Viscosity of hadron matter within relativistic mean-field based model with scaled hadron masses and couplings
aa r X i v : . [ nu c l - t h ] D ec Vis osity of hadron matter within relativisti mean-(cid:28)eld basedmodel with s aled hadron masses and ouplingsA.S. Khvorostukhin,1, ∗ V.D. Toneev,2, † and D.N. Voskresensky2, ‡ η ) and bulk ( ζ ) vis osities are al ulated in a quasiparti le relaxationtime approximation for a hadron matter des ribed within the relativisti mean-(cid:28)eldbased model with s aled hadron masses and ouplings. Comparison with results ofother models is presented. We demonstrate that a small value of the shear vis osityto entropy density ratio required for explaining a large ellipti (cid:29)ow observed at RHICmay be rea hed in the hadron phase. Relatively large values of the bulk vis osity arenoted in the ase of a baryon enri hed matter.PACS: 24.10.Nz, 25.75.-q I. INTRODUCTIONIn the past, transport oe(cid:30) ients for the nu lear matter were studied in [1, 2, 3, 4℄.Re ently, the interest in the transport oe(cid:30) ient issue has sharply been in reased in heavy-ion ollision physi s, see review-arti le [5℄. Values of the ellipti (cid:29)ow v observed at RHIC [6℄proved to be larger than at SPS. This (cid:28)nding is interpreted as that a quark-gluon plasma(QGP) reated at RHIC behaves as a nearly perfe t (cid:29)uid with a small value of the shearvis osity-to-entropy density ratio, η/s . The latter statement was on(cid:28)rmed by non-idealhydrodynami analysis of these data [7℄. Thereby, it was laimed [8, 9, 10℄ that a newstate produ ed at high temperatures is most likely not a weakly intera ting QGP, as it wasoriginally assumed, but a strongly intera ting QGP. The interest was also supported by a newtheoreti al perspe tive, namely, N = 4 supersymmetri Yang-Mills gauge theory using the ∗ hvorosttheor.jinr.ru; Joint Institute for Nu lear Resear h, Dubna, Russia † toneevtheor.jinr.ru; Joint Institute for Nu lear Resear h, Dubna, Russia ‡ voskrepisem.net; Mos ow Engineering Physi al Institute, Mos ow, RussiaAnti de-Sitter spa e/Conformal Field Theory (AdS/CFT) duality onje ture. Cal ulationsin this strongly oupled theory demonstrate that there is minimum in the η/s ratio [11℄: η/s ≈ / (4 π ) . It was thereby onje tured that this relation is in fa t a lower bound for thespe i(cid:28) shear vis osity in all systems [5℄ and that the minimum is rea hed in the hadron-quark transition riti al point (at T = T c ).In this paper, we ontinue investigation of the shear and bulk vis osities performed inRef. [12℄ within the quasiparti le model in the relaxation time approximation. We des ribethe hadron phase ( T < T c ) in terms of the quasiparti le relativisti mean-(cid:28)eld-based modelwith the s aling hadron masses and ouplings (SHMC) been su essfully applied to thedes ription of heavy ion ollision rea tions [13, 14℄, see se t.2. Then in se t.3 we al ulatethe shear and bulk vis osities and ompare our results with results of previous works. Inse t. 4 we formulate our on lusions.II. DESCRIPTION OF HADRON MATTER IN THE SHMC MODELA. Formulation of the modelWithin our relativisti mean-(cid:28)eld SHMC model [13, 14℄ we present the Lagrangian densityof the hadroni matter as a sum of several terms: L = L bar + L MF + L ex . (1)The Lagrangian density of the baryon omponent intera ting