Event-by-Event Efficiency Fluctuations and Efficiency Correction for Cumulants of Superposed Multiplicity Distributions in Relativistic Heavy-ion Collision Experiments
CChinese Physics C Vol. 42, No. 10 (2018) 104001
Event-by-event efficiency fluctuations and efficiency correction forcumulants of superposed multiplicity distributions in relativisticheavy-ion collision experiments * Shu He, Xiaofeng Luo Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China
Abstract:
We performed systematic studies on the effects of event-by-event efficiency fluctuations on efficiencycorrection for cumulant analysis in relativistic heavy-ion collision experiments. Experimentally, particle efficiencies ofevents measured under different experimental conditions should be different. For fluctuation measurements, the finalevent-by-event multiplicity distributions should be the superposed distributions of various type of events measuredunder different conditions. We demonstrate efficiency fluctuation effects using numerical simulation, in which weconstruct an event ensemble consisting of events with two different efficiencies. By using the mean particle efficiencies,we find that the efficiency corrected cumulants show large deviations from the original inputs when the discrepancybetween the two efficiencies is large. We further studied the effects of efficiency fluctuations for the cumulantsof net-proton distributions by implementing the UrQMD events of Au+Au collisions at √ s NN = 7 . V z ). When the efficiencies fluctuate dramatically within the studied event sample, the effects ofefficiency fluctuations have significant impacts on the efficiency corrections of cumulants with the mean efficiency. Wefind that this effect can be effectively suppressed by binning the entire event ensemble into various sub-event samples,in which the efficiency variations are relatively small. The final efficiency corrected cumulants can be calculated fromthe weighted average of the corrected factorial moments of the sub-event samples with the mean efficiency. Keywords:
QCD critical point, QCD phase diagram, heavy-ion collisions, higher moments
PACS:
DOI:
The major physics goals of heavy-ion collisionexperiments are to explore the phase structure ofstrongly interacting nuclear matter and to studythe properties of quark-gluon plasma (QGP) [1–8]. The QCD phase structure can be displayed ina two-dimensional phase diagram with the temper- ature T versus the baryon chemical potential µ B .Lattice QCD calculations show that the transitionfrom a hadronic phase to a QGP phase at zero µ B is a crossover [9] and QCD-based models suggestthat at larger µ B , the transition is of the first or-der [10, 11]. If these model calculations at finite µ B are correct, there should exist an endpoint of Received 16 April 2018 ∗ Supported by the MoST of China 973-Project No.2015CB856901, NSFC (11575069)1) E-mail: xfl[email protected] © a r X i v : . [ phy s i c s . d a t a - a n ] S e p hinese Physics C Vol. 42, No. 10 (2018) 104001 the first-order phase transition line, which is the so-called QCD critical point. Due to the sign problem,the first-principle Lattice QCD calculation becomesvery difficult at µ B > µ B , T ) can be created to accessbroad regions of the QCD phase diagram.One of the most important experimental meth-ods of searching for the critical point is the mea-surements of the event-by-event fluctuations of con-served quantities, such as the net-baryon ( B ) [24–26], net-charge number ( Q ) [27, 28] and net-strangeness ( S ) number [29–47] (And their proxyobservables net-kaon [48] and net-proton numberfluctuations). The fluctuation observables are sen-sitive to the correlation length ξ , which will divergenear the QCD critical point. The Solenoidal Trackerat the RHIC (STAR) experiment has measured thefluctuation of the net-proton multiplicity (which is aproxy to net-baryon) in Au + Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4, and 200 GeV,which is taken from the first phase of the RHICbeam energy scan program. The measured forthorder net-proton cumulants ration ( κσ = C /C )of 5% most central events show a non-monotonicenergy (or µ B ) dependence [49–51].To understand the underlying physics associatedwith this measurement, we need to perform care-ful studies on the background contributions, suchas the detector efficiency and acceptance effects,volume fluctuations, and other noncritical parame-ters [52–60]. Owing to the finite detector efficiency,efficiency correction is applied and plays a very im-portant role in cumulant analysis. Generally, theefficiencies are obtained by Monte Carlo (MC) em-bedding technique [61]. This allows for the deter-mination of the efficiency, which is the ratio of thematched MC tracks number and the number of in-put tracks. It contains the effects of both the re-constructed tracking efficiency and acceptance. Inprinciple, the properties of the efficiency, includ-ing fluctuations and acceptance, can be obtained from embedding. However, the embedding sampleis only with a limited number of events, and usu-ally, a small fraction of the real data. Thus, withlimited statistics of embedding data, it is difficultto capture every detail and property of the entiredata sample. The efficiencies are obtained by tak-ing the average within an event sample under dif-ferent experimental conditions, such as variation ofthe collision vertex position and the detector per-formance. The final event-by-event multiplicity dis-tributions should be the superposed distributionsof various types of events measured under differentexperimental conditions. For real data analysis, weusually use the mean efficiency to perform the effi-ciency corrections for cumulants. This is not prob-lematic if the mean efficiency is used, assuming theefficiency variation is relatively small. However, theproblem is that higher order cumulants are sensi-tive statistics and they are influenced by the bulkproperties of events. The average quantity of eventensemble will reduce the details of event-by-eventdiscrepancy, which could be crucial to the cumu-lants analysis. Experimentally, one needs to im-plement careful data quality assurance to performprecise measurement studies on efficiencies for datasamples.In our work, we demonstrate the effects of ef-ficiency fluctuations on efficiency correction for cu-mulants using the average efficiency and provide aneffective approach to suppress this effect in futuredata analysis. This is simulated by injecting particletracks from UrQMD events into the STAR detectoracceptance. The efficiency fluctuations result fromthe fluctuating collision vertex position and the set-ting different degree of the asymmetry of the TPCefficiencies. This paper is organized as follow. InSection 2, we will introduce the cumulant observ-ables and the efficiency correction to the cumulants.In Section 3, we will demonstrate the effects of us-ing the mean efficiency with numerical simulation.In Section 4, the effects of using mean efficiency areevaluated using events generated from the UrQMDmodel with a fast detector simulation. Finally, wewill end with a summary. The cumulants of conserved charge are sensi-tive probes to QCD phase transitions and the QCDcritical point, the fourth-order cumulant is propor-tional to the seventh-order of the correlation length C ∝ ξ . The cumulants C ∼ C