Shear viscosity of the gluon plasma in the stochastic-vacuum approach
aa r X i v : . [ h e p - ph ] M a y Shear vis osity of the gluon plasma in the sto hasti -va uumapproa hDmitri AntonovFakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, GermanyShear vis osity of the gluon plasma in SU(3) YM theory is al ulated nonpertur-batively, within the sto hasti va uum model. The result for the ratio of the shearvis osity to the entropy density, proportional to the squared hromo-magneti gluon ondensate and the (cid:28)fth power of the orrelation length of the hromo-magneti va -uum, falls o(cid:27) with the in rease of temperature. At temperatures larger than thede on(cid:28)nement riti al temperature by a fa tor of 2, this fall-o(cid:27) is determined by thesixth power of the temperature-dependent strong- oupling onstant and yields anasymptoti approa h to the onje tured lower bound of / (4 π ) , a hievable in N = 4 SYM theory. As a by-produ t of the al ulation, we (cid:28)nd a parti ular form of the two-point orrelation fun tion of gluoni (cid:28)eld strengths, whi h is the only one onsistentwith the Lorentzian shape of the shear-vis osity spe tral fun tion.I. INTRODUCTION AND PRELIMINARY ESTIMATESThe RHIC data on olle tive expansion dynami s of the quark-gluon plasma an be de-s ribed by the relativisti hydrodynami s applied to a system with very large initial pressuregradients [1℄. A ording to these data, parti les of di(cid:27)erent mass are emitted from the quark-gluon-plasma (cid:28)reball with a ommon (cid:29)uid velo ity, that is a signature of a hydrodynami expansion. Due to a large ellipti (cid:29)ow in non entral ollisions [2℄, an agreement betweenthe experimental data [2, 3℄ and the predi tions of relativisti hydrodynami s an only berea hed when the (cid:29)ow of the QGP-(cid:29)uid is treated as almost non-vis ous. This leads toan indi ation that, in the vi inity of the de on(cid:28)nement phase transition, the quark-gluonplasma produ ed in the RHIC experiments behaves more like an ideal quantum liquid ratherthan a weakly intera ting gas. The mean free path L mfp of a parton, whi h traverses su h aliquid, is mu h smaller than the inter-parti le distan e, whi h is of the order of the inversetemperature β = 1 /T , i.e. ( L liqmfp /β ) ≪ .One an onsider for omparison a weakly intera ting dilute-gas model of the quark-gluonplasma. There L gasmfp ∼ ( ρσ t ) − with ρ and σ t standing for the parti le-number density andthe Coulomb transport ross se tion, respe tively. Using the standard estimates ρ ∼ T and σ t ∼ g β ln g − , where g = g ( T ) is the perturbative (cid:28)nite-temperature QCD oupling,one obtains ( L gasmfp /β ) ∼ / ( g ln g − ) ≫ , that strongly ontradi ts the above-mentionedexperimental results. One an he k [4℄ that these results ould have only been reprodu edby the dilute-gas model if the perturbative transport ross se tion, σ t , were larger by anorder of magnitude. This in onsisten y of the weakly intera ting quark-gluon plasma withthe RHIC data initiated re ent al ulations of kineti oe(cid:30) ients in the strongly intera tingrelativisti plasmas.Among these oe(cid:30) ients, the one whose values de(cid:28)ne whether the plasma an be on-sidered as weakly or strongly intera ting is the shear vis osity η . It is related to the above ( L mfp /β ) -ratio via the estimate ( η/s ) ∼ ( L mfp /β ) , where s is the entropy density. A ord-ing to this relation, the shear-vis osity to the entropy-density ratio, η/s , be omes smallerwhen the plasma intera ts stronger. For instan e, for T ∼
