Efficiency correction for cumulants of multiplicity distributions based on track-by-track efficiency
EEfficiency correction for cumulants of multiplicity distributions based ontrack-by-track efficiency
Xiaofeng Luo ∗ and Toshihiro Nonaka † Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China
We propose a simplified procedure for the experimental application of the efficiency correctionon higher order cumulants in heavy-ion collisions. By using the track-by-track efficiency, we caneliminate possible bias arising from the average efficiencies calculated within the arbitrary binningof the phase space. Furthermore, the corrected particle spectra is no longer necessary for theaverage efficiency estimation and the time cost for the calculation of bootstrap statistical error canbe significantly reduced.
I. INTRODUCTION
Higher order cumulants of event-by-event conservedcharge fluctuations are important observables to searchfor the QCD critical point [1–6] in heavy-ion collisions.In the beam energy scan (BES) program at RHIC, theSTAR experiment has measured the cumulants up tothe 4th order of the net-charge and net-kaon multiplic-ity distributions [7, 8], and up to the 6th order of thenet-proton multiplicity distribution [9–11]. In the re-cent results of the net-proton multiplicity distribution,the non-monotonic behaviour of the fourth order fluc-tuation has been observed with respect to the collisionenergy [12], which is quite similar to the theoretical pre-diction with the critical point [13]. Since there are stilllarge uncertainties in low collision energies, the secondphase of the Beam Energy Scan program (BES-II) willbe carried out in 2019-2021 focusing on the collision en-ergy of √ s NN =7.7–19.6 GeV.One of the difficulties of measuring the higher order cu-mulants is the efficiency correction. It is known that thevalue of cumulants are artificially changed from the trueones [14, 15], due to the fact that detectors miss someparticles with the probability called efficiency. Some an-alytical formulas have been proposed to correct the mea-sured cumulants with the assumption that the responsefunction of the efficiency follows the binomial distribu-tion [14, 16–22]. Recently, a few attempts to under-stand and correct for the possible non-binomial efficien-cies have been discussed [11, 23, 24], but there are stilllarge systematic uncertainties arising from how to de-termine the detector-response functions. The efficiencycorrection with the binomial assumption is thus still im-portant. Hereafter, let us call it ”binomial correction”for simplicity. In the binomial correction, particles arecounted separately in the ”efficiency bin” where the ef-ficiency changes, which are substituted into the correc-tion formulas with the corresponding value of efficien-cies. Due to the huge calculation cost with large numberof efficiency bins by using the correction formulas basedon the factorial moments [14, 18, 19], more efficient for-mulas have been proposed in which factorial cumulantsare used in the derivation [20–22]. Experimentally, sin-gle particle efficiencies can be computed by the MC de-tector simulations in terms of the various experimental ∗ xfl[email protected] † [email protected] observables like centrality, multiplicity, vertex position,transverse momentum, rapidity, azimuthal angle and soon. Efficiency bins are then defined with respect to thetrack-wise variables in which the particles are counted.This binning is still arbitrary, which depends not onlyon the computing power for the detector simulations butalso on the homogeneousness of the detector in the ac-ceptance. Therefore, additional systematic studies willbe necessary on how many efficiency bins are enough.The most crucial thing is that we need to calculate theaveraged efficiency at each efficiency bin with weightedby the true spectra. This indicates that the traditionalefficiency correction based on average efficiency, can notbe performed until the corrected spectra of identified par-ticle is available. Fortunately, we found those difficultiescan be overcome by using the so called track-by-track ef-ficiency correction methods. By doing this, the