can be defined bymoments (cid:104) N (cid:105) , (cid:104) N (cid:105) , ... , (cid:104) N (cid:105) as: C = (cid:104) N (cid:105) C = (cid:10) N (cid:11) −(cid:104) N (cid:105) C =2 (cid:104) N (cid:105) − (cid:104) N (cid:105) (cid:10) N (cid:11) + (cid:10) N (cid:11) C = − (cid:104) N (cid:105) +12 (cid:104) N (cid:105) (cid:10) N (cid:11) − (cid:10) N (cid:11) − (cid:104) N (cid:105) (cid:10) N (cid:11) + (cid:10) N (cid:11) . (1)The variance σ , skewness S and kurtosis κ can bedefined as σ = C , S = C ( C ) / , κ = C C . The ratios of the cumulants can be directly com-pared to the thermodynamic susceptibilities, whichcan be computed in lattice QCD [30]. Sσ = C C = χ χ , κσ = C C = χ χ . The cumulants measured with detector efficienciescan be recovered by efficiency correction. For ex-ample, the mean value can be corrected by: (cid:104) N (cid:105) = (cid:104) N (cid:105) measure (cid:15) , (cid:10) N − ¯ N (cid:11) = (cid:104) N (cid:105) (cid:15) proton − (cid:10) ¯ N (cid:11) (cid:15) anti-proton . Typically, the efficiencies for proton and anti-protonare different. Thus, we should divide the mean valueby corresponding efficiency, respectively.The efficiency corrections for higher-order cu-mulants are not as straightforward. One can as-sume that the response function of detected parti-cles follows a binomial distribution with efficiencyparameter (cid:15) . We can then express the moments interms of factorial moments and/or the factorial cu-mulants [62–64]. The r th-order factorial momentsof a stochastic variable N can be defined from the expectation of its falling factorial as: F r = (cid:104) N ( N − ··· ( N − r +1) (cid:105) . and the factorial moments can be easily correctedfor the binomial efficiency. Suppose the measuredfactorial moment is f r with efficiency (cid:15) , we, there-fore, have: (Section 6.1) F r = f r (cid:15) r (2)With the efficiency-corrected factorial moments F to F r ,we can obtain the moments (cid:104) N r (cid:105)(cid:104) N r (cid:105) = r (cid:88) i =0 s ( r,i ) F r , (3)where the s is the Stirling numbers of the secondkind. With moments (cid:104) N r (cid:105) we can further obtainthe cumulants using equation (1). In the case wherethe net-proton cumulant is required, we should in-troduce a two-dimensional factorial moments andthe efficiency correction equation can be written as: F rs = f rs (cid:15) r p (cid:15) s ¯p , where the (cid:15) r p is the r th-order of the proton efficiencyand the (cid:15) s ¯p is the s th-order of anti-proton efficiency. f rs is defined as: f rs = (cid:104) n p ( n p − ··· ( n p − r +1) · n ¯p ( n ¯p − ··· ( n ¯p − s +1) (cid:105) , (4)where n p and n ¯p are the measured proton and anti-proton numbers, respectively. The conversion from F rs to (cid:10) N r p N s ¯p (cid:11) is (cid:10) N r p N s ¯p (cid:11) = r (cid:88) i =0 s (cid:88) i =0 s ( r,i ) s ( s,i ) F rs . The moments of the net-proton can be expressed as (cid:68) N k p − ¯p (cid:69) = (cid:68) ( N p − N ¯p ) k (cid:69) = k (cid:88) i =0 ( − i (cid:18) ki (cid:19)(cid:68) N k − i p N i ¯p (cid:69) . (5)It is straightforward to write the net-proton cumu-lants with equation (1). In this section, we discuss the factorial mo-ments of the superposed distribution. For example,if we have a distribution obtained by the mixtureof a Poisson distribution, Gaussian distribution, orsome other type of distribution, it is then left todetermine the relations between the factorial mo-ments of their superposed distribution and the sub-distributions. The probability density function ofthe superposed distribution can be expressed as:˜ P ( n )= (cid:88) a i P i ( n ) . (6)It describes the probability of detecting n particlesin an event, and the event may be from one of thevarious types. Therefore, ˜ P ( n ) is the summation ofthe probability of detecting n particles from the i thtype P i ( n ). a i is the weight of P i ( n ).With ˜ P ( n ), we can write down the generatingfunction ˜ G F ( s ) for factorial moments F r ˜ G F ( s )= ∞ (cid:88) n =0 ˜ P ( n ) s n = ∞ (cid:88) n =0 k (cid:88) i =0 a i P i ( n ) s n . (7)We can further write˜ G F ( s )= k (cid:88) i =0 a i ∞ (cid:88) n =0 P i ( n ) s n = k (cid:88) i =0 a i G ( i ) F . (8)We then have the relation between superposed fac-torial moments given by: ˜ F r and F ( i ) r ˜ F r = ∂ r ∂s r ˜ G F ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 = k (cid:88) i =0 a i ∂ r ∂s r G ( i ) F ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 = k (cid:88) i =0 a i F ( i ) r . (9)We find that ˜ F r is the weighted average of F ( i ) r .However, we will show that we cannot use theaverage of the cumulants from different types of dis- tributions. First, we note that the superposed cu-mulants ˜ C r is not the simple weighted average of C ( i ) r . Since the generating function of the cumu-lants K ( θ ) can be written as [65] K ( θ )= K fc (cid:16) e θ (cid:17) . (10)The K fc (cid:0) e θ (cid:1) is the generating function of the facto-rial cumulants, and we have: K fc (cid:16) e θ (cid:17) =ln G F (cid:16) e θ (cid:17) . (11)Therefore K ( θ )=ln G F (cid:16) e θ (cid:17) . (12)The generating function of the superposed distribu-tion is˜ K ( θ )=ln ˜ G F (cid:16) e θ (cid:17) =ln k (cid:88) i =0 a i G ( i ) F (cid:16) e θ (cid:17) . (13)The cumulants ˜ C r are given by:˜ C r = ∂ r ∂θ r ˜ K ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = ∂ r ∂θ r ln k (cid:88) i =0 a i G ( i ) F (cid:16) e θ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 (cid:54) = ∂ r ∂θ r k (cid:88) i =0 a i ln G ( i ) F (cid:16) e θ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 = k (cid:88) i =0 a i ∂ r ∂θ r K ( i ) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = k (cid:88) i =0 a i C ( i ) r . (14)Thus, if the individual distributions are different,the cumulant of the superposition of different dis-tributions is not the average of the cumulants of theindividual distributions.Suppose that G F (cid:0) e θ (cid:1) is the factorial mo-ment generating function after efficiency correc-tion. Therefore, the superposed cumulant gener- ating function is given as:˜ K ( θ )=ln ˜ G F (cid:16) e θ (cid:17) =ln k (cid:88) i =0 a i G ( i ) F (cid:16) e θ (cid:17) =ln k (cid:88) i =0 a i G F (cid:16) e θ (cid:17) =ln G F (cid:16) e θ (cid:17) = k (cid:88) i =0 a i ln G F (cid:16) e θ (cid:17) = k (cid:88) i =0 a i K ( i ) ( θ ) . (15)We find that this relation is only true when allthe G ( i ) F are equal. In order words, the average cu-mulant is only valid for the superposed distributionsof the same type.The statistical error for the superposed cumu-lants is given by the Delta theorem [63, 66]. In ourdiscussion, the detecting efficiency (cid:15) is taken as aconstant. Therefore, we have: V (cid:16) ˜ C r (cid:17) = r (cid:88) p,q ∂ ˜ C r ˜ f p ∂ ˜ C r ˜ f q Cov (cid:16) ˜ f p , ˜ f q (cid:17) = r (cid:88) p,q k (cid:88) i ∂ ˜ C r f ( i ) p ∂ ˜ C r f ( i ) q Cov (cid:16) f ( i ) p ,f ( i ) q (cid:17) . (16) Experimentally measured multiplicity distribu-tion can be treated as a superposed of distributionswith different efficiencies. For simplicity, we assumethat the response function of the detected efficiencyis a binomial distribution. This is a special case ofequation (6), where p i ( n ) is given by equation (A3)with a different efficiency (cid:15) i as: p i ( n )= ∞ (cid:88) N = n P ( N ) B N ( n,(cid:15) i ) (17) and the PDF for superposed distribution is:˜ p ( n )= k (cid:88) i =0 a i p i ( n )= k (cid:88) i =0 ∞ (cid:88) N = n a i P ( N ) B N ( n,(cid:15) i ) . (18)The generating function of the measured factorialmoments for each species of the distribution is givenby equation (A5) G ( i ) f ( s )= ∞ (cid:88) n =0 p i ( n ) s n = ∞ (cid:88) n =0 ∞ (cid:88) N = n P ( N ) B N ( n,(cid:15) i ) s n = ∞ (cid:88) N =0 P ( N )[1+ (cid:15) i ( s − N = ∞ (cid:88) N =0 P ( N ) s (cid:48) iN = G F ( s (cid:48) i ) , (19)where the s (cid:48) i =1+ (cid:15) i ( s − G f ( s )= k (cid:88) i a i G ( i ) f ( s )= k (cid:88) i a i G F ( s (cid:48) i ) . (20)We then have the relation between the measuredfactorial moments ˜ f r and the original factorial mo-ments from each species of event˜ f r = ∂ r ∂s r ˜ G f ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 = k (cid:88) i a i (cid:18) ∂s (cid:48) i ∂s (cid:19) r ∂ r ∂ ( s (cid:48) i ) r G F (cid:0) s (cid:48) i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) s =1 = k (cid:88) i a i (cid:15) ri ∂ r ∂ ( s (cid:48) i ) r G F (cid:0) s (cid:48) i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) i =1 = k (cid:88) i a i (cid:15) ri F r . (21)The mean efficiency (cid:104) (cid:15) (cid:105) should not be used forthe superposed distribution. It can be demon-strated in equation (21) by multiple 1 / (cid:104) (cid:15) (cid:105) to bothsides, and comparing it to the original superposed factorial moments˜ f r (cid:104) (cid:15) (cid:105) r − F r = k (cid:88) i a i (cid:15) ri (cid:104) (cid:15) (cid:105) r F r − F r = F r (cid:32) (cid:80) ki a i (cid:15) ri (cid:104) (cid:15) (cid:105) r − (cid:33) = F r (cid:18) (cid:104) (cid:15) r (cid:105)(cid:104) (cid:15) (cid:105) r − (cid:19) . (22)Since (cid:15) i is fluctuating, the last line is usually notequal to 0. Ideally, in order to obtain factorial mo-ments of the original distribution, we should per-form efficiency correction for each types of events,separately: F r = k (cid:88) i =0 a i f ( i ) r (cid:15) ri . (23)There are two methods to obtain the efficiency cor-rected cumulants for superposed distributions, fromdistributions with different efficiencies:1. Correct f ( i ) r to F ( i ) r and compute C ( i ) r . Then˜ C r = (cid:80) i a i C ( i ) r .2. Correct f ( i ) r to F ( i ) r , and ˜ F r = (cid:80) i a i F ( i ) r . Thencompute ˜ C r from ˜ F r .If we know the efficiencies of the different eventtypes in the superposed distribution, we should notethat after efficiency correction F (1) r = F (2) r = ··· = F ( k ) r . Therefore, we can demonstrate that methods1 and 2 are equivalent. However, in Section , wewill find that the efficiencies of each type of eventsare unknown. In this case, we use the mean (cid:15) ineach sub-event sample. Thus, the generating func-tions G ( i ) F in different event sample bins are still notthe same after efficiency correction. G ( i ) F is only anapproximation to the true G F . Thus, the correctionfor superposition of different types of distributionsshould be performed with the average factorial mo-ments (method 2). If the efficiency variation in eachbin is not small, then the weight average of the cu-mulants will introduce additional uncertainties. Wenote that this is similar to the case of using the tech-nique of centrality bin width correction (CBWC) to evaluate cumulants in a wide centrality bin to sup-press volume fluctuations [67]. Usually, the efficiency (cid:15) in equation (2) is ob-tained from MC embedding. It reflects the net con-tribution of the detector acceptance, tracking effi-ciency, and the other effects and it is obtained bytaking the average of the entire event ensemble. Formost situations, the true efficiency of each eventshould not fluctuate too far from this mean value.In these cases, the (cid:104) (cid:15) (cid:105) is a good approximation forcorrection. But we should be careful when efficien-cies of some events dramatically deviate from the (cid:104) (cid:15) (cid:105) . We can exclude bad events by rejecting eventswith unusual multiplicity or selecting events withina multiplicity range. It should be noted that with arelative large efficiency shift, the change in (cid:104) N (cid:105) canbe slight because the binomial distribution is wide.Thus, event selection becomes difficult.The problem of using mean efficiency to correctcumulants exists in reality. We can consider an ex-treme example in which an event ensemble mixestwo types of distinctive events. One type of eventhas efficiency (cid:15) , and the other type has efficiency (cid:15) .The mean efficiency eventually determined as theaverage of the two types of event ¯ (cid:15) . To model thisexample, we used a Monte Carlo simulation. Foreach event, the proton number N we input followsa Poisson distribution with parameter λ = 100. Inevents of type I, the detected proton number n fol-lows a binomial distribution B ( N,(cid:15) ). In events oftype II, the n follows B ( N,(cid:15) ). The event-by-eventproton number distribution from the simulation isshown in Fig. 1.In Figure. 1, we give the same original input dis-tribution with the number of events ( M ) for eachcase (the solid grey lines). We then divide theoriginal input events into two sub-event samples,which passes different efficiencies. The two typesof events are represented by type I and II with
50 1000.020.04 = 0.1 Total Events N =0.8 ˛ =0.8 ˛
50 1000.020.04 = 0.5 Total Events N
50 1000.020.040.06 = 0.9 Total Events N
50 100 0.020.04 =0.7 ˛ =0.8 ˛
50 100 0.020.04
After efficiency
50 100 0.020.040.06
50 100 0.020.04 =0.6 ˛ =0.8 ˛
50 100 0.020.04
Original
50 100 0.020.040.06
50 100 0.020.04 =0.5 ˛ =0.8 ˛
50 100 0.020.04
50 100 0.020.040.06
50 100 0.020.04 =0.4 ˛ =0.8 ˛
50 100 0.020.04
50 100 0.020.040.06
Proton Number E v en t N u m be r Fig.
1: (color online) Monte Carlo input (Original, gray line) and measured distribution with detectorefficiencies (dashed blue line). The statistics of event is 1.0 billion (10 ). In the first, second and thirdrows, events with efficiency (cid:15) (i.e., Events of type I) makes up 10%, 50% and 90% of the total eventnumber, respectively. In columns 1 to 5, the (cid:15) varies from 0.8 to 0.4, while the efficiency of type II is fixedat (cid:15) = 0.8. C = 0.1 Total Events N C Proton Cumulants = 0.8 ∈ C × C − × σ S − σ κ − × = 0.5 Total Events N × × × = 0.9 Total Events N × × × ∈ C = 0.1 Total Events N C ∈ Bin 1 efficiency = ∈ Bin 2 efficiency = C C σ S σ κ = 0.5 Total Events N Proton Cumulants = 0.8 ∈ = 0.9 Total Events N ∈ Fig.
2: (color online) Efficiency corrected cumulants and cumulant ratios. For each column, the eventsof type I makes up 10%, 50% and 90% of the total event number (10 ). The (cid:15) of the x -axis representsefficiency event type I and the efficiency of type II is fixed at (cid:15) = 0.8. Square markers (left): Resultcorrected with mean efficiency.
Circle markers (right): Result corrected independently by (cid:15) and (cid:15) (True efficiencies). C C C Ori. C -1000100 Fact. AverageCum. Average5 bins L ˛ C C C Ori. C L ˛ Fig.
3: (color online) Difference of the factorial moments average and the cumulants average. Simulationwith 5 bins (left) and 100 bins (right). The efficiencies of events are selected randomly in the range ( (cid:15) L , . M and M , respec-tively ( M + M = M ). The efficiency of type IIis fixed at (cid:15) = 0 .
