Centrality selection effect on higher-order cumulants of net-proton multiplicity distributions in relativistic heavy-ion collisions
Arghya Chatterjee, Yu Zhang, Jingdong Zeng, Nihar Ranjan Sahoo, Xiaofeng Luo
CCentrality selection effect on higher-order cumulants of net-proton multiplicitydistributions in relativistic heavy-ion collisions
Arghya Chatterjee, Yu Zhang, Jingdong Zeng, Nihar Ranjan Sahoo, and Xiaofeng Luo ∗ Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China Shandong University, Qingdao, Shandong 266237, China
We studied the centrality selection effect on cumulants (up to fourth order) and the cumulantsratios of net-proton multiplicity distributions in Au+Au collisions at √ s NN = 7.7, 19.6 and 200 GeVfrom UrQMD model. The net-proton cumulants are calculated with collision centralities by usingcharged particle multiplicity from different pesudorapidity ( η ) region. By comparing the resultsfrom various collision centralities, we found that the autocorrelation effects are not significant inthe results with collision centralities ”refmult-3” and ”refmult-2”, which are using mid-rapiditycharged particles but excluding (anti-)protons and analysis region, respectively. Furthermore, dueto the contributions of spectator protons, we observed poor centrality resolution when using chargedparticles at forward η region at low energies. This work can serve as a baseline for centrality selectionof future fluctuations analysis in relativistic heavy-ion collisions. I. INTRODUCTION
One of the major goal of high-energy heavy-ion colli-sion experiments is to explore the phase structure of thestrongly interacting QCD matter [1]. The QCD phasestructure can be represented as a function of tempera-ture ( T ) and baryon chemical potential ( µ B ) [2]. QCDbased model calculations predict that at large µ B thetransition from hadronic matter to Quark-Gluon Plasma(QGP) is of first order [3, 4]. The end point of the first or-der phase transition boundary is known as QCD criticalpoint (CP), after which there is no genuine phase tran-sition but a smooth crossover from hadronic to quark-gluon degrees of freedom [5, 6]. Many efforts has beenmade to find the signature and/or location of the CP,theoretically [7–18] and experimentally [19]. However,the location of the CP and even the existence of the CPhave not been confirmed yet. Experimental confirmationof the existence of the critical point will be a milestonefor the study of QCD phase structure.One of the foremost method for the critical pointsearch is through measuring the event-by-event higher-order fluctuations (called ’cumulants’) of conserved quan-tities, such as net-charge ( Q ), net-baryon ( B ) and net-strangeness( S ), because of their divergence nature nearthe critical point [20, 21]. Due to the limitation of mea-suring neutral particles, experimentally we measured thecumulants of net-proton, and net-kaon as a proxy of net-baryon and net-strangeness respectively. The STAR ex-periment at RHIC, over past few years have measuredthe higher order cumulants up to forth order of net-proton [22–25], net-charge [26] and net-kaon [27] multi-plicity distributions. Recently STAR has also reportedthe cross-cumulants between net-particles [28]. Theo-retically n -th order cumulants are related to the n -thorder thermodynamic susceptibilities as C n = V T χ n , ∗ xfl[email protected] where V and T are the system volume and tempera-ture, respectively. In order to compare the experimen-tal measurements with theoretical susceptibilities, dif-ferent cumulants ratios are constructed (like, C /C = χ /χ , C /C = χ /χ and C /C = χ /χ , etc.). Cu-mulants values also related with the correlation length( ξ ) of the matter created in the collisions as σ = C = ξ ; S = C /C / = ξ . ; κ = C /C = ξ [29, 30]. One ofthe characteristic signatures of the QCD CP is the diver-gence of correlation length which gives a non-monotonicvariation of these cumulant ratios as a function of µ B .The STAR experiment at RHIC has measured the cumu-lant ratios of net-proton, net-charge and net-kaon multi-plicity distributions in Au+Au collisions at broad rangeof collision energies from 200 GeV down to 7.7 GeV,which correspond to a chemical freeze-out µ B range from20 to 420 MeV. Interestingly, the forth-order