Pileup corrections on higher-order cumulants
JJ-PARC-TH-0220
Pileup corrections on higher-order cumulants
Toshihiro Nonaka, ∗ Masakiyo Kitazawa,
2, 3, † and ShinIchi Esumi ‡ Tomonaga Center for the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan J-PARC Branch, KEK Theory Center, Institute of Particle and Nuclear Studies,KEK, 203-1, Shirakata, Tokai, Ibaraki, 319-1106, Japan
We propose a method to remove the contributions of pileup events from higher-order cumulantsand moments of event-by-event particle distributions. Assuming that the pileup events are givenby the superposition of two independent single-collision events, we show that the true moments ineach multiplicity bin can be obtained recursively from lower multiplicity events. In the correctionprocedure the necessary information are only the probabilities of pileup events. Other terms areextracted from the experimental data. We demonstrate that the true cumulants can be reconstructedsuccessfully by this method in simple models. Systematics on trigger inefficiencies and correctionparameters are discussed.
I. INTRODUCTION
One of the ultimate goals of high energy physics experiments is to study the Quantum Chromo-Dynamics (QCD)phase diagram and especially the search for the QCD critical point [1]. It was suggested that the higher-orderfluctuation observables are sensitive to the critical point, and the phase transition from quark-gluon plasma phaseto the hadron-gas phase [2–5]. There have been lots of experimental efforts to measure the higher-order cumulantsof event-by-event net-particle distributions such as net-proton, net-charge and net-kaon multiplicity distributionsreported by ALICE [6], HADES [7], NA61 [8] and STAR collaborations [9–14]. In particular, the ratio of fourth to thesecond order cumulants of the net-proton distributions were presented to behave nonmonotonically as a function ofcollision energy with a strong enhancement at √ s NN = 7.7 GeV [13]. This result is qualitatively similar to a theoreticalmodel prediction [15], which would imply the existence of the critical point at low collision energy region. In order toestablish the signal from the critical point, it is important to investigate further lower collision energy region, wherethe signal is predicted to decrease again [15]. Such experiments are being carried out by the STAR collaboration withthe fixed-target mode instead of the collider mode at RHIC. In addition, future facilities focusing on low collisionenergies √ s NN <
10 GeV like CBM [16] and J-PARC-HI [17] experiments are also going to run with fixed-target mode.One major issue expected in fixed-target experiments is pileup events. When two collision events occur on the targetwithin a small space and and time interval, they are identified as a single event. These events are called the pileupevent. Usually, the rate of the pileup events is well suppressed and the effect is negligible for most of the measurements.Even if not, the effect would be removed from any averaged observables once the pileup probability is well understoodand estimated. Unfortunately, this is not the case for the higher order fluctuation observables. It was pointed out thatthe pileup events lead to a strong enhancement of the fourth order cumulant and moment at central collisions [18, 19].However, the correction method has not been known. Because the pileup events give rise to a fake enhancement, thiseffect makes it difficult to interpret the final results of the critical point search. In the future experiments at CBM andJ-PARC-HI, a proper understanding of the pileup events will be more crucial, because high collision rates which willbe achieved by these experiments would enhance the probability of the pileups. Development of a correction methodare urgent for proper understanding of upcoming experimental results.In the present work, we propose a method to correct higher-order moments and cumulants for the pileup effects.Assuming that the pileup events consist of independent single-collision events, we derive the relations to connect theexperimentally-observed moments including the pileup effects with the true moments. We also propose a systematicprocedure to obtain the true cumulants using these relations by a recursive reconstruction of moments from lowermultiplicity events. We then demonstrate the validity of this method by applying it to simple models. Systematics ontrigger inefficiencies and correction parameters are also discussed.This paper is organized as follows. In Sec. II, we explain the methodology for pileup corrections and derive correctionformulas. The method is demonstrated in Sec. III with the extreme cases and realistic situations. In Sec. IV we discusssystematics of our method. We then summarize this work in Sec. V. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ phy s i c s . d a t a - a n ] J un II. METHODOLOGYA. Pileup events
