Investigation of the elliptic flow fluctuations of the identified particles using the A Multi-Phase Transport model
Niseem Magdy, Xu Sun, Zhenyu Ye, Olga Evdokimov, Roy A. Lacey
aa r X i v : . [ nu c l - e x ] S e p Investigation of the elliptic flow fluctuations of the identified particles using the AMulti-Phase Transport model
Niseem Magdy, ∗ Xu Sun, Zhenyu Ye, Olga Evdokimov, and Roy A. Lacey Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA Department of Chemistry, State University of New York, Stony Brook, New York 11794, USA
A Multi-Phase Transport (AMPT) model is used to study the elliptic flow fluctuations of identi-fied particles using participant and spectator event planes. The elliptic flow measured using the firstorder spectator event plane is expected to give the elliptic flow relative to the true reaction planewhich suppresses the flow fluctuations. However, the elliptic flow measured using the second-orderparticipant plane is expected to capture the elliptic flow fluctuations. Our study shows that thefirst order spectator event plane could be used to study the elliptic flow fluctuations of the iden-tified particles in the AMPT model. The elliptic flow fluctuations magnitude shows weak particlespecies dependence and transverse momentum dependence. Such observation will have importantimplications for understanding the source of the elliptic flow fluctuations.
PACS numbers:Keywords: collectivity; correlation; shear viscosity
Many studies of the ultra-relativistic heavy-ion colli-sions at the Relativistic Heavy Ion Collider and the LargeHadron Collider show that an exotic state of matternamed Quark-Gluon Plasma (QGP) is created in thesecollisions. A large number of studies are focused on iden-tifying the dynamical evolution and the transport prop-erties of the QGP.In heavy-ion collisions, the produced particle az-imuthal anisotropy measurements have been used in vari-ous studies to show the viscous hydrodynamic response ofthe QGP to the initial energy density spatial distributionproduced in the early stages of the collisions [1–14]. Theazimuthal anisotropy of the particles emitted relative tothe reaction plane Ψ R can be described by the Fourierexpansion [15, 16] of the final-state azimuthal angle φ distribution, dNdφ ∝ ∞ X n =1 v n cos [ n ( φ − Ψ R )] , (1)The first Fourier harmonic, v , is the directed flow; v is called the elliptic flow, and v is the triangular flow,etc. A wealth of information on the characteristics of theQGP has been gained via the anisotropic flow studiesof directed and elliptic flow. [17–19], higher-order flowharmonics v n> [10, 20–23], flow fluctuations [24–26] anddifferent flow harmonics correlations [21, 27–31].Hydrodynamic studies suggest that anisotropic flowstems from the evolution of the medium in the presenceof initial-state anisotropies, determined by the eccentric-ities ε n . The v and v flow harmonics are recognized tobe linearly correlated to ε and ε , respectively [7, 28, 32–38]. Therefore for these flow harmonics, v n = κ n ε n , (2)where κ n encodes knowledge about the medium prop-erties such as the specific shear viscosity ( η/s ) of the QGP. Accurate extraction of η/s requires certain restric-tions on the initial-state models employed in such ex-tractions. Such constraints can be achieved via measure-ments of the flow harmonics and the event-by-event flowfluctuations [39]. Flow fluctuations could be arising fromseveral sources: one of which has attracted considerableattention is the initial eccentricity fluctuations [40–42].Recent theoretical studies have begun to take into ac-count initial conditions that include energy density fluc-tuations, initial flow [13, 37, 43], and the full shear stresstensor [44] at µ B = 0 and at µ B > v { } ,and the two-particles elliptic flow, v { } , is often used toestimate the strength of the elliptic flow fluctuations as afraction of the measured flow harmonic strength [51, 52].However, important caveats to studying the ellipticflow fluctuations using ( v { } /v { } ) for the identifiedhadrons are, first, the demand for high statistical power,and second, the multi-strange hadron identification pro-cess [53]. Consequently, the ratio of v { } /v { } is oflimited experimental use for carrying out these investi-gations for the multi-strange hadrons.In this work, we investigate an alternative validationscheme, which employs the use of the first-order specta-tor event plane ,Ψ SP1 , along with the second-order eventplane Ψ
EP2 to study the elliptic flow fluctuations of theidentified hadrons. Here, the underlying notion is that v SP2 (with respect to the spectator first-order event plane)will reduce the elliptic flow fluctuations due to the strongcorrelations between the Ψ
SP1 and the true reaction plane.Therefore, the ratio v SP2 /v EP2 is expected to reflect the el-liptic flow fluctuations.For RHIC highest energy and using the STAR detec-tor, we propose a similar investigation to be performedusing the first-order spectator event plane from specta-tor neutrons, measured by the zero-degree calorimeters(ZDC) [54] and the second-order event plane using thenew installed Event-Plane-Detector (EPD) [55]. Conse-quently, we think that conducting a similar experimentalstudy will reveal important information about the ellip-tic flow fluctuations and will shed light on the ICCINGscenario suggested in Reference [50].
