Radial and elliptic flow in Pb+Pb collisions at the Large Hadron
Collider
from viscous hydrodynamics
Abstract
A comprehensive viscous hydrodynamic fit of spectra and elliptic flow for charged hadrons and identified pions and protons from Au+Au collisions of all centralities measured at the Relativistic Heavy Ion Collider is performed and used as the basis for predicting the analogous observables for Pb+Pb collisions at the Large Hadron Collider at and 5.5 TeV. Comparison with recent measurements of the elliptic flow of charged hadrons by the ALICE experiment shows that the model slightly overpredicts the data if the same (constant) specific shear viscosity is assumed at both collision energies. In spite of differences in our assumptions for the equation of state, the freezeout temperature, the chemical composition at freezeout, and the starting time for the hydrodynamic evolution, our results agree remarkably well with those of Luzum [M. Luzum, Phys. Rev. C 83, 044911 (2011)], indicating robustness of the hydrodynamic model extrapolations. Future measurements of the centrality and transverse momentum dependence of spectra and elliptic flow for identified hadrons predicted here will further test the model and shed light on possible variations of the quarkgluon transport coefficients between RHIC and LHC energies.
pacs:
25.75.q, 12.38.Mh, 25.75.Ld, 24.10.NzI Introduction
The first measurement of elliptic flow in Pb+Pb collisions at the Large Hadron Collider (LHC) has just been reported Aamodt:2010pa . The elliptic flow coefficient characterizes the momentum anisotropy of final particle emission in noncentral heavyion collisions relative to the eventplane, defined by the beam direction and the minor axis of the nuclear overlap region in the collision. It describes the efficiency of the medium generated in the collision to generate from an initial spatial deformation of its density distribution an asymmetry in the final momentum distribution, through interactions of the medium constituents. This efficiency increases with the coupling strength between those constituents and becomes maximal for an infinitely strongly coupled medium. In this limit the mean free path of the constituents becomes as small as allowed by the uncertainty relation Danielewicz:1984ww , and the medium reaches very quickly a state of approximate local thermal equilibrium which allows to describe its evolution with fluid dynamics. For given initial spatial deformation of the collision fireball, ideal fluid dynamics (which assumes zero mean free path) is expected to generate the largest possible elliptic flow Heinz:2001xi . Shear viscosity, a consequence of nonzero mean free paths and limited from below by quantum mechanics Danielewicz:1984ww ; Policastro:2001yc , will lead to a suppression of Heinz:2002rs ; Teaney:2003kp .
Compelling evidence for fluid dynamical behavior of the collision fireballs created in ultrarelativistic heavyion collisions, with a very small ratio of shear viscosity to entropy density, , has been found in heavyion collisions at the Relativistic Heavy Ion Collider (RHIC) Arsene:2004fa ; Lacey:2006pn ; Romatschke:2007mq ; Song:2010mg . The new data from the LHC confirm this picture Aamodt:2010pa ; Luzum:2010ag ; Lacey:2010ej and agree, at least qualitatively, with hydrodynamic predictions of elliptic flow for Pb+Pb collisions at the LHC Abreu:2007kv ; Niemi:2008ta ; Kestin:2008bh ; Luzum:2009sb ; Bozek:2010wt ; Hirano:2010jg ; Schenke:2011tv .
The purpose of the present article is to explore how good this agreement is quantitatively, and to what extent the present and future LHC elliptic flow data can tell us novel facts about the transport behavior of hot QCD matter at temperatures that exceed those accessible at RHIC but are within reach at the LHC. Similar to the analyses in Niemi:2008ta ; Luzum:2009sb ; Luzum:2010ag ; Schenke:2011tv , but different from recent hybrid model studies in Hirano:2010jg ; Song:2011qa , we base our analysis on a purely hydrodynamic approach. While this ignores the fact that the late dilute hadronic stage of the expansion is very dissipative and not well described by fluid dynamics (neither ideal Hirano:2005xf nor viscous Song:2010aq ), the importance of the hadronic phase for the development of elliptic flow is expected to be much reduced at the LHC relative to RHIC Hirano:2007xd ; Hirano:2010jg . As in Refs. Luzum:2009sb ; Luzum:2010ag ; Schenke:2011tv ; Song:2011qa , but different from Refs. Niemi:2008ta ; Hirano:2010jg , we use viscous hydrodynamics with a nonzero (but constant, i.e. temperature independent) specific shear viscosity , adjusted to spectra and elliptic flow measurements at RHIC. Our fitted value (for CGC initial conditions, see below) is 25% larger than that used by Luzum and Romatschke Luzum:2009sb ; Luzum:2010ag but agrees well with the value for the quarkgluon plasma (QGP) viscosity recently extracted from RHIC data by using a hybrid viscous hydrodynamic + Boltzmann approach (VISHNU Song:2010mg ; Song:2010aq ). Calculations with such a hybrid approach are numerically much more demanding than purely hydrodynamic simulations; a generalization of the present analysis using VISHNU will follow soon and should further improve the reliability of the LHC predictions.