8. From column 1 to 5, we de-crease the efficiency of type I events from 0.8 to0.4. We find that the distinctive peaks of event-by-event distributions gradually emerged. The eventfraction of the total events for the type I sub-eventsample are varied as 0.1, 0.5, and 0.9. Then, weperform efficiency correction using the mean effi-ciency (cid:104) (cid:15) (cid:105) of each case. Since we can represent (cid:15) by (cid:15) = (cid:104) N (cid:105) measure / (cid:104) N (cid:105) input and so is (cid:15) , their aver-age can be written as: (cid:104) (cid:15) (cid:105) = M (cid:104) n (cid:105) + M (cid:104) n (cid:105) M (cid:104) N (cid:105) . where M is the number of events in type I and M is the number of events in type II. The measuredparticle number is denoted as (cid:104) n (cid:105) , and the inputparticle number is denoted as (cid:104) N (cid:105) .With the measured distributions (blue dashedlines in Fig. 1) and the mean efficiency (cid:104) (cid:15) (cid:105) , we cancalculate the efficiency corrected factorial moments(equation (2)) and the cumulants, which are shownin Fig. 2 (left) as blue square markers. We thentune the efficiency difference ∆ (cid:15) of the two types ofevents to determine how the efficiency corrected re-sults deviate from the original cumulants (markedas the solid gray lines). We found there is no is-sue in using the mean efficiency to correct C , sincethe results perfectly follow the solid lines (approxi- mately around 100.0 which is the Poisson parameter λ ) with the change of ∆ (cid:15) . However, the results startto deviate significantly for C , C and C . Obvi-ously, the correction failed even if the ∆ (cid:15) is as smallas 0.1 (When the event-by-event distribution showsno double peaks in Fig. 1).Therefore, we know that the event ensemblemixes two types of distinctive event which causesthe correction to fail for higher order cumulants.As such, it is necessary to determine whether the re-sults can be improved when we perform correctionson each type. In the following, we independentlycalculate the cumulants of two types of event andcorrect them using their own measured efficiencies. (cid:15) = (cid:104) N (cid:105) (1)measure (cid:104) N (cid:105) (1)input , (cid:15) = (cid:104) N (cid:105) (2)measure (cid:104) N (cid:105) (2)input . Therefore, we need to determine how to combine thecorrected result of different types of events. Thesimulation of two types of events is the simplestcase. Let’s consider the measured distribution froma combination of K types of events. We can find inSection that Equation (8) shows that the fac-torial moments f r of the superposed distribution isthe weighted average of the factorial moments f ( i ) r of each type. Therefore, the efficiency corrected fac-torial moments of the superposed distribution is: F r = k (cid:88) i =1 a i f ( i ) r (cid:15) k ( i ) . (24) The results are shown in Fig. 2 (right). As expected,the efficiency corrected cumulants follow the inputvalues perfectly in all three cases.As we have discussed in Section , the cu-mulants of the superposition of different distribu-tions (i.e., distribution corrected using mean effi-ciencies instead of true efficiencies) should be calcu-lated from averaged factorial moments. The aver-age of the cumulants will introduce additional devi-ation. We show the difference of the factorial mo-ments average and the cumulants average in Fig. 3.In this figure, the results of a numerical simula-tion with 100M events is presented. Instead of us-ing 2 different efficiencies in the previous simula-tion, each event is randomly assigned an efficiencynumber, which is uniformly distributed in the range( (cid:15) L , 0.8). To perform efficiency correction, the en-tire event sample is divided into sub-event sampleswith equal efficiency intervals between ( (cid:15) L , 0.8).The efficiency correction for each sub-event sam-ple is performed using the methods of factorial mo-ments average and the cumulants average, respec-tively. We can infer from the left panel of Fig. 3that with 5 efficiency bins (larger efficiency varia-tions in each bin), the efficiency corrected resultsfail to reproduce the higher-order input cumulantsfor both methods. However, for the fourth-order cu-mulant ( C ), the average of the factorial moments iscloser to the original, and the results of the cumu-lants average method exhibit large deviations. Inorder to reproduce the original cumulants, we haveto reduce the efficiency variation and use more ef-ficiency bins (with 100 bins in the right panel ofFig. 3). For finer efficiency bins, the factorial mo-ment generating function G ( i ) F with a mean efficiencybecomes closer to its true value G F . Moreover, theadditional uncertainties of the cumulant average aremuch smaller.In conclusion of this section, we demonstrate theeffects of using the mean efficiency in the efficiencycorrection for cumulants of multiple distributions.If the efficiency variation within the event sampleis large, it is incorrect to use the mean efficiency to perform the efficiency corrections. To perform pre-cise and reliable efficiency correction, one has care-fully bin the events into various sub-event samples,in which the efficiency variation is relatively small. In this section, we will examine whether or notthe failed correction in the last section can occur inreal experiments. In the STAR experiment, parti-cle identification and track reconstruction are per-formed with a time projection chamber [68] (TPC).The major structure of the TPC is a cylinder driftchamber with a high voltage electrode in the cen-ter. The two endcaps of the drift chamber are cov-ered with thin-gap, multiwire proportional cham-bers (MWPC). The particles that pass through theTPC will experience energy loss due to the ioniza-tion of the drifting electrons. By measuring the drifttime and the number of electrons collected at theendcaps, we can build the track of arrival particlesand calculate their energy loss d E/ d x . As shown inFig. 4, the voltage electrode in the center (CentralMembrane) divides the drift chamber into two sub-parts. Usually, the working conditions of the westand east side of the TPC endcaps are not essentiallythe same, which can result in different detection ef-ficiencies for the west and east TPC.The z -coordination of the primary vertex is de-scribed by an important event parameter V z . V z =0 indicates that the primary vertex is located at thelongitudinal center of the TPC. In the simulation,a positive V z indicates that the primary vertex islocated to the right. We also set a flat distribu-tion of V z within the range (-50 cm, 50 cm), whichis a similar case to RHIC BES at low energies. Inour discussion, V z distributions are important be-cause the efficiencies are unequal in the left and theright parts of the chamber. Since the particles fromevents with V z < V z < V z >
0. Obviously, we will arrive at the situation which V z ∈ ( − cm V z = TPCDrift ChamberDrift Chamber
Left RightCentral Membrane
Outer Radius 2.0 mInner Radius 0.5 mInner Radius 0.5 mOuter Radius 2.0 m E n d c a p E n d c a p E n d c a p E n d c a p length 4.2 m beam pipe Fig.
4: (color online) A sketch of the STAR TPC. V z > Left Right
Fig.
5: (color online) Geometry sketch in UrQMDsimulation. Each event from UrQMD has been as-signed a random V z .has been discussed in Section . In fact, the detect-ing efficiencies have been observed to change with V z in real experiments. To investigate the effect of fluctuation of V z ,we performed a fast simulation using the UrQMDmodel. UrQMD is a transport model that can simu-late nucleus-nucleus collision events [69]. The mainidea is to give each UrQMD event a random V z inthe range -50 < V z <