via σ, ω mean (cid:28)elds is asfollows: L bar = X b ∈{ bar } h i ¯Ψ b (cid:16) ∂ µ + i g ωb χ ω ω µ (cid:17) γ µ Ψ b − m ∗ b ¯Ψ b Ψ b i . (2)The onsidered baryon set is { b } = N (938) , ∆(1232) , Λ(1116) , Σ(1193) , Ξ(1318) , Σ ∗ (1385) , Ξ ∗ (1530) , and Ω(1672) , in luding antiparti les. The used σ -(cid:28)eld dependent e(cid:27)e tive massesof baryons are [13, 14, 15℄ m ∗ b /m b = Φ b ( χ σ σ ) = 1 − g σb χ σ σ/m b , b ∈ { b } . (3)In Eqs. (2), (3) g σb and g ωb are oupling onstants and χ σ ( σ ) , χ ω ( σ ) are oupling s alingfun tions.The σ -, ω -meson mean (cid:28)eld ontribution is given by L MF = ∂ µ σ ∂ µ σ − m ∗ σ σ − U ( χ σ σ ) − ω µν ω µν m ∗ ω ω µ ω µ , (4) ω µν = ∂ µ ω ν − ∂ ν ω µ , U ( χ σ σ ) = m N ( b f + c f ) , f = g σN χ σ σ/m N . There exist only σ and ω mean (cid:28)eld solutions of equations of motion. The mass terms ofthe mean (cid:28)elds are m ∗ m /m m = | Φ m ( χ σ σ ) | , { m } = σ, ω . (5)The dimensionless s aling fun tions Φ b and Φ m , as well as the oupling s aling fun tions χ m , depend on the s alar (cid:28)eld in ombination χ σ ( σ ) σ . Following [15℄ we assume approximatevalidity of the Brown-Rho s aling ansatz in the simplest form Φ = Φ N = Φ σ = Φ ω = Φ ρ = 1 − f. (6)The third term in the Lagrangian density (1) in ludes meson quasiparti le ex itations: π ; K, ¯ K ; η (547); σ ′ , ω ′ , ρ ′ ; K ∗± , (892) , η ′ (958) , φ (1020) . The hoi e of parameters and otherdetails of the SHMC model an be found in [13, 14℄.B. Thermodynami al quantitiesWithin SHMC model we al ulate di(cid:27)erent thermodynami al quantities in thermalequilibrium hadron matter at (cid:28)xed temperature T and baryon hemi al potential µ bar . InFig. 1 (left panel) we show the square of the sound velo ity c s = dP/dε ( P is pressure, ε isenergy density), as fun tion of temperature at zero baryon hemi al potential, µ bar = 0 , forthe SHMC model (solid line) and ompare this result with that for the ideal gas (IG) modelwith the same hadron set as in the SHMC model (long-dashed line), for the π + ρ mixture(dash-double dot) and for purely pion system (dash-dotted). As is seen from this (cid:28)gure,for the purely pion IG the c s monotonously in reases with in rease of the temperatureapproa hing the ultrarelativisti limit c s = 1 / at high temperatures. For the pion-rhomeson mixture, the c s exhibits a shallow minimum at T ∼ MeV. The minimum (inthe same temperature region) be omes more pronoun ed for multi- omponent systems (seedash urve). At T < ∼
50 MeV the pion ontribution is a dominant one, thereby all urves oin ide1. The urves for the SHMC model and the IG model al ulated with the samehadron set oin ide for T < ∼
100 MeV. At
T >
50 MeV heavier mesons start to ontributethat slows down the growth of pressure and then results in signi(cid:28) ant de rease of c s , ontraryto the ase of the one- omponent pion gas. Within the SHMC model c s gets pronoun edminimum at T ≃
180 MeV aused by a sharp de rease of the in-medium hadron masses atthese temperatures (see the right panel in Fig. 1, where e(cid:27)e tive masses of the nu leon, ω , ρ and σ ex itations are presented). The minimum of the sound velo ity (at T = T c ≃
180 MeV) an be asso iated with a kind of phase transition, e.g. with the hadron-QGP ross-over, asit follows from the detailed analysis of the latti e data, see [16℄.