correctedparticle spectra is no longer needed. We can also re-duce the potential systematic bias by using the averageefficiency correction method and the calculation cost forbootstrap statistical errors calculations.This paper is organized as follows. In Sec II, the effi-ciency correction with the binomial model is introduced.The correction formulas based on factorial cumulants willbe shown as well. In Sec. III, we discuss three difficultiesof the efficiency correction in the experimental applica-tions. We also use a numerical study to demonstrate oneof the issues. In Sec. IV, a simple solution using the track-by-track efficiency is shown, followed by the toy model tocheck the validity of the new method. II. EFFICIENCY CORRECTIONA. Cumulants and factorial cumulants
The m -th order cumulant of the probability distribu-tion function P ( N ) is defined as (cid:104) N m (cid:105) c = ∂ m ∂θ m K ( θ ) | θ =0 , (1) K ( θ ) = ln (cid:88) N e Nθ P ( N ) = ln (cid:104) e Nθ (cid:105) , (2)where K ( θ ) represents the cumulant generating function.Another quantity, factorial cumulants (cid:104) N m (cid:105) fc are also de-fined as (cid:104) N m (cid:105) fc = ∂ m ∂ m K f ( s ) | s =1 , (3) K f ( s ) = ln (cid:104) s N (cid:105) , (4) a r X i v : . [ phy s i c s . d a t a - a n ] A p r with K f ( s ) being the factorial-cumulant generating func-tion. Cumulants and factorial cumulants are connectedto each other, e.g, factorial cumulants are expressed interms of cumulants: (cid:104) N m (cid:105) fc = (cid:104) N ( N − · · · ( N − m + 1) (cid:105) c . (5) B. Binomial model
Let us assume that the probability distribution func-tion P ( N ) is observed as ˜ P ( n ) through detectors. Some of N generated particles are missed by the detectors, whichleads to the observation of n particles ( n ≤ N ). Thisfinite probability to measure particles characterized bythe detector is called efficiency. When the efficiencies for generated particles are independent each other, the detec-tion process can be described by the binomial distribution B ε,N ( n ): ˜ P ( n ) = (cid:88) N P ( N ) B ε,N ( n ) , (6) B ε,N ( n ) = N ! n !( N − n )! ε n (1 − ε ) N − n , (7)where ε represents the efficiency. It is known that in thissituation the relationship between factorial cumulants of P ( N ) and ˜ P ( n ) is given by [21, 22] (cid:104) n m (cid:105) fc = ε m (cid:104) N m (cid:105) fc . (8)Using Eq. (8) and extending to the M multi-variablecase P ( N ) = P ( N , N , ..., N M ), the formulas up to thefourth order cumulant are shown below: (cid:10) Q (cid:11) c = (cid:104) q (1 , (cid:105) c , (9) (cid:10) Q (cid:11) c = (cid:104) q , (cid:105) c + (cid:104) q (2 , (cid:105) c − (cid:104) q (2 , (cid:105) c , (10) (cid:10) Q (cid:11) c = (cid:104) q , (cid:105) c + 3 (cid:104) q (1 , q (2 , (cid:105) c − (cid:104) q (1 , q (2 , (cid:105) c + (cid:104) q (3 , (cid:105) c − (cid:104) q (3 , (cid:105) c + 2 (cid:104) q (3 , (cid:105) c , (11) (cid:10) Q (cid:11) c = (cid:104) q , (cid:105) c + 6 (cid:104) q , q (2 , (cid:105) c − (cid:104) q , q (2 , (cid:105) c + 4 (cid:104) q (1 , q (3 , (cid:105) c + 3 (cid:104) q , (cid:105) c +3 (cid:104) q , (cid:105) c − (cid:104) q (1 , q (3 , (cid:105) c + 8 (cid:104) q (1 , q (3 , (cid:105) c − (cid:104) q (2 , q (2 , (cid:105) c + (cid:104) q (4 , (cid:105) c − (cid:104) q (4 , (cid:105) c + 12 (cid:104) q (4 , (cid:105) c − (cid:104) q (4 , (cid:105) c , (12)with q ( r,s ) defined as q ( r,s ) = M (cid:88) i =1 ( a ri /ε si ) n i , (13)where M represents the number of efficiency bins, n i rep-resents the number of particles, and a i represents theelectric charge of particles in i th efficiency bin. The con-cept of the efficiency bin started to be taken into accountin Ref. [18] inspired by experimental requirement, whichwill be discussed in the next section. III. DIFFICULTIES IN THE EFFICIENCYCORRECTION
In this section, we clarify the three difficulties in the ex-perimental application of the efficiency correction. First,the details of the experimental procedures of the effi-ciency correction is explained to point out the difficulties.Second, a simple toy model is used to demonstrate oneof those.
