AA Summary of Bulk Dynamics from Quark Matter 2009
Derek Teaney a , b a Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA b RIKEN-BNL Research Center, Building 510A, Physics Department, Brookhaven National Laboratory, Upton, NY11973-5000, USA
Abstract
I review the recent progress in measuring elliptic flow in heavy ion collisions. These measure-ments show clearly how hydrodynamics starts to develop as the system size is increased fromperipheral to central collisions. During this transition, the momentum range described by hydro-dynamics increases as the system progresses from a kinetic to a hydrodynamic regime. Many ofthe systematic deviations from ideal hydrodynamics are reproduced e ff ortlessly once the shearviscosity is included. In order to extract the shear viscosity from the data, kinetic theory can beused to determine which aspects of the elliptic flow reflect the details of the microscopic interac-tions, and which aspects reflect the underlying transport coe ffi cients. I also review the identifiedhadron elliptic flow and the predictions of hydrodynamics for the LHC.
1. Overview
Perhaps the most important result from the Relativistic Heavy Ion Collider is the observationof strong elliptic flow [1, 2]. Elliptic flow is an asymmetry of particle production with respectto the reaction plane and has been measured as a function of transverse momentum, rapidity,and particle type. The interpretation of the observed flow which has been adopted by the heavyion community is that the elliptic flow is the hydrodynamic response to the collision geometry.Implicit in this interpretation of the observed flow is that the time scale for momentum relaxationnear the QCD phase transition is of order the quantum time scale τ R ∼ (cid:126) π T . (1)This estimate for the relaxation time is best expressed in terms of the shear viscosity to entropyratio. For instance, in the viscous Bjorken model the energy density at time τ o evolves as ded τ = − e + P τ o + ητ o , (2)where e is the energy density, P is the pressure, and η is the shear viscosity [3]. Comparing thesize of the viscous term to the ideal term, we conclude that hydrodynamics will provide a gooddescription of the observed flow when η ( e + P ) τ o (cid:28) . (3) Preprint submitted to Nuclear Physics A October 28, 2018 a r X i v : . [ nu c l - t h ] O c t sing e + P = sT , and an estimate for the temperature and τ o , this criterion reads0 . (cid:32) η/ s . (cid:33) (cid:32) τ o (cid:33) (cid:32)
300 MeV T o (cid:33) (cid:28) . (4)From this estimate we see that hydrodynamics will begin to be a good approximation for η/ s < ∼ . ffi cients of QCD.Many of the systematic trends seen in the elliptic flow data do support the hydrodynamicinterpretation of the observed flow. For example, as a function of centrality and transverse mo-mentum, the measured elliptic flow deviates from ideal hydrodynamics in a way characteristic ofviscosity. These experimental trends have been fully clarified only recently and the experimentalanalysis is now rather sophisticated. These developments were reported on at the Quark Matterconference and are reviewed in Section 2. The systematics of the recent flow measurements giveconfidence in the overall picture of the hydrodynamic expansion.Although these experimental trends support the notion of a hydrodynamic response, thereare several puzzling patterns in the elliptic flow data. For example, new data on the elliptic flowof the φ meson and the Ω − baryon are reviewed in Section 2. The di ff erences in the measuredflow between mesons and baryons is generally explained with the coalescence model, whichenjoys considerable phenomenological success. However, the coalescence model is theoreticallyunsatisfactory and is di ffi cult to realize in a dynamical model. The seemingly simple coalescencetrends seen in the elliptic flow data must be understood before the shear viscosity and othertransport properties can be reliably extracted from the heavy ion data.In addition to experimental progress, there has been substantial theoretical progress in classi-fying the form of viscous corrections, both with viscous hydrodynamics and with kinetic theory.A brief summary of some of the developments discussed at Quark Matter 2009 is presentedin Section 3. Most of these ideas presented in this summary are reviewed more completely inRef.[2], which was written shortly after the Quark Matter conference. Some sections from thislonger review have been copied for this brief summary.
