Efficiency correction for cumulants of multiplicity distributions based on track-by-track efficiency

TL;DR

This paper introduces a simplified method for correcting higher order cumulants in heavy-ion collision experiments using track-by-track efficiency, reducing bias, computational cost, and dependence on particle spectra.

Contribution

It presents a novel efficiency correction procedure that eliminates bias from phase space binning and reduces computational effort in cumulant analysis.

Findings

Bias from phase space binning is eliminated.

Computational time for bootstrap error estimation is reduced.

Correction method does not require particle spectra.

Abstract

We propose a simplified procedure for the experimental application of the efficiency correction on higher order cumulants in heavy-ion collisions. By using the track-by-track efficiency, we can eliminate possible bias arising from the average efficiencies calculated within the arbitrary binning of the phase space. Furthermore, the corrected particle spectra is no longer necessary for the average efficiency estimation and the time cost for the calculation of bootstrap statistical error can be significantly reduced.

Equations

$$⟨N^m{⟩}_{\mathrm{c}}$$

$$K(\theta )$$

$$⟨N^m{⟩}_{\mathrm{fc}}$$

$$K_{\mathrm{f}}(s)$$

$$⟨N^m{⟩}_{\mathrm{fc}}=⟨N(N-1)\cdots (N-m+1){⟩}_{\mathrm{c}}.$$

$$\tilde{P}(n)$$

$$B_{\epsilon ,N}(n)$$

$$⟨n^m{⟩}_{\mathrm{fc}}={\epsilon }^m⟨N^m{⟩}_{\mathrm{fc}}.$$

$$\displaystyle\bigl{<}Q\bigr{>}_{\rm c}$$

$$\displaystyle\bigl{<}Q^{2}\bigr{>}_{\rm c}$$

$$\displaystyle\bigl{<}Q^{3}\bigr{>}_{\rm c}$$

$$\displaystyle\bigl{<}Q^{4}\bigr{>}_{\rm c}$$

$$q_{(r,s)}=\sum _{i=1}^M(a_i^r/{\epsilon }_i^s)n_i,$$

$${\epsilon }_{\mathrm{sub},x}^{\pm }=⟨\sum _{i=5x}^{5(x+1)}{\epsilon }_i^{\pm }{N}_i^{\pm }/\sum _{i=5x}^{5(x+1)}{N}_i^{\pm }⟩,$$

$$q_{(r,s)}$$

$$=$$

$$q_{(r,s)}=\sum _{j=1}^{n_{\mathrm{tot}}}\frac{a_j^r}{{\epsilon }_j^s},$$

$$f(p_T)=p_T{e}^{-p_T/t}(0<p_T<2),$$

$$\epsilon (p_T)=\{\begin{array}{ll}a\times p_T^2+b\times p_T+c(0<p_T<1),& \\ d\times p_T+e(1<p_T<2),& \end{array}$$

Full text

I Introduction

Higher order cumulants of event-by-event conserved charge fluctuations are important observables to search for the QCD critical point Gupta et al. (2011); Ejiri et al. (2006); Stephanov (2009, 2011); Asakawa et al. (2009); Luo and Xu (2017) in heavy-ion collisions. In the beam energy scan (BES) program at RHIC, the STAR experiment has measured the cumulants up to the $4$th order of the net-charge and net-kaon multiplicity distributions Adamczyk et al. (2014a, 2018), and up to the $6$th order of the net-proton multiplicity distribution Adamczyk et al. (2014b); Luo (2016); Nonaka . In the recent results of the net-proton multiplicity distribution, the non-monotonic behaviour of the fourth order fluctuation has been observed with respect to the collision energy Luo (2015a), which is quite similar to the theoretical prediction with the critical point Stephanov et al. (1999). Since there are still large uncertainties in low collision energies, the second phase of the Beam Energy Scan program (BES-II) will be carried out in 2019-2021 focusing on the collision energy of $\sqrt{s_{\rm NN}}=$7.7–19.6 GeV.