A. Experimental application
Experimentally, single-particle efficiency can be deter-mined by the Monte-Carlo approach of the detector sim- ulation with respect to track-wise variable like transversemomentum, rapidity and azimuthal angle. This kind ofefficiency map can be computed precisely as long as thecomputing source permits. Usually, the efficiency map isdivided into various efficiency bins based on the track-wise variables and the averaged efficiency in each effi-ciency bin needs to be estimated by using the correctedspectra. In the current analysis of the net-proton fluc-tuations at the STAR experiment, the proton identifica-tion method is different between low and high p T regions,which leads to the step-like dependence of the efficiencyas a function of p T [12]. In this case, particles need tobe counted separately for the two p T region in which thevalues of the efficiency are different, then the true cumu-lants in entire p T regions can be reconstructed based onfactorial moments [19].If the number of efficiency bins M is large, it is moreefficient to apply the efficiency correction according toEqs. (9)–(12), which is based on factorial cumulants. Forthe statistical errors estimation, the bootstrap methodwould be the realistic way. We can obtain a new distri-bution by random sampling from the original distributionwith the same number of events to calculate cumulants.This procedure is repeated with 100 times, and the stan-dard deviation of the 100 cumulant values calculated fromthe new distributions are taken as the statistical error.In order to take into account the correlation between dif-ferent efficiency bins, the sampling would be performedbased on the M dimensional histogram.Below are three main difficulties in the experimentalimplementation of the efficiency correction.1. We expected that it can provide more precise ef-ficiency corrected cumulants when using the largenumber of efficiency bins. However, it is difficult toknow how many efficiency bins are enough.2. In order to obtain the averaged efficiency for eachefficiency bin, we need to consider the variation ofthe particle yields within the efficiency bin, whichmeans the corrected spectra is necessary. This isespecially crucial issue for new data set in new col-lision energy or collision system where the correctedspectra is not available.3. The calculation cost on the bootstrap increases as ∝ n M , which indicates that the statistical error es-timation will be difficult for many efficiency bins inview of the computing source. B. Toy model simulation
Let us discuss more about the first issue related to theapproximation of efficiency bins above by using a simpletoy model. We generated 50 independent binomial dis-tributions for positively and negatively charged particles, P ± i ( N ± i ) ( i = 1 , , ..., N + = 4, N − = 3 and ε ± = 0 . ε ± i )are randomly chosen within (0 . , .
0) with uniform dis-tribution. Particles N ± i are then randomly sampled withbinomial efficiencies ε ± i to define the measured particles n ± i . Hereafter, let us suppose the net-particle distributionwhich consists of 50 independent distributions given by P net ( N ) with N = (cid:80) i =1 ( N + i − N − i ). We define the m -thorder cumulant of P net ( N ) as ”true” cumulants. The effi-ciency correction is performed with M efficiency bins byusing the corresponding averaged efficiency. For instance,let us suppose the efficiency correction with M = 10 ef-ficiency bins. In this case, whole 50 distributions areequally divided into 10 sub-bins, each sub-bin contains5 