Beam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC
aa r X i v : . [ nu c l - e x ] J un Beam Energy Dependence of Higher Moments ofNet-proton Multiplicity Distributions in Heavy-ionCollisions at RHIC
Xiaofeng Luo ∗ (for the STAR Collaboration) Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, CentralChina Normal University, Wuhan 430079, ChinaE-mail: [email protected]
We present various order moments (Variance ( s ), Skewness( S ), Kurtosis( k )) of net-protonmultiplicity distributions in Au+Au collisions from the first phase of RHIC beam energy scan( √ s NN =7.7 −
200 GeV). The measurements are carried out by the STAR detector at mid-rapidity( | y | < .
5) and within transverse momentum range 0 . < p T < . / c . The product of mo-ments ( S s and ks ) are predicted to be sensitive to the correlation length and connected to ratiosof baryon number susceptibility. We observe deviations below Poisson expectations in the mostcentral collisions for all of the energies. These results are compared with a transport model to un-derstand the effects not related to critical physics. We also discuss the error estimation methodsand the techniques to suppress centrality resolution effect in the moment analysis. The 8th International Workshop on Critical Point and Onset of DeconfinementMarch 11 - 15, 2013Napa, CA, USA ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ eam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC
Xiaofeng Luo
1. Introduction
Exploring the Quantum Chromodynamics (QCD) phase structure is one of the most importantgoal of Beam Energy Scan (BES) program at Relativistic Heavy-Ion Collider (RHIC) [1]. The firstprinciple Lattice QCD calculation predicted that the transition from hadronic to partonic matter atzero m B is a smooth crossover [2], while at finite m B region is a first order phase transition [3]. Theend point of the first order phase transition boundary is so called the QCD Critical Point (CP). Thereare large uncertainties for Lattice QCD calculation in determining the first order phase boundaryas well as the QCD critical point [4] in the QCD phase diagram due to the sign problem at finite m B region [5]. In the first phase of the BES program, the Au+Au colliding energy was tuned from200 GeV down to 7.7 GeV and the corresponding baryon chemical potential ( m B ) coverage is fromabout 20 to 450 MeV [1]. This allows us to map a broad region of QCD phase digram (temperature( T ) versus baryon chemical potential ( m B )). Thus, it provides us a good opportunity to look for thefirst order phase boundary and search for the CP at RHIC.In heavy-ion collisions, moments (Variance ( s ), Skewness( S ), Kurtosis( k )) of conservedquantities, such as net-baryon, net-charge and net-strangeness, are predicted to be sensitive to thecorrelation length of the hot dense matter created in the collisions [6, 7, 8] and connected to the var-ious order susceptibilities computed in the Lattice QCD [9, 10, 11, 12] and Hardon Resonance Gas(HRG) [13] model. For instance, the higher order cumulants of multiplicity ( N ) distributions areproportional to the high power of the correlation length ( x ) as third order cumulant C = S s = < ( d N ) > ∼ x . and fourth order cumulant C = ks = < ( d N ) > − ( d N ) > ∼ x [6], where the d N = N − < N > and < N > is the mean multiplicities. The moment products, ks and S s , arealso related to the ratios of various order susceptibilities, such as ratios of baryon number suscep-tibilities can be compared with the experimental data as ks = c ( ) B / c ( ) B and S s = c ( ) B / c ( ) B . Theratios cancel out the volume effect. Theoretical calculations have shown that the experimentallymeasurable net-proton number (number of protons minus number of anti-protons) fluctuations mayreflect the fluctuations of the net-baryon number at CP [14]. Thus, the net-proton number fluctua-tions are measured as the approximation to the net-baryon fluctuations.The proceedings is organized as follows. In Section 2, we discuss the centrality resolutioneffect. In section 3, we present different methods of estimating the statistical errors for momentanalysis. In section 4, we present STAR preliminary measurements for higher moments of event-by-event net-proton multiplicity distributions from the first phase of the BES program at RHIC.Finally, we present a summary of the work.