50 cm. Then, we can assignefficiency to each particle base on its η and the V z of the event.We then simplify the geometry of the TPC intoa plane to emphasize the effects of interest. Sincewe can judge which part of the drift chamber theparticle will travel into by its pseudo-rapidity η , thetracks’ azimuthal angle ( φ ) can be omitted from ouranalysis. Therefore, our realm of interest can berepresented as shown in Fig. 5. In this figure, the V z of an event is randomly assigned. The angle ( θ )between a track and the beam pipeline can be de-rived from the pseudo-rapidity η . Owing to the magnetic field in TPC, particleshave a helix trajectory when passing through thechamber. However, we can simplify this motion asa straight line because we are only concerned withthe part of the chamber where the track will occur.It should be noted that tracks sometimes go throughthe central membrane. For the sake of clarity andsimplicity, we suppose that the entire track is in theleft/right part of the chamber if the end of the trackis in the left/right part of the chamber. Finally, wecan assign the detection efficiencies (cid:15) L to tracks inthe left chamber and (cid:15) R to tracks in the right cham-ber.In addition to the effect of V z fluctuation, the de-tecting efficiency is affected by the total multiplic-ity of charged particles. The multiplicity of chargedparticles is usually used as the reference to deter-mine the centrality. This implies that the detect-ing efficiency must be different from the central toperipheral collisions. Thus, we introduce a multi- -1 -0.5 0 0.5 1 h M ea . d N / d =0.3 L ˛ -1 -0.5 0 0.5 1 h O r i . d N / d -1 -0.5 0 0.5 1 E ff i c i en cy -1 -0.5 0 0.5 1 102030 =0.4 L ˛ -1 -0.5 0 0.5 1 10203040 -1 -0.5 0 0.5 1 0.20.40.60.8 -1 -0.5 0 0.5 1 102030 =0.5 L ˛ -1 -0.5 0 0.5 1 10203040 Proton -1 -0.5 0 0.5 1 0.20.40.60.8 -1 -0.5 0 0.5 1 102030 =0.6 L ˛ -1 -0.5 0 0.5 1 10203040 -1 -0.5 0 0.5 1 0.20.40.60.8 UrQMD 7.7 GeVAu+Au Collision5% most central -1 -0.5 0 0.5 1 102030 =0.7 L ˛ -1 -0.5 0 0.5 1 10203040 -1 -0.5 0 0.5 1 0.20.40.60.8 Whole Vz-50 6: (color online) d N/ d η distribution (row 1: measured, row 2: original input) and the measurementefficiency (row 3). From left to right columns, the detecting efficiency of tracks in the left part of the driftchamber arising from 0.3 to 0.7, while the efficiency of tracks in the right part of the chamber is fixed at0.8. The η dependence of the detecting efficiency can be represented by the ratio of the first and the secondrow.plicity dependence efficiency. The relation betweentotal multiplicity and efficiency can be expressed as: (cid:15) = (cid:15) − K R N mul , (25)where the (cid:15) and the K R are constant, and the N mul is the total multiplicity of charged particles within | η | < 1. The minus sign before K R indicates thedetecting efficiency decreases with increasing totalmultiplicity. This effect can be introduced in thesimulation by simply reducing the (cid:15) L and (cid:15) R by theminus of the factor K R N mul . For simplicity andclarity, we set the detecting efficiencies has no de-pendence of p T and the efficiencies of the p (¯ p ) arethe same. Since we are interested in the efficiencyfluctuations effects on net-proton cumulants, we didnot apply efficiencies to pions and kaons. In this work, we calculate the efficiency-corrected cumulants of The net-proton distributionsin Au + Au collisions at √ s NN = 7.7 GeV fromUrQMD and select 0–5% most central events fromthe dataset. The statistics of selecting events is 2.0 million. Collision centrality is determined by thecharged particles within | η | < N/ d η distribution for measured data (with V z fluc-tuation and detecting efficiencies) and the originalUrQMD data within a pseudo-rapidity coverage | η | < N/ d η distributions of protonswithin various V z ranges. In the columns from leftto right, we gradually increase the efficiency of thetracks in the left part of the chamber (correspond-ing to η < (cid:15) L = 0.3 to (cid:15) L = 0.7, while we fixthe efficiency in the right part of the chamber at (cid:15) R = 0.8). Thus, we can evaluate the efficiency fluctua-tion effects when we enlarge or reduce the differenceof (cid:15) L and (cid:15) R .We can learn from Fig. 6 that the proton d N/ d η distributions are asymmetry in the positive and neg-ative η regions, while the distributions of the origi-nal input are flat. The detecting efficiencies of par-ticles can be expressed as the ratio of the first andthe second row. We found that when we narrow z V -40 -20 0 20 40 P r o t on E ff i c i en cy =0.80 R ˛ =0.70 L ˛ UrQMDAu+Au 7.7 GeV5% most central y=0.2 D y=0.4 D y=0.8 D y=1 D Fig. 7: (color online) V z dependence of detectingefficiency within various rapidity coverage ∆ y . Thewider ∆ y coverage corresponds to a smoother slopein the figure. P r o t on E ff i c i en cy =0.7 L ˛ =0.6 L ˛ =0.5 L ˛ =0.4 L ˛ =0.3 L ˛ = 0.8 R ˛ UrQMD Simulation |y|<0.5 + K p RefMult-3 = Charged Fig. 8: (color online) Multiplicity dependence ofdetecting efficiency. The reference multiplicity inthe x -axis represents the multiplicity of charged π and K meson within | η | ¡ 1. The different lines (cid:15) L in the figure represent various (cid:15) in equation (25).the V z bin width, the slope from negative η to posi-tive η becomes steeper. On the contrary, a wider V z bin resulted in a smaller slope. This implies that awider V z bin mixes more distinctive events.The relations between rapidity coverage, V z andthe proton efficiency are shown in Fig. 7. In thisfigure, the mean proton efficiency within various ra-pidity coverage ∆ y is plotted as a function of V z .The efficiencies for V z > V z < 0, which is consistent with our setting. Whenthe ∆ y is small, particles are more likely to concen-trate in the left or right chamber. Therefore, theslope from V z < V z > y is larger; particles are more dispersed into differentchamber parts, which leads to a smooth transitionfrom V z < V z > π and K . Experimentally,instead of using wider centrality bins to calculatethe cumulants, the net-proton cumulants are calcu-lated in individual reference multiplicity bins to re-duce the volume fluctuation which arises from theuncertainty of the collision geometry. This is theso-called centrality bin width correction (CBWC) technique [67]. The reference multiplicity is equalto the multiplicity of charged π and K within | η | < 1. From the previous discussion, the mean efficiencyshould be different across centralities and referencemultiplicity bins. 10 20 3000.050.1 UrQMD 7.7 GeV Au+Au5% Most Central < 2 GeV/c T = . L ˛ = . L ˛ = . L ˛ = . L ˛ = . L ˛ =0.8 R ˛ æ p N Æ - æ p N Æ P r obab ili t y D i s t r i bu t i on Fig. 9: (color online) Event-by-event distributionof net-proton. From left to right, the detecting effi-ciency of tracks in the left part of the drift chamberthat arise from 0.3 to 0.7, while the efficiency oftracks in the right part of the chamber is fixed at0.8.In Fig. 6–8, we obtained the proton mean effi-ciency via simulation and thus we can perform theefficiency correction on the net-proton cumulants. We first examine the event-by-event distributionsof the measured net-proton number via simulation.In Fig. 9, we found that the shapes of the distri-butions show no significant change when we enlargethe difference between (cid:15) L and (cid:15) R . This result is dueto the continuous distributions of V z . Moreover, theevent-by-event distribution of the mean net-protonnumber is the superposition of events within thewhole V z range.We show the efficiency-corrected net-proton cu-mulants C to C (and their ratios Sσ = C /C , κσ = C /C ) within various pseudo-rapidity andrapidity coverages in Fig. 10–11. The mean efficien-cies of the proton and anti-protons are used in theefficiency correction. As we suppose in our previouswork, the cumulants in various rapidity or pseudo-rapidity acceptance have (cid:104) N p (cid:105) scaling behavior (or (cid:104) N p (cid:105) + (cid:104) N ¯p (cid:105) scaling) [70]. We also plot the cumu-lants and their ratios as functions of the mean total-proton number (cid:104) N p (cid:105) + (cid:104) N ¯p (cid:105) in Fig. 12. In this case,the cumulants within various ∆ η or ∆ y acceptanceshow a unified trend to