50 100 150 200 250 3000,00,10,20,30,40,5 IG with our hadron set IG of IG of + c s T, MeV bar =0
50 100 150 200 250 3000,00,20,40,60,81,0
N, , m * / m T, MeV n bar =0n bar =3 n èñ. 1: Left panel: The sound velo ity squared in hadron matter as fun tion of the temperature atzero baryon hemi al potential. Solid line (cid:21) al ulation within the SHMC model. Other notationsare given in the legend. Right panel: The temperature dependen e of e(cid:27)e tive masses of the nu leon, ω and ρ ex itations (solid line) and of the σ -meson ex itation (dashed line) al ulated within theSHMC model for two values of the baryon density.Note that in the Hagedorn-gas model [17℄ (for the Hagedorn mass m → ∞ ) one gets c s → at T = T c , whereas in the mass-trun ated Hagedorn-gas model the behavior very lose to that we have in ase of the IG model is observed.1 Note that within the SHMC model pions are treated as an ideal gas of free parti lesIn Fig. 2 we present the latti e data for the redu ed energy density and the pressuretogether with our SHMC model results. Following [14℄ we use suppressed oupling onstants g σb (ex ept for nu leons). This guarantees that even above T c up to the temperature T ∼
220 MeV the EoS omputed in the SHMC model is in agreement with the latti e data for the
100 200 300 400 500 600 7000246810121416 / T , P / T T, MeV bar =0 MeV3P èñ. 2: The redu ed triple pressure and the energy density at µ bar = 0 . Points are QCD latti eresult [16℄. The hadroni SHMC results are plotted by solid and dash lines, respe tively.pressure and energy density. At higher temperatures the SHMC model requires additionalmodi(cid:28) ations, although in reality the quark-gluon degrees of freedom should be taken intoa ount already for T > T c ∼ MeV.III. SHEAR AND BULK VISCOSITIES OF THE SHMC MODELA. Collisional vis osity, derivation of equationsSasaki and Redli h [12℄ derived expressions for the shear and bulk vis osities in the asewhen the quasiparti le spe trum is given by E ( ~p ) = p ~p + m ∗ ( T, µ ) . We perform a similarderivation, but in the presen e of mean (cid:28)elds. In the latter ase one should additionally takeinto a ount that quasiparti le distributions depend on the mean (cid:28)elds.In order to al ulate vis osity oe(cid:30) ients one needs an expression for spatial omponentsof the energy momentum tensor orresponding to the Lagrangian density (1): T ij = T ij MF + X b ∈{ bar } T ijb + X bos ∈{ ex } T ij bos , (7)where i, j = 1 , , and the mean-(cid:28)eld ontribution is as follows T ij MF = ∂ i σ ∂ j σ − ∂ i ω ∂ j ω + (cid:18) (cid:2) ∂ i σ ∂ j σ − ∂ i ω ∂ j ω + m ∗ σ σ − m ∗ ω ω (cid:3) + U ( σ ) (cid:19) g ij (8)with m ∗ σ and m ∗ ω given by Eq. (5).The quasiparti le (fermion and boson ex itation) ontribution is given by T ija = Z d Γ p ia p ja E a F a , a ∈ (bos ., bar) , E a = p ~p + m ∗ , Γ = ν a d p a (2 π ) , (9)where ν a is the degenera y fa tor.The quasiparti le distribution fun tion F b for baryon omponents in the presen e of mean(cid:28)elds ful(cid:28)lls the Boltzmann kineti equation [18℄, (cid:18) p µb ∂ µ − g ωb p µ ω µν ∂∂p νb + m ∗ b ∂ ν m ∗ b ∂∂p νb (cid:19) e F b = St e F b ; (10)with e F b ( p b , x b ) = δ ( p b − m ∗ b ) F b ( ~p b , x b ) .The lo al equilibrium boson or baryon distribution is given as follows: F loc . eq .a ( ~p a , x a ) = h e ( E a − ~p a ~u − µ a + t vec a X a ) /T ± i − , X a = g ωa χ ω ω , (11)where we suppressed ~u terms for | ~u | ≪ . Here the upper sign ( + ) is for fermions and ( − )is for bosons, and the ve tor parti le harge is t vec a = ± or ; g ωa = 0 only for a ∈ bar in ourmodel. Considering only slightly inhomogeneous solutions and using | ~u | ≪ we may dropthe terms ∝ ~u and ∝ ~u ∇ ω in the kineti equation (10). Then kineti equations for bosonand baryon omponents a quire ordinary quasiparti le form ∂F a ∂t + ∂E a ∂~p a ∂F a ∂~r a − ∂E a ∂~r a ∂F a ∂~p a = p µa E a ∂F a ∂x µa = StF a , (12)where p µa = ( E a ( ~p a , ~r a , σ, ω ) , ~p a ) . We used that ∂E a /∂~p a = ~p a /E a . Sin e al ulatingthe vis osity, we need only terms with velo ity gradients, we further put ∂E a /∂~r a =( ∂E a /∂µ a ) ~ ∇ a µ a + ( ∂E a /∂T ) ~ ∇ a T = 0 .In the relaxation time approximation StF a = − δF a /τ a , δF a = F a − F loc . eq .a . (13)Here τ a denotes the relaxation time of the given spe ies. Generally, it depends on thequasiparti le momentum ~p a and the quasiparti le energy E a ( ~p a ) .The averaged partial relaxation time ˜ τ a is related to the ross se tion as ˜ τ − a ( T, µ ) = X a ′ n a ′ ( T, µ ) (cid:10) v aa ′ σ taa ′ ( v aa ′ ) (cid:11) , (14)where n a ′ is the density of a ′ -spe ies, σ taa ′ = R d cos θ dσ ( aa ′ → aa ′ ) /d cos θ (1 − cos θ ) is thetransport ross se tion, in general, a ounting for in-medium e(cid:27)e ts and v aa ′ is the relativevelo ity of two olliding parti les a and a ′ in ase of binary ollisions. Angular bra ketsdenote a quantum me hani al statisti al average over an equilibrated system. In reality, the ross se tions entering the ollision integral and the orresponding relaxation time τ a in (13)may essentially depend on the parti le momentum. Thus, averaged values ˜ τ − a given by Eq.(14) yield only a rough estimate for the values τ − a whi h we a tually need for al ulation ofvis osity oe(cid:30) ients, see below Eqs. (21) and (22).In the relaxation time approximation from Eqs. (12), (13) we obtain δF a = − τ a E a p µa ∂F loc . eq .a ∂x µa , (15)and then the variation of the energy-momentum tensor (7) be omes: δT ij = − X a Z d Γ (cid:26) τ a p ia p ja E a p µa ∂ µ F a (cid:27) loc . eq . + δσ (cid:26) ∂T ij ∂σ (cid:27) loc . eq . + δω (cid:26) ∂T ij ∂ω (cid:27) loc . eq . . (16)Considering small deviations from the lo al equilibrium, we may keep in (16) only (cid:28)rst-orderderivative quasiparti le terms ∝ ∂ i , thus negle ting mean-(cid:28)eld ontributions ∝ ∂ i σ ∂ j σ and ∝ ∂ i ω ∂ j ω .The shear and bulk vis osities are as follows expressed through the variation of theenergy-momentum tensor: δT ij = − ζ δ ij ~ ∇ · ~u − η W ij , W kl = ∂ k u l + ∂ l u k − δ kl ∂ i u i . (17)To (cid:28)nd the shear vis osity, we put i = j in (17) and use that in this ase the variationof the se ond and third terms in (16) yields zero after integration over angles. To (cid:28)nd thebulk vis osity, we substitute i = j in (17) and use that T ii eq = 3 P eq . We put ~u = 0 in (cid:28)nalexpressions but retain gradients of the velo ity.Taking derivatives ∂F loc . eq .a /∂x µa in Eq. (15) we (cid:28)nd the variation of the total energy-momentum tensor as the fun tion of derivatives of the velo ity δT ij = X a Z d Γ p ia p ja T E a τ a F eq a (1 ∓ F eq a ) q a ( ~p ; T, µ bar , µ str ) (18)with q a ( ~p ; T, µ bar , µ str ) = ∂ k u l δ kl Q a − p k p l E a W kl , (19) Q a = − (cid:26) ~p a E a + (cid:16) ∂P∂n bar (cid:17) ǫ,n str h ∂ ( E a + X a ) ∂µ bar − t bar b i + (cid:16) ∂P∂n str (cid:17) ǫ,n bar h ∂ ( E a + X a ) ∂µ str − t str a i − (cid:0) ∂P∂ǫ (cid:1) n bar ,n str × (20) × h E a + X a − T ∂ ( E a + X a ) ∂T − µ bar ∂ ( E a + X a ) ∂µ bar − µ str ∂ ( E a + X a ) ∂µ str io . Finally, we obtain expressions for the shear vis osity η = T P a R d Γ τ a ~p a E a [ F eq a (1 ∓ F eq a )] , (21)and for the bulk vis osity ζ = − T P a R dΓ τ a ~p a E a F eq a (1 ∓ F eq a ) Q a . (22)At vanishing mean (cid:28)elds our results are redu ed to those derived in Ref. [12℄.B. Collisional vis osity in baryon-less matterIn the relaxation time approximation both shear and bulk vis osities for a omponent " a "depend on its relaxation ( ollisional) time τ a whi h should be parameterized or al ulatedindependently. Therefore to diminish this un ertainty it is legitimate at (cid:28)rst to (cid:28)nd theredu ed kineti oe(cid:30) ients (per unit relaxation time, assuming τ = const , i.e. τ = ˜ τ ).In Fig. 3 we demonstrate results of various al ulations for the redu ed shear (leftpanel) and bulk (right panel) vis osities s aled by the /T fa tor at µ bar = 0 . As we seefrom the (cid:28)gure, the redu ed shear vis osity of the massive pion gas (dashed line) be omesapproximately proportional to T for T > ∼