2. Centrality Resolution Effect (CRE)
The collision centrality and/or the initial collision geometry can be represented by many pa-rameters in heavy-ion collisions, such as impact parameter b , number of participant nucleons( N part ) and number of binary collisions ( N coll ). These initial geometry parameters are not inde-pendent but are strongly correlated with each other. Experimentally, the collision centrality is de-termined from a comparison between experimental measures such as the particle multiplicity andGlauber Monte-Carlo simulations [15]. Particle multiplicity, not only depends on the physics pro-cesses, but also reflects the initial geometry of the heavy-ion collision. This indicates that relation2 eam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC Xiaofeng Luo between measured particle multiplicity and initial collision geometry does not have a one-to-onecorrespondence and there are fluctuations in the particle multiplicity even for a fixed initial ge-ometry. Thus, it could have different initial collision geometry resolution (centrality resolution orvolume fluctuation) for different centrality definitions with particle multiplicity. This may affectmoments of the event-by-event multiplicity distributions. It is natural to expect that the more parti-cles are used in the centrality determination, the better centrality resolution and smaller fluctuationsof the initial geometry (volume fluctuations) we get [16]. |<0.5 h | |<1.0 h | |<1.5 h | |<2.0 h | part NAu+Au CollisionsNet-proton<0.8(GeV/c) T part N s S Figure 1: (Color Online) Centrality dependence of the momentsproducts S s of net-proton multiplicity distributions for Au+Aucollisions at √ s NN =7.7, 11.5, 19.6, 27, 39, 62.4, 200 GeV inAMPT string melting model. Different symbols represent dif-ferent collision centrality definition. |<0.5 h | |<1.0 h | |<1.5 h | |<2.0 h | part NAu+Au CollisionsNet-proton<0.8(GeV/c) T part N sk Figure 2: (Color Online) Centrality dependence of the momentsproducts ks of net-proton multiplicity distributions for Au+Aucollisions at √ s NN =7.7, 11.5, 19.6, 27, 39, 62.4, 200 GeV inAMPT string melting model. Different symbols represent dif-ferent collision centrality definition. To verify the CRE in the moment analysis, we use the charged kaon and pion multiplicity(as for the real data analysis to avoid autocorrelation effect) produced in the final state within | h | < . , , . | y | < . . < p T < . S s , ks ) of net-proton multiplicity distributions for four different h rangeof charged kaon and pion used to determine the centrality. We observe significant difference formoment products ( S s , ks ) for the different h range of the centrality definition. The behavior canbe understood as due to different contribution from volume fluctuations (increasing centrality reso-lution) arising from different centrality definition. When we increase the h range ( | h | < , . , ) ,the values of S s and ks will decreases, which indicates the centrality resolution effect will en-hance the moments values of net-proton distributions. On the other hand, the moment products( S s , ks ) are closer to the results with centrality directly determined by number of participant nu-3 eam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC Xiaofeng Luo cleons ( N part ) when the h range is larger. It confirms that the centrality resolution effect can besuppressed by having more particles to determine the centrality.Fig. 3 shows the energy dependence of moment product ( S s , ks ) of net-proton multiplicitydistributions for three different centralities ( − , − , − ) with four different h coverage in Au+Au collisions. We can find that the ks is more sensitive to the CRE than S s ,and the CRE has a larger effect in the peripheral collision as well as at low energies. Thus, weshould use a lager h coverage in the centrality definition for the real experimental moment analysisto reduce the centrality resolution effects. |<0.5 h | |<1.0 h | |<1.5 h | |<2.0 h | part N AMPT StringMelting
Net-protonAu+Au Collisions <0.8(GeV/c) T (GeV) NN s0-5% 30-40% 70-80% s S sk Figure 3: (Color online) Energy dependence of the moments products ( S s , ks ) of net-proton multiplic-ity distributions for Au+Au collisions at centralities (0 − , − , − √ s NN =7.7, 11.5,19.6, 27, 39, 62.4, 200GeV in AMPT string melting model. Different symbols represent different collisioncentrality definition.