METHOD
The current study is conducted with simulated eventsfor Au+Au collisions at √ s NN = 200 GeV, collected us-ing the AMPT [56] model with the string-melting mech-anism and hadronic cascade on. The AMPT model,which has been widely employed to study relativisticheavy-ion collisions [56–60, 60–62], includes four maindynamical components: initial condition, parton cascade,hadronization, and hadronic rescatterings. The initialconditions take into account soft string excitations andthe phase space distributions of minijet partons, whichare produced by the Heavy-Ion Jet Interaction Gener-ator model (HIJING) [63] in which the Glauber modelwith multiple nucleon scatterings are used to define theheavy-ion collisions initial state.The partons scatterings are handled according to theZhang’s Parton Cascade (ZPC) model [64], which containonly two-body elastic scatterings with a cross-section de-fined as: dσdt = 9 πα s µ s ) 1( t − µ ) , (3)where α s = 0.47 is the strong coupling constant, µ is thescreening mass and s and t are the Mandelstam variables.In the AMPT with the string-melting mechanism, the ex-cited strings and minijet partons are melted into partons.The partons scatterings will lead to local energy densityfluctuations, which are equivalent to the local transversedensity of participant nucleons.In the string-melting version and when partons stopinteracting with each other, a quark coalescence model isused to couple partons into hadrons. Consequently, the partonic matter is then converted into hadronic matterand the hadronic interactions are given by the A Rela-tivistic Transport (ART) model [65], which incorporatesboth elastic and inelastic scatterings for baryon–baryon,baryon–meson, and meson–meson interactions.In this work, the centrality intervals are definedby selecting the impact parameter distribution, thenthe AMPT events are analyzed using (i) the eventplane method and (ii) the multi-particle cumulant tech-nique [66–69]. Using both methods, particle of inter-est (POI) comes from pseudorapidities | η | <
1, whichmatches the STAR experiment pseudorapidity accep-tance, and with transverse momentum 0 . < p T < . /c .The second-order event plane (Ψ EP2 ), is estimated fromthe azimuthal distribution of final-state particles. The el-liptic flow that will be obtained using this method willthen be corrected with the corresponding event planeresolution (Res( Ψ
EP2 )) [16]. The Ψ
EP2 is reconstructedin a pseudorapidity range of 2 . < | η | < .