100 MeV. Naturally, this result is lose to thatobtained in the Gavin approximation [19℄ (dashed-double-dotted line in Fig. 3). The T s aling is violated for the π − ρ gas in the temperature interval under onsideration be ausethe ρ mass is not negligible even at T ∼
200 MeV. For ζ the approximate /T s alingproperty holds for the massive pion-rho gas at T > ∼
150 MeV. Note that ζ = c s = 1 / in this ase. For the massive pion gas ζ /T de reasesalready for T > MeV rea hing zero at large T similar to the massless gas. The redu ed Gavin approximation - IG ( / ) / T T, MeV ( / ) / T T, MeV bar =0 MeV èñ. 3: The redu ed (per unit relaxation time) T s aled shear (left panel) and bulk (right panel)vis osities as fun tion of the temperature al ulated within the SHMC model (solid lines) for thebaryon-less matter, µ bar =
0. Results are ompared with those for the massive pion gas (dashedlines), π − ρ mixture (short dashed lines) and with those for the massless pion gas (the Gavinapproximation [19℄, dot-dashed line), as well as for the IG model (open dots) with the same set ofspe ies as in the SHMC model.shear and bulk vis osities of a multi omponent system al ulated in our SHMC model (solidlines) and in the IG model with the same hadron set (open dots) do not ful(cid:28)ll the T s alinglaw. These models in lude large set of hadrons, due to that with the temperature in reasethe redu ed shear and bulk vis osities be ome signi(cid:28) antly higher than those for the piongas and the pion-rho gas models. An additional in rease of the redu ed vis osity withinSHMC model originates from signi(cid:28) ant mass de rease at temperatures near the riti altemperature. The bulk vis osity of a single- omponent pion system drops to zero both atlow and high temperatures and in the whole temperature interval ζ << η , that is frequentlyused as an argument for negle ting the bulk vis osity e(cid:27)e ts. However, the statement doesnot hold anymore for mixture of many spe ies. For example, at T ∼
150 MeV the η/ζ ratiois only about 3 in ase of the IG and SHMC models. Thus the bulk vis osity e(cid:27)e ts anplay a role in the des ription of the hadroni stage at high ollision energies, like at RHIC.Moreover, the bulk vis osity an be responsible for su h important e(cid:27)e t as (cid:29)ow anisotropy.0C. Collisional vis osity in baryon enri hed matter
50 100 150 20001234 ( / ) / T T, MeV
50 100 150 2000,1110 4 n ( / ) / T T, MeV èñ. 4: The SHMC model predi tions of the T s aled temperature dependen e of the redu edshear (left panel) and bulk (right panel) vis osities al ulated for hadron mixture at n bar = n and n (solid lines). Cal ulations performed in the IG based model with the same hadron set as in theSHMC model are demonstrated by dashed lines.For the ase of the multi- omponent hadron mixture within IG and SHMC models thetemperature dependen e of the redu ed T -s aled shear and bulk vis osities are shown atbaryon densities n bar = n and n ( n is the nu lear saturation density) in the left andright panels of Fig. 4, respe tively. The redu ed shear vis osity al ulated in the SHMCmodel (solid lines) is lose to that in the IG model with the same hadron set (dashedlines). Di(cid:27)eren es in the η/ ( τ T ) ratio for the IG and SHMC models appear only at hightemperatures T > ∼ MeV. At T < ∼ MeV the redu ed T -s aled bulk vis osity (rightpanel) in the IG based model proved to be larger than that in the SHMC model. Contrary,for larger T the redu ed bulk vis osity in the IG model be omes smaller than that in theSHMC model. Di(cid:27)eren es ome from the strong dependen e of the bulk vis osity ζ on thevalues of thermodynami al quantities (see Eqs.(20),(22)). Note that at T > ∼