the mean net-proton num-ber (or total-proton number). The original resultscomputed from the original input without detectingefficiency are shown as a solid gray line in the figure.The measured cumulants are represented by coloredmarkers. We found that the efficiency-corrected cu-mulants coincide with the original results when theefficiencies to the left and right parts of the chamberare close to each other. However, when the differ-ence between (cid:15) L and (cid:15) R is large (i.e., the case (cid:15) L = 0.5, (cid:15) R = 0.8), the efficiency correction failed forthe higher-order cumulants as was demonstrated inSection , and the deviations grow rapidly with thedifference.The simulation confirms that using the mean ef-ficiency for correction can produce inaccurate re-sults and the effect of efficiency fluctuations is notnegligible. Fortunately, the deviation is not neg-ligible only when the efficiency difference ∆ (cid:15) be-come unrealistically large compare to real experi-ments. In the case where the ∆ (cid:15) is less than 0.2,the efficiency-corrected cumulants coincide with theinput results within statistical uncertainties. V z bin correction The results shown in Fig. 6 suggest that wider V z bins mix up more distinctive events, thus we canperform the efficiency correction within smaller V z bins.The method to perform the correction of V z fluctuation is analogous to what was done in Sec-tion . We may suppose that events with different V z have their own mean efficiency. Therefore, themeasured distribution is the superposed distribu-tion from all V z ranges, and the superposed distri-bution from events with different efficiencies. Meanefficiencies of smaller V z bins can be obtained us-ing Monte Carlo embedding procedures. Supposethe efficiency-corrected factorial moments at V z is F r ( V z ), we can write the average result as equa-tion (24) ˜ F r = (cid:90) V z F r ( V z ) d V z . The statistical error is evaluated by propagation ofthe standard error.We first investigate the effect of V z bin correctionwith different ∆ (cid:15) and different V z bin width. Fig. 13shows that when ∆ (cid:15) = (cid:15) R − (cid:15) L is large, the correctedresults using the mean efficiency show large devia-tions from the input results. For the V z bin correc-tion, we set up 2 or 3, 4 V z bins with equal intervalin the range of (-50 cm, 50 cm) to perform efficiencycorrection. We found that the V z bin correction cansignificantly improve the results of high-order cu-mulants, especially in the case where ∆ (cid:15) is large.Finer V z bin is important when ∆ (cid:15) is large. Wealso learn from Fig. 13 that efficiency corrected 2 V z bins show great improvement. However, the ef-fect of V z bin correction may depend on the rapidityacceptance, since we found in Fig. 7 that a smallerrapidity acceptance ∆ y exhibits a steeper slope onthe V z dependence of efficiency. In Fig. 14 we show 2and 3 V z bins corrected results within various y cuts.When | y | cut is small, the result corrected by 3 binsis better than 2 bins. However, the discrepancy isnot significant for larger | y | cut. The reason is thatwhen | y | cut is large; particles are dispersed more C Ori.=0.5 L ˛ =0.7 L ˛ Net-Proton C |<1 h | <2.0 GeV/c T C =0.8 R ˛ C -2002040 UrQMD 7.7 GeVAu+Au Col. 5% s S sk | h | Fig. 10: (color online) Pseudo-rapidity dependence of efficiency corrected cumulants C to C (and theirratios Sσ = C /C , κσ = C /C ) of net-proton distributions. The mean efficiencies (cid:104) (cid:15) p (cid:105) , (cid:104) (cid:15) ¯p (cid:105) are used inthe efficiency corrections. C Ori.=0.5 L ˛ =0.7 L ˛ Net-Proton C UrQMD 7.7 GeVAu+Au Collision5% most central C =0.8 R ˛ |<1 h |<2 GeV/c T C -40-2002040 s S sk |y| Fig. 11: (color online) Rapidity dependence of efficiency-corrected cumulants C to C (and their ratios Sσ = C /C , κσ = C /C ) of net-proton distributions. The mean efficiencies (cid:104) (cid:15) p (cid:105) , (cid:104) (cid:15) ¯p (cid:105) are used in theefficiency corrections. C |y| | h |=0.5 L ˛ =0.7 L ˛ Net-Proton C UrQMD 7.7 GeVAu+Au Collision5% most central C =0.8 R ˛ <2 GeV/c T C Ori. s S sk æ p N Æ + æ p N Æ Fig. 12: (color online) Net-proton cumulants and cumulants ratios as functions of mean total-protonnumber (cid:104) N p (cid:105) + (cid:104) N ¯p (cid:105) . C C Ori.Mea. C =0.8 R ˛ <2 GeV/c T h | C -300-200-1000 UrQMD 7.7 GeVAu+Au Collision5% most central s S Net-Proton sk -6-4-202 Vz 2 bin corr.Vz 3 bin corr.Vz 4 bin corr. L ˛ Fig. 13: (color online) The efficiency-corrected net-proton cumulants with/without V z bin average in vari-ous ∆ (cid:15) . The x -axis is the efficiency (cid:15) L in the left chamber, and the (cid:15) R is fixed at 0.8. The result without V z bin average (full circle marker) show large deviations from original results. C Ori.Mea.Net-Proton C UrQMD 7.7 GeVAu+Au Collision5% most central C R ˛ =0.5 L ˛ C s S Vz 2 bins corr.Vz 3 bins corr. sk |y| Fig. 14: (color online) Rapidity dependence of efficiency-corrected net-proton cumulants with/without V z bin average. The precision of correction has been significantly improved by an average of 2 V z bins. Whenthe ∆ (cid:15) = (cid:15) R − (cid:15) L is larger, finer V z bins (3 bins) are required.evenly into two drifting chambers, so the efficiencydepends less on V z . In heavy ion collision, the cumulants of event-by-event multiplicity of conserved charge have beenused as sensitive observables to probe the QCDphase transition and the critical point. Since theexperiments have a finite acceptance and detectorefficiency, the measured distribution should be cor-rected for detecting efficiency in subsequent analy-sis. The particle efficiencies of events measured un-der different experimental conditions should be dif-ferent. We called this effect event-by-event effi-ciency fluctuations. The final event-by-event multi-plicity distributions should be the superposed dis-tributions of various type of events measured underdifferent conditions. However, the efficiency cor-rection is performed using the mean efficiency ofthe event sample. Mean efficiencies obtained fromMonte Carlo embedding procedures have been usedto do the efficiency correction for cumulants. The mean efficiencies reflect the net contribution of theacceptance, tracking efficiency and other effects. Wehave shown the relation between the factorial mo-ments of the superposed distribution and the facto-rial moments from individual distributions (i.e., thedistribution with different efficiencies). We deter-mined that the superposed factorial moments arethe weighted average of the individual factorial mo-ments, and the mean efficiency cannot restore theoriginal input since the efficiency fluctuates acrossthe various distributions. So we suggest that oneshould be very careful when binning the events intovarious sub-event samples, in which the efficiencyvariation is relatively small.We have done a numerical simulation whichcombined two types of events with different efficien-cies which revealed that a correction that is imple-mented using the mean efficiency can have a signifi-cant deviation from the original input. In addition,a more concrete simulation with the UrQMD modelindicates that similar effects can occur in real ex-periments. In the UrQMD simulation, we considerthe event-by-event fluctuation of z -coordination ofcollision vertex ( V z ). We also introduced the dif- ferent working conditions of detectors at the westand east endcap of the detector (TPC), which canlead to unequal detecting efficiencies of the tracksin the west and east subpart of the chamber. Weshow that the event-by-event efficiency fluctuationeffects can cause the efficiency-corrected cumulantsusing the mean efficiencies, to deviate from the orig-inal input. However, when the efficiency fluctuationis at the level of real experiments which is muchsmaller than our settings, the deviation can be ne-glected. We also attempted to reduce the efficiencyvariation by introducing the