2. Measurements
One of the best ways to test the hydrodynamic interpretation is to systematically observehow the response changes from small systems to large systems. Experimentally this can beaccomplished by colliding small nuclei such as CuCu and selecting peripheral collisions. Unfor-tunately measuring elliptic flow in these smaller systems is rather di ffi cult, and reliable, preciseresults have become available only recently.The di ffi culty in measuring flow in small systems stems from fluctuations. Especially inperipheral AuAu and CuCu collisions, there are fluctuations in the initial eccentricity of theparticipants. Thus rather than using the continuum approximation to categorize the geometry, itis better to implement a Monte-Carlo Glauber calculation and estimate the eccentricity using the“participant plane eccentricity”. This event by event eccentricity is denoted (cid:15) PP in the literature.Clearly the experimental goal is to extract the response coe ffi cient C relating the elliptic flow tothe eccentricity on an event by event basis v = C (cid:15) PP . (5)2 ! Data Corrected to
Figure 2: Elliptic flow v ( p T ) as measured by the STAR collaboration [12, 13] for di ff erent centralities. The measuredelliptic flow has been divided by the eccentricity. The curves are ideal hydrodynamic calculations based on Refs.[14, 15]rather than the viscous hydrodynamics discussed in much of this summary. to a strong dynamic response with growing system size. The interpretation adopted here is thatthis change is a consequence of a system transitioning from a kinetic to a hydrodynamic regime.There are several theoretical curves based upon calculations of ideal hydrodynamics[15, 16]which for p T < v /(cid:15) of this theory should be independent of system size orcentrality. This reasoning is borne out by the more elaborate hydrodynamic calculations shownin the figure. On the other hand, the data show a gradual transition as a function of increasingcentrality, rising towards the ideal hydrodynamic calculations in a systematic way. These trendsare captured by models with a finite mean free path[17].The data show other trends as a function of centrality. In more central collisions the linearlyrising trend, which resembles the ideal hydrodynamic calculations, extends to larger and largertransverse momentum. Viscous corrections to ideal hydrodynamics grow as (cid:18) p T T (cid:19) (cid:96) mfp L , (6)where L is a characteristic length scale. Thus these viscous corrections restrict the applicablemomentum range in hydrodynamics [18]. In more central collisions, where (cid:96) mfp / L is smaller,the transverse momentum range described by hydrodynamics extends to increasingly large p T .These qualitative trends are reproduced by the more involved viscous calculations [2].While many of the trends seen in Fig. 1 and Fig. 2 are reproduced and understood withviscous hydrodynamics, there are additional trends in the elliptic flow data which are only par-tially understood. For instance Fig. 3(a) shows the elliptic flow of identified particles π, K , p .4 he NCQ scaling is broken at KE T /n q ~1GeV. Different mechanism of recombination for pions and protons at intermediate p T ? Au + Au at 200 GeV, Run 2007 S. Huang, DNP 2008
Figure 3: (a) The elliptic flow as a function of transverse momentum for identified particles as measured by the PHENIXcollaboration [19]. (b) The elliptic flow of identified hadrons rescaled according to the quark coalescence model andplotted as a function of KE T = (cid:113) p T + M − M . At low momenta the separation amongst the di ff erent particle species is well reproduced by hy-drodynamics. However as the momentum is increased the proton elliptic flow equals, and thenexceeds, the pion elliptic flow. These systematic trends are seen in all collision systems andcentralities. The prevailing explanation is that constituent quarks coalesce to form hadrons at thephase boundary. This ansatz is supported by the observation that if the hadron momentum and v is divided by the valence quark content all of the v of the di ff erent hadron species lie along a sin-gle curve. This is illustrated in Fig. 3(b) which is plotted as a function of KE T = (cid:113) p T + M − M rather than transverse momentum to capture the hydrodynamic behavior at small momentum. Al-though constituent quark scaling works rather well, the theoretical support for quark coalescenceis small since it is di ffi cult to realize a coalescence mechanism in a dynamical simulation. Itnevertheless remains to find an alternative picture for the observed di ff erent flows of mesons andbaryons. At the Quark Matter conference the elliptic flow of identified particles was measuredaccurately out to rather large transverse momenta. At su ffi ciently large momenta the data deviatefrom a universal coalescence curve providing new insight into the hadronization dynamics in thisregion.To conclude this section, we turn to Fig. 4 which compares the elliptic flow protons and pionsto the flow of the multi-strange hadrons Ω − and φ . The important point is that the Ω − is nearlytwice as heavy as the proton and more importantly, does not have a strong resonant interactionanalogous to the ∆ . For these reasons the hadronic relaxation time of the Ω − is expected to bemuch longer than the duration of the heavy ion event. Nevertheless the Ω − shows nearly thesame elliptic flow as the protons. This provides fairly convincing evidence that the majority ofthe elliptic flow develops during a deconfined phase which hadronizes to produce a flowing Ω − baryon. 5 A n i s o t r op y P a r a m e t e r v π Ωφ S . S . S h i
200 GeV Au+Au M.B. collisions
STAR Preliminary (GeV/c) T Transverse Momentum p
Figure 4: A comparison of the elliptic flow of pions and protons to the elliptic flow of the multi-strange φ and Ω − [20].
