The Lee-Yang zeroes method is applied to study elliptic flow ($v_2$) in Au+Au
collisions at $\sqrt{s}=200A$~GeV, with the UrQMD model. In this transport
approach, the true event plane is known and both the nonflow effects and
event-by-event $v_2$ fluctuations exist. Although the low resolutions prohibit
the application of the method for most central and peripheral collisions, the
integral and differential elliptic flow from the Lee-Yang zeroes method agrees
with the exact $v_2$ values very well for semi-central collisions.Comment: 4 pages, 4 figure

Bleicher M

Borghini N

Borghini N Dinh P M Ollitrault J-Y

Horst Stöcker

Marcus Bleicher

Miller M Snellings R

Mrowczynski S

Tang A H

Teaney D Lauret J Shuryak E V

Xianglei Zhu

English

arXiv

Crossref

Elliptic flow analysis at RHIC with the Lee–Yang zeros method in a relativistic transport approach

The Lee-Yang zeroes method is applied to study elliptic flow (v_2) in Au+Au collisions at sqrt s =200 A GeV, with the UrQMD model. In this transport approach, the true event plane is known and both the nonflow effects and event-by-event v_2 fluctuations exist. Although the low resolutions prohibit the application of the method for most central and peripheral collisions, the integral and differential elliptic flow from the Lee-Yang zeroes method agrees with the exact v_2 values very well for semi-central collisions

