We study within a relativistic transport approach the impact of a temperature dependent shear viscosity to entropy density ratio, η/s(T), on the build-up of the elliptic flow, v2, a measure of the angular anisotropy in the particle production. Beam Energy Scan from √sNN = 62.4GeV at RHIC up to 2.76TeV at LHC has shown that the v2(pT ) as a function of the transverse momentum pT appears to be nearly invariant with energy. We show that such a surprising behavior is determined by a rise and fall of η/s(T) with a minimum at T ∼ Tc, as one would expect if the matter undergoes a phase transition or a cross-over

Csernai, L.P.

Greco, V.

Plumari, S.

English

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DOI 10.1393/ncc/i2014-11667-8Colloquia: IFAE 2013IL NUOVO CIMENTO Vol. 37 C, N. 1 Gennaio-Febbraio 2014Elliptic flow and shear viscosity from a beam energy scanS. Plumari(1)(2), V. Greco(1)(2) and L. P. Csernai(3)(1) Department of Physics and Astronomy, University of CataniaVia S. Sofia 64, I-95125 Catania, Italy(2) Laboratori Nazionali del Sud, INFN-LNS - Via S. Sofia 63, I-95125 Catania, Italy(3) Department of Physics and Technology, University of Bergen - Norwayricevuto l’1 Ottobre 2013Summary. — We study within a relativistic transport approach the impact ofa temperature dependent shear viscosity to entropy density ratio, η/s(T ), on thebuild-up of the elliptic flow, v2, a measure of the angular anisotropy in the particleproduction. Beam Energy Scan from√sNN = 62.4GeV at RHIC up to 2.76TeVat LHC has shown that the v2(pT ) as a function of the transverse momentum pTappears to be nearly invariant with energy. We show that such a surprising behavioris determined by a rise and fall of η/s(T ) with a minimum at T ∼ Tc, as one wouldexpect if the matter undergoes a phase transition or a cross-over.PACS 24.85.+p – Quarks, gluons, and QCD in nuclear reactions.PACS 25.75.Nq – Quark deconfinement, quark-gluon plasma production, and phasetransitions.PACS 25.75.-q – Relativistic heavy-ion collisions.1. – IntroductionThe main motivation for the ultra-relativistic heavy-ion collisions (uRHICs) programis to create a transient state of a quark-gluon plasma (QGP) [1]. The experimentalresults accumulated in these decades at the RHIC program at BNL and more recentlyat the LHC program at CERN have shown that the elliptic flow v2 = 〈cos(2ϕp)〉 =〈(p2x − p2y)/(p2x + p2y)〉, which is a measure of the anisotropy in the angular distributionof the emitted particle, is the largest ever seen in uRHICs [1, 2]. The comparison of theexperimental measured v2 and theoretical calculations within viscous hydrodynamics [3]or transport approach [4-6] have shown that such a matter has a very low shear viscosityto entropy density ratio η/s close the conjectured lower bound for a strongly interactingsystem in the limit of infinite coupling, η/s = 1/4π [7]. On the other hand, recentmeasurements of higher harmonics vn = 〈cos(nϕp)〉 with n > 2 have found resultswhich are compatible with a finite but not too large value of η/s ∼ 1–3 [8, 9]. However,a low value of η/s ∼ 0.1 in itself is not a direct signature of the creation of a QGPc© Societa` Italiana di Fisica 68ELLIPTIC FLOW AND SHEAR VISCOSITY FROM A BEAM ENERGY SCAN 69indeed from perturbative calculation one expects to have an η/s about one order ofmagnitude larger [10]. It is known from atomic and molecular physics that a minimumin η/s is expected close to the transition temperature as emphasized in the contextof QGP in refs. [11, 12]. For this reason not an average value for η/s, but rather aphenomenological estimate of its temperature dependence can give an information aboutthe phase transition of the matter created.It has been emphasized by both the STAR Collaboration at RHIC [13] and the ALICEat LHC [2] (in agreement with the measurements done also by CMS [14] and ATLAS [15])that surprisingly the v2(pT ) appears to be invariant in the very wide colliding energyrange of 62.4GeV ≤ √sNN ≤ 2.76TeV. It therefore appears a key question to answerthe reason for the invariance of v2(pT ). We show that the invariance of v2(pT ) in theenergy range 62.4GeV ≤ √sNN ≤ 2.76TeV is caused by a fall and rise of the η/s(T ) asone would expect if the created matter undergoes a phase transition.2. – Transport approach at fixed η/sIn this section we introduce a Relativistic Boltzmann Transport (RBT) at fixed η/sratio. The motivation to introduce this approach is twofold, on one hand it is possibleto make a direct comparison to viscous hydrodynamics and on the other hand we have atool to directly estimate the viscosity of the plasma in a wider range of η/s and pT andcompare it to hydrodynamics.To study the dynamics with a certain η/s(T ), we determine locally in space and timethe total cross section σtot according to