200 MeV and the estimatedtypi al mean free path L mfp ∼ . , one has ( η/s ) ∼ . . On the other hand, sin e themean momentum hange ∆ p of a parton, whi h propagates through the plasma over thedistan e L mfp , is of the order of T , the Heisenberg un ertainty prin iple forbids the ratio ( L mfp /β ) ∼ L mfp · ∆ p (and therefore also η/s ) to be vanishingly small. Up to now, theminimal value of / (4 π ) ≃ . for the shear-vis osity to the entropy-density ratio has beenfound in N = 4 SYM theory [5℄. It is thus hallenging to (cid:28)nd other QCD-motivated modelswhere this ratio would be that small. Re ently, it has been demonstrated [6℄ that su hlow values of the ( η/s ) -ratio an take pla e even in the perturbative YM plasma, due tothe bremsstrahlung gg ↔ ggg pro esses. In Ref. [7℄ it has been argued, though, that theperturbatively al ulated ollisional vis osity is anyhow larger (and therefore subdominant) ompared to the so- alled anomalous vis osity, whi h is generated by plasma instabilities.In this paper, we al ulate the ( η/s ) -ratio in the gluon plasma of SU(3) YM theorynonperturbatively. We obtain the shear vis osity from the Kubo formula, whi h relatesthe orresponding spe tral density ρ ( ω ) to the Eu lidean orrelation fun tion of the (1 , - omponent of the energy-momentum tensor T ( x , x ) . This method, proposed in Ref. [8℄,has been explored in Refs. [9, 10℄ with the aim to simulate shear vis osity on the latti e.Here we work in the ontinuum limit and parametrize the Eu lidean orrelation fun tion ofthe energy-momentum tensors by means of the sto hasti va uum model [11℄. This modelgeneralizes QCD sum rules by assuming the existen e of not only the gluon ondensate (cid:10) g ( F aµν ) (cid:11) but also of the (cid:28)nite va uum orrelation length µ − . This assumption is justi(cid:28)edby the latti e results on the exponential fall-o(cid:27) at large distan es of the two-point orrelationfun tion of gluoni (cid:28)eld strengths [13, 14℄, (cid:10) F aµν ( x ) F bλρ (0) (cid:11) ∼ e − µ | x | . By virtue of this (cid:28)nding,the model manages to quantitatively des ribe on(cid:28)nement; for instan e, the string tensionreads σ ∝ µ − (cid:10) g ( F aµν ) (cid:11) .Below we will use a (cid:28)nite-temperature generalization of the sto hasti va uum model,a essible by implementing the Eu lidean periodi ity of the x - oordinate. In the de on-(cid:28)nement phase of interest, su h a generalization yields for the spatial string tension σ s ( T ) a formula [15℄ pretty similar to its above-quoted va uum ounterpart. This formula reads σ s ( T ) ∝ µ − ( T ) (cid:10) g ( F aij ) (cid:11) T , where µ − ( T ) is the orrelation length of the hromo-magneti va uum, and (cid:10) g ( F aij ) (cid:11) T is the hromo-magneti gluon ondensate, whi h survives the de- on(cid:28)nement phase transition. The temperature dependen e of the two main ingredients ofthe model, µ ( T ) and (cid:10) g ( F aij ) (cid:11) T , an be extra ted from the results of the latti e simula-tions [13, 16℄.Sin e T = g F a µ F a µ , one a priori expe ts from the Kubo formula, where the h T (0) T ( x ) i - orrelator enters, that the shear vis osity η ∝ (cid:10) g ( F aij ) (cid:11) T . This is a gen-eral predi tion of the sto hasti va uum model for all the kineti oe(cid:30) ients, for examplefor the jet quen hing parameter [17℄. In fa t, a ording to the Kubo formula, all the kineti oe(cid:30) ients are proportional to the total s attering ross se tion of the propagating parton,whi h itself is proportional to (cid:10) g ( F aij ) (cid:11) T in this model [18, 19℄. Therefore, sin e the shearvis osity has the dimensionality of [mass℄ , one an on entirely dimensional grounds expe tfor it the following result: η ∝ µ − ( T ) (cid:10) g ( F aij ) (cid:11) T . At temperatures larger than the temperature of dimensional redu tion,