Zhu, Xianglei

Bleicher, Marcus

Stöcker, Horst

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/0601049 v1   17 Jan 2006Elliptic Flow Analysis at RHIC with the Lee-Yang Zeroes Method in a RelativisticTransport ApproachXianglei Zhu,1,2,3 Marcus Bleicher,2 and Horst St¨ ocker1,21Frankfurt Institute for Advanced Studies (FIAS),Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany2Institut f¨ ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨ at,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany3Physics Department, Tsinghua University, Beijing 100084, ChinaThe Lee-Yang zeroes method is applied to study elliptic ﬂow (v2) in Au+Au collisions at√s =200A GeV, with the UrQMD model. In this transport approach, the true event plane is knownand both the nonﬂow eﬀects and event-by-event v2 ﬂuctuations exist. Although the low resolutionsprohibit the application of the method for most central and peripheral collisions, the integral anddiﬀerential elliptic ﬂow from the Lee-Yang zeroes method agrees with the exact v2 values very wellfor semi-central collisions.PACS numbers: 25.75.Ld, 25.75.Dw, 25.75.GzAnisotropic ﬂow [1, 2, 3, 4], or more speciﬁcally, ellip-tic ﬂow (v2), which is the second Fourier harmonic [5] inthe transverse distribution of the emitted particles, is ex-pected to be sensitive to the early pressure gradients andtherefore to the equation of state (EOS) of the formedﬁreball in heavy-ion collisions [5, 6, 7, 8, 9, 10]. In addi-tion, the elliptic ﬂow of high pT particles is related to jetfragmentation and energy loss of the primordially pro-duced hard antiquark-quark pair when traveling throughthe hot QCD medium [11]. Therefore, understanding themicroscopic nature of the elliptic ﬂow is of great impor-tance for the study of the key questions of experimentaland theoretical relativistic heavy ion collisions, the trans-port coeﬃcients and the pressure, namely the equationof state.To measure ﬂow, experiments usually use the reactionplane method [12, 13] or the equivalent two-particle cor-relation method [12, 14, 15]. However, these two-particlecorrelation-based methods suﬀer from correlations un-related to the reaction plane. These additional contri-butions are usually dubbed nonﬂow eﬀects [16], such asthe overall transverse momentum conservation, small an-gle azimuthal correlations due to ﬁnal state interactions,resonance decays, jet production [17] and quantum cor-relations due to the HBT eﬀect [18]. In order to de-crease the contribution of the nonﬂow eﬀects to the ﬂowmeasurements, many-particle cumulant method was pro-posed [19]. In this method, usually 4- and 6-particle cu-mulants are used to estimate the collective ﬂow. Thismethod has been applied to the ﬂow analysis at RHIC,SPS and SIS/GSI [14, 15, 20, 21, 22, 23, 24, 25, 26].And it has also been tested in the UrQMD model in [27].It was found that the two-particle correlation method isstrongly aﬀected by nonﬂow eﬀects in the UrQMD model,such as jet production and resonance decays, while the4- and 6-particle cumulants give the true v2 values withbetter than < 5% accuracy for the semi-central collisionsat RHIC, even though this method is aﬀected severely bythe rather large v2 ﬂuctuations for the central and veryperipheral collisions [28]. More recently, the Lee-Yang ze-roes method was developed to analyze the collective ﬂow[29, 30, 31]. This new method enables the extraction ofthe “true” collective ﬂow from the genuine correlationbetween a large number of particles. It is expected togive the cleanest values of the genuine collective ﬂow. Itis, in fact, also easier to apply to data than the cumulantmethod. This method has been applied in the ﬂow anal-ysis at SIS/GSI [26]. Here it was found that the v1 andv2 values extracted from this new method do agree withthose from the higher order cumulant method.In the present paper, the Lee-Yang zeroes method willbe tested in the integral and diﬀerential v2 analysis ofUrQMD events at RHIC energy. The UrQMD model[32, 33] is well suited for the test: it has been shown in[27] that both the nonﬂow eﬀects and the v2 ﬂuctuations[36], which can aﬀect the ﬂow data, do exist naturallyin the model. Most importantly, the true reaction planeangle ΦR is known by construction in the UrQMD model.This allows for the direct calculation of the exact ellipticﬂow from its basic deﬁnition, i.e. v2 =  cos2(φ − ΦR) .Through the present analysis, it will be demonstratedthat the Lee-Yang zeroes method works well for the v2measurements at RHIC energy.In order to determine the “integrated” elliptic ﬂow,which is deﬁned asV2 =*M Xj=1ωj cos(n(φj − ΦR))+,the generating function of the Lee-Yang zeroes methodis introduced and deﬁned as [29, 30, 31]Gθ(ir) =*M Yj=1[1 + irωj cos(φj − θ)]+.In the above deﬁnitions, M is the number of particlesused to estimate the integral ﬂow in each event, φj isthe particle azimuthal angle, r is a positive real variable,θ is an arbitrary reference angle and ωj is the particle2weight. It is worth stressing that in our analysis, the 1/Mweight is used to decrease the eﬀect of event-by-event Mﬂuctuations. Therefore, V2 = v2 in our analysis. Fig. 1shows the amplitudes of the generating function |Gθ(ir)|as a function of r for θ = 0 at diﬀerent centralities. Forthe semi-central bins (e.g. 20-30%), with the increaseof r, |Gθ(ir)| decreases rapidly from the value of 1 atr = 0, then reaches a sharp minimum which is very closeto 0. This appearence of the sharp minimum indicatesthat there exists a zero of the generating function. Afterﬁnding the position rθ0 of the ﬁrst zero of |Gθ(ir)|, theintegral elliptic ﬂow is determined asvθ2 = j01/rθ0,where j01 = 2.40483 is the ﬁrst root of the Bessel functionJ0(x). The ﬁnal value of the integral v2 is the averageof vθ2 over 5 equally spaced values of θ from 0 to 4π/5,and the statistical uncertainty of the ﬁnal integral v2 isabout a factor of 2 smaller [30]. 