100 MeV and n bar > ∼ n the shear and bulk vis osities are getting omparable in magnitude. Growth ofthe relative importan e of ζ with in rease of temperature seems to be quite natural be ausethe bulk vis osity takes into a ount momentum dissipation due to inelasti hannels whi hnumber in reases with the temperature in rease.1D. Estimation of the relaxation timeThe relaxation time is de(cid:28)ned by Eq. (14). We implement free ross se tions in ase of theIG based model, similar to pro edure performed in Ref. [20℄. In ase of the SHMC model, thein-medium modi(cid:28) ation of ross se tions is in orporated by a shift of a (cid:16)pole(cid:17) of the ollisionenergy by the mass di(cid:27)eren e m a − m ∗ a a ording to pres ription of Ref. [21℄. Due to a la k ofmi ros opi al ulations this is the only modi(cid:28) ation whi h we do here. Important pe uliarityof the nu leon ontribution to the relaxation time at low temperature is asso iated with theparti ular role played by the Pauli blo king. It means that appropriate multi-dimensionalintegration should be arried out quite a urately with using quantum statisti al distributionfun tions. Cal ulations using the kineti Uehling-Uhlenbe k equations for the purely nu leonsystem in the non-relativisti approximation were performed in [3℄. For T < ∼ MeV anextrapolation expression has been obtained: ˜ τ NN ≃ T (cid:18) n bar n (cid:19) / (cid:20) . T n bar n (cid:21) + 38 T / (1 + 160 /T ) n n bar . (23)Thus the relaxation time demonstrates well known behavior T − , for T → .To simplify al ulations we use Eq. (23) for the partial nu leon relaxation time ˜ τ NN , to bevalid at low temperatures, smoothly mat hing it (at T ∼ MeV) with the partial nu leon ontribution al ulated following Eq. (14) for higher temperatures. We take into a ount thewhole hadron set involved into the SHMC model. The relaxation time for every omponentis evaluated a ording to Eq. (14).E. Collisional vis osity in heavy ion ollisionsAbove we have studied redu ed vis osities of the hadron matter at di(cid:27)erent temperaturesand baryon densities. In reality a hot and dense system being formed in a heavy-ion ollisionthen expands towards freeze-out state, at whi h the omponents stop to intera t with ea hother. Here we use the freeze-out urve T fr ( µ frbar ) extra ted from analysis of experimentalparti le ratios in statisti al model for many spe ies at the given ollision energy s / NN treatingthe freeze-out temperature T fr and hemi al potential µ frbar as free parameters [22, 23℄.In Fig. 5, vis osity oe(cid:30) ients per entropy density s are shown versus the freeze-outtemperature for Au + Au ollisions (whi h is unambiguously related to the freeze-out2 hemi al potential µ frbar [22℄ needed to al ulate thermodynami al quantities at the freeze-out). Dimensionless ratios of the vis osity to the entropy density η/s and ζ /s hara terizethe energy dissipation in the medium. As we see, the η/s ratio de reases monotonously within rease of the temperature, being higher than the lower bound / π but tending to it withfurther in rease of T fr . The value ζ /s exhibits a maximum at T fr ∼ MeV and then tends tozero with subsequent in rease of T fr . As has been emphasized above, at T > ∼
100 MeV valuesof the shear and bulk vis osities be ome quite omparable, ( η/s ) fr ≃ ζ /s ) fr .
40 60 80 100 120 140 160 1800,00,10,20,30,40,5 / s , / s T fr , MeV /s /s èñ. 5: Shear and bulk vis osities per entropy density al ulated in the SHMC model for entralAu+Au ollisions along the freeze-out urve (at T = T fr ) [22℄ for the baryon enri hed system. Thedotted line is the lower AdS/CFT bound η/s = 1 / π [11℄.In Fig. 6, the η/s ratio al ulated in our SHMC model (solid line) is plotted as a fun tionof the ollision energy √ s NN of two Au+Au nu lei. The result for the IG model with the samehadron set as in SHMC model is plotted by the dash-dotted line. We note that for √ s NN > ∼ the SHMC results prove to be very lose to the IG based model ones (with the same hadronset as in SHMC model), sin e the freeze-out density is rather small and the de rease of thehadron masses o urring in the SHMC model is not important. The results for the hadronhard ore gas model (the van der Waals ex luded volume model) [24℄ at two values of theparti le hard ore radius r are shown by dashed and short-dashed lines. In all ases for √ s NN > ∼ GeV the ratio η/s de reases along the hemi al freeze-out line with in reasing the ollision energy and then (cid:29)attens at √ s NN > ∼
