The Viscosity of Quark-Gluon Plasma at RHIC and the LHC

# The viscosity of quark-gluon plasma at RHIC and the LHC

## Abstract

The specific shear viscosity of quark-gluon plasma (QGP) can be extracted from elliptic flow data in heavy-ion collisions by comparing them with the dynamical model VISHNU which couples a viscous fluid dynamic description of the QGP with a microscopic kinetic description of the late hadronic rescattering and freeze-out stage. A robust method for fixing from the collision centrality dependence of the eccentricity-scaled charged hadron elliptic flow is presented. The systematic uncertainties associated with this extraction method are discussed, with specific attention to our presently restricted knowledge of initial conditions. With the extracted in this way, VISHNU yields an excellent description of all soft-hadron data from Au+Au collisions at top RHIC energy. Extrapolations to Pb+Pb collisions at the LHC, using both a purely hydrodynamic approach and VISHNU, are presented and compared with recent experimental results from the ALICE Collaboration. The LHC data are again well described by VISHNU, with the same value as at RHIC energies.

Hydrodynamics, quark-gluon plasma, heavy-ion collisions, eccentricity, shear viscosity, elliptic flow, hybrid approach
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address=Department of Physics, The Ohio State University, Columbus, Ohio 43026, USA address=Department of Physics, The Ohio State University, Columbus, Ohio 43026, USA address=Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

## 1 How to measure (η/s)QGP

Relativistic heavy-ion collisions create spatially deformed fireballs of hot, dense matter – in both non-central and (due to event-by-event shape fluctuations) central collisions. Hydrodynamics converts this initial spatial deformation into final state momentum anisotropies. Viscosity degrades the conversion efficiency of the fluid ( and are the directions transverse to the beam direction ); for given initial fireball ellipticity , the viscous suppression of the dynamically generated total momentum anisotropy is monotonically related to the specific shear viscosity . The observable most directly related to is the total charged hadron elliptic flow Heinz:2005zg (). Its distribution in depends on the chemical composition and -spectra of the various hadron species; the latter evolve in the hadronic stage due to continuously increasing radial flow (and so does ), even if (as expected at top LHC energy Hirano:2007xd ()) fully saturates in the QGP phase. When (as it happens at RHIC energies) does not reach saturation before hadronization, dissipative hadronic dynamics Hirano:2005xf () affects not only the distribution of over hadron species and , but even the final value of itself, and thus from which we want to extract . To isolate the QGP viscosity we therefore need a hybrid code that couples viscous hydrodynamics of the QGP to a realistic model of the late hadronic stage, such as UrQMDBass:1998ca (), that describes its dynamics microscopically. VISHNUSong:2010aq (), a hybrid of VISH2+1 (Viscous Israel-Stewart Hydrodynamics in 2+1 dimensions Song:2007fn ()) and UrQMD, is such a code.

## 2 (η/s)QGP at RHIC

The left panel in Fig. 1 shows that such an approach yields a universal dependence of the ellipticity-scaled total charged hadron elliptic flow, , on the charged hadron multiplicity density per overlap area, , that depends only on but not on the details of the initial state model that provides and Song:2010mg (). Pre-equilibrium flow and bulk viscous effects on these curves are small Song:2010mg ().

The QGP viscosity can be extracted from experimental data by comparing them with these universal curves. The right panels of Fig. 1 show this for MC-Glauber and MC-KLN initial state models (please see Song:2010mg () and references therein for a description of these models). In both cases the slope of the data Ollitrault:2009ie () is correctly reproduced; this is not the case for ideal nor for viscous hydrodynamics with constant . Due to the larger ellipticity of the MC-KLN fireballs, the magnitude of differs between the two models. Consequently, the value of extracted from this comparison changes by more than a factor 2 between them. Relative to the initial fireball ellipticity all other model uncertainties are negligible. Without constraining more precisely, cannot be determined to better than a factor 2 from elliptic flow data alone, irrespective of any other model improvements.1 Taking the MC-Glauber and MC-KLN models to represent a reasonable range of initial ellipticities, Fig. 1 gives for temperatures probed at RHIC.