the Chapmann-Enskog theory. For a pQCDinspired cross section, dσ/dt ∼ α2s/(t − m2D)2, typically used in parton cascade ap-proaches [4, 16,5], this gives:(1) η/s =115〈p〉 τη = 115〈p〉g(a)σtotρ,where a = mD/2T , with mD being the screening mass regulating the angular dependenceof the cross section σtot, while g(a) is the proper function accounting for the pertinentrelaxation time τ−1η = g(a)σtotρ associated to the shear transport coefficient and givenby g(a) = 50−1∫dyy6[(y2+3−1)K3(2y)− yK2(2y)]h(a2y−2) with Kn-s being the Besselfunctions and the function h is relating the transport cross section to the total oneσtr(s) = σtot h(m2D/s) and h(ζ) = 4ζ(1 + ζ)[(2ζ + 1)ln(1 + 1/ζ)− 2]. We have shown inref. [17] that eq. (1) correctly describes the η/s of the system.In this approach we solve the RBT equation with the constraint that η/s(T ) is fixedduring the dynamics of the collisions by means of eq. (1) in a way similar to [18], butwith an exact local implementation as described in detail in [4].3. – Effect of η/s(T ) on the elliptic flowIn our calculation the initial condition is longitudinal boost invariant flow. The initialdN/dη have been chosen in order to reproduce the final dNch/dη(b) at mid rapidityas observed in the experiments at RHIC and LHC energies. The partons are initiallydistributed according to the Glauber model in coordinate space. In the momentumspace the distribution is thermal up to pT = 2GeV and at larger pT we include thespectrum of non-quenched minijets according to standard NLO-pQCD calculations. Themaximum temperature, Tm0, in the center of the fireball is fixed by assuming that it70 S. PLUMARI, V. GRECO and L. P. CSERNAIFig. 1. – a) Different parametrizations for η/s. See the text for more details. b) v2(pT ) atmid rapidity for 20%–30% collision centrality. In the left panel it is shown the RHIC result forAu + Au at√s = 200GeV while in the right panel it is shown the LHC result for Pb + Pb at√s = 2.76TeV. The black dot dashed lines refer to 4πη/s = 1 during all the evolution of thefireball and without the freeze out condition while the orange solid lines are the calculationswith the inclusion of the kinetic freeze out and with 4πη/s = 1 in the QGP phase and finallythe blue dashed lines are the calculations with η/s(T ) ∝ T 2 in the QGP phase.scales with the collision energy according to the relation (τAT )−1dNch/dη ∝ T 3. Forthe initial time, τ0, we ensure that it satisfies the uncertainty relation between the initialaverage thermal energy and the initial time by Tm0τ0 ≈ 1, and we fix τ0 = 0.6 fm/c forAu + Au at√s = 200GeV as usually done in hydrodynamics. Thus we found that themaximum initial temperature is Tm0 = 290MeV, 340MeV and 560MeV respectively for√s = 62.4GeV, 200GeV and 2.76TeV. The local temperature scale according to theenergy density profile T (r) = Tm0((r)/(0))1/4.In the following we will discuss the impact of the temperature dependence of η/s, seefig. 1a. In the cross-over region that we identify as the region just below the knee inthe /T 4 curve [19], i.e. at  < 1.5GeV/fm3 and/or T < T0 = 1.2Tc, the η/s increaseslinearly at decreasing T matching the estimates from chiral perturbation theory for ahigh temperature meson gas [20] (triangles in fig. 1a). In fig. 1a it is also shown a com-parison with the estimate of η/s extrapolated from heavy-ion collisions at intermediateenergies [21] (HIC-IE diamonds). At higher temperature due to the large error bars inthe lQCD results for η/s it is not possible to infer a clear temperature dependence in theQGP phase. Therefore for T > 1.2Tc we have considered two cases: one with a lineardependence 4πη/s = T/T0 (red solid line in fig. 1a) in agreement with the indication oflQCD calculation in ref. [22] (up-triangles) and quasi-particle model prediction (orangeband) where η/s ∼ Tα with α ≈ 1–1.5 [23]. The other one with a quadratic dependence4πη/s = 3.64(T/T0 − 1) + (T/T0)2 (blue dashed line) resembling the lQCD in quenchedapproximation in ref. [24] as given also in [25]. We also consider a common case of aconstant η/s at its conjectured minimum value 1/4π. In order to discuss the differenteffects on the v2(pT ) of an η/s(T ) at different collision energies we compare the v2(pT )obtanined using three different temperature parametrizations of η/s(T ) for Au + Au at√s = 200GeV (left panel) and Pb+Pb at√s = 2.76TeV (right panel). As shown in theleft panel of fig. 1b, at RHIC the v2 is sensitive to the value of η/s in the cross over regionand the effect of the freeze out (the increase of η/s at low temperature) is to reduce thev2 of about of 20%, from black dot dashed line to orange solid line. At LHC energies,right panel of fig. 1b, the scenario is different, we have that v2 is less sensitive to theELLIPTIC FLOW AND SHEAR VISCOSITY FROM A BEAM ENERGY