5, whichmatches the STAR experiment EPD acceptance, and0 . < p T < . /c :Ψ EP2 = 12 tan − (cid:20) P ω i sin(2 φ i ) P ω i cos(2 φ i ) (cid:21) , (4)where φ i is the final-state azimuthal angle of particle i ,and ω i is its weight. The weight is chosen to be equalto p T . Also, the first order spectator plane Ψ SP1 is con-structed using the AMPT spectator x and y position in-formation. Using the spectator or the event planes wecan give the elliptic flow as: v EP2 = h cos (cid:0) φ i − Ψ EP2 ) (cid:1) i Res (Ψ EP2 ) , (5) v SP2 = h cos (cid:0) φ i − Ψ SP1 ) (cid:1) i Res (Ψ SP1 ) , (6)where Res( Ψ EP2 ) and Res( Ψ
SP1 ) represent the resolu-tion of the event planes. The event planes resolution iscalculated using the two-subevent method [16].On the other hand, the standard (subevents) cumulantmethods framework is discussed in References [66–69]. Inthe standard cumulant method, the n -particle cumulantsare constructed using particles from the | η | < . h v n i = hh cos( n ( ϕ − ϕ )) ii , (7) h v n i = hh cos( nϕ + nϕ − nϕ − nϕ ) ii , (8)where, hh ii represents the average over all particles in asingle event, and then in average over all events, n is theharmonic number and ϕ i expresses the azimuthal angle
0 20 40 60 v (a) AMPT Au+Au 200 GeV v { } v SP2 v EP2 (b) STAR Data Au+Au 200 GeV v { } v SP2 v EP2 R a ti o Centralty (%) (c) v { } /v EP2 v SP2 /v EP2
0 20 40 60
Centralty (%) (d) v { } /v EP2 v SP2 /v EP2
Fig.
1: The charged particles centrality dependence of v SP2 and v EP2 are compared to the four-particles elliptic flow(hashed band) for Au+Au collisions at √ s NN = 200 GeVfrom the A Multi-Phase Transport (AMPT) model panel ( a ).The charged particles centrality dependence of v SP2 and v EP2 are compared to v { } for Au+Au collisions at √ s NN = 200GeV from the STAR experiment [18, 72] panel ( b ). The el-liptic flow fluctuations represented by the ratios v SP2 /v EP2 and v { } /v EP2 are presented in panels ( c , d ). of the i th particle. Then the four-particle elliptic flowharmonic can be given as: v { } = 2 h v i − h v i . (9)In general, when the flow fluctuation σ is smaller thanthe true reaction plan elliptic flow h v i one can write [70,71]: v SP2 = h v i (10) v EP2 = h v i + 0 . σ h v i . (11)Then the ratio v SP2 /v EP2 can be used to estimate thestrength of the elliptic flow fluctuations as a fraction ofthe measured flow harmonic (large value of v SP2 /v EP2 in-dicates less fluctuations whereas a smaller value indicateslarge fluctuations), v SP2 v EP2 = h v ih v i + 0 . σ h v i = 11 + 0 . (cid:18) σ h v i (cid:19) (12)The reliability of this elliptic flow fluctuations extrac-tion will depend on the strength of the correlations be-tween the spectator plane and the reaction plane. RESULTS AND DISCUSSION
Panel (a) of Figure 1 compares the centrality depen-dence of the four-particle elliptic flow ( v { } ) with the el-
0 1 2 3 4 v (a)AMPT Au+Au 200 GeV (0-40 %) v { } v SP2 v EP2 R a ti o p T (GeV/c) (b) v { } /v EP2 v SP2 /v EP2
Fig.
2: The charged particles p T dependence of v SP2 and v EP2 are compared to the four-particles elliptic flow (hashed band)panel ( a ). The ratios v SP2 /v EP2 and v { } /v EP2 are presentedin panel ( b ) for Au+Au collisions at √ s NN = 200 GeV fromthe AMPT model. liptic flow measured with respect to the event plane ( v EP2 )and spectators plane ( v SP2 ). The comparison of the v { } and the v EP2 shows larger v EP2 magnitudes for v { } . Bycontrast, the values for v SP2 show good agreement with v { } . Qualitatively, one expects such patterns due tothe respective flow fluctuations contributions to v { } and v EP2 . The experimental measurements for chargehadrons reported by the STAR experiment, shown in Fig-ure 1b [18, 72], also show good agreement between v { } and v SP2 ( v ZDC2 ), consistent with the AMPT simulations.Here, no attempt was made to improve the agreementbetween the model and the experimental results by vary-ing the model parameters to influence the flow magnitudeand its associated fluctuations [73–76]. We defer such aninvestigation to a future study. The ratio v SP2 /v EP2 , pre-sented in panel (c) from AMPT, and data panel (d) servesas a metric for elliptic flow fluctuations. The v SP2 /v EP2 decrease from central to peripheral collisions, consistentwith the patterns expected when initial-state eccentricityfluctuations dominate. Note, however, that other sourcesof fluctuations could contribute.The transverse momentum dependence of the v { } , v EP2 and v SP2 are shown in Figure 2. This differentialcomparison further reflects the effect of the elliptic flowfluctuations on the v EP2 which is highlighted in the ratio v Centralty (%) (a) v EP2 π kp
0 20 40 60 v Centralty (%)
AMPT Au+Au 200 GeV (b) v SP2 φΛΣ + Σ - v SP / v E P Centralty (%) (c) Ξ Ξ - Ω - K Fig.