T > T ∗ , µ ( T ) and (cid:10) g ( F aij ) (cid:11) T are proportional to the orresponding power of the only dimensionful parameterpresent in the YM a tion at su h temperatures, g T , i.e. µ ( T ) ∝ g T, (cid:10) g ( F aij ) (cid:11) T ∝ ( g T ) . On the other hand, the entropy density s ( T ) ∝ T , so that ηs ∝ g ( T ) at T > T ∗ . In this paper, we quantitatively answer the naturally arising question of whether or not thisfun tion manages to get below the (4 π ) − -threshold at temperatures T . T c , whi h area essible experimentally and on the latti e.The outline of the paper is as follows. In Se tion II, by assuming an exponential fall-o(cid:27)for the two-point orrelation fun tion of the energy-momentum tensors h T (0) T ( x ) i , weobtain from the Kubo formula an integral equation for the spe tral density ρ ( ω ) of η . InSe tion III, by using for ρ ( ω ) a Lorentzian-type ansatz, with the same orrelation lengthas that of h T (0) T ( x ) i , we explore this equation for the ases of large and small | k | 's,where k is the number of a Matsubara mode. The solution in the large- | k | limit yields therange of variation of the numeri al parameter α , whi h enters the initial parametrizationof h T (0) T ( x ) i . The solution in the small- | k | limit an only oin ide with the large- | k | solution for a single value of α from this range. This (cid:28)xes α ompletely and makes further al ulations straightforward. In Se tion IV, we analyti ally al ulate the shear vis osity η .In Se tion V, we use this result for η to numeri ally (cid:28)nd the ratio η/s . Also in Se tion Vwe ompare the al ulated nonperturbative spe tral density ρ ( ω ) with the perturbative one,whi h dominates at large ω 's. In Se tion VI, we dis uss the results of the paper, as well aspossible further developments. In Appendix A, we illustrate the separation of perturbative ontributions to the Kubo formula from the nonperturbative ones.II. KUBO FORMULA FOR THE SPECTRAL DENSITYShear vis osity η an be de(cid:28)ned through the relation η = π dρdω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 , (1)where the spe tral density ρ ( ω ) is a solution to the following integral equation, alled Kuboformula [8, 10℄ Z ∞ dωρ ( ω ) cosh (cid:2) ω (cid:0) x − β (cid:1)(cid:3) sinh( ωβ/
2) = Z d x + ∞ X n = −∞ h T (0) T ( x , x − βn ) i . (2)Here the sum on the RHS runs over winding modes. We use the Fourier de omposition onthe LHS of Eq. (2): cosh (cid:2) ω (cid:0) x − β (cid:1)(cid:3) sinh( ωβ/
2) = 2 T · ω + ∞ X k = −∞ e iω k x ω + ω k , (3)where ω k = 2 πT k is the k -th Matsubara frequen y. The idea is to have a similar de ompo-sition also on the RHS of Eq. (2). For the implementation of this idea, the following hainof equalities is important: + ∞ X k = −∞ e iω k x [1 + ( ω k /M ) ] α = 1Γ( α ) + ∞ X k = −∞ Z ∞ dλλ α − e − λ [ ω k /M ) ] + iω k x == M β √ π Γ( α ) Z ∞ dλλ α − e − λ + ∞ X n = −∞ e − M x − βn )24 λ == M β π Γ( α ) Z ∞ dλλ α − e − λ Z d x + ∞ X n = −∞ e − M [ x x − βn )2 ] λ == M β α +1 π Γ( α ) Z d x + ∞ X n = −∞ K − α (cid:16) M p x + ( x − βn ) (cid:17)h M p x + ( x − βn ) i − α . Here α > is some parameter, (cid:16) Γ (cid:17) and (cid:16) K − α (cid:17) stand for the Gamma and the Ma Donaldfun tions, respe tively. We assume that, at T = 0 , h T (0) T ( x ) i = N ( α ) (cid:10) G (cid:11) K − α ( M | x | )( M | x | ) − α , (4)where h G i ≡ (cid:10) g ( F aµν ) (cid:11) and N ( α ) > is a numeri al oe(cid:30) ient, whi h will be determined.Then, in the de on(cid:28)nement phase ( T > T c ) of interest, the above hain of equalities yields RHS of Eq . (2) = 2 T · π α Γ( α ) N ( α ) (cid:10) G (cid:11) T M α − ∞ X k = −∞ e iω k x ( ω k + M ) α , where h G i T ≡ (cid:10) g ( F aij ) (cid:11) T . Note that, for temperatures T > T ∗ , only the ( k = 0) -termin the sum should be onsidered, sin e the dimensionally redu ed theory is a theory of thezeroth Matsubara mode.By using also Eq. (3), we an now rewrite Eq. (2) in terms of Fourier modes as Z ∞ dωρ ( ω ) ωω + ω k = π α Γ( α ) N ( α ) (cid:10) G (cid:11) T M α − ( ω k + M ) α . (5)To solve this equation, we use the parametrization ρ ( ω ) = ωf ( ω ) , where f ( ω ) is some even fun tion su(cid:30) ient for the onvergen e of the integral at large ω 's.Motivated by earlier works [8, 10, 12℄, we hoose it in a Lorentzian-type form f ( ω ) = C ( ω + M ) α + . (6)Here C = C ( T ) > is some fun tion, whi h will be determined. Apparently, the ω -integration in Eq. (5) onverges for any hoi e of α > . Parametrization of the spe tral density through Eq. (6) guarantees furthermore that bothsides of Eq. (5) have the same leading large- | k | behavior. It also implies [12℄ that M = M ( T ) is the momentum s ale below whi h perturbation theory breaks down. For this reason, M ( T ) should be of the order of the inverse orrelation length of the hromo-magneti va uum, µ ( T ) .Shear vis osity an (cid:28)nally be obtained by means of Eq. (1) as η = πCM α +1 . (7)We now solve Eq. (5) subsequently in the following ases: | k | ≫ and | k | ∼ .III. CONTRIBUTIONS TO η FROM HIGH AND LOW MATSUBARA MODESA. ( | k | ≫ - asePlugging the Lorentzian-type ansatz (6) into Eq. (5), we obtain LHS of Eq . (5) = C Z ∞ dω ω ( ω + ω k )( ω + M ) α + == Cω αk Z ∞ dx x ( x + 1)( x + ξ k ) α + , where x ≡ ω/ω k and ξ k ≡ M/ω k . At | k | ≫ , one has | ξ k | ≪ , (8)and the latter integral yields LHS of Eq . (5) = Cω αk (cid:26) ξ αk (cid:20) √ π Γ( α − α + ) ξ k + O ( ξ k ) (cid:21) + π πα ) + O ( ξ k ) (cid:27) . (9)We see that, if α < , then the leading term of the expansion is ∝ π πα ) , i.e. it is k -independent, whereas otherwise the leading term of the expansion be omes k -dependent.For this reason, we restri t ourselves to α < only. Furthermore, sin e αξ k ≪ as well, one an expand in powers of ξ k also the RHS ofEq. (5) to obtain RHS of Eq . (5) = π α Γ( α ) N ( α ) h G i T ω αk M α − [1 + O ( ξ k )] . (10)Equations (9) and (10) yield the fun tion C : C = π α +1 Γ( α ) N ( α ) sin( πα ) (cid:10) G (cid:11) T M α − . A ordingly, Eq. (7) yields for the shear vis osity η (cid:12)(cid:12)(cid:12) | k |≫ = π α +1 Γ( α ) N ( α ) sin( πα ) h G i T M . (11)The parametri dependen e of this expression on h G i T and M is indeed the one followingfrom the elementary dimensional analysis made in Introdu tion.B. ( | k | ∼ - aseConsider | k | 's su(cid:30) iently small for the inequality | ω k | ≪ M, i . e . | ξ k | ≫ , (12)to hold. Disregarding terms O ( ω k /M ) and higher, one has LHS of Eq . (5) ≃ CM α Z ∞ dz ( z + 1) α + = √ π α )Γ (cid:0) α + (cid:1) CM α , where z ≡ ω/M , while RHS of Eq . (5) ≃ π α Γ( α ) N ( α ) h G i T M . We obtain from these two equations C = π / α +1 Γ (cid:16) α + 12 (cid:17) N ( α ) (cid:10) G (cid:11) T M α − and, a ording to Eq. (7), η (cid:12)(cid:12)(cid:12) | k |∼ = π / α +1 Γ (cid:16) α + 12 (cid:17) N ( α ) h G i T M . (13)In parti ular, at T > T ∗ , where only the ( k = 0) -mode should be onsidered, this resultbe omes exa t. PSfrag repla ements η (cid:12)(cid:12) | k | ≫ / η (cid:12)(cid:12) | k | ∼ α Figure 1: The ratio η (cid:12)(cid:12) | k |≫ η (cid:12)(cid:12) | k |∼ given by Eq. (25).IV. η FROM THE CORRELATION FUNCTION h T (0) T ( x ) i We determine now the parameters N ( α ) and M , whi h enter the orrelation fun tion (4).This orrelation fun tion reads h T (0) T ( x ) i = (cid:10) g F a µ (0) F a µ (0) F b ν ( x ) F b ν ( x ) (cid:11) == (cid:10)(cid:10) g F a µ (0) F a µ (0) F b ν ( x ) F b ν ( x ) (cid:11)(cid:11) + (cid:10) g F a µ (0) F a µ (0) (cid:11) (cid:10) g F b ν ( x ) F b ν ( x ) (cid:11) ++ (cid:10) g F a µ (0) F b ν ( x ) (cid:11) (cid:10) g F a µ (0) F b ν ( x ) (cid:11) + (cid:10) g F a µ (0) F b ν ( x ) (cid:11) (cid:10) g F a µ (0) F b ν ( x ) (cid:11) , (14)where double angular bra kets denote a onne ted (or irredu ible) average. We use theGaussian-dominan e hypothesis [11℄, whi h allows one to disregard this onne ted aver-age. For the two-point orrelation fun tion of gluoni (cid:28)eld strengths we use the standardparametrization [11, 19℄ (cid:10) g F aµν ( x ) F bλρ ( x ′ ) (cid:11) = ( δ µλ δ νρ − δ µρ δ νλ ) h G i N c − δ ab D ( x − x ′ ) , (15)where D ( x ) is a dimensionless fun tion mediating the on(cid:28)ning intera tion. In thisparametrization, we have disregarded a small ontribution of non- on(cid:28)ning non-perturbativeintera tions [17℄, [26℄. By using Eq. (15), we obtain for the orrelation fun tion (14): h T (0) T ( x ) i ≃ (cid:10) g F a µ (0) F a µ (0) (cid:11) (cid:10) g F b ν ( x ) F b ν ( x ) (cid:11) + (cid:10) g F a µ (0) F b ν ( x ) (cid:11) (cid:10) g F a µ (0) F b ν ( x ) (cid:11) + + (cid:10) g F a µ (0) F b ν ( x ) (cid:11) (cid:10) g F a µ (0) F b ν ( x ) (cid:11) = h G i N c − D ( x ) . (16)The dimensionless fun tion D ( x ) is usually hosen in the form D ( x ) = e − µ | x | . (17)Inserting this expression into the formula for the string tension in the fundamental repre-sentation, σ f = h G i Z d xD ( x ) , (18)one an de(cid:28)ne the gluon ondensate in terms of σ f and the va uum orrelation length µ asfollows [17, 19℄: (cid:10) G (cid:11) = 72 π σ f µ . (19)To obtain for the orrelator h T (0) T ( x ) i the fun tional form given by the RHS ofEq. (4), we modify parametrization (17) to D ( x ) = A ( α ) s K − α (2 µ | x | )(2 µ | x | ) − α , (20)where A ( α ) is a numeri al normalization fa tor. At | x | & µ − , the new fun tion (20) falls o(cid:27)with the same exponent as Eq. (17). To obtain the normalization fa tor A ( α ) , we substituteEq. (20) into relation (18), whi h holds for any fun tion D ( x ) . Using further expression (19),we obtain A ( α ) = 4 R ∞ dz p z α K − α ( z ) . (21)The orrelator (16) now reads h T (0) T ( x ) i = A ( α )576 (cid:10) G (cid:11) K − α (2 µ | x | )(2 µ | x | ) − α , (22)where the fun tion A ( α ) is given by Eq. (21), and we have (cid:28)xed N c = 3 . Comparing Eq. (22)with the original de(cid:28)nition (4), we on lude that N ( α ) = A ( α )576 and M = 2 µ. Equations (11) and (13) yield now ontributions to η from high and low Matsubara modes: η (cid:12)(cid:12)(cid:12) | k |≫ = π α Γ( α ) sin( πα ) [ A ( α ) h G i T ] µ ( T ) (23)0 T/T c PSfrag repla ements s ( T ) / T Figure 2: Entropy density s ( T ) in the units of T obtained from the latti e values for the pressure p lat [16℄ ( ourtesy of F. Kars h).and η (cid:12)(cid:12)(cid:12) | k |∼ = π / α Γ (cid:16) α + 12 (cid:17) [ A ( α ) h G i T ] µ ( T ) . (24)The ratio of these results, η (cid:12)(cid:12)(cid:12) | k |≫ η (cid:12)(cid:12)(cid:12) | k |∼ = Γ( α ) sin( πα ) √ π Γ (cid:16) α + (cid:17) , (25)in the interval < α < of interest is plotted in Fig. 1. It equals to unity at α = 1 / , i.e.,at this value of α , our results for shear vis osity be ome k -independent, as they should be.This yields the prin ipal analyti result of the present paper: η ( T ) = π / √ A (1 / h G i T ] µ ( T ) , (26)where A (1 / ≃ . . Remarkably, ansatz (6) at α = 1 / takes the onventional Lorentzianform. In the next Se tion, we will evaluate the ratio η/s numeri ally.1V. NUMERICAL EVALUATIONFollowing Ref. [16℄, we assume the value T c = 270 MeV in SU(3) YM theory. We use thetwo-loop running oupling [16℄ g − ( T ) = 2 b ln T Λ + b b ln (cid:18) T Λ (cid:19) , where b = 11 N c π , b = 343 (cid:18) N c π (cid:19) , Λ = 0 . T c , and N c = 3 for the ase under study. We also assume for µ ( T ) and for the spatial stringtension in the fundamental representation, σ f ( T ) , the following parametrizations [15, 17℄: µ ( T ) = µ · T c < T < T ∗ , g ( T ) Tg ( T d . r . ) T d . r . at T > T ∗ , (27) σ f ( T ) = σ f · T c < T < T ∗ , h g ( T ) Tg ( T d . r . ) T d . r . i at T > T ∗ , (28)where µ = 894 MeV [13℄ and σ f = (0 .