Let us first clarify the definition of the pileup events and assumptions to be made in the present study.First, in experimental analyses the pileup events are removed using various methods, some of which are carried outby offline analysis. For example, suppose a correlation plot between number of particles measured by two detectorsin different acceptances. The normal single-collision events are expected to appear as a band having a positive ornegative slope. On the other hand, the pileup events would appear as additional bands having different slopes and/oroffsets, while there could be be some uncorrelated components. The pileup events are then removed by cutting outlierevents outside the correlated band. However, there will be a finite probability that the band is contaminated by thepileup events due to randomness. These events cannot be removed by this analysis. We call these residual eventsremaining after various cutting as the pile up events, and investigate the correction of their effects.Second, in the following discussion two distribution functions play crucial roles. One of them is the “multiplicitydistribution”, i.e. the distribution of the number of produced particles measured at mid- or forward-rapidity. Themultiplicity is sometimes used to define centrality. The other distribution is the “particle distribution”. In the event-by-event analysis, we focus on the distribution of the number of a specific particle or charge, N , such as the net-protonnumber or net-charge, represented by the probability distribution function P ( N ), and study its cumulants. Throughoutthis study, we consider the distribution of a single variable N to simplify the discussion, but it is straightforward toextend the following method to deal with the multi-particle distributions, P ( N , N , · · · ), and the mixed cumulants ofvarious particle species.The distribution P ( N ) depends on the multiplicity. Throughout this paper, we denote the experimentally-observeddistribution function including the pileup effects at multiplicity m as P m ( N ), while the distribution of N in true single-collision events are represented as P t m ( N ). We suppose that m and N would be measured at different acceptances toreduce the auto-correlation effects between m and N . This means that a collision event with m = 0 can take placeand have with nonzero N .Third, except for Sec. II C we consider the pileup events composed of two single-collision events. As discussed inSec. II C, it is possible to extend the following analysis to include the pileup events with more than two single-collisions.The probability of those events, however, are usually well suppressed and explicit consideration of their effects are notneeded. The important assumption taken throughout this paper is that two single-collision events included in a pileupevent are independent. B. Pileup correction
Let us suppose that the pileup events occur with the probability α m at the m th multiplicity bin. Then, theprobability to find N particles of interest at multiplicity m with the pileup effects is given by P m ( N ) = (1 − α m ) P t m ( N ) + α m P pu m ( N ) , (1)where P t m ( N ) and P pu m ( N ) are the probability distribution functions of N for the true (single-collision) and pileupevents, respectively. The pileup events are further decomposed into the “sub-pileup” events given by the superpositionof two single-collision events with multiplicities i and j satisfying m = i + j as P pu m ( N ) = (cid:88) i,j δ m,i + j w i,j P sub i,j ( N ) , (2) P sub i,j ( N ) = (cid:88) N i ,N j δ N,N i + N j P t i ( N i ) P t j ( N j ) , (3)where P sub i,j ( N ) represents the probability distribution of N in the sub-pileup events labeled by ( i, j ), and w i,j is theprobability to observe the sub-pileup events among the