net-protoncumulant ratio ( κσ = C /C ) for most central 0-5% col-lisions shows a non-monotonic variation as a function ofcollision energy [24].To understand the underlying physics associated withthis measurement, we need to perform careful studies onthe background contributions, such as the effects frominitial volume fluctuations, the detector efficiency andthe effects of centrality selection. Some of the effectsare discussed before [31–36]. Collision centralities can bequantified by impact parameter ( b ) or number of partic-ipant nucleons ( N part ). Unfortunately, in experiment wecannot directly measure such geometrical variables. Asthe particle multiplicities depend on initial geometry, sothe collision centrality in heavy-ion collisions is usuallydetermined by the charge particle multiplicities. The cen-trality resolution is determined by the multiplicities andkinematics of the charged particles used in the centralitydefinition. As the bad centrality resolution will intro-duce larger volume fluctuations and enhance the higherorder cumulants, a good centrality resolution of chargedparticle centrality definition is very important for fluc-tuation analysis. On the other hand, there is so called a r X i v : . [ nu c l - e x ] J u l iautocorrelation effect [31], which indicates that values ofhigher order cumulants will be suppressed if the chargedparticles involved in centrality definition are also used inthe cumulant calculations. To avoid the autocorrelation,particles from different kinematic region are proposed todefine the collision centralities. Experimentally, a ded-icate Event Plane Detector (EPD) [37] has been builtand installed in the froward region (2 . < | η | < .
1) ofthe STAR experiment. The EPD will be used for eventplane and centrality determination in the second phaseof the Beam Energy Scan program (BES-II,2019-2021)at RHIC. It has been proposed that the centrality selec-tion by using Event Plane Detector (EPD) will stronglysuppress the effect of autocorrelation in the fluctuationanalysis [38]. In this work, we will demonstrate the varia-tion of net-proton cumulants values by selecting centrali-ties from different central as well as forward region usingUrQMD model.The paper is organized as follows. In section II, webriefly discuss the UrQMD model used for this analy-sis. In section III, we introduce the observables presentedhere. The centrality selection is discussed in section IV.In section V, we present cumulants ( C - C ) of net-protonmultiplicity distributions for different centrality defini-tion in Au+Au collisions at √ s NN = 7.7, 19.6 and 200GeV using UrQMD model. Finally in section VI, wepresent a summary of this work. II. THE URQMD MODEL
The Ultra Relativistic Quantum Molecular Dynamic(UrQMD) is a microscopic transport model [39, 40]. Inthis model, the space-time evolution of the fireball isstudied in terms of excitation of color strings which frag-ment further into hadrons, the covariant propagation ofhadrons and resonances which undergo scatterings and fi-nally the decay of all the resonances. UrQMD model hasbeen quit successful and widely applied towards heavy-ion phenomenology [40, 41]. Previously, this model hasbeen used to compute several cumulants and studied dif-ferent effects of experimental limitations [31, 32, 42–48].The choice of acceptance window plays an important roleto such studies. The initial distributions of net-baryon( N B ) in rapidity is a consequence of the baryon stoppingphenomenon which strongly depends on collision energy.As a result, the mid-rapidity region for high √ s NN isfree of N B while, at lower √ s NN , most of the N B aredeposited in mid-rapidity region. This collision energydependence baryon stopping phenomenon is dynamicallyincluded in the UrQMD model. More details about theUrQMD model can be found in the reference [39, 40]. Inthis study, we have used six million events per beam en-ergy for Au+Au collisions at √ s NN = 7.7, 19.6 and 200GeV. Using this simulated events, we measure the cu-mulants of event-by-event net-proton ( N p − ¯ p ) multiplicitydistributions within the kinematic acceptance | y | < . . < p T < . c . The same kinematic accep- tance have been used in the net-proton cumulant analysisby STAR experiment [24]. III. OBSERVABLES
