tm β b , (2) ζ = T α a . (3)The quantities α , β and γ are known functions of the temperature andindependent of the particle interaction. In general, by introducing the di-mensionless quantity z = m/T, (4)the enthalpy per particle h = K ( z ) K ( z ) (5)3nd the adiabatic coefficient γ = c p c V (6)these scalar function are given by γ = − h = − K ( z ) K ( z ) , (7) β = − γγ − { K ( z ) /K (2) } d z − (8) α = 32 (cid:20) zh (cid:18) γ − (cid:19) + γ (cid:21) , (9)where K n ( z ) denotes the modified Bessel function of the second kind of order n . The quantities a , b and c are interaction independent and formallygiven by c = 16 (cid:18) ω (2)2 − z − ω (2)1 + 13 z − ω (2)0 (cid:19) , (10) b = 8 (cid:16) ω (2)1 + z − ω (2)0 (cid:17) , (11) a = 2 ω (2)0 , (12)where the relativistic omega integrals are in general defined as ω ( s ) i ( z ) = 2 πz K ( z ) Z ∞ d u sinh u cosh i u K j (2 z cosh u ) × Z ∞ d θ sin θ d σ ( u, θ )dΩ (1 − cos s θ ) , j = 52 + 12 ( − i . (13)The particle interaction is encoded in the differential cross section d σ ( u, θ ) / dΩ.Here θ denotes the scattering angle in the center of momentum frame. Thevariable u is directly connected to the relative momentum q and total mo-mentum Q of the two colliding particles by q = m sinh u and Q = 2 m cosh u. (14)One has to evaluate the omega integrals in order to calculate the transportproperties of the system. The results, in particular, the behavior of the crosssection as function of u and θ , are of course model dependent. The simplest4ase, i.e. constant cross section, which correspond to an assumption of hardspheres, has been investigated in Refs. [12, 20]. In following we considerthe more general case, where the cross section is momentum independent,i.e. d σ/ (dΩ) = d σ ( θ ) / dΩ and we evaluate the omega integrals under thisassumption. The momentum independence of the differential cross section leads to asimplification in Eq.(13). Performing the the integration the u integration,using x = cosh u , yields ω ( s ) i ( z ) = 2 πz K ( z ) W ( s ) Z ∞ d x ( x − x i K j (2 zx ) , (15)where the angular integral W ( s ) is given by W ( s ) = Z ∞ d θ sin θ d σ ( θ )dΩ (1 − cos s θ ) . (16)We note here, that for the calculation of the transport coefficients omegaintegrals for s = 2 have to be computed. Then, one can directly connect theintegral W (2) with the (viscous)transport section defined by [21] σ t ≡ Z dΩ d σ dΩ sin θ. (17)One obtains the relation W (2) = σ t π (18)and the relativistic omega integral may be written as ω (2) i ( z ) = σ t K ( z ) z Z ∞ d x ( x − x i K j (2 zx ) . (19)By replacing zx → x and using the binomial formulae we obtain ω (2) i ( z ) = σ t K ( z ) z − i − X k =0 (cid:18) k (cid:19) ( − k z k Z ∞ z d x x i +6 − k K j (2 x ) . (20)Because the limit z → ω (2) j around z = 0 leads to a systematic evaluation of the transport5oefficients in this approximation. A detailed investigation of this expansionallows to estimate the importance of the leading order terms as well as tosystematically compare the different transport coefficients at the same orderin z . In fact, in the ultrarelativistic limit, the integral in Eq.(20) becomesindependent of z , because the lower bound can be replaced in this expres-sion by zero. This replacement allows to perform the integration analyticallyusing Z ∞ d x x µ K ν (2 x ) = 14 Γ[( µ + ν + 1) / µ − ν + 1) /