3: The identified particles centrality dependence of theelliptic flow harmonic with respect to participant and spec-tator event planes panels ( a , b ) respectively. The elliptic flowfluctuations represented by the ratio v SP2 /v EP2 are presentedin panel ( c ) for Au+Au collisions at √ s NN = 200 GeV fromthe AMPT model. between v EP2 and v { } . Also a good agreement (withinthe errors) has been observed between the v { } and v SP2 .The ratio v SP2 /v EP2 , presented in panel (b) presents thestrength of the elliptic flow fluctuations which shows no p T dependence, consistent with the preliminary STARmeasurements [77].The centrality dependence of the identified particles v EP panel (a), v SP panel (b) and v SP /v EP panel (c) areshown in Figure 3 for Au+Au collisions at √ s NN = 200GeV from the AMPT model. The results of v EP and v SP show the mass ordering effect on the observed magnitude.This mass ordering effect, which cancels out for the ratio v SP /v EP , presented in panel (c) indicates the dominationof the initial-state eccentricity fluctuations in the AMPTmodel.Figure 4 compares the p T dependence of the identi-fied particles v EP panel (a), v SP panel (b) and v SP /v EP panel (c) for 0 −
40% Au+Au collisions at √ s NN = 200GeV from the AMPT model. The ratios v SP /v EP panel(c) (elliptic flow fluctuations) show week sensitivity tothe p T increase. The v EP and v SP vs. p T show the ex-pected mass ordering dependence, which cancels out forthe ratio v SP /v EP vs. p T , presented in panel (c), whichfurther suggests that the elliptic flow fluctuations in theAMPT model are governed by initial-state fluctuations. CONCLUSIONS
In summary, we studied the centrality and transversemomentum dependence of the identified particles v SP , v EP and the elliptic flow fluctuations presented by theratio v SP /v EP using the AMPT model. The magnitudeof the elliptic flow fluctuations is observed to increase v p T (GeV/c) (a) v EP2 π kp
0 1 2 3 4 v p T (GeV/c) AMPT Au+Au 200 GeV (0-40 %) (b) v SP2 φΛΣ + Σ - v SP / v E P p T (GeV/c) (c) Ξ Ξ - Ω - K Fig.
4: The identified particles p T dependence of the ellip-tic flow harmonic with respect to participant and spectatorevent planes panels ( a , b ) respectively. The elliptic flow fluc-tuations represented by the ratio v SP2 /v EP2 are presented inpanel ( c ) for Au+Au collisions at √ s NN = 200 GeV from theAMPT model. from central to mid-central collisions, consistent with thepatterns expected from the initial-state eccentricity fluc-tuations; a weak p T dependence is also observed. Thecentrality and p T dependence of the identified particles v EP and v SP show the expected mass ordering. However,the elliptic flow fluctuations show no particle species de-pendence. The integrated and differential elliptic flowfluctuation results indicate the domination of the effectof the initial-state eccentricity fluctuations as expectedin the AMPT model. It is suggested that similar inves-tigations of experimental data could display importantinsight on the ICCING scenario in heavy-ion collisions. Acknowledgments
The authors thank Jacquelyn Noronha-Hostler for theuseful discussion and Emily Racow for the languagecheck. This research was funded by the US Departmentof Energy under contract DE-FG02-94ER40865 (NM,XS, ZY and OE) and DE-FG02-87ER40331.A008 (RL). ∗ Electronic address: [email protected][1] Ulrich W. Heinz and Peter F. Kolb, “Early ther-malization at RHIC,”
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