In statistics, any distribution can be characterized bydifferent order moments or cumulants ( C n ) and can beexpressed via generating function [49] as, C n = ∂ n ∂α n K ( α ) | α =0 , (1)where K ( α ) = ln( M ( α )) and M ( α ) = (cid:104) e αN (cid:105) are thecumulant and moment generating functions, respectively. N is the event-by-event net-quantity (here net-protonnumber, N p = N p − N ¯ p ) and (cid:104) ... (cid:105) represents an aver-age over events. Then the various order cumulants canbe expressed as, C = (cid:104) N (cid:105) , (2) C = (cid:104) ( δN ) (cid:105) , (3) C = (cid:104) ( δN ) (cid:105) , (4) C = (cid:104) ( δN ) (cid:105) − (cid:104) ( δN ) (cid:105) , (5)where δN = N − (cid:104) N (cid:105) represents the deviation of N fromits average value. (cid:104) ( δN ) m (cid:105) is also called the m -th ordercentral moment. Thermodynamically the cumulants areconnected to the corresponding susceptibilities by C n = ∂ n ln( Z ( V, T, µ )) ∂µ n = V T χ n , (6)The cumulant ratios between different orders can beconstructed to cancel the volume term. This cumulantsratios are measured experimentally and compared to thesusceptibility ratios [19, 50], σ M = C C = χ χ , Sσ = C C = χ χ , κσ = C C = χ χ , (7)With above definitions, we have studied various cumu-lants (up to forth order) and cumulant ratios of event-by-event net-proton multiplicity distributions from UrQMDmodel with different centrality selection.In heavy-ion collisions, we cannot directly measure thegeometrical variables, such as impact parameter. Thecollision centrality in heavy-ion collisions is usually deter-mined through charged particle multiplicities, in whichthe smallest centrality bin is a single multiplicity value.However, for better statistical significance, we report thecumulant results for a wider centrality bins, like 0-5%(most central) or 70-80% (peripheral). But, this par-ticle multiplicities not only reflects the initial geometrybut also depends on different physics process. This alsoii - - - (a) 7.7 GeV h dd N - - -
10 110 (b) 19.6 GeV h STARTPC STAREPD - - -
10 110 (c) 200 GeV - +h + Q=h + P + Q-P
FIG. 1. (Color online) The dN/dη distributions for charged particle, proton and (number of charged particles - protons number)multiplicity in minimum bias Au+Au collisions at √ s NN = 7.7, 19.6 and 200 GeV. The bands at different color correspond todifferent pesudorapidity ( η ) regions for centrality selection. correspond that the measured observables N ch and geo-metrical variable ( b ) is not one-to-one correspondence. Afixed N ch may come from different initial geometry. Thisvariation even become large when we use wider 5% or10% centrality class. To reduced the variation for widercentrality bins, a so called Centrality Bin Width Cor-rection (CBWC) technique is applied in cumulant anal-ysis [31]. The techniques for this corrections are follows.We first calculate different cumulants ( C n ) in each bin ofunit multiplicity and then weight the cumulants by thenumber of events in each bin over a desired centralityclass. The method can be expressed as, C n = (cid:80) i n i C n i (cid:80) i n i = (cid:88) i ω i C n i , (8)where C n i is the cumulant value measured in the i th mul-tiplicity bin. n i and ω i (= n i / (cid:80) i n i ) are the numberof events and the weight factor for i th multiplicity bin.It was shown that the CBWC can effectively suppressthe volume fluctuations within a wide centrality bin [31].However, even CBWC is applied, there could be stillresidual volume fluctuations if the centrality resolutionis not good. Another centrality selection related artifactis so called autocorrelation effect, which is due to the cor-relations between particles used in the centrality selectionand the cumulant calculations. For example, a typical au-tocorrelation effect is caused by the fact that some of theparticles involved in the cumulant analysis are also usedfor the centrality selection. In the STAR experiment,to avoid the autocorrelation effect in net-proton andnet-charge fluctuation measurements, collision centrali-ties are carefully selected, the so called ”refmult-3” and”refmult-2”, respectively. In refmult-3 definition, the col-lision centrality is determined by the measured charged particle multiplicities ( N ch ) within | η | < . . < | η | < .