2] (21)for µ ± ν + 1 > ω (2) i ( z ) = σ t K ( z ) z i +4 3 X k =0 (cid:18) k (cid:19) ( − k z k Γ[( µ + ν +1) / µ − ν +1) / . (22)We emphasize here, that this result is a fractional expression, where thenumerator is given by a polynomial in z . The denominator contains themodified Bessel function K ( z ), whose expansion at z = 0 can not be ex-pressed as a polynomial as shown in Appendix A. For K ( z ) the leadingterms of the expansion are K ( z ) = 2 z (cid:20) − z + − z ln z + 132 c EM z + O ( z ln z ) (cid:21) , (23)where the constant c EM is connected to the gamma of Euler-Mascheroni γ EM by c EM = 2 ln 2 + 32 − γ EM ≈ . . (24)We obtain for the omega integrals the the following expansion in zω (2) i ( z ) = σ t
16 Γ[( i + j + 7) / i − j + 5) / × (cid:18) z − i + i + j + 10 i + 1 i − j i + 25 z − (cid:19) + o ( z − i ln z ) , (25)where only the first two leading terms are considered. Combining this ex-pression with the analytic expression for the quantities α , β and γ givenin Eqs.(7-9) we can analytically evaluate the transport coefficients.6 .2. Evaluation of transport coefficients in ultrarelativistic limit For the calculation of the transport coefficients in the ultrarelativisticlimit we also need the asymptotic expansion for the scalar function α , β and γ . Using Eqs.(7-9), they can be expressed in term of the Bessel function K ( z ) and K ( x ) In addition to the asymptotic expansion K ( z ) given inEq.(23), we need the corresponding expression for K ( z ), K ( z ) = 8 z (cid:20) − z + 164 z + O ( z ln z ) (cid:21) . (26)Then we obtain for the (relevant) scalar quantities γ = 1600 z (cid:18) z (cid:19) + O ( z ln z ) , (27) β = 144 (cid:18) − z (cid:19) + O ( z ln z ) , (28) α = 136 z + O ( z ln z ) . (29)The dynamical quantities a , b and c defined in Eqs.(10)-(12) can beevaluated in the ultrarelativistic limit using Eq.(25). We obtain c = 200 σ t z (cid:18) z + O ( z ln z ) (cid:19) , (30) b = 288 σ t z (cid:0) O ( z ln z ) (cid:1) , (31) a = 3 σ t (cid:18) − z + O ( z ln z ) (cid:19) . (32)Finally, using Eqs.(1)-(3) we can write the transport coefficients as η = 4 m σ t z (cid:18) z + O ( z ln z ) (cid:19) , (33) κ = 43 σ t (cid:18) − z + O ( z ln z ) (cid:19) , (34) ζ = mz σ t (cid:0) O ( z ln z ) (cid:1) . (35)7sing the definition of z for the leading term of the shear viscosity we recoverthe known expression [9, 21] η = 4 T σ t (cid:18) z + O ( z ln z ) (cid:19) , (36)We note that the leading term is temperature dependent, but independentof the mass. For the heat conductivity κ the leading term does not dependon z . Nevertheless, a implicit temperature dependence can be encoded viathe transport cross section. Additionally, we note that in this ultrarelativis-tic approach the bulk viscosity is automatically suppressed in comparison tothe other transport coefficient as expected from general physical considera-tions [22].
3. Results
For the final calculation using Eqs.(33-35) we have to specify an interac-tion. We follow Ref. [9] to calculate the σ t from the interaction W used todescribe within a virial expansion the three-flavor thermodynamics propertiesof the QGP from the lattice [23]. We use an effective quark-quark potential W motivated in Ref. [16] and inspired by a phenomenological model whichincludes non-perturbative effects from dimension two gluon condensates [24].The effective potential between the quarks reads W ( r, T ) = (cid:18) π
12 1 r + C N c T (cid:19) e − M ( T ) r , (37)where C = (0 . is the non-perturbative dimension two condensate and M ( T ) the Debye mass estimated as M ( T ) = p N c / N f / gT = ˜ gT. (38)Setting the coupling parameter ˜ g = 1 .
