aa r X i v : . [ nu c l - e x ] A ug Proc. 24th Winter Workshop onNuclear Dynamics (2008) 000–000 ¯ on Nuclear Dynamics ¯¯ South Padre, Texas,USAA¯ pril 5–12, 2008
Flow Results and Hints of Incomplete Thermalization
Aihong Tang for the STAR Colllaboration Brookhaven National Lab, PO BOX 5000, Upton, NY. USA
Abstract.
We classified v measurements according to their sensitivities w.r.t.to two planes, namely, reaction plane and participant plane. Likewise, in v /ǫ scaling, we showed that one needs to choose a ǫ that is sensitive to the sameplane as that v is sensitive to. We presented our v /ǫ as a function of centralityand transverse momentum. We studied the ratio of v /v . We discussed theapplicable range for hydrodynamics, as well as implications to an incompletethermalization. Keywords: flow, v , hydrodynamics, thermalization, RHIC, STAR PACS:
1. Introduction
In ultra-relativistic heavy ion collisions, spectators pass through each other quickly,and the system begins its evolution with what is left behind - the overlap regionof two nuclei. The pressure gradient convert the spacial anisotropy, quantified byeccentricity ǫ , of the overlap region into anisotropy in momentum space, quantifiedby azimuthal anisotropy v . v is defined as the second Fourier coefficient in thedescription of particles distribution w.r.t. the reaction plane [ 1], and it is largelydetermined by the collective motion in the in-plane direction. v at RelativisticHeavy Ion Collider (RHIC) is reported to be large and, for the first time in heavyion collisions, can be described by ideal hydrodynamics [ 2]. Theoretical calculationshows that in order to explain the large v observed at RHIC, one has to assumethat the shear viscosity is extremely small [ 3]. That is one of the important reasonsfor which scientists think that a perfect liquid has been formed in relativistic heavyion collisions [ 4].Since the announcement of the discovery of a perfect liquid, our understandingof the matter created at RHIC has continued to advance. Elliptic flow analyseshave been extended to great details by all four experiments at RHIC. At the sametime, due to different techniques used, it becomes an increasingly amount of workto understand/compare results across different experiments and, sometimes, even A. Tang et al.different analyses within the same experiment. Therefore it becomes important tounderstand what is the relation between various v measurements. For the physicsside, in order to quantify how perfect the liquid is, it is necessary to re-examinethe hydrodynamics limit. In this paper, we try to study the hydrodynamic behav-ior under a more general context, namely, the transport approach which recovershydrodynamics when mean free path is extremely small if compared to the systemsize [ 5]
2. Choosing the right v and ǫ pairs The ratio of v /ǫ reflects how well the initial anisotropy is converted into mo-mentum anisotropy [ 6]. This conversion process is directly affected by dynam-ics of the system, e.g. Equation Of State, thermalization etc. It is importantto measure this quantity as accurate as possible. However, there exist many v measurements, for example, v measured by event plane method ( v { EP } ), by cu-mulants ( v { } , v { } ), by Lee-Yang zero method ( v { LYZ } ), and by using eventplane reconstructed with Shower Maximum Detectors at Zero Degree Calorimeters( v { ZDCSMD } ), etc. Likewise there exist many ǫ calculations. In this section,we will try to make connections between various v and ǫ methods, and make thejustification for the right combination of them.Define ε = { ε x , ε y } = * σ y − σ x σ x + σ y + part , (cid:28) σ xy σ x + σ y (cid:29) part , (1)where σ x = (cid:10) x (cid:11) −h x i , σ y = (cid:10) y (cid:11) −h y i , and σ xy = h xy i−h y i h x i , and the average istaken over the coordinates of the participants in a given event. With this definition, ε x is the eccentricity of reaction plane (defined by the impact parameter), and ε x is also called ε RP . ε y has a distribution centered at zero with finite width. Theparticipant eccentricity measures the asymmetry in the participant plane (definedby the principle axis of the ellipsoid), and is given by ε part = q ε x + ε y ≡ ε P P
When both ε x and ε y has a Gaussian distribution, to the first order this istrue and it is supported by Glauber Monte Carlo simulations, then the probabilitydensity function for ε P P is dndε part = ε part σ ε I (cid:18) ε part h ε RP i σ ε (cid:19) exp − ε part + h ε RP i σ ε ! ≡ BG( ε part ; h ε RP i , σ ε ) , (2)With this p.d.f., one can show that ε part { } = h ε RP i [ 7]. Similarly, under theassumption that v is proportional to the initial system eccentricity (this is nottrue over a broad centrality range, but for a fine centrality bin, it still ensures agood Gaussian for v and s ( ≡ h sin2( φ − Ψ RP ) i ), which is what is required for thederivation of BG formula shown in Eq.( 2) ), one can find that v { } = h v { RP }i [ 7,ints of Incomplete Thermalization 38]. That means that, v { } and ε part { } are measurements sensitive to the reactionplane, not the participant plane. Indeed, that explains the reason that STAR’s v { ZDCSMD } agrees with v { } , as shown in the left panel of Fig.1. v { ZDCSMD } is measured with the first order event plane reconstructed by spectator neutronsthus is more sensitive to v in the reaction plane (not participant plane). Because % Most Central ( % ) v AuAu: 200 GeV S T AR p r e li m i n a r y Standard {2} v {4} v {ZDC-SMD} v max /n ch n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 v event plane method4th-order cumulantq-fit Fig. 1.