0. In BES-II, a dedicate forward EventPlane Detector (EPD) with coverage 2 . < | η | < . η ) distributions( dN/dη ) of charged particle, proton ( p ) and N ch − p mul-tiplicities for the min-bias Au+Au collisions at √ s NN =7.7, 19.6 and 200 GeV. The bands at different η re-gions correspond to the acceptance of STAR Time Pro-jection Chamber (TPC) and EPD, respectively. Theparticle pseudorapidity ( η ) distribution is not uniformthrough out the acceptance for all energies. At the for-ward pseudo-rapidity region from 2 to 5 unit, one canfind lots of spectator contributions at low energies. Wecan see two peaks structures around | η | = 7-8 for 200GeV, which correspond to the spectator protons. As wego towards low beam energies, the peaks shifted towardscentral η region, like for 7.7 GeV the peak is around | η | = 3.5. For the η window from 2.1 to 5.1 unit at 200GeV, the charged particles are mostly contributed fromthe produced particles. However, if we go towards lowerbeam energies, charged particles in that range are dom-inated by spectator protons, as most of the spectatorprotons are around beam rapidity,Figure 2 shows the beam rapidity for different center ofmass energies and the corresponding η values at p T = 0 . STAR TPC range: 0<| η |<1STAR EPD range: 2.1<| η |<5.1 P s eudo -r ap i d i t y ( η ) Proton Rapidity (Y)
Beam rapidity at √ s NN (GeV/c) η at p T = 0.2 GeV/c FIG. 2. (Color online) Beam rapidity values as a function ofcenter of mass energy. Central and forward detector regionfor centrality selection using charged-particle multiplicity rep-resented in band. The relation of p T , pseudo-rapidity ( η ) andrapidity ( y ) is p T = m / (cid:112) sinh η/ sinh y −
1, where the m is the particle rest mass. GeV/c. We found that at low energies between 3 and27 GeV, the protons with beam rapidity and p T = 0 . η coverage of the STAR EPD(2 . < | η | < . IV. CENTRALITY SELECTION
In this work we select centralities using charge particlemultiplicities from different η region. The definitions ofdifferent centrality selections are listed in Table I. Wefurther subdivide the forward region ”Fwd-All” rangein three region : (a) Fwd-1, pseudorapidity acceptance2 . < | η | <
3, (b) Fwd-2 within 3 < | η | <
4, and (c)Fwd-3 within 4 < | η | <
5. Figure 3 shows the min-imum bias charged particle multiplicity distributions at √ s NN = 7.7, 19.6 and 200 GeV from different acceptanceregions as listed in Table I. We select 9 different central-ity classes: 0-5% (top central), 5-10%, 10-20%, 20-30%, 30-40%, 40-50%, 50-60%, 60-70% and 70-80% from thearea percentile of the multiplicity distributions. We canfind that at √ s NN = 7.7 and 19.6 GeV the multiplicitydistributions at forward region behaves differently thancentral region. This is mainly caused by the spectatorprotons, which are positively correlated with impact pa-rameter and are opposite to the trend of the producedcharged particle multiplicity distributions. As shown inthe Fig. 3, if we exclude the protons from forward region,then the trend of the distributions looks like a hose-tailshaped similar to the central ones.Figure 4 shows the two dimension correlation plotsbetween the charged particle multiplicity distributionsin different acceptance and the impact parameter. Itwas found that at lower energies the multiplicities within2 . < | η | < . √ s NN = 7.7, 19.6 and 200 GeV us-ing different centrality definitions. To compare the cen-trality resolution between different centrality definitions,we define a quantity Φ(= σ b X /σ b centrality-b ) as shown inFig. 