pileup events at the m th multiplicity bin. The sum over i and j runs non-negative integers. Obviously w i,j satisfies w i,j = w j,i and (cid:88) i,j δ m,i + j w i,j = 1 . (4)From Eq. 4 one also finds w , = 1.The pileup probabilities α m and w i,j are related to the multiplicity distribution of the single-collision events. Let T ( m ) be the multiplicity distribution, i.e. probability that a collision event with multiplicity m occurs for all single-collision events. When all sub-pileup events are rejected by an experimental analysis with the same resolution, theprobability to find a sub-pileup event labeled by ( i, j ) among all collision events is given by αT ( i ) T ( j ), where α denotesthe probability to find a pileup event among all collision events. Also, the probability to find an event with multiplicity m without distinction between single-collision and pileup events is given by (1 − α ) T ( m ) + α (cid:80) i,j δ m,i + j T ( i ) T ( j ). Wethus have w i,j = αT ( i ) T ( j ) (cid:80) i,j δ m,i + j αT ( i ) T ( j ) , (5) α m = α (cid:80) i,j δ m,i + j T ( i ) T ( j )(1 − α ) T ( m ) + α (cid:80) i,j δ m,i + j T ( i ) T ( j ) . (6)Therefore, in this case α m and w i,j are completely determined from α and T ( m ). We, however, note that Eqs. 5and 6 might not hold in realistic experimental cases if the probability distribution of the pileup rejection is differentfrom the multiplicity distribution. We thus do not use Eqs. 5 and 6 explicitly in the rest of this section. In realexperiments, w i,j can directly be estimated by some reasonable assumptions within the experimental simulation. Themodels employed in Sec. III satisfy Eqs. 5 and 6.From Eqs. 1, 2 and 3, the moment generating function [20] for events at multiplicity m is expressed as G m ( θ ) = (cid:88) N e Nθ P m ( N )= (1 − α m ) G t m ( θ ) + α m (cid:88) i,j δ m,i + j w i,j G sub i,j ( θ ) , (7)with G sub i,j ( θ ) = G t i ( θ ) G t j ( θ ) , (8)where G t m ( θ ) = (cid:80) N e Nθ P t m ( N ) is the moment generating function of P t m ( N ). The r th order moment of the observeddistribution P m ( N ) is then given by (cid:104) N r (cid:105) m = (cid:88) N N r P m ( N ) = d r dθ r G ( θ ) | θ =0 = (1 − α m ) (cid:104) N r (cid:105) t m + α m (cid:88) i,j δ m,i + j w i,j (cid:104) N r (cid:105) sub i,j , (9)with (cid:104) N r (cid:105) t m = (cid:80) N N r P t m ( N ) and (cid:104) N r (cid:105) sub i,j = (cid:88) N N r P sub i,j ( N ) = r (cid:88) k =0 (cid:18) rk (cid:19) (cid:104) N r − k (cid:105) t i (cid:104) N k (cid:105) t j . (10)The right-hand sides in Eq. 10 up to fourth order are written as (cid:104) N (cid:105) sub i,j = (cid:104) N (cid:105) t i + (cid:104) N (cid:105) t j , (11) (cid:104) N (cid:105) sub i,j = (cid:104) N (cid:105) t i + (cid:104) N (cid:105) t j + 2 (cid:104) N (cid:105) t i (cid:104) N (cid:105) t j , (12) (cid:104) N (cid:105) sub i,j = (cid:104) N (cid:105) t i + (cid:104) N (cid:105) t j + 3 (cid:104) N (cid:105) t i (cid:104) N (cid:105) t j + 3 (cid:104) N (cid:105) t i (cid:104) N (cid:105) t j , (13) (cid:104) N (cid:105) sub i,j = (cid:104) N (cid:105) t i + (cid:104) N (cid:105) t j + 4 (cid:104) N (cid:105) t i (cid:104) N (cid:105) t j + 4 (cid:104) N (cid:105) t i (cid:104) N (cid:105) t j + 6 (cid:104) N (cid:105) t i (cid:104) N (cid:105) t j . (14)We note that Eq. 10 is alternatively expressed using cumulants in a compact form as [20] (cid:104) N r (cid:105) sub i,j, c = (cid:104) N r (cid:105) t i, c + (cid:104) N r (cid:105) t j, c , (15)where (cid:104) N r (cid:105) sub i,j, c and (cid:104) N r (cid:105) t j, c are the cumulants of the sup-pileup and true distributions, respectively.Substituting Eq. 10 into Eq. 9, one obtains formulas connecting (cid:104) N r (cid:105) m and (cid:104) N r (cid:105) t m . It is notable that in theseformulas the observed moment (cid:104) N r (cid:105) m is given by the combination of the true moments (cid:104) N r (cid:48) (cid:105) t m (cid:48) with r (cid:48) ≤ r and m (cid:48) ≤ m .The true moments (cid:104) N r (cid:105) t m are obtained from the observed moments (cid:104) N r (cid:105) m by solving Eqs. 9 and 10. This procedurecan be carried out recursively starting from m = 0 and r = 1, and by increasing m and r . To see this, it is convenientto rewrite Eqs. 9 and 10 as (cid:104) N r (cid:105) t m = (cid:104) N r (cid:105) m − α m