10 GeV, sin e freeze-out at su h high ollisionenergies already o urs at almost onstant value of T fr ≈ MeV. The shear vis osity of3
IG (SHMC hadron set) r=0.3 fm r=0.5 fm / s s , GeV èñ. 6: The ratio of the shear vis osity to the entropy density al ulated for entral Au+Au ollisionsalong the hemi al freeze-out urve [22℄ within the SHMC model as a fun tion of the ollision energy s / NN (solid line). Dashed and short-dashed urves are the results of the ex luded volume hadron gasmodel [24℄ with hard- ore radii r = r = ∝ √ mT /r . Sin e Fermistatisti al e(cid:27)e ts are not in luded within this model, the shear vis osity, η , de reases withde rease of T . Nevertheless the η/s ratio in reases and diverges at low energy/temperature,as the onsequen e of a more sharp de rease of the entropy density ompared to η , see Fig. 6.As follows from the (cid:28)gure, the smaller r is, the higher η/s is in the given ex luded volumemodel. For √ s NN > ∼ and r ≃ . fm the η/s ratio is expe ted to be lose to the values omputed in the IG and SHMC models.Re ently an interesting attempt has been undertaken in [25℄ to extra t the shear vis osityfrom the 3-(cid:29)uid hydrodynami al analysis of the ellipti (cid:29)ow in the AGS-SPS energy range.An overestimation of experimental v values in this model was asso iated with dissipativee(cid:27)e ts o urring during the expansion and freeze-out stages of parti ipant matter evolution.The resulting values of η/s vary in interval η/s ∼ − in the onsidered domain of √ s NN ≈ − GeV ( orresponding to temperatures T ≈ − MeV) [25℄. Authors onsider their result as an upper bound on the η/s ratio in the given energy range. Notethat mentioned values are mu h higher than those whi h follow from our estimations givenabove and presented in Figs. 5 and 6.4Other mi ros opi estimate of the share vis osity to the entropy density ratio for therelativisti hadron gas based on the UrQMD ode was performed in Ref. [26℄ where 55baryon spe ies and their antiparti les and 32 meson spe ies were in luded. The full kineti and hemi al equilibrium is a hieved at T =
130 and 160 MeV, respe tively. The extra tedratio η/s > ∼ η/s = 1 / π . Analyzingtheir result authors [26℄ on lude that the dynami s of the evolution of a ollision at RHICis dominated by the de on(cid:28)ned phase (exhibiting very low values of η/s ) rather than by thehadron phase. Note however that in-medium e(cid:27)e ts in the hadron phase are not in ludedinto onsideration in the UrQMD model though, namely, these e(cid:27)e ts result in the requiredde rease of the η/s ratio in our SHMC model.IV. CONCLUSIONSIn this paper, we derived expressions for the shear and bulk vis osities in the relaxation-time approximation for a hadron system des ribed by the quasiparti le relativisti mean-(cid:28)eld theory with s aling of hadron masses and ouplings (SHMC). The EoS of the SHMCmodel fairly well reprodu es global properties of hot and dense hadron matter in ludingthe temperature region near T c provided all oupling onstants g σb are strongly suppressedex ept for nu leons. Thus obtained kineti oe(cid:30) ients are ompared with those al ulatedin other models of the hadron matter.With in reasing freeze-out temperature T fr (for entral Au+Au ollisions), the η/s ratioundergoes a monotonous de rease approa hing values lose to the AdS/CFT bound at T ∼ T c MeV, while the ζ /s ratio exhibits a maximum at T fr ∼