30 one describes the recent three-flavorQCD lattice data for all thermodynamic quantities [23] in the temperaturerange up to 5 T c very well.The transport cross section is given by [25, 26] σ t (ˆ s ) = σ z (1 + ˆ z ) [(2ˆ z + 1) ln(1 + 1 / ˆ z ) − , (39)with the total cross section σ (ˆ s ) = 9 πα (ˆ s ) / µ . Here α V = α V ( T ) is theeffective temperature-dependent coupling constant and and ˆ z ≡ M ( T ) / ˆ s .8or simplicity, we will assume σ to be energy independent and neglect itsweak logarithmic dependence on ˆ s in the relevant energy range and set ˆ s ≈ T [27].In this study we consider only elastic cross sections. An extension toinelastic scattering processes for an ultrarelativistic Boltzmann gas and itsapplication for the calculation of the shear viscosity has been recently pre-sented [28]. Following Refs. [9, 29] we can extract the coupling α V from theinteraction W . We define the coupling in the so-called qq-scheme, α qq ( r, T ) ≡ − π r d W ( r, T )d r . (40) T/T c σ t / σ t ( ) Figure 1: (Color online) The transport cross section normalized to its value at the criticaltemperature σ t /σ (t
0) as a function of the temperature expressed in units of the criticaltemperature T c . The coupling α qq ( r, T ) then exhibits a maximum for fixed temperature ata certain distance denoted by r max . By analyzing the size of the maximumat r max we fix the temperature dependent coupling α V ( T ) as α V ( T ) ≡ α qq ( r max , T ) . (41)9ith these results we obtain a temperature dependent transport cross section σ t ( T ).In Fig. 1 the transport cross section normalized to its value at the criticaltemperature, is shown as a function of the temperature expressed in units ofthe critical temperature T c . We note a strong decrease of the cross sectionwith temperature. This can be not neglected in a consistent evaluation ofthe transport coefficients. T/T c -5051015202530 t r a n s po r t c o e ff i c i e n t s ζ /T κ /T η /T Figure 2: (Color online) The scaled sheer viscosity η/T (red line), heat conductivity κ/T (blue line) and bulk viscosity ζ/T (green line) as function of the temperature expressedin units of the critical temperature T c . In order to investigate the importance of the different coefficients wescale them by the appropriate power of the temperature in order to obtaina dimensionless quantity. Therefore we show in Fig. 2 the ratios η/T (redline), κ/T (blue line) and ζ /T (green line) as a function of the temperatureexpressed in units of the critical temperature T c . Several features becomeevident: i) For the shear viscosity and the heat conductivity an increase with thetemperature is found for T c . T . . T c .10 T/T c ζ Figure 3: (Color online) The bulk viscosity ζ (solid line) as function of the temperatureexpressed in units of the critical temperature T c . In comparison the bulk viscosity usingthe temperature independent transport cross section σ (0)t . ii) At higher temperatures η/T and κ/T become constant. This behaviorof the shear viscosity is also present in very different approaches, suchas AdS/CFT [30], the quasiparticle approximation including the quarkselfenergy [31, 32] and in the weak coupling estimate from Ref. [33, 34].For the heat conductivity that is also found in perturbative calcula-tions [14, 35], where κ scales with the temperature square. iii) The bulk viscosity is completely negligible in comparison to the othercoefficients. This justifies the omission of ζ in several hydrodynamicaland transport calculations [36, 37, 38].In Fig. 3 we show ζ as function of the temperature expressed in units ofthe critical temperature T c (solid green line). To investigate the role of thetemperature dependent transport cross section we compare with the resultsobtained using the constant cross section σ (0)t (green dashed line) In thecalculation with the full cross section the suspected enhancement of the bulkviscosity is missing. In contrast, the interplay between cross section and z leads to a broad maximum around 3 . T c . This is clearly a consequence of11he temperature dependence of σ t : namely, using the constant value σ (0)t forthe transport cross section, we find a decreasing behavior, that suggests thepossibility of a maximum of the bulk viscosity at T c . It is not surprisingthat our ultrarelativistic model can not describe the peak of ζ at the criticaltemperature. This maximum is a consequence of the hadronic correlations at T c [39] that are non included in this approach. To include such correction onecan add to this parameter free derivation of the cross section an estimationof the hadronic transport cross section and the omega integrals for the wholetemperature range have to be calculated. Therefore, analytic expressions forthe coefficients can not performed [20].Alternatively, an estimate for the correlation term of the the bulk viscositycan be given using the expression ζ cor = − A ( c − / η , with A = 4 .
558 or A = 4 .
935 [40] or A = 2 [41] respectively.Another important aspect of our results regarding the sheer viscosity andthe heat conductivity is the relative importance of the next to leading ordercontributions. In Fig. 4 the percent deviation ∆ η of the sheer viscosity (red T/T c D e v i a ti on [ % ] η κ Figure 4: (Color online) The percentage deviation sheer viscosity ∆ η (red line) and of theheat conductivity ∆ κ (blue line) as function of the temperature expressed in units of thecritical temperature T c . line) and ∆ κ of the heat conductivity (blue line) have been investigated. The12ercent deviations are defined by∆ X = | X − X (NLO) | X × , (42)where X = η, κ and X (NLO) denotes the next to leading order contribution.Evidently, the deviations from the leading order terms are very small andthe higher order contributions can be neglected. In this sense, the calcula-tions of the ratio sheer viscosity to entropy density performed in Ref. [9],where the leading term of the ultrarelativistic expansion and the entropywithin this virial expansion model have been used, is unchanged by the higherterm corrections.