C ( r ) m − α m + 2 α m w m, , (16)with C ( r ) m = µ ( r ) m + (cid:88) i,j> δ m,i + j w i,j (cid:104) N r (cid:105) sub i,j , (17)and µ ( r ) m = w m, r − (cid:88) k =0 (cid:18) rk (cid:19) (cid:104) N r − k (cid:105) t0 (cid:104) N k (cid:105) t m ( m > , r − (cid:88) k =1 (cid:18) rk (cid:19) (cid:104) N r − k (cid:105) t0 (cid:104) N k (cid:105) t0 ( m = 0) . (18)Up to the fourth order, the explicit forms of µ ( r ) m are µ (1) m = 0 , (19) µ (2) m = 2 w m, (cid:2) (cid:104) N (cid:105) t0 + 2 (cid:104) N (cid:105) t m (cid:104) N (cid:105) t0 (cid:3) , (20) µ (3) m = 2 w m, (cid:2) (cid:104) N (cid:105) t0 + 3 (cid:104) N (cid:105) t m (cid:104) N (cid:105) t0 + 3 (cid:104) N (cid:105) t m (cid:104) N (cid:105) t0 (cid:3) , (21) µ (4) m = 2 w m, (cid:2) (cid:104) N (cid:105) t0 + 4 (cid:104) N (cid:105) t m (cid:104) N (cid:105) t0 + 4 (cid:104) N (cid:105) t m (cid:104) N (cid:105) t0 + 6 (cid:104) N (cid:105) t m (cid:104) N (cid:105) t0 (cid:3) , (22)for m > µ (1)0 = 0 , (23) µ (2)0 = 2 (cid:104) N (cid:105) t (cid:104) N (cid:105) t , (24) µ (3)0 = 6 (cid:104) N (cid:105) t (cid:104) N (cid:105) t , (25) µ (4)0 = 8 (cid:104) N (cid:105) t (cid:104) N (cid:105) t + 6 (cid:104) N (cid:105) t (cid:104) N (cid:105) t . (26)To obtain the true moments (cid:104) N r (cid:105) t m , we first use the fact that C (1)0 = 0, which leads to (cid:104) N (cid:105) t0 = (cid:104) N (cid:105) / (1 + α ). Next,Eqs. 17 and 18 shows that the correction factors C ( r )0 at m = 0 are given only by the moments (cid:104) N r (cid:48) (cid:105) t0 with r (cid:48) < r .One thus can obtain (cid:104) N r (cid:105) t0 recursively from lower order up to any higher orders. Similarly, one can obtain the truemoments at multiplicity m = 1 from lower order moments up to any order using the fact that the correction factor C ( r )1 consists of (cid:104) N r (cid:48) (cid:105) t m (cid:48) with r (cid:48) ≤ r and m (cid:48) ≤ m . By repeating the same procedure one can obtain the true momentsfor all multiplicities.An important remark here is that this procedure can be carried out in almost data-driven way. Only thing we needis the probabilities w i,j and α m , which would be determined by simulations. C. Pileups composed of more than two single-collision events
So far, we considered the pileup events composed of two single-collision events. It is not difficult to extend theseresults to include the pileup events composed of three single-collision events. In this case, Eq. 2 is modified as P pu m = (cid:88) i,j δ m,i + j w i,j P sub i,j ( N ) + (cid:88) i,j,k δ m,i + j + k w i,j,k P sub i,j,k ( N ) , (27)where P sub i,j,k ( N ) represents the probability distribution of N on the sub-pileup events composed of three single collisionswith multiplicities i , j , and k , and w i,j,k is the probability of the sub-pileup event. From the independence of theindividual collisions, P sub i,j,k ( N ) is given by P sub i,j,k = (cid:88) N i ,N j ,N k δ N,N i + N j + N k P t i ( N i ) P t j ( N j ) P t j ( N j ) . (28)Then, it is straightforward to derive the relations like Eqs. 9 and 16. These results allow us to obtain the true moments (cid:104) N r (cid:105) t m recursively from small m as before. In this way, pileups with arbitrary many single-collision events can betaken into account in principle. III. MODEL
In this section we apply the procedure introduced in the previous section to the pileup correction in simple modelsand demonstrate that the true cumulants are successfully obtained.
A. Multiplicity distributions
Let us first generate a realistic multiplicity distribution with pileup events. We employ the Glauber and two-component model for this purpose. Two gold nuclei are collided in the Glauber model, where the pp cross section ischosen to be 33 mb. The number of participant nucleons, N part , and binary collisions, N coll , are obtained. In order topropagate N part and N coll to the multiplicity, we define the number of sources, N sc as N sc = (1 − x ) N part + xN coll , (29)where x is the fraction of the hard component. We choose x = 0 . N sc based on the negative binomial distribution: P µ,k ( N ) = Γ( N + k )Γ( N + 1)Γ( k ) · ( µ/k ) N ( µ/k + 1) N + k , (30)where µ is the mean value of particles generated from one source, and k corresponds to the inverse of width of thedistribution. µ = 1 . k = 1 . α = 0 .