Ii Hydrodynamic fit of RHIC Au+Au data
In this work, we employ (2+1)d viscous hydrodynamics Song:2007fn with the lattice QCD based equation of state s95pPCE Huovinen:2009yb ; Shen:2010uy , which accounts for chemical freezeout before thermal decoupling at MeV, to simulate the expansion of the collision fireball. From the analysis Song:2010mg of charged hadron spectra and elliptic flow in GeV Au+Au collisions at RHIC we take over a value of (corresponding to MCKLN initial conditions, see below) for the effective specific shear viscosity of the strongly interacting fluid. Using the insights obtained from the systematic parameter study presented in Shen:2010uy , we initialize the hydrodynamic expansion at time fm/ and decouple at MeV at both RHIC and LHC energies. For Au+Au collisions at RHIC energies these parameters allow for a good global description of the hadron spectra and differential elliptic flow (see below). Lacking strong theoretical or phenomenological guidance how to adjust their values for Pb+Pb collisions at the LHC, we here decided to keep them unchanged.
At thermalization time , we assume that the shear stress
tensor is given by its NavierStokes value (where
is the symmetric and traceless velocity shear tensor),
and that the initial expansion flow is entirely longitudinal with Bjorken
profile and zero transverse flow velocity. In Milne
coordinates this corresponds to an initial flow
4velocity . Kinetic freezeout
is implemented by converting the hydrodynamic output to particle
spectra with the CooperFrye prescription Cooper:1974mv
on a decoupling surface of constant temperature . We use
a quadratic ansatz
Teaney:2003kp for the viscous deviation from local thermal
equilibrium of the local phasespace distribution function on the
freezeout surface. Our final hadron spectra include decay products
from strong decays of all particles and resonances up to 2 GeV mass
Sollfrank:1990qz , using the resonance decay program from the
AZHYDRO package.^{1}^{1}1AZHYDRO is available at the URL
http://www.physics.ohiostate.edu/~froderma/.
For the initial density profile we here use a MonteCarlo version
Drescher:2006pi ; Drescher:2007ax of the KharzeevLevinNardi
KLN model (MCKLN).^{2}^{2}2The Monte Carlo code is
available at URL
http://www.aiu.ac.jp/~ynara/mckln/.
The specific implementation used in this work is described in
Hirano:2009ah ; Hirano:2010jg . The model gives for each event the
gluon density distribution in the transverse plane. We assume it to
thermalize by time , and convert the gluon density into entropy
density Hirano:2005xf . Over one million Monte Carlo events are
recentered to the beam axis and rotated in the transverse plane such
that their minor axis aligns with the impact parameter (i.e. their
“participant plane” coincides with the reaction plane). After sorting
them into collision centrality bins according to their number of
participating (“wounded”) nucleons, we average them to obtain a smooth
average initial entropy density which is then converted to energy
density using the equation of state. Using this smooth energy density
as weight, we compute the initial eccentricity
of the reaction zone; these represent
the corresponding mean values of events in this centrality
class.^{3}^{3}3Note that about 10% larger overlap areas are
obtained when using the entropy density as weight
Hirano:2010jg ; Song:2010mg , whereas for all but the most
central collisions the eccentricities of the energy and entropy
densities are nearly identical Qiu:2011iv . and overlap area
The KLN model involves a couple of parameters that need to be adjusted to obtain the correct final charged hadron multiplicity in central Au+Au collisions at RHIC. In Hirano:2009ah this adjustment was performed for ideal fluid dynamics (which conserves entropy) coupled to a hadron cascade. The model then correctly predicts the charged hadron multiplicities at all other collision centralities. In our viscous hydrodynamic model, viscous heating produces additional entropy, leading to somewhat larger final multiplicities. We thus perform an iterative renormalization of the initial entropy density profile until the measured charged hadron multiplicity in the most central GeV Au+Au collisions at RHIC is once again reproduced. The lower panel of Fig. 1 shows that, after this renormalization, the model again correctly describes the measured Back:2004dy centrality dependence of charged hadron production. The centrality dependence of viscous entropy production (which is relatively larger in peripheral than in central collisions Song:2007fn ) is (at least at RHIC energies) sufficiently weak to not destroy the agreement of the model with experimental observations.