44 GeV) . Equation (19), extrapolated to (cid:28)nitetemperatures, yields for the hromo-magneti gluon ondensate h G i T [15, 17℄: (cid:10) G (cid:11) T = 72 π σ f ( T ) µ ( T ) = (cid:10) G (cid:11) · T c < T < T ∗ , h g ( T ) Tg ( T d . r . ) T d . r . i at T > T ∗ . The value of T ∗ an be estimated from the equation σ f ( T ∗ ) = σ f , where σ f ( T ) = [0 . g ( T ) T ] is the high-temperature parametrization of the fundamentalspatial string tension [16℄. Solving this equation numeri ally, one obtains [20℄ T ∗ = 1 . T c . The entropy density s = s ( T ) an be obtained by the formula s = ∂p lat /∂T , where weuse for the pressure p lat the orresponding latti e values from Ref. [16℄. In Fig. 2, we plot s ( T ) in the units of T . The temperature dependen e of the ratio η/s is determined by thefun tion h G i T / [ µ ( T ) s ( T )] . One an he k numeri ally that, at T & T c where s/T isnearly onstant, h G i T / [ µ ( T ) s ( T )] = O (cid:0) g ( T ) (cid:1) , as was mentioned in Introdu tion.In Fig. 3, we plot the ratio η/s , with η given by Eq. (26), as a fun tion of temperature.Also in Fig. 3, we plot the onje tured lower bound for this ratio, equal to (4 π ) − ≃ . ,2 T/T c Our resultN=4 SYM
PSfrag repla ements η / s Figure 3: Cal ulated values of the ratio η/s as a fun tion of temperature. Also shown is the onje tured lower bound of (4 π ) − for this quantity, realized in N = 4 SYM.whi h is realized in N = 4 SYM [5℄. This bound is indeed not rea hed by our values,although they get very lose to it at the highest temperature T = 4 . T c where the latti edata for the pressure (and therefore also for s ) are available.Furthermore, we ompare numeri ally the obtained nonperturbative spe tral density, ρ ( ω, T ) = C ( T ) ωω + 4 µ ( T ) , where C ( T ) = (cid:16) π (cid:17) / A (1 / h G i T µ ( T ) , (29)with the perturbative one, whi h at the tree level reads [9, 10℄ ρ pert ( ω, T ) = 120 π θ ( ω − ˜ ω ( T )) ω tanh ω T . Note that ρ pert ( ω, T ) = 0 only at ω > ˜ ω ( T ) , where ˜ ω ( T ) ≃ . T [10℄. For this reason, ρ pert ( ω, T ) in any ase does not a(cid:27)e t the al ulated η , whi h is de(cid:28)ned a ording to Eq. (1)by the values of the spe tral density at ω → . Figure 4 illustrates the full spe tral density ρ full = ρ + ρ pert as a fun tion of ω/T at T = T c . At ω < ˜ ω ( T ) , ρ full is given by theobtained result (29), while at ω > ˜ ω ( T ) the perturbative part ρ pert takes it over. Were ρ pert nonvanishing down to ω = 0 , it would dominate over ρ already at ω > . T . That is thereason why, by ω = ˜ ω ( T ) , ρ pert signi(cid:28) antly ex eeds ρ , as one an see from the gap in the3values of ρ full at this value of ω . Qualitatively, the same ρ full and the relation between ρ and ρ pert persist with the in rease of temperature.Finally, in Appendix A, we illustrate the orresponden e between the splitting of ρ full = ρ + ρ pert and the splitting of h T (0) T ( x ) i full = h T (0) T ( x ) i + h T (0) T ( x ) i pert . This orresponden e allows one to isolate the ontribution, whi h ρ pert brings about to the ω -integral in the full Kubo formula.VI. DISCUSSION AND OUTLOOKIn this paper, we have applied Kubo formula to a nonperturbative al ulation of theshear vis osity η in SU(3) YM theory. With the use of the sto hasti va uum model,the h T (0) T ( x ) i - orrelator entering Kubo formula has been expressed in terms of thetemperature-dependent hromo-magneti gluon ondensate (cid:10) g ( F aij ) (cid:11) T and the orrelationlength of the hromo-magneti va uum µ − ( T ) . As was expe ted ( f. Introdu tion), η turnsout to be ∝ µ − ( T ) (cid:10) g ( F aij ) (cid:11) T , where the numeri al fa tor is given by Eq. (26). At tem-peratures