05. In this way, 10 million Au+Au collision events are processed. We note that in this model the pileupprobabilities w i,j and α m are given by Eqs. 5 and 6 by construction.The resulting multiplicity distribution is shown by the black line in Fig. 1. The blue squares show the multiplicitydistribution from single-collision events, while those from pileup events are shown by the red circles. It is found that,due to the pileup events, the measured distribution has the tail on top of the distribution from the single-collisionevents. The inset panel shows α m , i.e. the ratio of the pileup events at multiplicity m . From the panel one finds that α m grows with increasing m . This behavior suggests that the effect of pileup events are more problematic in centralcollisions rather than peripheral collisions.In Fig. 2, we plot the multiplicity distribution of single-collision events T ( m ) and the number of sub-pileup events( i, j ) normalized by total simulated events, αT ( i ) T ( j ). From these results w i,j and α m are constructed according toEqs. 5 and 6. These parameters are used in the following two subsections. m N u m be r o f e v en t s - % - % - % =0.05 a MeasuredPileupTrue m p il eup / m ea s u r ed m a FIG. 1. The multiplicity distribution generated from the Glauber and two component model. The black line includes thecontribution from pileup events with α = 0 .
05 (measured distribution). The red open circles are the distribution from pileupevents, and the blue squares are from the normal single collision events. The bands indicate 0-5%, 5-10% and 70-80% centralities.The inset panel shows the ratio of pileup to measured distributions as a function of multiplicity ( α m ). B. Simple case
In this and next subsections, we discuss the pileup correction for two model distributions P t m ( N ) with the multiplicitydistribution obtained in Sec. III A. In this subsection, we consider a simple model where the particle number N obeysthe Poisson distribution with the mean value of 10 at all the multiplicity bin. We emphasize that this model is totallyimpractical, because 10 particles on average are created at both m = 0 and m = 300. However, this model is suitableto demonstrate the validity of the recursive correction procedures. The more realistic model will be discussed in thenext subsection.Figure 3 shows the particle distribution for the first 4 multiplicity bins ( m = 0, 1, 2, 3). The red circles show pileupevents, and the blue squares show the single-collision events. The measured distribution given by the sum of these - - - =0.05 a T(i) x T(j) i , j w Normalized Counts - - - - M u l t i p li c i t y : j T(j)
NormalizedCounts
Multiplicity : i N o r m a li z ed C oun t s5 - - - - T(i) N o r m a li z ed C oun t s FIG. 2. Correlation between multiplicities i and j from two collisions which forms pileup events. Z-axis is normalized by totalnumber of events. The inset panel shows the expanded plot at x < y <
7. Projected histograms into x and y axes areshown as well. distributions is shown by the black solid lines, which are found to have a bump structure at N (cid:38)
20 due to pileupevents. Other colored lines show the sub-pileup events for all possible combinations of ( i , j ) with m = i + j . In the caseof m = 0, there is only one combination for sub-pileup, ( i, j ) = (0 , w , = 1. As shown in Fig. 3 (b), (c) and (d), in the case of m ≥
1, the pileup distributionconsists of multiple sub-pileup events with ( i, j ) = ( m, · · · , (0 , m ).In Fig. 4, we show the cumulants (cid:104) N r (cid:105) m, c at the m th multiplicity bin. In the figure, the cumulants are plotted forthe true single-collision distribution obtained by the simulation, measured distribution with the pileup effects, and thecorrected results. Statistical uncertainties are estimated by bootstrap. True cumulants are (cid:104) N rm (cid:105) = 10 by definition.The measured cumulants have strong deviations from this value due to the pileup events [18, 19]. It is notable thatthe measured cumulants especially for r = 1 and 2 behave similar to what we have already seen in α m in the insetpanel in Fig. 1. Because the particles are generated according to the Poisson distributions having the same meanvalue, the effects from the pileup events only depend on the pileup probability. Corrected cumulants are found to beconsistent with the true value (cid:104) N r (cid:105) m, c = 10 within statistics, which indicates that our method does work well. Largepoint-by-point variations are due to the increased statistical uncertainties after the corrections. C. Realistic case
Next, we move on to more realistic case, where the mean value of N increases with increasing N part . We againemploy the Poisson distribution for N , but in contrast to the previous subsection we assume that the mean value variesdepending on N part as (cid:104) N (cid:105) = 0 . N part . We employ the same pileup probability α = 0 .