V z bin average method,which can significantly improve the precision of theefficiency correction. The event-by-event efficiency fluctuation imple-mented in our simulation, is not the only sourcethat can be present in real experiments. For ex-ample, other sources may exist such as the vari-ation of the detector performance as a functionof time and bad events outliers in the multiplic-ity distributions. To obtain high precise and reli-able efficiency-corrected cumulants, it is necessaryto careful study event selection and classificationto ensure the event-by-event efficiency fluctuationsis small. This work presents a simple but effectivemethod to improve the precision of efficiency correc-tion for cumulant analysis in relativistic heavy-ioncollision experiments. Appendices A Efficiency correction for factorial moments The factorial moments generating function is G F ( s )= ∞ (cid:88) n =0 P ( n ) s n (A1)where the n is the value of the stochastic variable and the P ( n ) is the probability density function. The summationcan also be expressed as: G F ( s )= (cid:104) s n (cid:105) Therefore the r -th factorial moment of n is given as: F r = ∂ r ∂s r G F ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 = ∞ (cid:88) n = r P ( n ) n ( n − n − ··· ( n − r +1)= (cid:104) n ( n − ··· ( n − r +1) (cid:105) (A2)The probability of detecting n particles ( p ( n )) is givenby the Binomial distribution B N ( n, (cid:15) ), where N is the number of input particles and the (cid:15) is the efficiency. p ( n )= ∞ (cid:88) N = n P ( N ) B N ( n,(cid:15) )= ∞ (cid:88) N = n P ( N ) (cid:18) Nn (cid:19) (cid:15) n (1 − (cid:15) ) N − n (A3)The generating function of the measured factorial mo-ments is then G f ( s )= ∞ (cid:88) n =0 p ( n ) s n = ∞ (cid:88) n =0 ∞ (cid:88) N = n P ( N ) (cid:18) Nn (cid:19) (cid:15) n (1 − (cid:15) ) N − n s n = ∞ (cid:88) N =0 P ( N ) N (cid:88) n =0 (cid:18) Nn (cid:19) (cid:15) n (1 − (cid:15) ) N − n s n = ∞ (cid:88) N =0 P ( N ) N (cid:88) n =0 (cid:18) Nn (cid:19) ( (cid:15)s ) n (1 − (cid:15) ) N − n (A4)The last line can be simplified using the Binomial theo-rem to give: N (cid:88) n =0 (cid:18) Nn (cid:19) ( (cid:15)s ) n (1 − (cid:15) ) N − n =[ (cid:15)s +(1 − (cid:15) )] N G f ( s )= ∞ (cid:88) N =0 P ( N )[1+ (cid:15) ( s − N = ∞ (cid:88) N =0 P ( N ) s (cid:48) N = G F ( s (cid:48) ) (A5)where s (cid:48) = [1+ (cid:15) ( s − f r and the origi-nal factorial moments F r f r = ∂ r ∂s r G f ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 = ∂ r ∂s r ∞ (cid:88) N =0 P ( N )[1+ (cid:15) ( s − N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =1 = ∞ (cid:88) N = r (cid:15) r P ( N ) N ( N − ··· ( N − r +1)= (cid:15) r F r (A6) Multivariate factorial moments In the report, we use multivariate factorial momentsto describe the net-proton number. The net-proton fac-torial moments has 2 dimensions which describe the pro-ton and anti-proton number respectively. The generatingfunction of q -dimensional factorial moments is an exten-sion to equation (A1) G F ( t )= (cid:89) q ∞ (cid:88) n q =0 P q ( n q ) t n q q (A7)Therefore F r = (cid:89) q ∂ r q ∂t r q q ∞ (cid:88) n q =0 P q ( n q ) t n q q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t q =1 = (cid:42)(cid:89) q ( t q ) r q (cid:43) (A8)where the symbol ( t q ) r q is a falling factorial( t q ) r q = t q ( t q − t q − ··· ( t q − r q +1)The detecting probability density function for each kindof particle is identical to equation (A3) p q ( n q )= ∞ (cid:88) N q = n q P q ( N q ) B N q ( n q ,(cid:15) q ) (A9) Therefore, the generating function for measured factorialmoments is given by equation (A4) G f ( t )= (cid:89) q ∞ (cid:88) n q =0 p q ( n q ) t n q q = (cid:89) q ∞ (cid:88) n q =0 ∞ (cid:88) N q = n q P q ( N q ) B N q ( n q ,(cid:15) q )= (cid:89) q P q ( N q )[1+ (cid:15) q ( t q − N q − n q (A10)Thus, the efficiency correction relation is similar to equa-tion (A6) f r = (cid:32)(cid:89) q (cid:15) r q q (cid:33) F r (A11)The conversation from q -dimensional factorial momentsto q -dimensional moments is similar to equation (3) (cid:42)(cid:89) q N r q q (cid:43) = r (cid:88) i =0 ··· r q (cid:88) i q =0 s ( r ,i ) ··· s ( r q ,i q ) F r ,r ,...,r q (A12)With q -dimensional moments, we can write down themoments of any combination of q kinds of particles, forexample, the moments of net-proton number is given as: (cid:104) ( N − N ) r (cid:105) = (cid:42) r (cid:88) i =0 (cid:18) ri (cid:19) ( − i N r − i N i (cid:43) = r (cid:88) i =0 (cid:18) ri (cid:19) ( − i (cid:10) N r − i N i (cid:11) = r (cid:88) i =0 (cid:18) ri (cid:19) ( − i r − i (cid:88) k =0 i (cid:88) k =0 s ( r − i,k ) s ( i,k ) F k k (A13)The cumulants of the net-proton number is given byequation (1) directly.104001-18hinese Physics C Vol. 42, No. 10 (2018) 104001 References Science , 332:1525–1528, 2011.2 J. Cleymans, H. Satz, E. Suhonen, and D. W. vonOertzen. Strangeness Production in Heavy Ion Collisionsat Finite Baryon Number Density. Phys. Lett. , B242:111–114, 1990.3 N. J. Davidson, H. G. Miller, R. M. Quick, and J. Cley-mans. Chemical equilibration in heavy ion collisions. Phys. Lett. , B255:105–109, 1991.4 Hai-Ling Lao, Fu-Hu Liu, Bao-Chun Li, and Mai-YingDuan. Kinetic freeze-out temperatures in central and pe-ripheral collisions: Which one is larger? Nucl. Sci. Tech. ,29:82, 2018.5 L. Adamczyk et al. Beam-Energy Dependence of the Di-rected Flow of Protons, Antiprotons, and Pions in Au+AuCollisions. Phys. Rev. Lett. , 112(16):162301, 2014.6 J. Brachmann, S. Soff, A. Dumitru, Horst Stoecker, J. A.Maruhn, W. Greiner, L. V. Bravina, and D. H. Rischke.Antiflow of nucleons at the softest point of the EoS. Phys.Rev. , C61:024909, 2000.7 Huichao Song, You Zhou, and Katarina Gajdosova. Col-lective flow and hydrodynamics in large and small systemsat the LHC. Nucl. Sci. Tech. , 28(7):99, 2017.8 Koichi Hattori and Xu-Guang Huang. Novel quantumphenomena induced by strong magnetic fields in heavy-ion collisions. Nucl. Sci. Tech. , 28(2):26, 2017.9 Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K.Szabo. The Order of the quantum chromodynamics tran-sition predicted by the standard model of particle physics. Nature , 443:675–678, 2006.10 B. J. Schaefer and M. Wagner. QCD critical region andhigher moments for three flavor models. Phys. Rev. ,D85:034027, 2012.11 Masayuki Asakawa, Shinji Ejiri, and Masakiyo Kitazawa.Third moments of conserved charges as probes of QCDphase structure. Phys. Rev. Lett. , 103:262301, 2009.12 G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo. TheQCD phase diagram at nonzero quark density. JHEP ,04:001, 2011.13 Mikhail A. Stephanov. QCD phase diagram and the crit-ical point. Prog. Theor. Phys. Suppl. , 153:139–156, 2004.[Int. J. Mod. Phys.A20,4387(2005)].14 Jiunn-Wei Chen, Jian Deng, Hiroaki Kohyama, and LanceLabun. Robust characteristics of nongaussian fluctuationsfrom the NJL model. Phys. Rev. , D93(3):034037, 2016.15 V. Vovchenko, D. V. Anchishkin, M. I. Gorenstein, andR. V. Poberezhnyuk. Scaled variance, skewness, and kur-tosis near the critical point of nuclear matter. Phys. Rev. ,C92(5):054901, 2015.16 Volodymyr Vovchenko, Mark I. Gorenstein, and HorstStoecker. van der Waals Interactions in Hadron Reso- nance Gas: From Nuclear Matter to Lattice QCD. Phys.Rev. Lett. , 118(18):182301, 2017.17 Wenkai Fan, Xiaofeng Luo, and Hongshi Zong. Identify-ing the presence of the critical end point in QCD phasediagram by higher order susceptibilities, 2017.18 Kenji Fukushima. Hadron resonance gas and mean-fieldnuclear matter for baryon number fluctuations. Phys.Rev. , C91(4):044910, 2015.19 Frithjof Karsch and Krzysztof Redlich. Probing freeze-outconditions in heavy ion collisions with moments of chargefluctuations. Phys. Lett. , B695:136–142, 2011.20 P. Braun-Munzinger, B. Friman, F. Karsch, K. Redlich,and V. Skokov. Net-proton probability distribution inheavy ion collisions. Phys. Rev. , C84:064911, 2011.21 R. V. Gavai and Sourendu Gupta. Lattice QCD predic-tions