Left: v measurements as a function of centrality. This plot is from [ 9].Right: v of Au+Au collisions at √ s NN = 130GeV. This plot is made based ondatapoints from [ 10]the p.d.f. of the magnitude of the flow vector, the q − distribution [ 10], shares analmost identical formula as Eq. (2), the v obtained from fitting the q − distributionshould be equivalent to v { } , as confirmed by experimental data in the right panelof Fig.1.To summarize this section, we find that v { } , v { ZDCSMD } and v { q − dist } are sensitive to anisotropy in the reaction plane, they should be scaled with the stan-dard eccentricity or 4-particle cumulant eccentricity. Other v measurements thatare based on two particle correlations, that includes v { } , v { EP } and v { scalarProduct } [10], etc., should be scaled with participant eccentricity or the 2-particle cumulanteccentricity.
3. Hints of Incomplete Thermalization
The large data set from run IV Au+Au collisions at √ s NN = 200GeV allows usto extend v { } measurement to large p t and in fine centrality bins (see Fig. 2).For the reason stated in the previous section, we scale v { } by initial standardeccentricities. The left plot of Fig. 2 shows v { } scaled by the eccentricity fromColor Glass Condensate (CGC) [ 11], and the right plot shows v { } scaled by theMonte Carlo Glauber eccentricity for wounded nucleons. As expected, for the CGCcase, the magnitude of v /ǫ is lower if compared to the ratio in which a Glaubereccentricity is used. For both cases, we see the ratio rises from peripheral events tocentral events, indicating that stronger flow has been developed in central collisions.We also notice that the p t where v reaches its maximum increases from peripheral A. Tang et al.collisions to central collisions, which is consistent with the expectation that theapplicable range for hydrodynamics extends to large pt in central collisions. Notethat v /ǫ shows sign of saturation in central collisions for the CGC case but notmuch for the Glauber case. This is explained by [ 11] as the following: in veryperipheral collisions, due to little asymmetry in the saturation scales, the CGCeccentricity approaches the same value as that in the Glauber model, but in centralcollisions, CGC predicts a larger eccentricity than the Glauber model when thereis a large asymmetry in the local saturation scales of the collisions partners, alonga path in impact-parameter direction away from the origin. Note that v { } is notsuitable for this study because it is more susceptible to nonflow at large p t . (GeV/c) t p0 1 2 3 4 5 6 7 8 C G C ˛ / { } v
60% - 70%50% - 60%40% - 50%30% - 40%20% - 30%10% - 20% 5% - 10%
AuAu 200 GeV STARPreliminary (GeV/c) t p0 1 2 3 4 5 6 7 8 s t d ˛ / { } v
60% - 70%50% - 60%40% - 50%30% - 40%20% - 30%10% - 20% 5% - 10%
AuAu 200 GeV STARPreliminary
Fig. 2. v scaled by initial CGC eccentricity (left) and Glauber eccentricity (right)as a function of p t . This plot is from [ 12].To understand how well hydrodynamics describes STAR’s v , we investigatedthe behavior of v under the contex of the transport model, which will be reducedto hydrodynamics when the mean free path is much smaller than the system size [5]. In such approach, the dependence of v /ǫ on particle’s density in the transverseplane (1 /SdN/dy ) can be described by: v ǫ = h v ǫ i hydro
11 +
K/K = h v ǫ i hydro
11 + (cid:16) σ c s c S dNdy (cid:17) − K (3)Where K is Knudsen number defined by the mean free path divided by the systemsize(sometimes it is more convenient to use K − which means number of collisionsa particle encounters before it escapes), and K is a constant can be determinedthrough transport calculations. In our study, we take K = 0 . S is to take into account the differentdefinition of S between STAR ( S = π p h x i h y i ) and [ 11]. (( S = 4 π p h x i h y i ). (cid:2) v ǫ (cid:3) hydro and σ are free parameters that have to be determined from fitting thedata. In this approach hydrodynamic limit of v /ǫ can be never reached, but canbe only asymptotic approached.Fig. 3 shows v { } scaled by CGC initial eccentricity and Glauber initial ec-centricity for wounded nucleons, as a function of particle density in the transverseints of Incomplete Thermalization 5 ) -2 ˛ / { } v ˛ wounded N ˛ CGC