The ability of the MCKLN model to describe the centrality dependence of charged hadron production without additional parameters is the main reason for choosing it over the MCGlauber model as our basis for extrapolation from RHIC to LHC energies. It was recently shown ALbacete:2010ad that this centrality dependence is robust against running coupling corrections Balitsky:2006wa ; Albacete:2007sm ; Kovchegov:2006vj in the BalitskyKovchegov evolution (on which the KLN model is based) which were found to hardly affect its shape. They do, however, modify the collision energy dependence of particle production, with the LHC Pb+Pb data being better described if running coupling corrections are included Aamodt:2010cz . Our version of the MCKLN model does not include running coupling corrections,^{4}^{4}4The rcBK code in ALbacete:2010ad includes running coupling corrections but it has not been renormalized to take into account viscous entropy production at RHIC energies. and we must normalize the initial entropy density profile for Pb+Pb collisions at the LHC separately from Au+Au collisions at RHIC. Without such an independent renormalization, we overpredict the measured charged multiplicity from central TeV Pb+Pb collisions Aamodt:2010pb ; Aamodt:2010cz by about 10%.
After renormalization we obtain the solid lines bounding the shaded region in the upper panel of Fig. 1, with the lower (upper) bound corresponding to Pb+Pb collisions at 2.76 (5.5) GeV, respectively. The data in that panel are from the ALICE Collaboration for Pb+Pb at GeV Aamodt:2010pb ; Aamodt:2010cz . (For GeV Pb+Pb collisions we assumed (corresponding to ), based on an extrapolation of Fig. 3 in Ref. Aamodt:2010pb .) One sees that, even without running coupling corrections, but including viscous entropy production, the MCKLN model does a good job in describing the measured centrality dependence of charged hadron production in Pb+Pb collisions at the LHC. This gives hope that the successful description of the centrality dependence of hadron spectra and elliptic flow at RHIC energies (see below and Song:2010mg ) translates into a reliable prediction of the corresponding centrality dependences in Pb+Pb collisions at the LHC.
Figures 2 and 3 establish our baseline for the extrapolation to LHC energies. In Fig. 2 we show our purely hydrodynamic fit (obtained with parameters , , , and set as described above^{5}^{5}5Note that our value fm/ is 45% smaller than the value of 1.05 fm/ used for in the VISHNU simulations in Song:2010mg . The earlier evolution of hydrodynamic transverse flow arising from this smaller value compensates for the lack of a highly dissipative hadronic phase in the purely hydrodynamic approach. Hadronic dissipation leads to a significant broadening in particular of the proton spectra during the hadronic stage which (given the constraints from the elliptic flow data which prohibit us from simply lowering ) viscous hydrodynamics with temperatureindependent cannot replicate.) of the hadron spectra measured in GeV Au+Au collisions at RHIC. Fig. 2a shows the midrapidity transverse momentum spectra per unit pseudorapidity for unidentified charged hadrons from the STAR Adams:2003kv and PHENIX Adler:2003au experiments compared with the hydrodynamical model. Figs. 2b,c show a similar comparison for the spectra per unit rapidity of identified pions and protons from STAR Adams:2003xp ; :2008ez and PHENIX Adler:2003cb . In the experimental spectra, protons from weak decays were removed; STAR quotes a large systematic error associated with this feeddown correction, and within that large error band the two data sets agree with each other, even if the central values of the STAR proton data appear to be up to 50% higher than PHENIX data. Our results agree well with the STAR protons for GeV/ but overpredict the PHENIX protons by up to a factor 2.
Figure 3 shows the hydrodynamically calculated differential elliptic flow for unidentified charged hadrons in comparison with STAR data BaiThesis ; :2008ed , for four centrality classes ranging from semicentral to midperipheral collisions ( centrality). With , viscous hydrodynamics gives an excellent description of the STAR data, even up to 3 GeV/ in transverse momentum (i.e. beyond the range where the hydrodynamic description is expected to begin to break down, due to the increasing influence of hard production processes and large uncertainties in the viscous correction to the local phasespace distribution at kinetic freezeout Shen:2011kn ). Looking more carefully one sees that our model slightly overestimates the elliptic flow at low GeV while underestimating it in the high region, GeV.