All calculations in Fig. 1 and following below were done in “single-shot” mode, where the ensemble of fluctuating Monte Carlo initial states was first averaged in the participant plane Song:2010mg () to obtain a smooth average initial density profile and then evolved just once through the hydrodynamic stage. Event-by-event evolution of each fluctuating initial state separately and performing the ensemble average only at the end may produce somewhat less elliptic flow and thus slightly reduce the values extracted from comparison with the data Qiu:2011iv (). The magnitude of this reduction depends on Qiu:2011fi () but is not expected to exceed (0.2-0.3)/4 Qiu:2011iv ().

VISHNU with for MC-Glauber and for MC-KLN provides an excellent description of all aspects of soft ( GeV) hadron production (-spectra and differential for all charged hadrons together as well as for individual identified species) in 200  GeV Au+Au collisions at all but the most peripheral collision centralities Song:2011hk (). As an example we show in Fig. 2 -spectra and differential elliptic flow for identified pions and protons (resonance decay contributions are included). Such a level of theoretical control is unprecedented.2

## 3 (η/s)QGP at the LHC

The successful comprehensive fit of soft hadron spectra and elliptic flow in Au+Au collisions at RHIC shown in Fig. 2 and elaborated on in more detail in Refs. Song:2011hk (); Shen:2011eg () allows for tightly constrained LHC predictions. Fig. 3 shows such predictions for both pure viscous hydrodynamics VISH2+1Shen:2011eg () and VISHNUSong:2011qa ().

A straightforward extrapolation with fixed overpredicts the LHC values by 10-15%; a slight increase of from 0.16 to 0.20 (for MC-KLN) gives better agreement with the ALICE data Aamodt:2010pa (). However, at LHC energies becomes sensitive to details of the initial shear stress profile Shen:2011eg (), and no firm conclusion can be drawn yet whether the QGP turns more viscous (i.e. less strongly coupled) at higher temperatures. Furthermore, ALICE Floris:2011ru () has noted a discrepancy between the ratio measured in Pb+Pb collisions at the LHC and the value observed by STAR in Au+Au collisions at RHIC. The latter has a strong influence on the value of the chemical decoupling temperature implemented in the model. We use  MeV which nicely fits the normalization of the proton spectra from STAR but overpredicts those from PHENIX by a factor (left panel in Fig. 2). The ratio measured by ALICE at the LHC agrees with the PHENIX value measured at RHIC (see Fig. 7 in Floris:2011ru ()) but is smaller by a factor than what is implemented in the LHC predictions from Refs. Shen:2011eg (); Song:2011qa (). Correspondingly, our predictions of the spectra for Pb+Pb@LHC Shen:2011eg () overpredict the measured spectra by this factor Floris:2011ru (). Reducing the ratio to the measured value will reduce the charged hadron elliptic flow Hirano:2002ds (). To to the larger radial flow, this reductions is stronger at the LHC than at RHIC. This may account for the overprediction of for at the LHC seen in the left panel of Fig. 3.

The right panel of this figure shows that, at fixed  GeV, increases from RHIC to LHC for pions but decreases for all heavier hadrons. The similarity at RHIC and LHC of for the sum of all charged hadrons noted in Ref. Aamodt:2010pa () thus appears accidental. As a result of this shift of the elliptic flow to larger for heavier particles, which is caused by the stronger radial flow at the LHC, the mass-splitting between the curves for different mass hadrons grows from RHIC to LHC. This predicted growth has been confirmed by ALICE (see Fig. 6 in Collaboration:2011yba ()).