distributions. The number of measured particles arecounted at each sub-bin n ± sub ,x , ( x = 1 , , ..., ε ± sub ,x = (cid:42) x +1) (cid:88) i =5 x ε ± i N ± i / x +1) (cid:88) i =5 x N ± i (cid:43) , (14)where the bracket represents the event average. Then n ± sub ,x and ε ± sub ,x are substituted into Eqs. (9)–(13) event-by-event to calculate cumulants . The efficiency correc-tion has been done with different number of efficiencybins for M =1, 2, 5, 10, 25, 50 to check how many ef-ficiency bins are needed to obtain the true cumulants.Figure 1 shows the corrected C , C and C as a functionof the number of efficiency bins, where the true value ofcumulants are shown in red squares. We find that theresults with M = 50 are consistent with the true cumu-lants, which is because 50 efficiency bins are assumed inthe toy model. It is also found that using wide efficiencybins is clearly incorrect and M =25 efficiency bins is stillnot enough. FIG. 1. C , C and C as a function of efficiency bins. The true value of cumulants are shown in red squares. Since the notation of electric charge is explicitly included inEq. (13), we need following modification for substitution: M → M , a i = +1 ( i ≤ M ), a i = − i > M ) IV. SOLUTIONA. Track-by-track efficiency
Let us suppose infinite number of efficiency bins M →∞ . Equation (13) is then written by q ( r,s ) = ∞ (cid:88) i =1 ( a ri /ε si ) n i (15)= a r ε s n + a r ε s n + · · · + a ri ε si n i + · · · , (16)Since we consider that the width of the efficiency bin isnow zero, each efficiency bin contains up to one parti-cle. Thus, the summations for n i = 0 vanish (in otherwords, efficiency bins containing no particles don’t needto be taken into account), and we immediately find thatEq. (16) is equivalent to the summation with respect tothe total number of particles n tot in one event, which isgiven by q ( r,s ) = n tot (cid:88) j =1 a rj ε sj , (17)which is connected with Eq. (13) via n tot = (cid:80) Mi =1 n i .It is found that no variable related to the efficiency binappears in Eq. (17). What we need to take care of isonly the track-by-track efficiency, which indicates thatthe analytical formula of the efficiency with respect totrack-wise variables can be directly used to determine thetrack-by-track efficiency . Accordingly, we don’t need toestimate the averaged efficiency at each efficiency bin,so the corrected spectra is no longer necessary for theefficiency correction. The first two difficulties have beensolved. Hereafter, let us call the correction in Eq. (13)”bin-by-bin” method, and call the track-wise correctionin Eq. (17) ”track-by-track” method. B. Method validation
In this sub-section, we employ a toy model to demon-strate the validity of the track-by-track efficiency method,and also discuss the rest one problem regarding how toestimate the statistical errors. We start from two Gaussdistributions, one is for positively charged particles, andthe other is for negatively charged particles. Figure 2–(a)shows the even-by-event correlation histogram betweenpositively N + and negatively N − charged particles. Foreach particle p T is allocated according to the spectragiven by f ( p T ) = p T e − p T /t (0 < p T < , (18)where t = 0 .
26 and 0 .
24 for positively and negativelycharged particles, respectively. p T dependent efficiencyis assumed to be the convolution of the 1st and 2nd poly-nomial functions given by ε ( p T ) = (cid:40) a × p T + b × p T + c (0 < p T < ,d × p T + e (1 < p T < , (19)where ( a, b, c, d ) =( − . , . , . , . , .
65) and( − . , . , . , . , .