In Shen:2010uy we noted a tension in trying to simultaneously fit within a purely viscous hydrodynamic approach the proton spectra and the charged hadron differential elliptic when using EOS s95pPCE. Even a temperature dependent that has a larger shear viscosity in the hadronic phase could not resolve this tension: in Shen:2011kn two of us found that the RHIC Au+Au hadron spectra are insensitive to a temperaturedependent increase of the shear viscosity in the hadron gas phase, as was previously seen in Niemi:2011ix . Figs. 2 and 3 demonstrate that this problem is largely resolved when using the data (Fig. 3) instead of (see Fig. 8 further below): We obtain an excellent description of the differential elliptic flow, together with an acceptable description (within large experimental uncertainties) of the spectra.
Overall, the viscous fluid dynamic description of the hadron spectra and charged hadron elliptic flow shown here is of similar quality as the hybrid model description with VISHNU presented in Song:2010mg . Since purely hydrodynamic simulations are numerically much less costly than calculations with VISHNU, we will now use them to generate a broad range of predictions for soft hadron production in Pb+Pb collisions at the LHC.
Iii Predictions for Pb+Pb collisions at the LHC
As discussed above, the extrapolation from RHIC to LHC is done keeping and fixed. When comparing the resulting viscous hydrodynamic predictions with experimental data from the recently started LHC heavyion collision program, we will search for indications from experiment that would motivate changing these parameters. First results for spectra Aamodt:2010jd as well as both the differential and integrated elliptic flow of unidentified charged hadrons Aamodt:2010pa have already been published and will be compared with the theoretical predictions below. Additional experimental information on spectra and elliptic flow of identified hadrons will become available soon; the relevant hydrodynamic predictions are presented in this section.
In Fig. 4 we show the transverse momentum spectra for all charged hadrons, as well as for identified pions and protons, for minimum bias collisions of Au+Au at RHIC and Pb+Pb at the LHC.^{6}^{6}6To simulate minimum bias collisions, we compute the spectra for the centrality classes shown in Figs. 2(b) and 6 and average them. Any additional observables, such as the minimum bias elliptic flow in Fig. 8 below, are calculated from these minimum bias spectra. For RHIC we compare with data from the PHENIX Collaboration Adler:2003cb . The upper and lower bounds of the shaded areas are predictions for minimum bias Pb+Pb collisions at collision energies of 5.5 and 2.76 TeV per nucleon pair, respectively. The LHC spectra are visibly flatter than at RHIC energies, reflecting stronger radial flow. For central collisions ( centrality), the fireball lifetime increases from Au+Au at RHIC to Pb+Pb at LHC by about 19% and 24%, respectively, for 2.76 and TeV collision energy; for peripheral collisions at centrality, the corresponding lifetime increases are even larger (34% and 41%, respectively). The average radial flow velocity increases in central collisions ( centrality) by 5 and 7%, respectively, and in peripheral collisions ( centrality) by 9 and 11%.
Figure 5 shows the integrated charged hadron elliptic flow as a function of collision centrality for Au+Au collisions at RHIC and Pb+Pb collisions at the LHC. At RHIC energy, our results (lower red line) overestimates the STAR data by about 11% in midcentral collisions, but agrees nicely with except for the most peripheral collisions.^{7}^{7}7In very peripheral collisions, the fireball lifetime decreases dramatically, cutting short the buildup of anisotropic hydrodynamic flow and thereby prohibiting from saturating. In addition, viscous effects are stronger in the small fireballs created in peripheral collisions than in the larger central collision fireballs. Both effects together cause the theoretical values to decrease sharply at large collision centralities, in apparent conflict with the experimental data. The experimental and measurements are, however, contaminated by nonflow effects, in particular in very peripheral collisions. Once nonflow effects are corrected for :2011vk , the experimental values decrease at large collision centralities much in the same way as predicted by hydrodynamics. At first sight the overprediction of the integrated at RHIC is surprising, given the excellent description of the differential elliptic flow shown in Fig. 3. The apparent paradox is resolved by observing that the hydrodnamically computed charged hadron spectra shown in Fig. 2 are somewhat harder than measured, thereby giving too much weight in the integral to the range GeV/ where is large.^{8}^{8}8The agreement with the data is fortuitous and should, in fact, not happen since the measured includes a positive contribution from eventbyevent fluctuations Ollitrault:2009ie while our hydrodynamic calculation yields the average elliptic flow which is smaller.