3. Modeling Elliptic Flow with Hydrodynamics and Kinetic Theory
Many of the trends seen in the data (with the possible exception of quark coalescence) arereproduced by hydrodynamics and kinetic theory. Generally the kinetic theory estimates for theshear viscosity are consistent with the estimates from viscous hydrodynamics. Specifically unless η/ s < ∼ . P ( e ), and the shear viscosity and bulk viscosities, η ( e ) and ζ ( e ). In the sensethat kinetic theory provides a reasonable guess as to how the surface to volume ratio influencesthe forward evolution, these models can be used to estimate the shear viscosity, and the estimatemay be more reliable than the hydrodynamic models. More importantly, by comparing the resultsof di ff erent microscopic models one can determine which features of the heavy ion data areuniversal ( i . e . only depend on η ( e ) , P ( e ) and ζ ( e ) ).There were a number of promising e ff orts to reproduce hydrodynamic results from kinetictheory reported at the conference. First there was an e ff ort to reproduce hydrodynamic shockswith kinetic theory by the Frankfurt group. Fig. 5(a) shows a kinetic theory simulation of theshock tube problem. Clearly the BAMPS code is capable of reproducing the correct hydro-dynamic limit in detail. The deviations of the BAMPS code from ideal hydro are beautifullyreproduced by viscous hydro. This gives a great deal of confidence in the BAMPS code and inthe viscous hydrodynamic code vSHASTA. Additional results from the BAMPS simulation ofelliptic flow were presented in the poster session.A similar approach to the hydrodynamic limit was reported by Huovinnen and Molnar [21],and Gombeaud [11]. In particular Fig. 5(b) shows a simulation of the MPC code which alsomakes a direct comparison with viscous hydrodynamics for the Bjorken expansion of an idealmassless gas with constant cross section. As the Knudsen number K ≡ σ/π R dN / dy is in-creased, the simulation approaches Israel-Stewart hydro and ultimately the Navier-Stokes limit.6 larger η/ s value results in a finite transition layer where the quantities change smoothly rather than discontinuously as in the case of a perfect fluid. Furthermore a non-zero viscosity, if large enough, impedes the formation of a shock plateau and a clear separation of the shock front from the rarefaction fan. P / P z [fm]BAMPS ! /s = 0.01vSHASTA ! /s = 0.01BAMPS ! /s = 0.1vSHASTA ! /s = 0.1 0.00.10.20.30.40.50.6 -3 -2 -1 0 1 2 3 v z [fm] Figure 2: (Color online) Same as in Fig. 1. Results are obtained using BAMPS and vSHASTA.
In Fig. 2 we compare the results from BAMPS and vSHASTA for η/ s = .
01 and 0 .
1. We see a perfect agreement for η/ s = .
01, whereas for larger value of η/ s = . the region of the shock front and rarefaction wave are found. The reason for the di ff erence is that in these regions the local Knudsen number K θ = λ mfp ∂ µ u µ [14] is large and thus the applicability of IS equations is questionable. Transport calculations do not su ff er from that drawback.
4. Time scale of formation of shock waves P / P z (fm) ! /s = 0.1 initial conditiont = 0.64 fm/ct = 3.20 fm/ct = 6.40 fm/c 0.00.10.20.30.40.5 -6 -4 -2 0 2 4 6 v z (fm) Figure 3: (Color online) Same as in Fig. 1. Results are obtained using BAMPS for η/ s = . The formation of a shock wave takes a certain amount of time, as demonstrated in Fig. 3 for η/ s = .