6, where the ”X” is different centrality definitions us-ing N ch as discussed before. Here, σ b X is the variance inimpact parameter distribution in a centrality class from”X” centrality definition whereas the σ b centrality-b repre-sents the variance in impact parameter distribution withthe centrality defined by the impact parameter ( b ) itself.So larger values in Φ corresponds to a poorer resolutionthan smaller Φ values.We find that the resolution in ”refmult-3” definitionis better for all energies followed by ”refmult-2” and”refmult-1”. At √ s NN = 7.7 and 19.6 GeV, the resolutionbecomes poorer as we go towards larger η region. Thisis due to the spectator contributions. It is also observedthat the resolution get improved if we select centralityfrom forward region by excluding protons. But still Φvalue is large at 7.7 GeV because of smaller number ofproduced particles in that region. We can also observedthat the centrality resolutions are always better in centralcollisions than those from peripheral collisions. V. RESULTS
In this study, we compare the cumulants and theirratios of event-by-event net-proton multiplicity distri-butions within the kinematic acceptance | y | < . . < p T < . c for different centrality definitionsas discussed in the previous section. Figure 7 shows theevent-by-event net-proton multiplicity distributions forAu+Au collisions at √ s NN = 7.7, 19.6 and 200 GeV forthree centralities (0-5%, 30-40% and 60-70%). We canfind that the mean and width are larger for central thanperipheral collisions. The mean values of net-proton dis-tributions shifted towards zero as the energy increases. Identify Definitioncentrality-b impact parameterrefmult-1 N ch within | η | < . N ch within 0 . < | η | < . N ch − p within | η | < . N ch within 2 . < | η | < . N ch within 2 . < | η | < . N ch within 3 . < | η | < . N ch within 4 . < | η | < . − p N ch − p within 2 . < | η | < . - - - - - - (a) 7.7 GeV N o r m a li z ed d i s t r i bu t i on s - - - - - -
10 110 (b) 19.6 GeV
Charge particle multiplicity refmult-1refmult-2refmult-3Fwd-All - - - - - -
10 110 (c) 200 GeV
Fwd-1Fwd-2Fwd-3Fwd-All - p
FIG. 3. (Color online) Normalized distributions for charged particle multiplicities in different η -window in Au+Au collisionsat √ s NN = 7.7, 19.6 and 200 from UrQMD model.FIG. 4. (Color online) Correlations between multiplicities in different η window used for the centrality definitions and impactparameter in Au+Au collisions at √ s NN = 7.7, 19.6 and 200 GeV from UrQMD model. i FIG. 5. (Color online) The impact parameter (b) distributions in different centrality definitions for three different centralityclasses ((a) 0-5%, (b) 20-30% and (c) 60-70%) at √ s NN = 7.7, 19.6 and 200 GeV. -
80 60 -
70 50 -
60 40 -
50 30 -
40 20 -
30 10 -
20 5 -
10 0 - Φ = σ X / σ c en t r a li t y - b -
80 60 -
70 50 -
60 40 -
50 30 -
40 20 -
30 10 -
20 5 -
10 0 - Centrality class (%) -
80 60 -
70 50 -
60 40 -
50 30 -
40 20 -
30 10 -
20 5 -
10 0 -
200 GeV refmult-1refmult-2refmult-3Fwd-1Fwd-2Fwd-3Fwd-AllFwd-All - p
FIG. 6. (Color online) The centrality dependence of the Φ(= σ b X /σ b centrality-b ) of impact parameter distributions for Au+Aucollisions at √ s NN = 7.7, 19.6 and 200 GeV in UrQMD model with different centrality definitions. ii
10 (1) 7.7 GeV (a) 0-5 %0 20 40 6010
10 (2) 19.6 GeV N u m be r o f e v en t s
10 (3) 200 GeV
10 (b) 30-40 %0 20 40 6010 Net-proton
10 (c) 60-70 %0 20 40 6010
10 < 2.0 GeV/c T
10 refmult-1refmult-2refmult-3Fwd-AllFwd-All - p
FIG. 7. (Color online) Event-by-event distributions of net-proton multiplicity distribution for Au+Au collisions at √ s NN =7.7, 19.6 and 200 GeV for different centrality selection methods. At 200 GeV, the net-proton distributions from all thecentrality sets are very similar. At 7.7 GeV, the net-proton distributions for ”Fwd-All” centrality case lookscompletely different. This is mainly caused by the distor-tion of the spectator protons in the centrality definition”Fwd-All”. As shown in Fig. 5, due to the positive cor-relations between the number of spectator protons andthe impact parameter, the