At LHC energy ( TeV) our integrated lies between and values measured by the ALICE Collaboration Aamodt:2010pa . Again, we overpredict the integrated by about . We note that from RHIC to LHC the hydrodynamically computed integrated in midcentral to midperipheral collisions increases by about 30%, in agreement with the experimental observations. This is due to reduced viscous suppression effects in the larger and denser fireballs created at the LHC and a longer fireball lifetime which allows the momentum flow anisotropy to approach saturation more closely than at lower energies Kolb:2003dz ; Hirano:2007xd . In very peripheral collisions, even at LHC energies such a saturation of does not happen; this is the reason why in Fig. 5 the integrated is seen to decrease at large collision centralities, both at RHIC and LHC.
In Fig. 6 we present hadron transverse momentum spectra for Pb+Pb collisions at LHC energies, for a range of collision centralities. In panel (a) we compare the hydrodynamic predictions with first data from the ALICE experiment Aamodt:2010jd . Overall, the theoretical description of these experimental data is of similar quality as for Au+Au collisions at RHIC (see Fig. 2). In the most central collisions, the hydrodynamical model describes the charged hadron spectrum somewhat better than at RHIC, whereas in the very peripheral collisions the hydrodynamic spectra are too flat, presumably due to large viscous shear pressure effects. Future comparison with the measured spectra at other collision centralities and for identified hadrons, shown here in panels (b) and (c) as predictions, should shed further light on the origin of the discrepancy in peripheral collisions.
Figure 7a shows a comparison of the hydrodynamically generated differential for charged hadrons with the ALICE data Aamodt:2010pa , for four different collision centralities. For the most peripheral of these, we also show the measured for comparison. The hydrodynamic predictions agree nicely with the data at low GeV/, but overshoot the experimental values by at larger , especially in the more peripheral bins. In the centrality bin, the theoretical prediction happens to agree nicely with even though the latter should be shifted upward by flow fluctuations that are not included in the theoretical calculation. We note that the theoretical overshoot is less severe in the VISHNU hybrid model (see Fig. 3 in Song:2011qa ) than in the purely hydrodynamic simulations shown here. This suggests that the excess of over the measured values at GeV/ in Fig. 7a may be caused by an inadequate description of the late hadronic stage and its freezeout.
We can summarize Figs. 2a, 3, 5, 6, and 7a by observing that the hydrodynamic model overpredicts the integrated charged hadron by at both RHIC and LHC, but for different reasons: at RHIC the differential elliptic flow is correctly reproduced while the inverse slope of the theoretical spectra is slightly too large, while the LHC spectra are described a bit better (at least in the most central collisions where published data are available) but the slope of at the LHC is slightly overpredicted.
Panels (b) and (c) of Fig. 7 give predictions for the differential of identified pions and protons. Please note the different shape of the proton from that of the pions at low : radial flow pushes the proton elliptic flow to larger values of . Comparing the curves for and TeV, we see that this “radial push” of the proton increases with collision energy, so for higher the rise of is shifted to larger transverse momenta, while at fixed GeV/ the proton elliptic flow decreases with increasing collision energy. This happens only for heavy hadrons but not for the much lighter pions (see panel (b)).
In Figs. 8 and 9 we pursue this theme further, by directly comparing the differential elliptic flows at RHIC and LHC energies. In Fig. 8 we show results for minimum bias collisions; the RHIC predictions are compared with available data from STAR Adams:2004bi . We see that at low , the elliptic flow for unidentified charged hadrons (which are strongly pion dominated) and for identified pions increases from RHIC to LHC whereas the opposite is true for protons. At higher ( GeV/), on the other hand, increases for both pions and protons as we increase the collision energy. Fig. 9 shows this for a few more hadron species, for the and centrality bins: the heavier the hadron, the stronger a push of towards higher is observed. At sufficiently large , is larger at LHC than at RHIC for all particle species, but at low this holds only for pions whereas all heavier hadrons show a decrease of from RHIC to LHC at fixed . As the hadron rest mass grows, the crossing point where the decrease of at fixed with rising collision energy turns into an increase shifts to larger values. In view of Fig. 9, the experimental observation Aamodt:2010pa that for unidentified charged hadrons hardly changes at all from RHIC to LHC appears accidental:^{9}^{9}9Contrary to the claim made in Lacey:2010ej , the observation that the ratio between measured at LHC and at RHIC is approximately independent of cannot be directly used to conclude that does not change from RHIC to LHC. If that argument were correct, this ratio should be independent of not only for the sum of all charged hadrons, but also for each identified hadron species separately. Our hydrodynamic calculations show that the latter does not hold even if remains unchanged from RHIC to LHC. The increase of at fixed for pions is balanced by a corresponding decrease for all heavier hadrons leaving, as it happens, no visible net effect once all charged hadrons are lumped together.