One of the difficulties of measuring the higher order cumulants is the efficiency correction. It is known that the value of cumulants are artificially changed from the true ones Bzdak and Koch (2012); Nonaka et al. (2016), due to the fact that detectors miss some particles with the probability called efficiency. Some analytical formulas have been proposed to correct the measured cumulants with the assumption that the response function of the efficiency follows the binomial distribution  Bialas and Peschanski (1986); Kitazawa and Asakawa (2012); Bzdak and Koch (2012, 2015); Luo (2015b); Kitazawa (2016); Nonaka et al. (2017); Kitazawa and Luo (2017). Recently, a few attempts to understand and correct for the possible non-binomial efficiencies have been discussed Bzdak et al. (2016); Nonaka ; Nonaka et al. (2018), but there are still large systematic uncertainties arising from how to determine the detector-response functions. The efficiency correction with the binomial assumption is thus still important. Hereafter, let us call it ”binomial correction” for simplicity. In the binomial correction, particles are counted separately in the ”efficiency bin” where the efficiency changes, which are substituted into the correction formulas with the corresponding value of efficiencies. Due to the huge calculation cost with large number of efficiency bins by using the correction formulas based on the factorial moments Bzdak and Koch (2012, 2015); Luo (2015b), more efficient formulas have been proposed in which factorial cumulants are used in the derivation Kitazawa (2016); Nonaka et al. (2017); Kitazawa and Luo (2017). Experimentally, single particle efficiencies can be computed by the MC detector simulations in terms of the various experimental observables like centrality, multiplicity, vertex position, transverse momentum, rapidity, azimuthal angle and so on. Efficiency bins are then defined with respect to the track-wise variables in which the particles are counted. This binning is still arbitrary, which depends not only on the computing power for the detector simulations but also on the homogeneousness of the detector in the acceptance. Therefore, additional systematic studies will be necessary on how many efficiency bins are enough. The most crucial thing is that we need to calculate the averaged efficiency at each efficiency bin with weighted by the true spectra. This indicates that the traditional efficiency correction based on average efficiency, can not be performed until the corrected spectra of identified particle is available. Fortunately, we found those difficulties can be overcome by using the so called track-by-track efficiency correction methods. By doing this, the corrected particle spectra is no longer needed. We can also reduce the potential systematic bias by using the average efficiency correction method and the calculation cost for bootstrap statistical errors calculations.

This paper is organized as follows. In Sec II, the efficiency correction with the binomial model is introduced. The correction formulas based on factorial cumulants will be shown as well. In Sec. III, we discuss three difficulties of the efficiency correction in the experimental applications. We also use a numerical study to demonstrate one of the issues. In Sec. IV, a simple solution using the track-by-track efficiency is shown, followed by the toy model to check the validity of the new method.

II Efficiency correction

II.1 Cumulants and factorial cumulants

The $m$-th order cumulant of the probability distribution function $P(N)$ is defined as

[TABLE]

where $K(\theta)$ represents the cumulant generating function. Another quantity, factorial cumulants $\langle N^{m}\rangle_{\rm fc}$ are also defined as

[TABLE]

with $K_{\rm f}(s)$ being the factorial-cumulant generating function. Cumulants and factorial cumulants are connected to each other, e.g, factorial cumulants are expressed in terms of cumulants:

[TABLE]

II.2 Binomial model

Let us assume that the probability distribution function $P(N)$ is observed as $\tilde{P}(n)$ through detectors. Some of $N$ generated particles are missed by the detectors, which leads to the observation of $n$ particles ($n\leq N$). This finite probability to measure particles characterized by the detector is called efficiency. When the efficiencies for generated particles are independent each other, the detection process can be described by the binomial distribution $B_{\varepsilon,N}(n)$:

[TABLE]

where $\varepsilon$ represents the efficiency. It is known that in this situation the relationship between factorial cumulants of $P(N)$ and $\tilde{P}(n)$ is given by Nonaka et al. (2017); Kitazawa and Luo (2017)

[TABLE]

Using Eq. (8) and extending to the $M$ multi-variable case $P({\bm{N}})=P(N_{1},N_{2},...,N_{M})$, the formulas up to the fourth order cumulant are shown below:

[TABLE]

with $q_{(r,s)}$ defined as

[TABLE]

where $M$ represents the number of efficiency bins, $n_{i}$ represents the number of particles, and $a_{i}$ represents the electric charge of particles in $i$th efficiency bin. The concept of the efficiency bin started to be taken into account in Ref. Bzdak and Koch (2015) inspired by experimental requirement, which will be discussed in the next section.

III Difficulties in the Efficiency Correction

In this section, we clarify the three difficulties in the experimental application of the efficiency correction. First, the details of the experimental procedures of the efficiency correction is explained to point out the difficulties. Second, a simple toy model is used to demonstrate one of those.