0 0.2 0.4 0.6 0.8 1 0  50  100  150  200  250  300|Gθ(ir)|rUrQMD(2.2) Au+Au s1/2=200 AGeV 0-5%5-10%10-20%20-30%30-40%40-50%50-60%FIG. 1: (Color online) |Gθ(ir)| as a function of r for θ = 0 atdiﬀerent centralities.Fig. 2 shows the Lee-Yang zeroes results on the cen-trality dependence of the integral v2. For the integralv2 analysis, all particles in the pseudorapidity region|η| < 2.5 are used and the number of particles from eventto event ﬂuctuates in each centrality bin. The centralitiesin our analysis are selected according to the same geomet-rical fractions of the total cross section (0-5%,5-10%,10-20%,20-30%,30-40%,40-50%,50-60%,60-70%) as used bythe STAR experiment [14]. Here, however, we use im-pact parameter cuts instead of multiplicity cuts. Morethan 1.3   106 minimum bias events are used in the inte-gral v2 analysis.The v2 values extracted from the Lee-Yang zeroesmethod with 1/M weight (solid circles) agrees with theexact v2 (open triangles) very well for the semi-centralbins (about 5-40%), as has been seen in Fig. 2. How-ever, the Lee-Yang zeroes method does not work wellfor the most central (0-5%) and peripheral (>40%) binsin the present model study: the resolution parameter χ,which is related to the event-plane resolution [35], decides 0 1 2 3 4 5 6 0  10  20  30  40  50  60  70v2(%) of all particles (|η|<2.5)% Most CentralUrQMD(2.2) Au+Au s1/2=200AGeVv2{∞} 1/M w.v2{∞} unit w.v2FIG. 2: (Color online) The UrQMD model results for integralv2 values for Au+Au collisions at√s = 200A GeV. Resultsfrom the Lee-Yang zeroes method (v2{∞}) with unit weightand 1/M weight are compared to the exact v2 in diﬀerentcentrality bins. (see text for details.) 0 2 4 6 8 10 12 0  2  4  6  8  10v2(%) (|η|<2.5)PT (GeV/c)UrQMD(2.2)Au+Au s1/2=200AGeV (20-30%)v2{∞}v2FIG. 3: The UrQMD model results for v2(pT) in semi-central(20-30%) Au+Au collisions at√s = 200A GeV. Results fromthe Lee-Yang zeroes method (v2{∞}) are compared to theexact v2.whether the method can be applied or not [29, 30, 31]. Ifthis parameter is too small, i.e., χ ≤ 0.5, the statisticalerror will be too large, and the method is inapplicable. χis roughly given by v2√M. Therefore, it will be smallerfor the most central bin where v2 is small, and for pe-ripheral bins where M is small. For the feasibility testwith the UrQMD model, χ is about 1 in the semi-centralbin (20-30%), while it is close to 0.5 in the most central(0-5%) and peripheral bins (>40%). In fact, as is shownin Fig. 1, the ﬁrst minimum of |Gθ(ir)| is not compat-ible with a zero for most central (0-5%) and peripheral(>40%) bins. Therefore, the Lee-Yang zeroes methodcan not be used in most central (0-5%) and peripheral(>40%) bins from the UrQMD model.We have also tried the Lee-Yang zeroes method withunit weight. These results are shown as open circles in3Fig. 2. For the semi-central bins (10-40%), it slightlyundershoots the exact v2, which indicates that the ze-roes method with unit weight is not completely free fromthe eﬀect of M ﬂuctuations. However, the eﬀect of Mﬂuctuations is small. Experimentally, the centrality isusually selected according to the multiplicity M of theevents. In this case, the M ﬂuctuations will be narrower,and the eﬀect of M ﬂuctuations on the zeroes methodwill be weaker. The present Lee-Yang zeroes analysisbased on the M selected centralities (not shown in thisLetter) indeed tells that the integral v2 values from the1/M weight and unit weight agree well with each otherand the M ﬂuctuations have almost negligible eﬀects. 0 1 2 3 4 5-4 -2  0  2  4v2(%)ηUrQMD(2.2)Au+Au s1/2=200AGeV (20-30%)v2{∞}v2FIG. 4: The UrQMD model results for v2(η) in semi-central(20-30%) Au+Au collisions at√s = 200A GeV. Results fromthe Lee-Yang zeroes method (v2{∞}) are compared to theexact v2.After determining rθ0, the diﬀerential v2 can be esti-mated from Eq. (9) in [31]. For the test of the diﬀerentialﬂow, we use more than 6   105 semi-central events (withimpact parameters from 6.7 to 8.3 fm, corresponding toabout 20% to 30% of the total cross section). From theabove results on the integral v2, we know that the Lee-Yang zeroes method reproduces the exact v2 in this cen-trality bin with better than 2% accuracy, However it isstill necessary to see whether it reproduces the diﬀeren-tial v2 correctly. Especially at large transverse momenta(pT), nonﬂow contributions are expected to be large. Fig.3 and Fig. 4 shows the comparisons of the Lee-Yang ze-roes results for v2(pT) and v2(η) to the exact v2 values. Itis clear that the results from the Lee-Yang zeroes methodagree with the exact v2 values very well for the wholepT- and η-range. This shows that the Lee-Yang zeroesmethod indeed reproduced exactly the genuine ellipticﬂow values for these centrality bins very well.To summarize, we have applied the Lee-Yang zeroesmethod in the v2 analysis of the UrQMD model at200A GeV RHIC Au+Au collisions. Only for the semi-central bins (5-40%), is this method actually applicable.Both the integral and diﬀerential v2 from the Lee-Yangzeroes method agree well with the exact values in thesebins. The UrQMD model includes many of the nonﬂowcorrelations which are also seen in the real data. Hence,these good agreements show that the Lee-Yang zeroesmethod is not sensitive to the nonﬂow correlations andv2 ﬂuctuations for the semi-central bins and indeed givesthe genuine elliptic ﬂow there. It should be noted that thecurrent UrQMD model underpredict the RHIC v2 databy about 40%. Therefore, in the experiments at RHICor LHC, the zeroes method will work over a somewhatbroader range of centrality bins.AcknowledgmentsThis work was triggered by a discussion with Dr. A.Poskanzer at LBL. The authors thank Drs. A. Poskanzer,J.-Y. Ollitrault and R. Snellings for stimulating discus-sions. We are grateful to the Center for the ScientiﬁcComputing (CSC) at Frankfurt for the computing re-sources. 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Elliptic flow analysis at RHIC with the Lee-Yang Zeroes method in a relativistic transport approach

Elliptic Flow Analysis at RHIC with the Lee-Yang Zeroes Method in a
  Relativistic Transport Approach

Elliptic Flow Analysis at RHIC with the Lee-Yang Zeroes Method in a Relativistic Transport Approach

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