AuAu 200 GeV
STARPreliminary ˛ Fitted Limit with standard ˛ Fitted Limit with CGC
Fig. 3. v { } scaled by initial eccentric-ities, as a function of centrality. Fittedhydrodynamic limits for v /ǫ are indi-cated by horizontal lines. CGC eccen-tricity and overlap area S are from [ 11].plane, with fits to Eq. 3. The fitted hydrodynamic limits for the ratio of v /ǫ are0 .
23 and 0 .
36, for CGC case and Glauber case, respectively. For the same reasonmentioned above, we see that the curve shows a hint of saturation for the CGC case,but not for the Glauber case, due to the relatively larger ǫ for CGC case in centralcollisions. It is interesting to see that, for central Au+Au collisions, the ratio of v /ǫ is about 20 −
30% away from hydrodynamic limits. It means that, there isstill significant room for flow to grow before the system saturates at hydrodynamiclimits.From the simple observation that both v and v are proportional to K − for small K − , one expects that v /v decreases with K − , reaching a minimumwhen the hydrodynamical regime is reached. For this reason the ratio of v /v hasbeen argued as a probe to test the degree of thermalization. Fig. 4 shows STAR’smeasurement of v /v as a function of transverse momentum. The major systematicuncertainty in this measurement comes from v [ 12]. In this analysis, the inducedsystematic error from v uncertainty is estimated by studying the difference between v { } and v measured with event plane constructed by tracks from STAR’s ForwardTime Projection Chamber (FTPC). This advanced study reduces the previously-reported systematic error at QM06 conference by 40% relatively. The dashed linesare ratio come out of calculations by solving Boltzmann equations with MonteCarlo simulation, with different Knudsen number K . When the Knudsen number issmall, it recovers the hydrodynamic limit as indicated the solid line. The plot showsthat the system exhibits, again, significant deviation from ideal hydrodynamic limit( K <<
K > .
4. Conclusion
To summarize, we found that v { } , v { ZDCSMD } and v { q − dist } are all sensi-tive to azimuthal correlation w.r.t the reaction plane, not the participant plane,thusthey should be scaled by eccentricities that are sensitive to reaction plane too. Thatincludes standard eccentricity and 4-particle cumulant eccentricity. For v methodsthat are based on two particle correlations, they are sensitive to the azimuthal cor-relation w.r.t the participant plane, and they need to be scaled by the corresponding A. Tang et al. (GeV/c) t p0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 / v v [ [ /v vHydroBoltzmann K= 0.1Boltzmann K= 0.5Boltzmann K= 1.0 AuAu 200 GeV 20 - 30%
STARPreliminary
Fig. 4. v { EP } /v { } as a function of p t . This plot is from [ 12] with an ad-vanced estimation on systematics repre-sented by half box-brackets.eccentricities that are sensitive to the participant plane. That includes participanteccentricity or 2-particle cumulant eccentricity. We found that from peripheral tocentral Au+Au collisions flow increases, and the applicable range for hydrodynam-ics extends to larger p t . However, v /ǫ and v /v shows significant deviation fromideal hydrodynamic limit, when that limit is extracted from fitting the data itselfwith a Boltzmann equation motivated formula. Our study shows that although ingeneral hydrodynamic does a good job in terms of describing v at RHIC, there arefeatures that are not consistent with a complete thermalization and they cannot beeasily dismissed. References
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