In Refs. Song:2010mg it was argued that a robust method for extracting the QGP shear viscosity is to fit the collision centrality dependence of the eccentricityscaled charged hadron elliptic flow with a viscous hydrodynamic + hadron cascade hybrid code. In that study it was found that, at fixed collision energy,^{10}^{10}10We recently checked that this multiplicity scaling carries over to other collision systems such as Cu+Cu at the same collision energy SSH . plotting against the charged hadron multiplicity density per unit overlap area, , yields “universal” curves that depend only on the QGP shear viscosity but not on the model for the initial energy density distribution (in particular its eccentricity). In Fig. 10 we show such a plot for GeV Au+Au collisions at RHIC together with Pb+Pb collisions at two LHC energies. The four panels show this scaling in terms of distributions in pseudorapidity (, left column) or rapidity (, right column), and also compare it for our default choice of using the initial energy density as weight for the calculation of the average eccentricity and overlap area (top row) with what one obtains by evaluating these quantities with the initial entropy density instead (as is done in Refs. Hirano:2010jg ; Song:2010mg ) (bottom row). We see that, independent of these choices of representation, the universality of vs. or does not carry over to different collision energies (at least not for the purely hydrodynamic simulations studied in the present work): At the same multiplicity density or , more peripheral higher energy collisions produce less elliptic flow per initial eccentricity than more central lower energy collisions. At fixed , the difference between GeV Au+Au and TeV Pb+Pb collisions (red circles vs. green upward triangles in Fig. 10) is as large as the difference between and for Au+Au collisions at fixed GeV (red circles vs. black squares).
We note that the tendency in Fig. 10 of higher energy collisions producing less at fixed than lower energy ones contradicts the opposite tendency observed by Hirano et al. in Fig. 3 of Ref. Hirano:2010jg where an ideal hydro + hadron cascade hybrid code was employed.^{11}^{11}11The careful reader will notice that for 200 GeV Au+Au collisions, our maximal values for in Fig. 10b are significantly larger than those shown in Fig. 3 of Ref. Hirano:2010jg . This is due to a lower normalization of the initial entropy density in Hirano:2010jg , corresponding to instead of our in central Au+Au collisions (T. Hirano, private communication). The authors of Hirano:2010jg presented strong arguments that their observation of larger at fixed in higher energy collisions is not related to their use of a hadron cascade for describing the late hadronic stage. Our opposite finding, on the other hand, is supported by the earlier purely hydrodynamic scaling studies presented in the last two works of Song:2007fn whose authors came to the same conclusion as we do here. At present this discrepancy remains unresolved; we suspect, however, that the origin of the difference between our work and that of Hirano et al. could be in their use of a more realistic (3+1)dimensional hydrodynamic evolution Monnai:2011ju , although in the earlier ideal fluid hydrodynamical studies at the full RHIC energy, the differences between boost invariant and nonboost invariant results were small Heinz:2001xi ; Hirano:2002ds . Possible consequences of the violation of boostinvariance in RHIC and LHC heavyion collisions are presently being studied SSH .
Before moving on, let us comment on the different shape at the highmultiplicity end of the curves shown in Fig. 10 when using entropy instead of energy density as the weight for calculating the initial eccentricity overlap area : It is caused by the different centrality dependence of the energy and entropy density weighted eccentricities in nearcentral collisions observed in Ref. Qiu:2011iv whose authors showed that in the most central collisions (where is dominated by eventbyevent shape fluctuations) the entropyweighted participant eccentricity decreases faster with decreasing impact parameter than the energyweighted one.