impact parameter distributionfrom ”Fwd-All” contains more peripheral events (large bvalues) in 0-5% centrality class than that of 60-70% cen-trality class. One needs to keep in mind that the rawnet-proton distributions shown in Fig. 7 are not directlyused to calculate various order cumulants and needs toapply the CBWC to suppress volume fluctuations in awide centrality bin.Figure 8 shows the centrality dependence of cumulants( C to C ) of net-proton multiplicity distributions inAu+Au collisions at √ s NN = 7.7, 19.6 and 200 GeV. Thecollision centralities are represented by the average num-ber of participant nucleons ( (cid:104) N part (cid:105) ). We use a MonteCarlo Glauber model [51, 52] to estimate N part similarto conventional cumulant analysis [22–24, 26–28]. Thestatistical uncertainties are obtained using analytical er-ror propagation method [53–55]. The statistical uncer-tainties mainly depends on the variance of the respectivedistributions and the number of events. All the cumu-lants show a linear dependence as a function of (cid:104) N part (cid:105) .However C and C at √ s NN = 7.7 GeV show an oppo-site trend for ”Fwd-All” centrality case. This is because at 7.7 GeV within 2 . < | η | < . N ch percentile corresponds to peripheral collisions notcentral collisions. However, if we substract protons from”Fwd-All” centrality definitions then the C matches toother cases. We also observed that the cumulants val-ues based on ”refmult-2” and ”refmult-3” centrality def-initions are consistent for all three energies. For lowerbeam energies ( √ s NN = 7.7 and 19.6 GeV) higher or-der cumulants ( C and C ) from ”Fwd-All” and ”Fwd-All-p”centrality definition are deviated from the cumu-lants using ”refmult-2” and ”refmult-3” centrality defi-nition. This is because of the poor centrality resolutionin ”Fwd-All” and ”Fwd-All-p”centrality definitions dueto smaller multiplicity distribution and/or spectator con-tribution. For √ s NN = 19.6 GeV, we found the valuesof higher order cumulants from ”refmult-1” are smallerthan the results from ”Fwd-All-p” centrality definition.This is caused by the autocorrelation effect in ”refmult-1” centrality, as the centrality resolution of ”Fwd-All-p”is better than the case of ”refmult-1”. Meanwhile, at 19.6GeV, we found the higher order cumulants from ”refmult-2” and ”refmult-3” centrality cases are smaller than theresults from ”Fwd-All-p”. We will discuss more later inthis chapter (Fig. 10-12) showing that this is due to bet-ter centrality resolution in the refmult-2/refmult-3 thanthe ”Fwd-All-p” case and not caused by the autocorre-lation effects in refmult-2 and refmult-3 centrality defini-tions. At √ s NN = 200 GeV, the cumulants from forwardiii (a) C (b) C (c) C y/20y/2 -200 0 200 400 (d) C y/50y/2 C u m u l an t s ( C n )
200 GeV T <2.0 GeV/c|y|<0.5 Average Number of Participant Nucleons
FIG. 8. (Color online) Centrality dependence of cumulants ( C ∼ C ) of net-proton multiplicity distributions within | y | < . . < p T < . c for Au+Au collisions at √ s NN = 7.7, 19.6 and 200 GeV for different centrality selection methods.The third and fourth order cumulant from ”Fwd-All” and ”Fwd-All-p” centrality definitions at √ s NN =7.7 GeV are scaledwith different factors to compare with other cases. (a) C /C y/5 1 2 3 4 (b) C /C y/5 0 5 10 (c) C /C y/50y/2 1 1.5 2 C u m u l an t r a t i o s ( C m / C n ) T <2.0 GeV/c|y|<0.5 5 10 15 20 0 100 200 300
200 GeV net-protonAverage Number of Participant Nucleons