4. Conclusions
We presented the calculation of the transport coefficients of the QGP byrevisiting the first order of the first Chapman-Enskog approximation. Weformulated them in terms of the relativistic omega integrals. Their evalua-tion in the ultrarelativistic limit is explictly shown by assuming that the crosssection is independent on the relative and total momentum of two collidingparticles. The well known features of the ultrarelativistic limit are recov-ered, i.e. the suppression of the bulk viscosity in comparison to the othercoefficients. In particular, the expected peak of ζ at the critical temperatureis missing, because the hadronic correlations at T c have not been included.However, the overall suppression justifies neglecting of the bulk viscosity inseveral calculations for the evolution of the QGP [4, 36, 37, 38]. For the sheerviscosity and heat conductivity we observe an increasing behavior with thetemperature. At high temperatures, they scale as power law in T . Addition-ally, we find that all results in this ultrarelativistic expansion are dominatedby the leading order contribution. Therfore, for the ratio sheer viscosity toentropy density we recover the results of our previous estimation [9].For further work, the limitation of the ultrarelativistic limit has to be re-laxed and the coefficients can be evaluated in the whole temperature range.This allows the including of hadronic correlations at T c in the calculations of ζ . Additionally, we can implement the temperature dependent transport co-efficients calculated here as well as our realistic equation of state in dissipativehydrodynamical calculation. In this way we can systematically investigatethe evolution of the QGP in the heavy-ion collisions following the lines ofRef. [4]. 13cknowledgment: I thank Stefan Strauss and Hendrik van Hees for usefuldiscussions and suggestions. This work is supported by Deutsche Forschungs-gemeinschaft and by the Helmholtz International Center for FAIR within theLOEWE program of the State of Hesse. A. Modified Bessel functions of second kind
The Bessel functions [42], first defined by the mathematician DanielBernoulli and generalized by Friedrich Bessel, are canonical solutions w ( z )of Bessel’s differential equation z d w d z + x d w d z + ( z − ν ) w = 0 (43)for an arbitrary real or complex number ν which indicate the order of theBessel function. In general, ν is integer or half-integer. The solutions calledBessel functions of the first kind are denoted as J ν ( z ) and are holomorphicfunctions of z throughout the z -plane cut along the negative real axis. It ispossible to define the function by its Taylor series expansion around z = 0 J ν ( x ) = 12 z ∞ X k =0 ( − z ) k k ! Γ( ν + k + 1) , (44)where Γ( z ) is the Gamma function defined byΓ( z ) = Z ∞ t z − e − t d t . (45)By replacing z by ± iz one obtains the modified Bessel equation z d w d z + x d w d z + ( z + ν ) w = 0 . (46)The solutions are called modified Bessel functions of first and second kind, I ν ( z ) and K ν ( z ) respectively, and are connected to the functions J ν ( z ) by I ν ( z ) = i − ν J ν ( iz ) , (47) K ν ( x ) = π I − ν ( x ) − I ν ( x )sin( νπ ) (48)Unlike the ordinary Bessel functions, which are oscillating as functions of areal argument, the modified Bessel functions are exponentially growing and14ecaying functions, respectively. For real arguments x , the function I ν ( x )goes to zero at x = 0 for ν > x = 0 for ν = 0, like theordinary Bessel function J ν ( x ). Analogously, K ν ( x ) diverges at x = 0.There are many integral representations of these functions, using integralsalong the real axis as well as contour integrals in the complex plane. In thefollowing we focus on the modified Bessel functions of the second kind. Veryuseful integral representations are K ν ( z ) = √ π ( z ) ν Γ( ν + ) Z ∞ d t e − zt ( t − ν − , ℜ ν > − , | ph z | < π (49) K ν ( z ) = Z ∞ d t e − z cosh t cosh( νt ) , | ph z | < π. (50)Like the ordinary Bessel function, the modified Bessel function of second kindcan be expressed as a power series. In the case of integer order the expansionreads K n ( z ) = 12 ( 12 z ) − n n − X k =0 ( n − k − k ! ( − z ) k + ( − n +1 ln( 12 z ) I n ( z )+ ( − n