Iv Temperature dependent
Shear viscosity is known to suppress the buildup of elliptic flow. Naively, the systematic overprediction of in Pb+Pb collisions at the LHC seen in Fig. 7a, together with the excellent description of the same quantity in Au+Au collisions at RHIC seen in Fig. 3, thus suggests that the fireball liquid formed in LHC collisions might be slightly more viscous (i.e. possess larger average ) than at RHIC energies Niemi:2011ix ; Song:2011qa . In this section we present some results using a temperature dependent specific shear viscosity, , that were motivated by such considerations.
Figure 11 illustrates the following three trial functions explored in this section:
(1)  
(2)  
(3) 
Here MeV is the chemical decoupling temperature and stands for the “transition temperature” at which the hadronization of quarks and gluons is complete.
In principle, the value of should exhibit a minimum near and increase again in the hadronic phase below Csernai:2006zz ; Chen:2007jq ; Kapusta:2008vb . The authors of Niemi:2011ix pointed out, however, that at the full LHC collision energy of 5.5 TeV the behavior of at temperatures below has very little effect on the final hadron spectra and their elliptic flow. At 2.76 TeV the effect on elliptic flow was moderate, and negligible on the spectra. Here we will concentrate on qualitative aspects of effects arising from a temperature dependent growths of in the high temperature region that can be explored at LHC energies but is beyond the reach of RHIC, and continue to set at for simplicity.
As pointed out in Niemi:2011ix , the spectra and elliptic flow in Au+Au collisions at RHIC energies are most sensitive to the average value of in the temperature region below 220230 MeV. We have checked that altering at higher temperatures as shown in Fig. 11 has little influence on the results at RHIC energies shown in Sec. II.
Figure 12 illustrates the influence of a linear temperature dependence of as in Eq. (1) on the centrality dependence of charged hadron production. The solid black line is the same as shown in the upper part of Fig. 1 where it forms the lower bound of the shaded region; it corresponds to constant and NavierStokes initial conditions for the shear stress tensor, at fm/. The dashed and dashdotted lines in Fig. 12 use with either NavierStokes (dashed) or zero (dashdotted) initial conditions for . These last two lines were normalized to the ALICE point for the most central Pb+Pb collisions ( Aamodt:2010pb ), whereas the black line was normalized to our best guess before the ALICE data became available (, corresponding to ). The centrality dependence is then controlled by the predictions from the MCKLN model, modified by viscous entropy production during the hydrodynamic evolution.
We see that even a relatively modest temperature dependent increase of in the QGP phase leads to a significantly stronger nonlinearity in the dependence of charged particle production on the number of wounded nucleons. The reason is that an increase of with temperature leads to more viscous heating in central collisions (which probe higher initial temperatures and such larger effective shear viscosities) than in peripheral ones (whose initial temperatures are lower). Since the entropy production rate is given by
(4) 
this effect is stronger for NavierStokes initial conditions (where is proportional to the velocity shear tensor which at early times diverges like ) than for zero initial shear stress (where starts from zero and approaches its NavierStokes value only after several relaxation times when, due to its decay, it has already decreased to much smaller values).^{12}^{12}12For reference we list the fractions of the finally measured entropy in the most central and most peripheral centrality bins shown in Fig. 12 that are generated by viscous heating during the hydrodynamic expansion: Constant : () and 33% (); with : () and 15% (); with : () and 49% ().
If one were to postulate the validity of the MCKLN model as the correct description of the initial particle production, the ALICE data shown in Fig. 12 would exclude a temperature dependence of as given in Eqs. (1) and (2) for NaverStokes initial conditions. While we are not prepared to make such a statement on the basis of Fig. 12 alone, we believe that it is important to point out this relatively strong sensitivity of the centrality dependence of to the transport properties of the expanding fireball medium and to emphasize the constraints it thus places on possible models for the QGP shear viscosity.