T & T c , the al ulated ratio η/s falls o(cid:27) as O (cid:0) g ( T ) (cid:1) , as it should do in thedimensionally-redu ed theory. Numeri ally, up to the temperature T = 4 . T c , where thelatti e data on bulk thermodynami quantities are still available, the obtained values of theratio η/s stay above the onje tured lower bound of (4 π ) − , whi h is rea hed in N = 4 SYM.Formally, our result (26) persists even at higher temperatures, being extrapolated towhi h it yields for the ( η/s ) -ratio values smaller than (4 π ) − . One should, however, re-alize that the monotoni fall-o(cid:27) of η/s with temperature, stemming from the relation η ∝ µ − ( T ) (cid:10) g ( F aij ) (cid:11) T , is prede(cid:28)ned by our al ulational method, whi h ombines Kuboformula with the sto hasti va uum model. In fa t, all the kineti oe(cid:30) ients derivable inthis way should be ∝ (cid:10) g ( F aij ) (cid:11) T ( f. Introdu tion). In parti ular, this is the ase for thebulk vis osity ζ [12℄, whi h an be obtained from the orrelation fun tion (cid:10) g F a µν (0) F b λρ ( x ) (cid:11) ≃ (cid:10) G (cid:11) (cid:20) N c − D ( x ) (cid:21) . (Here (cid:16) ≃ (cid:17) stands for (cid:16)Gaussian approximation(cid:17).) On general grounds [23℄, one indeed ex-pe ts a monotoni fall-o(cid:27) of the ( ζ /s ) -ratio with temperature, as was on(cid:28)rmed by expli it al ulations [12℄. However, on the same general grounds [23℄, for the ( η/s ) -ratio in question4 PSfrag repla ements ω/T ρ f u ll ( ω , T ) Figure 4: The full spe tral density ρ full ( ω, T ) as a fun tion of ω/T at T = T c . At ωT < . , ρ full is entirely nonperturbative and given by Eq. (29), while at ωT > . the perturbative part ρ pert isdominating.one expe ts the existen e of a minimum at temperatures lose to T c [27℄ and a subsequentin rease with the further in rease of temperature. Indeed, at least at the temperatureas high as . T c , g ( T ) be omes smaller than unity, and the weakly-intera ting dilute-gasmodel of the gluon plasma gradually sets in. As mentioned in Introdu tion, in the dilute-gasmodel [25℄ ( η/s ) ∼ / ( g ln g − ) , i.e. this ratio in reases with temperature. The sto hasti va uum model, on the other hand, being appli able at strong oupling, orre tly yields theexpe ted fall-o(cid:27) of the ( η/s ) -ratio at temperatures . T c , but annot reprodu e its in reaseat mu h higher temperatures.We would also like to emphasize an interesting fa t, whi h has been realized by the end ofthe al ulation. We have started with the general α -dependent Lorentzian-type ansatz (6)for the spe tral density ρ ( ω ) . By using it in the Kubo formula, we have ome to the on lusion that only for the single value, α = 1 / , this ansatz provides the Matsubara-modeindependen e of ρ ( ω ) . For this value of α , Eq. (6) takes the onventional Lorentzian form.In this way, also the fun tion D ( x ) from Eq. (15) is de(cid:28)ned unambiguously as D ( x ) = A (1 / s K / (2 µ | x | )(2 µ | x | ) / , A (1 / an be found after Eq. (26).Note (cid:28)nally that we have used in our al ulation the Gaussian-dominan e hypothesis [11℄,whi h allows one to disregard in Eq. (14) the onne ted four-point orrelation fun tionof gluoni (cid:28)eld strengths ompared to the pairwise produ ts of the two-point orrelationfun tions. The same approximation was used in Ref. [21℄ for the al ulation of topologi alsus eptibility via the four-point orrelation fun tion. This approximation an be relaxed inthe following way. Consider a parametrization for the nonperturbative part of the onne tedfour-point orrelation fun tion suggested in Ref. [22℄: (cid:10)(cid:10) g F a µ ν ( x ) F a µ ν ( x ) F a µ ν ( x ) F a µ ν ( x ) (cid:11)(cid:11) == (cid:10) G (cid:11) n f a a b f a a b ( ε µ ν µ ν ε µ ν µ ν − ε µ ν µ ν ε µ ν µ ν ) ++ f a a b f a a b ( ε µ ν µ ν ε µ ν µ ν − ε µ ν µ ν ε µ ν µ ν ) ++ f a a b f a a b ( ε µ ν µ ν ε µ ν µ ν − ε µ ν µ ν ε µ ν µ ν ) −− h δ a a δ a a ( δ µ µ δ ν ν − δ µ ν δ µ ν ) ( δ µ µ δ ν ν − δ µ ν δ µ ν ) ++ δ a a δ a a ( δ µ µ δ ν ν − δ µ ν δ µ ν ) ( δ µ µ δ ν ν − δ µ ν δ µ ν ) ++ δ a a δ a a ( δ µ µ δ ν ν − δ µ ν δ µ ν ) ( δ µ µ δ ν ν − δ µ ν δ µ ν ) io ˜ D ( z , . . . , z ) , (30)where z = x − x , z = x − x ,..., z = x − x are relative oordinates. For the onne ted orrelation fun tion entering Eq. (14) this parametrization yields (cid:10)(cid:10) g F a µ (0) F a µ (0) F b ν ( x ) F b ν ( x ) (cid:11)(cid:11) = 2( N c − (cid:18) N c − (cid:19) (cid:10) G (cid:11) ˜ D (0 , x, x, x, x, . Similarly to Eq. (20), for the fun tion ˜ D too one an have a parametrization ompatiblewith Eq. (4): ˜ D (0 , x, x, x, x,
0) = B ( α ) K − α (4 µ | x | )(4 µ | x | ) − α . The normalization fa tor B ( α ) should now be determined simultaneously with the normal-ization fa tor A ( α ) from a system of equations for two observables, whi h both depend onthe fun tions D and ˜ D . Natural observables of this kind are the string tension and thetopologi al sus eptibility. The ontribution of the fun tion ˜ D to the string tension has al-ready been evaluated in Ref. [22℄. Further analysis of the outlined extension of the Gaussianapproximation is, however, not the purpose of the present paper.6A knowledgmentsI am grateful to Frithjof Kars h, Hans-Jürgen Pirner and Arif Shoshi for helpful dis ussions,and to Yoshimasa Hidaka and Olaf Ka zmarek for the useful orresponden e and omments.I also thank Frithjof Kars h for providing the details of the latti e data from Ref. [16℄. Thiswork has been supported by the German Resear h Foundation (DFG), ontra t Sh 92/2-1.Appendix A. Mat hing perturbative ontributions in the Kubo formula.For this Appendix, we promote Eq. (2) to the full Kubo formula, i.e. repla e ρ by ρ full = ρ + ρ pert and h T (0) T ( x ) i by h T (0) T ( x ) i full = h T (0) T ( x ) i + h T (0) T ( x ) i pert . Onthe LHS of su h a full Kubo formula, onsider the integral ontaining ρ pert . To fa ilitate the ω -integration, we approximate the perturbative part of the spe tral density by the fun tion ρ pert = N ω down to ω = 0 and determine the oe(cid:30) ient N . The integral emerging on theLHS of the Kubo formula then reads Z ∞ dω ω [cosh( ωx ) coth( ωβ/ − sinh( ωx )] = ∞ X n =0 (cid:20) βn + x ) + 1[ βn + ( β − x )] (cid:21) == 24 ( x + ∞ X n =1 (cid:20) βn + x ) + 1( βn − x ) (cid:21)) . ( A. To obtain the last expression, we have extra ted the ( n = 0) -term from the sum ∞ P n =0 1( βn + x ) and shifted n by 1 in the sum ∞ P n =0 1[ βn +( β − x )] . Note that x ∈ [0 , β ] , and the obtainedexpression has singularities at x = 0 and x = β [in the ( n = 1) -term℄. These singularitiesare identi al, sin e cosh (cid:2) ω (cid:0) x − β (cid:1)(cid:3) = cosh( ωβ/ both at x = 0 and x = β .We will demonstrate now that this expression orresponds to the ontribution, whi h h T (0) T ( x ) i pert brings about to the RHS of the full Kubo formula. The UV-(cid:28)nite part ofthis perturbative orrelation fun tion an be written as h T (0) T ( x ) i pert = A ( N c − g x , where the value of the numeri al onstant A depends on the regularization s heme applied.Thus, the sum emerging on the RHS of the Kubo formula reads + ∞ X n = −∞ x + ( βn + x ) ] . d x (cid:28)rst, we have Z ∞ dx x [ x + ( βn + x ) ] = π | βn + x | . The part of the sum with negative winding modes reads − X n = −∞ | βn + x | = − X n = −∞ − βn − x ) = ∞ X n =1 βn − x ) . 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