05. The centrality is definedby dividing the multiplicity distribution of single-collision events (see Fig. 1 for corresponding regions in multiplicitydistributions). Figure 5 shows the particle number distributions for 0-5%, 10-20%, 40-50% and 70-80% centralities.The pileup distributions are found inside the true distribution at peripheral collisions, while the pileup distributionsin central collisions appear as a long tail in the measured distributions. Thus, large effects on cumulants are expectedin central collisions in this simulation [19].Cumulants for each multiplicity bin are averaged in each centrality by using event statistics as a weight [21], whichare shown in Fig. 6 as a function of centrality. The centrality is 0 − − − −
80% from x = 0to x = 8. In this case, significant deviation on measured cumulants are observed only in the central collisions, as wasexpected from Fig. 5. Corrected cumulants are consistent with true cumulants even at 0 − N part are shown by the dotted lines. The difference between markers and lines seen especially for higher-order cumulants i j m;i,jsub P 0 0(N) m P (N) mpu
P (N) mt P =0.05 a (a) m=0 i j m;i,jsub P 0 1 1 0 (b) m=1 i j m;i,jsub P 0 2 1 1 2 0 (c) m=2 i j m;i,jsub P 0 3 1 2 2 1 3 0 (d) m=3 N N u m be r o f E v en t s FIG. 3. Particle number distributions, P m ; i,j ( N ), in the simple model in Sec. III B for the first 4 multiplicity bins, (a) m = 0,(b) m = 1, (c) m = 2 and (d) m = 3. The red circles shows the pileup events, and blue squares are for single-collision events.The black solid line is for measured events. Distributions for sub-pileups are shown in colored or dotted lines. =0.05 a (a) r=1 TrueMeasuredCorrected (b) r=2 - (c) r=3 - (c) r=4 m c æ r N Æ FIG. 4. Cumulants up to the 4th-order as a function of multiplicity in the simple model in Sec. III B. True values of cumulantsare shown by the blue squares, and the measured value of cumulants (including pileup events) are shown by the black circles.The red stars show the results corrected for pileups. indicate the residual volume (participant) fluctuations even after the centrality bin width averaging [21, 23]. Thishappens because we let N part fluctuate event by event based on the Glauber model and the mean value of Poissondistribution is defined as the function of N part . It should be noted that the location of the kink structure at x = 1 ∼ N part distribution have minimum or maximum values [23],observed in true and corrected cumulants would depend on the model and the binning of the centrality. Interestingly,the measured cumulants including pileup events look rather qualitatively normal (linear) compared to the true andcorrected cumulants. This would imply that the pileup events could accidentally hide the characteristic kink structurearising from the volume fluctuations. One should always be careful if the effects from the volume fluctuations areremoved from the measurements. Otherwise, the final results could be spoiled by sizable effects of both pileup andvolume fluctuations. P(N)(N) pu P(N) t P =0.05 a (a) 70-80% (b) 40-50% (c) 10-20% (d) 0-5% N N u m be r o f E v en t s FIG. 5. Particle number distributions for (a) 70-80%, (b) 40-50%, (c) 10-20% and (d) 0-5% centralities in the realistic modelin Sec. III C. The red circles shows the pileup events, and the blue squares are for single-collision events. The black solid line isfor measured events. (a) r=1 =0.05 a True (WO/VF)TrueMeasuredCorrected (b) r=2 (c) r=3 (d) r=4
Measured/TrueCorrected/True
Centrality c æ r N Æ R a t i o c æ r N Æ R a t i o FIG. 6. Cumulants up to the 4th-order as a function of centrality, 0-5%, 5-10%, 10-20%, · · · , and 70-80% centralities from x = 0 to 8. True values of cumulants are shown by the open blue squares, and the measured values of cumulants (includingpileup events) are shown by open black circles. The red open stars show results corrected for pileups. The ratios of measuredand corrected results to the true cumulants are shown by the filled markers in lower panels. The true values excluding volumefluctuations are shown in dotted lines. IV. SYSTEMATICSA. Trigger inefficiency
An important procedure of the pileup correction is the recursive solving of moments from the lowest multiplicity eventat m = 0. At such super-peripheral collisions, however, the event itself cannot be triggered due to small multiplicityand the detector threshold to reject backgrounds. The event efficiency is thus reduced in peripheral collisions, whichis known as “trigger inefficiency”. It is possible that these effects at smaller multiplicity events accumulate in therecursive procedure and give rise to a large systematic deviation on the reconstructed cumulants at large m .To check this problem, in this subsection the events for m <
