A closer look at the influence of tubular initial conditions on two-particle correlations
aa r X i v : . [ nu c l - t h ] D ec A closer look at the influence of tubular initial conditions ontwo-particle correlations
R.P.G.Andrade, F.Grassi, Y.Hama, and W.-L.Qian
Instituto de F´ısica, Universidade de S˜ao Paulo, SP, Brazil (Dated: Oct. 2009)In a recent paper, the hydrodynamic code NEXSPheRIO was used in conjunctionwith STAR analysis methods to study two-particle correlations as function of ∆ η and ∆ φ . The various structures observed in data were reproduced. In this work, wediscuss the origin of these structures as well as present new results. I. INTRODUCTION
One of the most striking results in relativistic heavy ion collisions is the existence of somestructures in the two-particle correlations [1, 2, 3]. One structure has a long (pseudo)rapidityextent [4] and a narrow azimuthal extent. The other may have a long (pseudo)rapidity extentand has a single or double hump in azimuth. In order that two-particles A and B emittedat some proper time τ f.out appear as correlated, the process that correlated them must haveoccurred[5, 6] at some proper time τ ≤ τ f.out exp( −| y A − y B | / early times in the nuclearcollisions.These two-particle azimuthal correlations data have motivated many theoretical investi-gations (for a short critical review see e.g.[7]). In many of these approaches, the mechanismis closely related to jet quenching and the response of the medium to the deposited energy.However, 1) the ridge structure is also seen at low transverse momentum[8, 9]; 2) recentexperimental data seem to indicate no correlation between high pt trigger and associatedridge particles [10]. In another class of models, it is suggested that the combined effect of alongitudinal structure in the initial conditions (IC) and transverse expansion is responsiblefor the ridge [5, 6, 11]. In this line, we have studied the two-particle correlation by usinga hydrodynamic code NEXSPheRIO[12]. Both the near-side and double-hump away-sidestructures were reproduced. In this work, we discuss how exactly these structures appearas well as present new results. II. RIDGES AND PEAK IN NEXSPHERIO
The NeXSPheRIO code uses IC from the microscopic code NeXus [13] and solves thehydrodynamic equations with the SPheRIO code [14], on an event by event basis. Figure1 shows a typical example of IC with tubular structures along the collision axis. In ourmodel, this is what causes the long (pseudo)rapidity extent of the two-particle correlations.We note that in figure 1, the near-side structure is composed of a ridge and a peak, as seenexperimentally. In our approach, the IC constructed by “thermalizing” NeXus output[14]do not explicitly involve jets, these are however not totally forgotten in the IC as they leavesome localized region with higher transverse fluid velocity as shown in figure 2. This regionis not correlated with a tube so peak and ridge are independent.
FIG. 1: NeXSPheRIO initial energy density in the transverse (left) and reaction (center) planesfor a central Au+Au collision at 200 GeV A. Two-particle correlation (right).
We also note in figure 1 the existence not only of the near-side ridge but also of the double-hump away-side ridge. This result was obtained in [12] using the same methods as in theSTAR analysis, in particular elliptic flow was removed using ZYAM. In this work, a totallydifferent analysis is done, in particular the directions of the different event planes (which iscalculable in our model[15]) are aligned, then the mixed event contribution is obtained andsubtracted from the raw correlation. By doing this, the effects of η distribution shape andflow are removed in a single step.The fact that this new analysis leads to similar results as in [12] reinforces both ap-proaches. However it is still unclear what causes in detail the near and away-side structuresas well as what fixes the position of the double-hump in the away-side ridge. -8-6-4-2 0 2 4 6 8-8 -6 -4 -2 0 2 4 6 8 y [f m ] x [fm] (GeV/fm ) τ = 1fm η = 0c 14.00 12.00 10.00 8.00 6.00 4.00 2.00 1.00 0.50 0.10 0.05 0.01 0.005v -8-6-4-2 0 2 4 6 8-8 -6 -4 -2 0 2 4 6 8 η x [fm] (GeV/fm ) τ = 1fmy = -6.08fmc 2.00 1.00 0.50 0.10 0.05 0.01v FIG. 2: NeXSPheRIO initial fluid velocity for a Au+Au collision at 200 GeV A in the 25-35%centrality window.
III. ORIGIN OF THE RIDGES IN A SIMPLIFIED MODEL
To investigate the origin of the ridges, we use a simplified two-dimensional model. Thismodel consists of a slice of matter which initially has a high energy density spot in a smoothbackground. This slice subsequently undergoes transverse expansion and boost-invariantlongitudinal expansion. More details are given in [16]. The single particle angular distribu-tion has not a single peak as one might expect but two peaks located on both sides of theposition of the tube as seen in figure 3 (left). This double peak structure is observed for alltransverse momenta at more or less the same position [16] and its location is in agreement (cid:1)(cid:2) (cid:0)(cid:3) (cid:4) (cid:5) (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) φ φ (cid:29)(cid:30) (cid:31) !" δ φ m δφ FIG. 3: Single (left) and two (right) particle angular distribution in the simplified model. with data (for a central collision). The reason, as seen in figure 4 is that the effect of thetube is to deflect the otherwise isotropic radial flow.From figure 3 (left), we can guess how the two-particle angular correlation will be. Thetrigger particle is more likely to be in one of the two peaks. We first choose the left-hand sidepeak. The associated particle is more likely to be also in this peak i.e. with ∆ φ = 0 or inthe right-hand side peak with ∆ φ ∼ +2. If we choose the trigger particle in the right-handside peak, the associated particle is more likely to be also in this peak i.e. with ∆ φ = 0 orin the left-hand side peak with ∆ φ ∼ −
2. So the final two particle angular correlation musthave a large central peak at ∆ φ = 0 and two smaller peaks respectively at ∆ φ ∼ ±
2. Figure3 (right) shows that this is indeed the case. The peak at ∆ φ = 0 corresponds to the near-side ridge and the peaks at ∆ φ ∼ ± -5-4-3-2-1 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 y [f m ] x [fm] (GeV/fm )12.010.0 9.0 8.0 5.0 2.0 FIG. 4: Temporal evolution of energy density for the simplified model (left). Trajectories of thefluid cells around the tube (right).