III.1 Experimental application

Experimentally, single-particle efficiency can be determined by the Monte-Carlo approach of the detector simulation with respect to track-wise variable like transverse momentum, rapidity and azimuthal angle. This kind of efficiency map can be computed precisely as long as the computing source permits. Usually, the efficiency map is divided into various efficiency bins based on the track-wise variables and the averaged efficiency in each efficiency bin needs to be estimated by using the corrected spectra. In the current analysis of the net-proton fluctuations at the STAR experiment, the proton identification method is different between low and high $p_{T}$ regions, which leads to the step-like dependence of the efficiency as a function of $p_{T}$ Luo (2015a). In this case, particles need to be counted separately for the two $p_{T}$ region in which the values of the efficiency are different, then the true cumulants in entire $p_{T}$ regions can be reconstructed based on factorial moments Luo (2015b).

If the number of efficiency bins $M$ is large, it is more efficient to apply the efficiency correction according to Eqs. (9)–(12), which is based on factorial cumulants. For the statistical errors estimation, the bootstrap method would be the realistic way. We can obtain a new distribution by random sampling from the original distribution with the same number of events to calculate cumulants. This procedure is repeated with 100 times, and the standard deviation of the 100 cumulant values calculated from the new distributions are taken as the statistical error. In order to take into account the correlation between different efficiency bins, the sampling would be performed based on the $M$ dimensional histogram.

Below are three main difficulties in the experimental implementation of the efficiency correction.

We expected that it can provide more precise efficiency corrected cumulants when using the large number of efficiency bins. However, it is difficult to know how many efficiency bins are enough. 2. 2.

In order to obtain the averaged efficiency for each efficiency bin, we need to consider the variation of the particle yields within the efficiency bin, which means the corrected spectra is necessary. This is especially crucial issue for new data set in new collision energy or collision system where the corrected spectra is not available. 3. 3.

The calculation cost on the bootstrap increases as $\propto n^{M}$, which indicates that the statistical error estimation will be difficult for many efficiency bins in view of the computing source.

III.2 Toy model simulation

Let us discuss more about the first issue related to the approximation of efficiency bins above by using a simple toy model. We generated 50 independent binomial distributions for positively and negatively charged particles, $P^{\pm}_{i}(N^{\pm}_{i})$ $(i=1,2,...,50)$. The parameters of the binomial distributions are selected to be $N^{+}=4$, $N^{-}=3$ and $\varepsilon^{\pm}=0.8$ in Eq. (7). Efficiencies for each distribution ($\varepsilon^{\pm}_{i}$) are randomly chosen within $(0.5,1.0)$ with uniform distribution. Particles $N^{\pm}_{i}$ are then randomly sampled with binomial efficiencies $\varepsilon^{\pm}_{i}$ to define the measured particles $n^{\pm}_{i}$. Hereafter, let us suppose the net-particle distribution which consists of 50 independent distributions given by $P_{\rm net}(N)$ with $N=\sum_{i=1}^{50}(N^{+}_{i}-N^{-}_{i}$). We define the $m$-th order cumulant of $P_{\rm net}(N)$ as ”true” cumulants. The efficiency correction is performed with $M$ efficiency bins by using the corresponding averaged efficiency. For instance, let us suppose the efficiency correction with $M=10$ efficiency bins. In this case, whole 50 distributions are equally divided into $10$ sub-bins, each sub-bin contains $5$ distributions. The number of measured particles are counted at each sub-bin $n^{\pm}_{{\rm sub},x}$, $(x=1,2,...,10)$. Also the averaged efficiency at each sub-bin is given by

[TABLE]

where the bracket represents the event average. Then $n^{\pm}_{{\rm sub},x}$ and $\varepsilon^{\pm}_{{\rm sub},x}$ are substituted into Eqs. (9)–(13) event-by-event to calculate cumulants 111 Since the notation of electric charge is explicitly included in Eq. (13), we need following modification for substitution: $M\rightarrow 2M$, $a_{i}=+1\;\;(i\leq M)$, $a_{i}=-1\;\;(i>M)$ . The efficiency correction has been done with different number of efficiency bins for $M=$1, 2, 5, 10, 25, 50 to check how many efficiency bins are needed to obtain the true cumulants. Figure 1 shows the corrected $C_{2}$, $C_{3}$ and $C_{4}$ as a function of the number of efficiency bins, where the true value of cumulants are shown in red squares. We find that the results with $M=50$ are consistent with the true cumulants, which is because 50 efficiency bins are assumed in the toy model. It is also found that using wide efficiency bins is clearly incorrect and $M=$25 efficiency bins is still not enough.