1. At early times a shock has not yet developed, the profile looks like a free streaming of particles. But at later times we observe the creation of a shock plateau. Formally we define the time of formation of the shock plateau when the maximum of the velocity distribution v ( z ) reaches the value v plat of the ideal-fluid solution in Fig. 1. From the right panel of Fig. 3, we see that this happens at t = . / c. PASI HUOVINEN AND DENES MOLNAR PHYSICAL REVIEW C , 014906 (2009) Navier-StokestransportIS hydro η/s ≈ const ideal hydro K = 1 K = 2 K = 3 K = 6 . τ/τ p L / p T σ = const ideal hydro K = 1 K = 2 K = 3 K = 6 . K = 20 τ/τ p L / p T FIG. 2. Same as Fig. 1, but for an initial pressureanisotropy R p ( τ ) = .
476 ( ξ = − . σ = const scenario, the NS curve for K = for the σ = const scenario. For K =
1, the anisotropy from IShydro starts to fall rapidly below the transport above τ > ∼ τ ,and it is a factor of ∼ τ ∼ τ . Clearly,the system cannot stay near equilibrium when the rate ofscatterings equals the expansion rate. With increasing K ,the undershoot becomes smaller and gradually vanishes as K → ∞ . The difference is only ∼
10% already at K = K ≈ η s /s eq ≈ const. The situation of courseimproves because in this case K increases with time. For K =
1, IS hydro undershoots the pressure anisotropy fromthe transport only by ∼ R p = τ → ∞ ). About 10% accuracy is achieved already for K = K =
3, IS hydro is accurate to a few percent.Moreover, the above findings hold for a wide range ofinitial conditions, including large initial pressure anisotropies,as shown in Figs. 2 and 3. These figures are for the samecalculation but with R p ( τ ) = .
476 and 1 . ξ = − .
423 and 0 . ∼
10% accuracy) provided K > ∼
3, even for the most pessimistic constant cross sectionscenario. If η s /s eq = const, only K > ∼ ξ ( τ ) = R IS p ( τ ) = − τ − τ )3 τ + O (( τ − τ ) ) (64) for any initial value and evolution scenario for κ . Fromcovariant transport, on the other hand (see Appendix D2), R transp p ( τ ) = − τ − τ )5 τ + O (( τ − τ ) ) . (65)That is, pressure anisotropy develops, universally, 20% fasterfrom the transport than from IS hydrodynamics (if theevolution starts from equilibrium).This illustrates a limitation of the hydrodynamic descriptionof transport solutions. Similar discrepancies were observedin Ref. [8] in the early evolution of differential elliptic flow Navier-StokestransportIS hydro η/s ≈ const ideal hydro K = 1 K = 2 K = 3 K = 6 . τ/τ p L / p T σ = const ideal hydro K = 1 K = 2 K = 3 K = 6 . K = 20 τ/τ p L / p T FIG. 3. Same as Fig. 1, but for an initial pressureanisotropy R p ( τ ) = .
693 ( ξ = . σ = const scenario, the Navier-Stokes curve for K = Figure 5: (a) The pressure relative to the initial pressure for shock tube initial conditions. The three curves are for theBAMPS parton cascade model, viscous hydrodynamics (vSHASTA), and ideal hydrodynamics [22]. (b) The ratio oflongitudinal and transverse pressures in kinetic theory, Israel Stewart hydrodynamics, and the Navier-Stokes equations,for a Bjorken expansion of a massless ideal gas with constant cross section. The simulations are compared as a functionof Knudsen number, K ≡ σ/π R dN / dy [21]. Given that the kinetic code and the viscous hydrodynamic simulations agree reasonably, the MPCcode can be used to reliably extract the shear viscosity from the heavy ion data.When viscous hydrodynamics is extended to a second order there are additional relaxationtimes (e.g. τ π ) beyond the shear viscosity, η ( e ). Just as transport models should be approximatelyindependent of the details of the microscopic interactions, results from viscous hydrodynamicsshould be approximately independent of the precise way in which the second order terms are im-plemented. This is indeed the case [23, 24, 25]. Additional results from viscous hydrodynamicswill be discussed more completely by P. Romatschke in this volume [26].