IV. COMPARISON OF NEXSPHERIO RESULTS WITH EXPERIMENTAL DATA
Now that the origin of the ridges is clarified, we return to a comparison of NeXSPheRIOresults with data. Since, as already explained, the jets are “thermalized” in our model, a n o pqrstuvwxyz{|}~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142)(cid:143)(cid:144) ∆ η (cid:145)(cid:146)(cid:147) ∆ φ (cid:148) ∆φ (cid:149)(cid:150)(cid:151)(cid:152)(cid:153)(cid:154)(cid:155)(cid:156)(cid:157)(cid:158) (cid:159)(cid:160) ¡ ¢£⁄¥ƒ§¤' “ «‹›fifl(cid:176)–†‡·(cid:181)¶ • ‚„”»…‰(cid:190)¿(cid:192)`´ˆ ˜ ¯˘˙¨(cid:201)˚ FIG. 5: NeXSPheRIO dependence of the ridges on the p t cutoffs for Au+Au collisions at 200AGeV. precise quantitative comparison at this stage is not possible. Nevertheless several qualitativecomparisons can be done.For fixed p trigt and increasing p assoct , the near-side and away-side peaks decrease as seenin figure 5 for central collisions (this is generally expected since the number of associatedparticles decreases). On the other side, for fixed p assoct and increasing p trigt , the peaks increase.This behavior is in agreement with data (fig. 36 in [17]).When going from central to peripherical collisions, the near-side ridge decreases and theaway-side ridge changes from double to single hump, as seen in figure 6 and in conformitywith data(fig. 36-38 in [17]). FIG. 6: Two-particle correlation, as computed with our NeXSPheRIO code, for different centralitywindows of Au+Au collisions at 200A GeV(a: 0-5%, b: 5-10%, c: 10-20%, d: 20-30%, e: 30-40%,f: 40-50%). p trigt > . p assoct > . The correlation can be studied as a function of the trigger particle angle with relationto the event plane. In figure 7 for a mid-central window, the away-side ridge changes fromsingle peak for in-plane trigger to double peak for out-of-plane trigger. For central collisions(not shown), it is always double-peaked. This is in accordance with data (fig.1 in [18]).Some other qualitative features can also be mentioned. The near-side ridge seems to bepresent for untriggered correlation as found experimentally [8, 9] but our analysis methodis different from the experimental one and this result is still being checked. As explainedabove, the near-side ridge is independent of the jet peak, also in agreement with data [10].Due to its origin, we expect the near-side ridge to have similar composition as the bulk and
FIG. 7: Two-particle correlation, from NeXSPheRIO, for 20-30% centrality Au+Au collisions at200A GeV, for different φ s (a: φ s = 0 o − o , b: φ s = 15 o − o , c: φ s = 30 o − o , d: φ s = 45 o − o ,e: φ s = 60 o − o , f: φ s = 75 o − o . p trigt > . GeV and p assoct > . GeV. its spectra should be a little harder than bulk (see figure 3 left and figgure 5 of [16]), also inline with data [10, 19]. We expect a similar behavior for the away-side ridge in accordancewith data when available [10, 20]. Finally, for non-central collisions, due to their origin, weexpect asymmetry in the ridges azimuthal correlation as function of the trigger angle withrespect to the event plane; for the near-side ridge this has been observed in [21].
V. CONCLUSION
In conclusion, the hydrodynamic expansion starting from fluctuating tubular IC producesthe various structures observed in the two-particle correlations [12]. We showed in this paperexplicitly how this happens, using a simplified single tube model: this tube deflects matterin two directions which results in the near-side and away-side ridges. In addition we showthat NeXSPheRIO code reproduces several other observed characteristics of the two-particlecorrelations.Other models have been suggested to explain some of theses data. Three particle correla-tions might provide a way to discriminate: some models such as the Mach cone one predictthat the associated particles emerge at two angles leading to figure 8 (left) while others suchas the deflected jet model result in particles (if not flying with the trigger) emerging at oneangle giving rise to figure 8 (right). Such a figure is also expected for the simplified one tubemodel and approximately for NeXSPheRIO central collisions. We are working on a preciseprediction and comparison with data [22] .
FIG. 8: Schematic three particle correlations as function of the angles of the two associated particleswith respect to the trigger particle.
It seems that an interesting observable would be 2+1 correlations (first associated particlefixed in a narrow kinematical interval), which requires less statistics and leads to a cleanprediction. For example choosing the trigger from one peak in figure 3 (left) and the firstassociated particle from the other peak, the second associated particle will be from any ofthe two peaks. This means that in two particle correlation between the second associatedparticle and the trigger presented in terms of ∆ η and ∆ φ , there would be two stripes. Incontrast, the Mach-cone model would lead to three stripes. VI. ACKNOWLEDGMENTS
We thank for discussions with Takeshi Kodama, Jun Takahashi, Bernardo Tavares,Fuqiang Wang, Guoliang Ma, Paul Sorensen, Klaus Werner. We acknowledge funding fromFunda¸c˜ao de Amparo `a Pesquisa de Estado de S˜ao Paulo, FAPESP, and Conselho Nacionalde Desenvolvimento Cientit´ıfico e Tecnol´ogico, CNPq. [1] J. Putschke for the STAR collaboration, Nucl.Phys.
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