55) for positively and nega-tively charged particles, respectively. Each particle isthen sampled by the corresponding value of efficiencydetermined in Eq. (19). The resulting correlation be-tween measured positively n + and negatively n − charged particles is shown in Fig. 2–(b), and p T spectra is shownin Fig. 2–(d).Since efficiency changes continuously with respect to p T , it is cumbersome to use the bin-by-bin correction for-mulas in Eq. 13. Instead, we substitute the value of effi-ciency for each measured particle determined by Eq. (19)into Eq. 17 to calculate q ( r,s ) at each event.The statistical errors can be estimated by the bootstrapmethod. As was mentioned in Sec. III, in the case ofbin-by-bin correction method with M efficiency bins, thebootstrap sampling would be performed based on the M dimensional histogram. But now there is no longer theefficiency bin, sampling can be simply done based on thespectra as follows:1. Resample n and n randomly from Fig. 2–(b)2. Allocate p T for each particle based on the measuredspectra in Fig. 2–(d)3. Apply the efficiency correction to obtain efficiencycorrected cumulants by using the known efficiencycurve in Fig. 2–(c)4. Repeat 1–3 with 100 times and take the standarddeviation as the statistical error.Above procedures are repeated with 100 times indepen-dently in order to check the validity of the correction andits statistical error. Results up to the fourth order areshown in Fig. 3. It is found that the data points are dis-tributed around the true value, so the correction methodusing track-by-track efficiency works well. The proba-bility of data points touching the true value within thestatistical error is shown in the top right of each panel.We find it comparable with the 1 σ nature of the Gaus-sian, which indicates the validity of the bootstrap. Wealso checked that the calculation cost only depends onthe number of particles ∝ n tot , which is due to the factthat we don’t have to consider particles which was notmeasured, while the bins containing no particles neededto be taken into account in bin-by-bin method. On theother hand, we can also use the Delta theorem [25, 26] tocalculate the statistical errors of the efficiency correctedcumulants, which will be implemented in the future dataanalysis.It should be noted that one would need to considerhow to treat the momentum resolution in real experi-ments. Since we use p T of individual particles for theefficiency correction, the momentum resolution might di-rectly affect the cumulants. This effect can be studiedby smearing p T for each particle with the known value ofmomentum resolution. Finally, we note that bin-by-binmethod (even single efficiency bin) using the averagedefficiency should also work in this toy model. This isbecause only one probability distribution function is con-sidered, which indicates we assume that the underlyingphysics is identical for whole p T region for each electriccharge [21, 22]. Dependence of the efficiency on event-wise variable can be alsoincluded if necessary.
FIG. 2. Correlation of the positively and negatively charged particles for the (a) true number distribution and (b) measurednumber distribution, (c) efficiencies as a function of p T and (d) true and measured p T spectra.FIG. 3. Efficiency corrected cumulants up to the 4th order with statistical errors estimated by the bootstrap. Data points arecalculated from 100 independent samples. Red lines represent the true value of cumulants. The number shown in right top ofeach panel shows the possibility of data points touching the true value out of 100 points. V. SUMMARY
In this paper, we pointed out three difficulties arisingin the experimental application of the current efficiencycorrection method based on the arbitrary binning withrespect to track-wise variables. It was shown that thosedifficulties can be addressed by using the track-by-trackefficiency in the efficiency correction formula based onfactorial cumulants. We don’t need to worry about howmany efficiency bins are enough to calculate the efficiencycorrected cumulants by using the analytical parameteri-zation of the efficiency directly. Thus, the averaged ef-ficiency doesn’t need to be estimated and the cumulantanalysis can be proceeded without the corrected spec-tra. Furthermore, the calculation cost for the statisticalerror estimation have been significantly reduced. Experi-mentally, single particle efficiency of the detectors woulddepend on transverse momentum, rapidity and azimuthalangle in one event. We assume that efficiencies for indi-vidual particles are independent, which leads to the bi-nomial response function of the single particle efficiency.One thing we have to work hard is to parametrize the sin- gle particle efficiency as a function of track-wise variableswith the best precision as long as the computing sourceallows. On the other hand, one should also study the pos-sible non-binomial efficiency effects in the experimentalcondition. Finally, we emphasize that the method shownin this paper could serve as one of the most precise andefficient way for the cumulant efficiency correction withbinomial response function. It will play an importantrole for the QCD critical point search and can be appliedfor the cumulant analysis in the future heavy-ion colli-sion experiments, such as the BES-II program at RHIC,experiments at FAIR and NICA facilities.
VI. ACKNOWLEDGEMENT
This work is supported by the MoST of China973-Project No. 2015CB856901, the National Natu-ral Science Foundation of China under Grants (No.11575069, 11828501, 11890711 and 11861131009), Funda-mental Research Funds for the Central Universities No.CCNU19QN054 and China Postdoctoral Science Founda-tion funded project 2018M642878. [1] S. Gupta, X. Luo, B. Mohanty, H. G. Ritter, and N. Xu,Science , 1525 (2011), arXiv:1105.3934 [hep-ph].[2] S. Ejiri, F. Karsch, and K. Redlich, Phys. Lett.
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