3. Statistical Error Estimation
Since we don’t know exactly the underlying probability distributions for proton and anti-proton, it is not accurate to estimate the statistical error with standard error propagation with respectto the number of proton and anti-proton. Several statistical methods (Delta theorem, Bootstrap [18],Sub-group) of error estimation in the moment analysis and their comparisons will be discussed bya Monte Carlo simulation. For simplicity, skellam distribution is used to perform the simulation.If protons and anti-protons multiplicity follow independent Poissonian distributions, the net-protonmultiplicity will follow the skellam distribution, which is expressed as: P ( N ) = ( M p M p ) N / I N ( p M p M p ) exp [ − ( M p + M p )] , where I N ( x ) is a modified Bessel function, M p and M p are the measured mean values of protonsand anti-protons. If the net-proton follows the skellam distribution, then we have, S s = C / C =( M p − M p ) / ( M p + M p ) and ks = C / C =
1, which then provides the Poisson expectations for4 eam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC
Xiaofeng Luo the moment products. To perform the simulation, we set the two mean values of the skellamdistributions as M p = 4.11, M p = 2.99. Then, we generate random numbers as per the skellam dis-tribution. The details of Delta theorem error estimation method for moment analysis can be foundin [19]. The bootstrap method [18] is based on the repeatedly sampling with the same statisticsof the parent distribution and the statistical errors can be obtained as the root mean square of theobservable from each sample. In the sub-group method, we randomly divide the whole sample intoseveral sub-groups with same statistics and the errors are obtained as the root mean square of theobservable from each sub-group. In our simulation, we set 200 bootstrap times and 5 sub-groupsfor bootstrap and sub-group methods, respectively. Fig. 4 shows the error estimation comparison -2024 -2024 Delta TheoremBootstrapSub-group -2024 Order s k Figure 4: (Color online) ks for 50 samples that independently and randomly generated from the skellamdistribution with different number of events (0.01, 0.1, 1 million). The dashed lines are expectations value 1for the skellam distribution. between Delta theorem, Bootstrap and Sub-group methods for ks of skellam distribution. Foreach method, fifty independent samples are sampled from the skellam distribution with statistics0.01 , 0.1 and 1 million, respectively. The probability for the value staying within ± s aroundexpectation is about 68 .
3% and it means error bars of 33 out of 50 points should touch the expectedvalue(dashed line at unity) in Fig. 4. We find that the results from the Delta theorem and Boot-strap method show similar error values and satisfies the above criteria, while the random sub-groupmethod over estimates the statistical errors. It indicates that the Delta theorem and Bootstrap errorestimation methods for the moment analysis are reasonable and can reflect the statistical propertiesof moments.
4. Results and Discussion
The results presented here are obtained from the Au+Au collisions at √ s NN =7.7, 11.5, 19.6,27, 39, 62.4 and 200 GeV in the first phase of the BES program at RHIC and p + p collisions at √ s NN =62.4 and 200 GeV. The protons and anti-protons are identified at midrapidity ( | y | < .