As mentioned in footnote 2, the purely hydrodynamic simulations based on VISH2+1 with constant fail to correctly reproduce the centrality dependence of the proton elliptic flow . Especially in central collisions, is overpredicted at small (see Fig. 2 in Krzewicki:2011ee ()), i.e. the radial flow pushing the elliptic flow to higher (and thereby reducing at low ) as generated by the model is not strong enough in central collisions. The same problem is seen in Fig. 4 for the extrapolation of VISH2+1 to LHC energies: whereas the pion and kaon elliptic flows at  GeV/ are well described at all collision centralities, there appears to be a lack of radial flow in central and semi-central collisions such that the proton is not pushed towards larger as strongly as seen in the data. This problem disappears in the more peripheral bins, indicating an incorrect centrality dependence of the balance between radial and elliptic flow in the VISH2+1 model.

As was the case at RHIC energies, this problem is removed when replacing the hydrodynamic description of the late hadronic phase with low by a microscopic kinetic description using VISHNU (see Fig. 5). The calculations shown in Fig. 5 were done primarily to understand systematic differences between the predictions for spectra and elliptic flow from VISH2+1 and VISHNU. For this reason they were performed with the same value in the QGP phase, even though the VISHNU calculations for identified hadrons from Au+Au at RHIC Song:2011hk () had shown a slight preference for the smaller value . Fig. 5 shows that the problem with the lack of radial flow in central collisions that was seen in Fig. 4 has been resolved: for the 5%-10% and 10%-20% centrality bins, VISHNU describes the differential up to  GeV/ almost perfectly, for all three particle species. Looking closely, one observes a slight (6%) underprediction of for all three particle species. This underprediction gets stronger in more peripheral collisions (reaching 9% at 50%-60% centrality) – a clear sign that we have slightly overestimated . (Since, for fixed , viscous effects increase in inverse proportion to the fireball size Song:2007fn (), an overestimate of will lead to an underprediction of that grows with the impact parameter of the collisions.) We are confident that, after reducing to 0.16, the data will be well described at all collision centralities. Corresponding simulations are in progress.

## 4 Conclusions

The hybrid model VISHNU, which describes the evolution of the dense and strongly coupled quark-gluon plasma phase macroscopically using viscous fluid dynamics and that of the dilute late hadronic rescattering and freeze-out stage microscopically using a kinetic approach, provides a comprehensive quantitative description of the bulk matter created in relativistic heavy-ion collisions at RHIC and LHC. Transverse momentum spectra and elliptic flow of soft charged hadrons, pions, kaons, and protons are well reproduced at all collision centralities, with a QGP shear viscosity if MC-KLN initial conditions are used. So far the data yield no evidence for a change of between RHIC and LHC that would reflect the different temperature ranges probed. Overall, the QGP liquid created in heavy-ion collisions at the LHC appears to be as strongly coupled as at RHIC energies.

Acknowledgments: This work was supported by the U.S. Department of Energy under grants No. DE-AC02-05CH11231, DE-SC0004286 and (within the framework of the JET Collaboration) No. DE-SC0004104. Extensive computing resources provided by the Ohio Supercomputing Center are gratefully acknowledged.

### Footnotes

1. It has been suggested Shen:2011zc (); Qiu:2011fi (); :2011vk (); Adare:2011tg () that the ambiguity between the MC-Glauber and MC-KLN ellipticities which lies at the origin of this uncertainty can be resolved by simultaneously analyzing elliptic and triangular flow, and .
2. We note that the purely hydrodynamic model VISH2+1 does almost equally well, with for MC-KLN initial conditions Shen:2011eg (), except for the centrality dependence of the differential elliptic flow for protons. We will see a similar failure of VISH2+1 for Pb+Pb collisions at the LHC further below. The main difference to VISHNU is that, in order to generate enough radial flow at freeze-out, VISH2+1 must be started earlier ( instead of 1.05 fm/) because it lacks the highly dissipative hadronic phase that generates additional radial flow in VISHNU at late times (in the VISH2+1 simulations is held constant at 0.2 until hadronic freeze-out). The variation with collision centrality of the final balance between radial and elliptic flow turns out to be correct in VISHNU (where more of the radial flow develops later) but incorrect in VISH2+1 (where more of it is created early).

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