for shapes of event distributions along the freezeoutcurve in heavy-ion collisions. Phys. Lett. , B696:459–463,2011.22 S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti,and K. K. Szabo. Freeze-out parameters: lattice meetsexperiment. Phys. Rev. Lett. , 111:062005, 2013.23 A. Bazavov et al. Freeze-out Conditions in Heavy IonCollisions from QCD Thermodynamics. Phys. Rev. Lett. ,109:192302, 2012.24 Kenji Morita, Bengt Friman, and Krzysztof Redlich. Crit-icality of the net-baryon number probability distributionat finite density. Phys. Lett. , B741:178–183, 2015.25 Lijia Jiang, Pengfei Li, and Huichao Song. Multiplicityfluctuations of net protons on the hydrodynamic freeze-out surface. Nucl. Phys. , A956:360–364, 2016.26 Lijia Jiang, Pengfei Li, and Huichao Song. Correlatedfluctuations near the QCD critical point. Phys. Rev. ,C94(2):024918, 2016.27 L. Adamczyk et al. Beam energy dependence of momentsof the net-charge multiplicity distributions in Au+Au col-lisions at RHIC. Phys. Rev. Lett. , 113:092301, 2014.28 Paolo Alba, Wanda Alberico, Rene Bellwied, MarcusBluhm, Valentina Mantovani Sarti, Marlene Nahrgang,and Claudia Ratti. Freeze-out conditions from net-proton and net-charge fluctuations at RHIC. Phys. Lett. ,B738:305–310, 2014.29 Xiaofeng Luo and Nu Xu. Search for the QCD CriticalPoint with Fluctuations of Conserved Quantities in Rel-ativistic Heavy-Ion Collisions at RHIC : An Overview. Nucl. Sci. Tech. , 28(8):112, 2017.30 S. Ejiri, F. Karsch, and K. Redlich. Hadronic fluctuationsat the QCD phase transition. Phys. Lett. , B633:275–282,2006.31 Misha A. Stephanov, K. Rajagopal, and Edward V.Shuryak. Event-by-event fluctuations in heavy ion colli-sions and the QCD critical point. Phys. Rev. , D60:114028,1999.32 M. A. Stephanov. Non-Gaussian fluctuations near theQCD critical point. Phys. Rev. Lett. , 102:032301, 2009. 33 M. A. Stephanov. On the sign of kurtosis near the QCDcritical point. Phys. Rev. Lett. , 107:052301, 2011.34 Adam Bzdak, Volker Koch, and Nils Strodthoff. Cumu-lants and correlation functions versus the QCD phase di-agram. Phys. Rev. , C95(5):054906, 2017.35 Xiaofeng Luo. Search for the QCD Critical Point byHigher Moments of Net-proton Multiplicity Distributionsat STAR. Nucl. Phys. , A904-905:911c–914c, 2013. [Cen-tral Eur. J. Phys.10,1372(2012)].36 Bengt Friman. Probing the QCD phase diagram withfluctuations. Nucl. Phys. , A928:198–208, 2014.37 Masakiyo Kitazawa and Masayuki Asakawa. Revealingbaryon number fluctuations from proton number fluctu-ations in relativistic heavy ion collisions. Phys. Rev. ,C85:021901, 2012.38 Masakiyo Kitazawa and Masayuki Asakawa. Relation be-tween baryon number fluctuations and experimentally ob-served proton number fluctuations in relativistic heavyion collisions. Phys. Rev. , C86:024904, 2012. [Erratum:Phys. Rev.C86,069902(2012)].39 B. Friman, F. Karsch, K. Redlich, and V. Skokov. Fluc-tuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC. Eur. Phys.J. , C71:1694, 2011.40 M. Cheng et al. Baryon Number, Strangeness and Elec-tric Charge Fluctuations in QCD at High Temperature. Phys. Rev. , D79:074505, 2009.41 Xiaofeng Luo. Exploring the QCD Phase Structure withBeam Energy Scan in Heavy-ion Collisions. Nucl. Phys. ,A956:75–82, 2016.42 M. M. Aggarwal et al. An Experimental Exploration ofthe QCD Phase Diagram: The Search for the CriticalPoint and the Onset of De-confinement, 2010.43 Ameng Zhao, Xiaofeng Luo, and Hongshi Zong. BaryonNumber Fluctuations in Quasi-particle Model. Eur. Phys.J. , C77(4):207, 2017.44 S. Jeon and V. Koch. Fluctuations of particle ratios andthe abundance of hadronic resonances. Phys. Rev. Lett. ,83:5435–5438, 1999.45 Masayuki Asakawa, Ulrich W. Heinz, and Berndt Muller.Fluctuation probes of quark deconfinement. Phys. Rev.Lett. , 85:2072–2075, 2000.46 Ji Xu. Energy Dependence of Moments of Net-Proton,Net-Kaon, and Net-Charge Multiplicity Distributions atSTAR. J. Phys. Conf. Ser. , 736(1):012002, 2016.47 Ji Xu, Shili Yu, Feng Liu, and Xiaofeng Luo. Cumu-lants of net-proton, net-kaon, and net-charge multiplicitydistributions in Au + Au collisions at √ s NN =7.7 , 11.5,19.6, 27, 39, 62.4, and 200 GeV within the UrQMD model. Phys. Rev. , C94(2):024901, 2016.48 L. Adamczyk et al. Collision Energy Dependence of Mo-ments of Net-Kaon Multiplicity Distributions at RHIC,2017.49 M. M. Aggarwal et al. Higher Moments of Net-proton Multiplicity Distributions at RHIC. Phys. Rev. Lett. ,105:022302, 2010.50 L. Adamczyk et al. Energy Dependence of Moments ofNet-proton Multiplicity Distributions at RHIC. Phys.Rev. Lett. , 112:032302, 2014.51 Xiaofeng Luo. Energy Dependence of Moments of Net-Proton and Net-Charge Multiplicity Distributions atSTAR. PoS , CPOD2014:019, 2015.52 Swagato Mukherjee, Raju Venugopalan, and Yi Yin. Uni-versal off-equilibrium scaling of critical cumulants in theQCD phase diagram. Phys. Rev. Lett. , 117(22):222301,2016.53 Swagato Mukherjee, Raju Venugopalan, and Yi Yin. Realtime evolution of non-Gaussian cumulants in the QCDcritical regime. Phys. Rev. , C92(3):034912, 2015.54 Marlene Nahrgang, Marcus Bluhm, Paolo Alba, ReneBellwied, and Claudia Ratti. Impact of resonance re-generation and decay on the net-proton fluctuations in ahadron resonance gas. Eur. Phys. J. , C75(12):573, 2015.55 Adam Bzdak, Volker Koch, and Vladimir Skokov. Baryonnumber conservation and the cumulants of the net protondistribution. Phys. Rev. , C87(1):014901, 2013.56 Bo Ling and Mikhail A. Stephanov. Acceptance depen-dence of fluctuation measures near the QCD critical point. Phys. Rev. , C93(3):034915, 2016.57 Boris Berdnikov and Krishna Rajagopal. Slowing out-of-equilibrium near the QCD critical point. Phys. Rev. ,D61:105017, 2000.58 Shu He, Xiaofeng Luo, Yasushi Nara, ShinIchi Esumi,and Nu Xu. Effects of Nuclear Potential on the Cumu-lants of Net-Proton and Net-Baryon Multiplicity Distri-butions in Au+Au Collisions at √ s NN = 5GeV. Phys.Lett. , B762:296–300, 2016.59 Marlene Nahrgang, Tim Schuster, Michael Mitrovski,Reinhard Stock, and Marcus Bleicher. Net-baryon-, net-proton-, and net-charge kurtosis in heavy-ion collisionswithin a relativistic transport approach. Eur. Phys. J. ,C72:2143, 2012.60 Miki Sakaida, Masayuki Asakawa, and Masakiyo Ki-tazawa. Effects of global charge conservation on timeevolution of cumulants of conserved charges in relativisticheavy ion collisions. Phys. Rev. , C90(6):064911, 2014.61 B. I. Abelev et al. Systematic Measurements of IdentifiedParticle Spectra in pp,d + Au and Au+Au Collisions fromSTAR. Phys. Rev. , C79:034909, 2009.62 Adam Bzdak and Volker Koch. Local Efficiency Cor-rections to Higher Order Cumulants. Phys. Rev. ,C91(2):027901, 2015.63 Xiaofeng Luo. Unified description of efficiency correctionand error estimation for moments of conserved quantitiesin heavy-ion collisions. Phys. Rev. , C91(3):034907, 2015.64 Toshihiro Nonaka, Masakiyo Kitazawa, and ShinIchi Es-umi. More efficient formulas for efficiency correction ofcumulants and effect of using averaged efficiency. Phys. Rev. , C95(6):064912, 2017.65 Masakiyo Kitazawa and Xiaofeng Luo. Properties anduses of factorial cumulants in relativistic heavy-ion colli-sions. Phys. Rev. , C96(2):024910, 2017.66 Xiaofeng Luo. Error Estimation for Moments Analysis inHeavy Ion Collision Experiment. J. Phys. , G39:025008,2012.67 Xiaofeng Luo, Ji Xu, Bedangadas Mohanty, and Nu Xu.Volume fluctuation and auto-correlation effects in the mo-ment analysis of net-proton multiplicity distributions inheavy-ion collisions. J. Phys. , G40:105104, 2013. 68 W. R. Leo. Techniques for Nuclear and Particle PhysicsExperiments: A How to Approach . Berlin, Germany:Springer (1987) 368 p, 1987.69 S. A. Bass et al. Microscopic models for ultrarelativisticheavy ion collisions. Prog. Part. Nucl. Phys. , 41:255–369,1998. [Prog. Part. Nucl. Phys.41,225(1998)].70 Shu He and Xiaofeng Luo. Proton Cumulants and Corre-lation Functions in Au + Au Collisions at √ s NN =7.7-200GeV from UrQMD Model. Phys. Lett. , B774:623–629,2017., B774:623–629,2017.