5) andwithin the transverse momentum range 0 . < p T < . dE / dx ) of charged particles measured by the Time Projection Chamber (TPC) of STAR detector.5 eam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC Xiaofeng Luo
To suppress autocorrelation effects between measured net-proton fluctuations and centrality definedusing charged particles, a new method of centrality selection is used in the net-proton momentanalysis. The new centrality is determined from the uncorrected charge particle multiplicity byexcluding the protons and anti-protons within pseudorapdity | h | < . (a) 7.7 GeV (b) 11.5 GeV (c) 19.6 GeV (d) 27 GeV STAR Preliminary0 100 200 300 (e) 39 GeV (f) 62.4 GeV (g) 200 GeV s k )/Poisson s (S Au+Au Collisions
Net-proton <0.8 (GeV/c) T |y|<0.5 part N Figure 5: (Color Online) Centrality dependence of S s /Poisson and ks of net-proton distributions forAu+Au collisions at √ s NN =7.7, 11.5, 19.6, 27, 39, 62.4 and 200 GeV. The error bars are statistical and capsare systematic errors. Figure 5 shows the ratios of the cumulants, which are connected to the moment products as S s = C / C and ks = C / C . The various order cumulants ( C − C ) can be obtained from thenet-proton multiplicity distributions and corrected for the finite centrality bin width effect [20]. Itis observed that the ks and the S s values normalized to Poisson expectations are below unityfor √ s NN above 11.5 GeV and above unity for 7.7 GeV in Au+Au collisions. The ks showslarger deviation from Poisson expectations than S s . The statistical error shown in Fig. 5 areobtained by the Delta theorem method and the systematical errors are estimated by varying thetrack quality condition and particle identification criteria. The data presented here may allow us toextract freeze-out conditions in heavy-ion collisions using QCD based approaches [21].Figure 6 shows the energy dependence of ks and S s of net-proton distributions for fourcentralities (0-5%, 5-10%, 30-40% and 70-80%) in Au+Au collisions. The bottom panel of Fig.6shows S s values normalized to the corresponding Poisson expectations. The ks and normalized S s values are close to the Poisson expectations for Au+Au collisions at √ s NN =39 , 62.4 and 200GeV. They show deviation from Poisson expectations for the 0-5% central Au+Au collisions below √ s NN =39 GeV. The UrQMD model [22] results are also shown in the Fig. 6 for 0-5% centrality tounderstand the non-CP effects, such as baryon number conservation and hadronic scattering. TheUrQMD calculations show a monotonic decrease with decreasing beam energy.For the preliminary experimental results shown here and also in the QM2012 proceedings [23],the collision centralities are determined from the uncorrected charge particle multiplicity by exclud-ing the protons and anti-protons within pseudorapdity | h | < .
5. Based on the model simulationresults, there could be significant centrality resolution effects in mid-central and peripheral col-6 eam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC
Xiaofeng Luo
Au+Au Poisson0-5% <0.8 (GeV/c),|y|<0.5 T p+p STAR Preliminary s S s k (GeV) NN s ) / P o i ss on s ( S Figure 6: (Color online) Energy dependence of ks and S s for net-proton distributions for four collisioncentralities (0-5%, 5-10%, 30-40% and 70-80%) measured at STAR. The results are compared to UrQMDmodel calculations and p + p collisions at √ s NN =62.4 and 200 GeV. The lines in top panel are the Poissonexpectations and in the bottom panel shows the S s normalized to the corresponding Poisson expectations. lisions at low energies. The amount of particles used in the centrality determination still can beincreased by extending the pseudorapdity coverage to the | h | <
1, the current acceptance limit ofthe STAR TPC. Thus, to obtain more precise results with less centrality resolution effect, we willuse the pseudorapdity coverage | h | < | h | < | h | < .
7. This upgrade is expected to becompleted in the year 2017, which can be used for data taking in the second phase of BES atRHIC. It will allow us to define the centralities with a much wider h range to further suppress thecentrality resolution effects.
5. Summary
We have presented the beam energy ( √ s NN =7.7 −
200 GeV) and centrality dependence for thehigher moments of net-proton distributions in Au+Au collisions from the first phase of the BESprogram at RHIC. It is observed that the ks and S s values are close to the Poisson expectationfor Au+Au collisions at √ s NN =39 , 62.4 and 200 GeV. They show deviation from Poisson expecta-tions in the 0-5% central Au+Au collisions below √ s NN =39 GeV. The UrQMD calculations show amonotonic decrease with decreasing beam energy. We also need more statistics to get precise mea-7 eam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC Xiaofeng Luo surements below 19.6 GeV and additional data at √ s NN =15 GeV. These are planned for the secondphase of the BES program at RHIC. The centrality resolution effect in moment analysis has beenpointed out and large h coverage will be used in centrality definition to suppress this effect. Threestatistic error estimation methods and their comparisons in moment analysis have been discussedthrough a Monto Carlo simulation. Acknowledgments
The work was supported in part by the National Natural Science Foundation of China undergrant No. 11205067 and 11135011. CCNU-QLPL Innovation Fund(QLPL2011P01) and ChinaPostdoctoral Science Foundation (2012M511237).
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