SCAN 71Fig. 2. – v2(pT ) at mid rapidity for 10%–20% collision centrality. The thin solid line, the dashedline and the thick solid line refer to: Au+Au at√s = 62.4GeV and√s = 200GeV and Pb+Pbat√s = 2.76TeV, respectively. On the left panel, it is shown the result for an η/s(T ) witha quadratic dependence, (blue dashed line in fig. 1a), while on the right panel it is shown theresult for a linear dependence, (red solid line in fig. 1a). In the right panel symbols refers todata for v2[4] measured by STAR and ALICE collaborations [2, 13].increase of η/s at low temperature in the cross over region. The effect of large η/s inthe cross over region at LHC energies is to reduce the v2 by less than 5% in the low pTregion, from black dot dashed line to orange solid line. Therefore, in general the effect ofthe kinetic f.o. is to reduce the elliptic flow and for decresing collision energy the v2(pT )becomes more sensitive of the value of η/s at low temperature. As we can see comparingthe blue dashed lines with the orange solid one in the left panel and in the right panelof fig. 1b the increase of η/s in the QGP phase affects more the system created at LHC.This means that the v2 developed at LHC energies is more sensitive to the value of η/sin the QGP phase.The different impact of η/s(T ) on v2(pT ) is determined by the different initial tem-perature and consequent lifetime of the stage at T > Tc. In fact at RHIC energies such alifetime is about 4–6 fm/c in contrast to about 10 fm/c at LHC. Therefore at RHIC theelliptic flow has not enough time to fully develop in the QGP phase, while at LHC thelifetime is long enough to let the v2 develop almost completely in the QGP phase and theincrease of η/s(T ) at low T becomes irrelevant, see fig. 1b. From this observation one hasthe hint that an invariant v2(pT ) with the collision energy as shown in the experimentscan be caused by the specific T -dependence of η/s that balances the suppression due tothe viscosity above and below Tc where a minimum in η/s should occur. In fig. 2 we showthat it is possible to discriminate the T -dependence of η/s in the QGP phase (T > TC).As we can see in the left panel of fig. 2 the rapidly increasing η/s(T ) in the QGP phaseaffects the system created at LHC and this generates again a larger splitting of the v2(pT )among the different energies. This means that also a strong T -dependence in the QGPphase is in contrast with the observed v2(pT ) invariance, essentially it would make thesystem at LHC too much viscous. Finally we consider 4πη/s = T/T0, according to thesolid red line in fig. 1a. The results in the right panel of fig. 2 are also compared withthe experimental results for the v2[4] measured at RHIC and LHC energy, data takenby [2, 13]. We can see that in such case there is an almost perfect invariance of v2(pT )(within a 5%) in agreement with what is observed in the experimental data shown bythe different symbols in fig. 2 (right panel). The main effect is that with a mild increaseof η/s(T ) at T > T0 the elliptic flow at LHC energies goes up reaching the higher v2(pT )obtained at lower energies.72 S. PLUMARI, V. GRECO and L. P. CSERNAITherefore a comparative analysis at different beam energies is able to reveal an infor-mation on the η/s(T ), namely the necessity of a “U” shape of η/s(T ) with a minimumslightly above TC which is tyical when the matter undergoes a cross-over [11].4. – ConclusionsWe have investigated within a transport approach the effect of a temperature depen-dence η/s(T ) on the build up of v2 from RHIC to LHC energies. We find that for thedifferent beam energies considered the suppression of the elliptic flow due to the viscos-ity of the medium has a different damping coming from the hadronic or QGP phasesdepending on the average energy of the system. As shown the elliptic flow at LHC is lessaffected by the hadronic phase and it becomes more sensitive to the QGP properties. Inour analysis, we have shown that going from RHIC to LHC energies it is possible to havea nearly invariant v2(pT ) only if the η/s has a fall and rise with a minimum around thetransition which is a typical behaviour of a phase transition or a cross-over. For detailssee [26].REFERENCES[1] Adams J. et al., Nucl. Phys. A, 757 (2005) 102.[2] Aamodt K. et al., Phys. Rev. Lett., 105 (2010) 252302.[3] Romatschke P. and Romatschke U., Phys. Rev. Lett., 99 (2007) 172301.[4] Ferini G., Colonna M., Di Toro M. and Greco V., Phys. Lett. B, 670 (2009) 325.[5] Xu Z. and Greiner C., Phys. Rev. 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Elliptic flow and shear viscosity from a beam energy scan

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Elliptic flow and shear viscosity from a beam energy scan

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