10 110 N o r m a li z ed d i s t r i bu t i on s Charge particle multiplicityrefmult-3Fwd-1Fwd-1 - pFwd-All - p |<5.1) h -p (1.4<| ch N |<5.1) h -p (1.2<| ch N FIG. 10. (Color online) Normalized distributions for chargedparticle multiplicities in different η -window in Au+Au colli-sions at √ s NN = 19.6 from UrQMD model. centrality definitions (”Fwd-All” and ”Fwd-All-p”) areconsistent with the results from the centralities defined atcentral region (”refmult-2” and ”refmult-3”). This com-parison indicates that the autocorrelation effects in thecentrality definitions of ”refmult-2” and ”refmult-3” arenot significant within statistical uncertainties in Au+Aucollisions at √ s NN = 200 GeV within UrQMD model cal-culations.Figure 9 shows the (cid:104) N part (cid:105) dependence of cumulantratios ((a) C /C = σ /M , (b) C /C = Sσ and (c) C /C = κσ ) of net-proton multiplicity distributions inAu+Au collisions at three different collision energies. At √ s NN = 200 GeV, all the cumulants ratios in differentcentrality selection sets are consistent with each other.As we go towards lower energies, the effect of centralityselection start to play important role and cumulant ratiosare deviating from each other. The results in centralitydefinition sets from forward region show more deviatesdue to the poor centrality resolution caused by spectatorproton contributions.As shown in Fig. 8 and 9, the higher order cumulantsand cumulant ratios for ”refmult-2” and ”refmult-3” cen-trality cases are smaller than the results from forwardregion centrality definition ”Fwd-All-p”. We argue thisis due to the better centrality resolution of ”refmult-2”and ”refmult-3” centrality definitions than the case of”Fwd-All-p”. In Fig. 10, we show the charged particlemultiplicity distributions in various η -window in Au+Aucollisions at √ s NN = 19.6 from UrQMD model. We foundthat the charged particle multiplicity from ”refmult-3”centrality is much larger than the forward centrality def- inition ”Fwd-All-p”. This will cause larger volume fluc-tuations with ”Fwd-All-p” centrality definition than the”refmult-3” case. To justify this argument, two new cen-tralities were defined for Au+Au collisions at √ s NN =19.6 GeV from UrQMD with wider η range in the forwardregion, which are charged particle multiplicities (exclud-ing protons) within 1 . < | η | < . . < | η | < . C /C and C /C at non-central Au+Au collisions be-tween ”refmult-3” and ” N ch − p (1 . < | η | < . η distribution in the forward region andcan distort the centrality resolution even with a smallfraction.Figure 12 shows the energy dependence of the cumu-lant ratios ( C /C , C /C and C /C ) of net-proton mul-tiplicity distributions in Au+Au collisions at √ s NN = 7.7,19.6 and 200 GeV for three centrality definition methods.The ”centrality-b” is used to represent the centrality def-inition by using impact parameter. The centrality binwidth correction has been applied to suppress volumefluctuations within wide centrality bin as discussed insection III. For ”centrality-b”, we first calculated cumu-lants in each 0.1 fm bin and then weight the cumulantsby the number of events in each 0.1 fm bin over a desiredcentrality class as discussed in equation 8. Based on theUrQMD model study, we found that the results in 0-5%most central Au+Au collisions from the ”refmult-3” and”refmult-2” centralities are consistent with the resultsfrom ”centrality-b” definition, which is directly relatedto the initial collision geometry. This comparison furtherconfirms that the ”refmult-3” and ”refmult-2” centralitydefinitions are robust to be used for studying the net-proton fluctuations in heavy-ion collisions. VI. SUMMARY
The cumulants of net-proton multiplicity distributionsare important observables to probe the signature of theQCD critical point in heavy-ion collisions. In this work,we studied the centrality dependence cumulants (up tofourth order) and the cumulants ratios ( C /C , C /C and C /C ) of net-proton multiplicity distributions at (a) C C u m u l an t s ( C n ) (b) C (c) C (d) C (e) C /C C u m u l an t r a t i o s ( C m / C n )
0 100 200 300
Net-proton
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