4. Outlook
Clearly there is a strong convergence between kinetic and hydrodynamic simulations of heavyion reactions. These simulations reproduce many trends observed in increasingly precise mea-surements of elliptic flow. This convergence strongly suggests that the hydrodynamic interpre-tation of the observed flow is correct. One of the striking tests of hydrodynamic predictions isthe saturation of elliptic flow at high energy. From RHIC to the LHC, hydrodynamics predictsan increase in the flow which is significantly less than a naive extrapolation from lower energies.This is illustrated in Fig. 6 and will be one of the first tests of the hydrodynamic paradigm at theLHC.
Acknowledgments
This summary was a result of extensive discussions at the Quark Matter meeting with PeterArnold, Aihong Tang, Kevin Dusling, Raimond Snellings, Paul Romatschke, Peter Petreczky,Mikko Laine, Ste ff an Bass, Vincenzo Greco, Denes Molnar, Fuqiang Wang, Sergei Voloshin.This work was supported in part by by the U.S. Department of Energy under an OJI grant DE-FG02-08ER41540 and as a RIKEN and Sloan Fellow.7 igure 6: Figure presented by W. Busza providing an estimate for the elliptic flow at the LHC by extrapolating trendlines from lower energy to higher energy (solid black lines) [27]. The dashed red lines shows an estimate based on idealhydrodynamics [28]. References [1] See for example, S. A. Voloshin, A. M. Poskanzer and R. Snellings, arXiv:0809.2949 [nucl-ex].[2] D. A. Teaney, “Viscous Hydrodynamics and the Quark Gluon Plasma,” arXiv:0905.2433 [nucl-th]; prepared for“Quark-gluon plasma. Vol. 4,” eds. R.C. Hwa, X.N. Wang.[3] P. Danielewicz and M. Gyulassy, Phys. Rev. D , 53 (1985).[4] B. Alver et al. [PHOBOS Collaboration], Phys. Rev. Lett. , 242302 (2007) [arXiv:nucl-ex / ff ect on elliptic flow arXiv:nucl-ex / , 260 (2006) [arXiv:nucl-th / , 537 (2008) [arXiv:0708.0800 [nucl-th]].[9] B. Alver et al. , Phys. Rev. C , 014906 (2008) [arXiv:0711.3724 [nucl-ex]].[10] H. J. Drescher, A. Dumitru, C. Gombeaud and J. Y. Ollitrault, Phys. Rev. C , 024905 (2007) [arXiv:0704.3553[nucl-th]].[11] C. Gombeaud these proceedings; see also C. Gombeaud, T. Lappi and J. Y. Ollitrault, arXiv:0907.1392 [nucl-th].[12] The data presented are from the thesis Dr. Yuting Bai.[13] See for example, B. I. Abelev et al. [STAR Collaboration], Phys. Rev. C , 054901 (2008) [arXiv:0801.3466[nucl-ex]].[14] P. Huovinen and P. V. Ruuskanen, Ann. Rev. Nucl. Part. Sci. , 163 (2006) [arXiv:nucl-th / , 58 (2001) [arXiv:hep-ph / , 232 (2001) [arXiv:hep-ph / et al. , “Heavy IonCollisions at the LHC - Last Call for Predictions,” J. Phys. G , 054001 (2008).[18] D. Teaney, Phys. Rev. C , 034913 (2003) [arXiv:nucl-th / et al. [PHENIX Collaboration],Phys. Rev. Lett. , 162301 (2007) [arXiv:nucl-ex / , 014906 (2009) [arXiv:0808.0953 [nucl-th]].[22] I. Bouras et al. , Phys. Rev. Lett. , 032301 (2009) [arXiv:0902.1927 [hep-ph]].[23] M. Luzum and P. Romatschke, Phys. Rev. C , 034915 (2008) [arXiv:0804.4015 [nucl-th]].[24] K. Dusling and D. Teaney, Phys. Rev. C , 034905 (2008) [arXiv:0710.5932 [nucl-th]].[25] H. Song and U. W. Heinz, Phys. Rev. C , 064901 (2008) [arXiv:0712.3715 [nucl-th]].[26] P. Romatschke, these proceedings; see also Ref.[23].[27] W. Busza these proceedings, arXiv:0907.4719 [nucl-ex].[28] D. Teaney, J. Lauret and E. V. Shuryak, arXiv:nucl-th / ibid , Phys. Rev. Lett. , 4783 (2001), 4783 (2001)