85 MeV. In a broad temperatureinterval the η/s and ζ /s ratios are not small and vis ous e(cid:27)e ts an be noti eable. Thevis osity values at the freeze-out an be transformed into dependen e on the olliding energy √ s NN (for entral Au+Au ollisions). When the ollision energy de reases, the η/s goes up.The high-energy (cid:29)attening of the √ s NN dependen e o urs at quite low η/s < . . It impliesthat a small value of η/s required for explaining a large ellipti (cid:29)ow observed at RHIC ouldbe rea hed in the hadroni phase. This might be an important observation whi h we havedemonstrated within the SHMC model.5The v analysis indi ates to di(cid:27)erent values of η/s for peripheral and entral ollisions.Therefore, it would be interesting to perform hydrodynami al ulations using the T − µ bar dependent transport oe(cid:30) ients rather than onstant ones. The need of su h an approa hwas re ently emphasized in [27℄. Further we will use the SHMC model EoS with the derivedtransport oe(cid:30) ients for this purpose.A knowledgementsWe are grateful to K.K. Gudima, Y.B. Ivanov, Y.L. Kalinovsky and E.E. Kolomeitsevfor numerous dis ussions and valuable remarks. This work was supported in part by theBMBF/WTZ proje t RUS 08/038, the RFFI grants 08-02-01003-a and 10-02-91333 ÍÍÈÎ-à, the Ukrainian-RFFI grant (cid:157) 09-02-90423-óêð--a, the DFG grant WA 431/8-1 and theHeisenberg-Landau grant.[1℄ J.L. Anderson and H.R. Witting, Physi a 74 466, 489 (1973).[2℄ V.M. Galitsky, Yu.B. Ivanov and V.A. Khangulian, Sov. J. Nu l. Phys. 30 401 (1979).[3℄ P. Danielewi z, Phys. Lett. B146, 168 (1984); L. Shi and P. Danielewi z, Phys. Rev. C68064604 (2003).[4℄ R. Hakim and L. Mornas, Phys. Rev. C47 2846 (1993).[5℄ J.I. Kapusta, arXiv:0809.3746[nu l-th℄.[6℄ S.S. Adler et al. (PHENIX Collaboration), Phys. Rev. Lett. 91 182301 (2003); J. Adams et al.(STAR Collaboration), Phys. Rev. Lett. 92 052302 (2004).[7℄ D. Teaney, Phys. Rev. C68 034913 (2005); P. Romats hke and U. Romats hke, Phys. Rev.Lett. 99 172301 (2007); M. Luzum and P. Romats hke, Phys. Rev. C78 034915 (2008);K. Dusling and D. Teaney, Phys. Rev. C77 034905 (2008); H. Song and U.W. Heinz, Phys. Lett.B658 279 (2008); H. Song and U.W. Heinz, Phys. Rev. C77 064901 (2008); A.K. Chaudhuri,arXiv:0801.3180 [nu l-th℄.[8℄ G. Poli astro, D.T. Son and A.O. Starinets, Phys. Rev. Lett. 87 081601 (2001).[9℄ A. Peshier and W. Cassing, Phys. Rev. Lett. 94 172301 (2005).[10℄ E.V. Shuryak, Nu l. Phys. A750 64 (2005); M. Gyulassy and L. M Lerran, Nu l. Phys. A75030 (2005); U.W. Heinz, arXiv:nu l-th/0512051.6[11℄ P. Kovtun, T.D. Son and O.A. Starinets, JHEP 0310, 064 (2003); Phys. Rev. 94 111601 (2005).[12℄ C. Sasaki and K. Redli h, Phys. Rev. C79 055207 (2009).[13℄ A.S. Khvorostukhin, V.D. Toneev and D.N. Voskresensky, Nu l. Phys. A791 180 (2007).[14℄ A.S. Khvorostukhin, V.D. Toneev and D.N. Voskresensky, Nu l. Phys. A813 313 (2008).[15℄ E.E. Kolomeitsev and D.N. Voskresensky, Nu l. Phys. A759 373 (2005).[16℄ F. Kars h, arXiv:hep-lat/0601013.[17℄ P. Castorina, J. Cleymans, D. E. Miller and H. Satz, arXiv:0906.2289 [hep-ph℄.[18℄ Yu.B. Ivanov, Nu l. Phys. A474 669 (1987).[19℄ S. Gavin, Nu l. Phys. A435 826 (1985).[20℄ M.Prakash, M. Prakash, R. Venugopalan and G. Welke, Phys. Reps., 227 321 (1993).[21℄ E.L. Bratkovskaya and W. Cassing, Nu l. Phys. A807 214 (2008).[22℄ J. Cleymans, H. Oes hler and K. Redli h, Phys. Rev. C73 034905 (2006).[23℄ A. Androni , P. Braun-Munzinger and J. Sta hel, Nu l. Phys. 772 167 (2006).[24℄ M.I. Gorenstein, M. Hauer and O.N. Moroz, Phys. Rev. C77 024911 (2008).[25℄ Yu.B. Ivanov, I.N. Mishustin, V.N. Russkikh and L.M. Satarov, arXiv:0907.4140 [nu l-th℄.[26℄ N. Demir and S.A. Bass, Phys. Rev. Lett. 102 172302 (2009); arXiv:0907.4333 [nu l-th℄.[27℄ A. K. Chaudhuri, arXiv:0910.0979 [nu l-th℄.7ÀííîòàöèÿÂÿçêîñòü àäðîííîé ìàòåðèè â ðåëÿòèâèñòñêîé ìîäåëèñðåäíåãî ïîëÿ ñî ñêåéëèíãîì àäðîííûõ ìàññ èêîíñòàíò ñâÿçèÀ.Ñ. Õâîðîñòóõèí, Â.Ä. Òîíååâ è Ä.Í. ÂîñêðåñåíñêèéÑäâèãîâàÿ ( η ) è îáúåìíàÿ( ζζ