We now turn to the discussion of the influence of a possible temperature dependence of on the charged hadron spectra and elliptic flow. Figure 13 shows LHC predictions for TeV Pb+Pb collisions of centrality. To ensure comparability of the different cases studied in this figure we simply normalized the initial entropy density profile such that we always obtain , i.e. the same value that we had obtained before for constant at this centrality. We first note that for constant , we don’t observe any significant difference in the charged hadron spectra and elliptic flow between zero and NavierStokes initialization for . Turning to the temperaturedependent parametrizations , we note that for zero initialization of (solid lines) our results agree with those reported in Niemi:2011ix : An increase of at higher QGP temperatures leads to somewhat harder charged hadron spectra (i.e. somewhat stronger radial flow, caused by the larger tranverse effective pressure gradients at early times) and a suppression of the differential elliptic flow (due to an increase of the timeaveraged effective shear viscosity of the fluid). It is interesting to observe the hierarchy of the curves in Fig. 13 corresponding to the three parametrizations (1)–(3): For the spectra, all three dependent viscosities lead to almost identical hardening effects on the spectral slope, while for the differential elliptic flow the curves are ordered not according to the values at the initial central fireball temperature (see Table 1), but according to their hierarchy in the suppression for the three functions suggest that, at this beam energy and collision centrality, the buildup of elliptic flow is dominated by the QGP transport properties at MeV dominate the generation of Niemi:2011ix .) MeV. (At RHIC energies, the transport properties for MeV range. In fact, the observed magnitudes of the viscous
model  (fm)  (MeV)  
















For NavierStokes initial conditions (dashed lines in Fig. 13), the increase in radial flow caused by an increase of at high temperature is stronger and the viscous suppression is weaker than for zero initial . This is caused by the much larger initial shear stress tensor components in the NS case, compared to the case of where approaches its (by that time already much smaller) NavierStokes limit only after several relaxation times Song:2007fn . The increase of with temperature generates a steeper initial transverse effective pressure gradient (since grows faster than the entropy density when increases with temperature), and this generates stronger radial flow. It also causes a larger spatial eccentricity of the initial effective pressure profile which (when compared to the case of ) generates stronger elliptic flow. In fact, we found that for earlier starting times (where the NavierStokes values for are even larger), the quadratic parametrization with NS initial conditions can lead to more elliptic flow than a constant , in spite of the larger mean viscosity of the fluid.
We conclude from this exercise that a firm determination whether or not the ALICE data point towards a temperaturedependent growth of with increasing , as expected from perturbative QCD Arnold:2003zc and (perhaps) from lattice QCD Meyer:2007ic , is not possible without a better understanding of the initial conditions for the energy momentum tensor (in particular the shear stress components) at the beginning of the hydrodynamic evolution. Whereas generically larger viscosities cause a suppression of the elliptic flow, temperaturedependent viscosities can influence the initial effective pressure profile and its eccentricity in a way that counteracts this tendency and, for some models such as NavierStokes initial conditions, can even overcompensate it.
V Conclusions
Based on an successful global fit of soft hadron production data in GeV Au+Au collisions at RHIC with a pure viscous hydrodynamic model with CooperFrye freezeout, presented in Sec. II, we generated hydrodynamic predictions for the spectra and differential elliptic flow of unidentified charged hadrons and identified pions and protons for Pb+Pb collisions at the LHC. Where available, these predictions were compared with available experimental data from the ALICE Collaboration. Our extrapolation from RHIC to LHC energies was based on the assumption that the QGP shear viscosity does not change with increasing fireball temperature and stays fixed at the value extracted from the RHIC data, assuming MCKLN initial conditions. The start time for the hydrodynamic evolution and the freezeout temperature were held fixed, too. We found that, using the beam energy scaling implicit in the MCKLN model, such an extrapolation gives a good description of the centrality dependence of charged hadron production and the charged hadron spectra in central PbPb collisions, but overpredicts the slope of the differential elliptic flow and the value of its integrated value by about in midcentral to midperipheral collisions. In the most peripheral collisions, the predicted charged hadron spectra are too flat, and the integrated elliptic flow is too small compared to the experimental data. A preliminary study of possible temperature dependent variations of in the hightemperature region explored for the first time at the LHC remained unconclusive but pointed to a clear need for better theoretical control over the initial conditions for the hydrodynamic energymomentum tensor, in particular its shear stress components. The development of detailed dynamical models for the prethermal evolution of the collision fireball and their matching to the viscous hydrodynamic stage is a matter of priority for continued progress towards quantifying the transport properties of the quarkgluon plasma at different temperatures and densities.
Acknowledgements.
We would like to thank R. Snellings and A. Tang for providing us with tables of the experimental data from the ALICE experiment and for helpful discussions. This work was supported by the U.S. Department of Energy under contracts DEAC0205CH11231, DEFG0205ER41367, DESC0004286, and (within the framework of the JET Collaboration) DESC0004104. P.H.’s research was supported by the ExtreMe Matter Institute (EMMI) and by BMBF under contract no. 06FY9092.References
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