20 are artificially reduced by the arbitrary functionof the multiplicity, and the pileup correction is not applied for this region. In other words, we regard the observedmoments (cid:104) N r (cid:105) m as the true moments (cid:104) N r (cid:105) t m for m <
20, and perform the correction only for m ≥
20. The model inSec. III C is employed.Figure 7 shows the ratios of measured and corrected cumulants to the true cumulants as a function of multiplicity.The averaged results for centrality bins 0-5, 5-10, 10-20, ... 50-60% are also shown. As the correction is not appliedfor m <
20, the corrected results are identical with measured values. On the other hand, the corrected cumulants for m ≥
20 are quickly approaching to the true value, which shows that the correction works well regardless of incorrectcorrection factors in peripheral collisions. This is because the sub-pileup moments (the second term in Eq. 9) haveless contributions from peripheral collisions due to the tiny production rate of particles of interest. Since it dependson how significant the production of particles of interest is in peripheral collisions compared to central collisions, wewould propose to check the results by changing the starting point of the recursive corrections, and implement it as apart of systematic uncertainties in final results. (a) r=1 (b) r=2 (c) r=3 Measured/TrueCorrected/TrueMeasured/True (Centrality)Corrected/True (Centrality) (d) r=4 m t c æ r N Æ R a t i o t o FIG. 7. The ratios of measured (black circles) and corrected (red stars) cumulants with respect to the true cumulants asfunctions of multiplicity. The bands represent the statistical uncertainties. The results averaged into 0-5, 5-10, 10-20,... and50-60% centralities are shown in blue squares and green crosses.
B. Correction parameters
The new method relies on the probabilities w i,j and α m , and other terms are all extracted from data. Hence, thesystematic uncertainties would come from how precisely those parameters are determined in the simulations.To check how the uncertainty of w i,j and α m affects the final result, we again employ the model in Sec. III C andperform the pileup correction using wrong pileup probabilities, α (1 + p ), with α = 0 .
05. We vary the value of p from -10% to 10% and determine the values of w i,j and α m according to Eqs. 5 and 6. The pileup correction isthen performed with these wrong probabilities. Figure 8 shows cumulants up to the 4th order at 0-5% centrality asfunctions of p . It can be found that the results are overcorrected for p >
0, while the corrections are not enough for0 p <
0. Further, higher-order cumulants get more affected by wrong values of the correction parameters as seen in thelarger slope of the fitted functions. We would propose to consider those variations as systematic uncertainties on finalresults. - - =-0.19 c (a) r=1 =0.05, 0-5% centrality a - - =-0.74 c (b) r=2 - - =-3.85 c (c) r=3 - - =-26.18 c (d) r=4 p c æ r N Æ FIG. 8. Cumulants up to the 4th-order corrected with a wrong pileup probability , α (1 + p ), as a function of p . Statisticaluncertainties are shown in bands. The red dotted lines are the polynomial fit functions with c p + c . The values of the fitparameter c are shown in the panel. The scale of y-axis is set to ±
15% with respect to the cumulant values at p = 0 for allpanels. V. SUMMARY
In this paper, we proposed a method to correct the effect of the pileup events on the higher-order moments andcumulants. The method can be derived by decomposing pileups into various combinations of sub-pileup events in termsof moments. The moments for sub-pileup events can be reconstructed assuming that the pileups are the consequencesof the superposition between two independent events. We utilized the fact that the pileup changes the total multiplicity.The correction formulas are expressed by the sub-pileup moments and the moments from the lower multiplicity events,thus solvable from the lowest multiplicity events. Two models are performed with the same mean values of particledistributions for all multiplicity events, and with the N part -dependent mean values. The method works correctly forboth cases. The method can deal with the pileup events for more than two single-collisions. The effect of triggerinefficiencies needs to be carefully checked by changing the starting point of the recursive corrections. The systematicuncertainties will be reduced by determining the pileup probability precisely.Finally, we remark that one has to make sure that the detector efficiencies are corrected in a proper way [24–35]before performing the pileup correction. VI. ACKNOWLEDGEMENT
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