IV Solution

IV.1 Track-by-track efficiency

Let us suppose infinite number of efficiency bins $M\rightarrow\infty$. Equation (13) is then written by

[TABLE]

Since we consider that the width of the efficiency bin is now zero, each efficiency bin contains up to one particle. Thus, the summations for $n_{i}=0$ vanish (in other words, efficiency bins containing no particles don’t need to be taken into account), and we immediately find that Eq. (16) is equivalent to the summation with respect to the total number of particles $n_{\rm tot}$ in one event, which is given by

[TABLE]

which is connected with Eq. (13) via $n_{\rm tot}=\sum_{i=1}^{M}n_{i}$. It is found that no variable related to the efficiency bin appears in Eq. (17). What we need to take care of is only the track-by-track efficiency, which indicates that the analytical formula of the efficiency with respect to track-wise variables can be directly used to determine the track-by-track efficiency 222Dependence of the efficiency on event-wise variable can be also included if necessary.. Accordingly, we don’t need to estimate the averaged efficiency at each efficiency bin, so the corrected spectra is no longer necessary for the efficiency correction. The first two difficulties have been solved. Hereafter, let us call the correction in Eq. (13) ”bin-by-bin” method, and call the track-wise correction in Eq. (17) ”track-by-track” method.

IV.2 Method validation

In this sub-section, we employ a toy model to demonstrate the validity of the track-by-track efficiency method, and also discuss the rest one problem regarding how to estimate the statistical errors. We start from two Gauss distributions, one is for positively charged particles, and the other is for negatively charged particles. Figure 2–(a) shows the even-by-event correlation histogram between positively $N^{+}$ and negatively $N^{-}$ charged particles. For each particle $p_{T}$ is allocated according to the spectra given by

[TABLE]

where $t=0.26$ and $0.24$ for positively and negatively charged particles, respectively. $p_{T}$ dependent efficiency is assumed to be the convolution of the $1$st and $2$nd polynomial functions given by

[TABLE]

where (a,b,c,d)=$$(-0.2,0.4,0.6,0.15,0.65) and $(-0.2,0.2,0.65,0.1,0.55)$ for positively and negatively charged particles, respectively. Each particle is then sampled by the corresponding value of efficiency determined in Eq. (19). The resulting correlation between measured positively $n^{+}$ and negatively $n^{-}$ charged particles is shown in Fig. 2–(b), and $p_{T}$ spectra is shown in Fig. 2–(d).

Since efficiency changes continuously with respect to $p_{T}$, it is cumbersome to use the bin-by-bin correction formulas in Eq. 13. Instead, we substitute the value of efficiency 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 bootstrap method. As was mentioned in Sec. III, in the case of bin-by-bin correction method with $M$ efficiency bins, the bootstrap sampling would be performed based on the $M$ dimensional histogram. But now there is no longer the efficiency bin, sampling can be simply done based on the spectra as follows:

Resample $n_{1}$ and $n_{2}$ randomly from Fig. 2–(b) 2. 2.

Allocate $p_{T}$ for each particle based on the measured spectra in Fig. 2–(d) 3. 3.

Apply the efficiency correction to obtain efficiency corrected cumulants by using the known efficiency curve in Fig. 2–(c) 4. 4.

Repeat 1–3 with 100 times and take the standard deviation as the statistical error.

Above procedures are repeated with 100 times independently in order to check the validity of the correction and its statistical error. Results up to the fourth order are shown in Fig. 3. It is found that the data points are distributed around the true value, so the correction method using track-by-track efficiency works well. The probability of data points touching the true value within the statistical error is shown in the top right of each panel. We find it comparable with the 1$\sigma$ nature of the Gaussian, which indicates the validity of the bootstrap. We also checked that the calculation cost only depends on the number of particles $\propto n_{\rm tot}$, which is due to the fact that we don’t have to consider particles which was not measured, while the bins containing no particles needed to be taken into account in bin-by-bin method. On the other hand, we can also use the Delta theorem Luo (2012); Luo et al. (2013) to calculate the statistical errors of the efficiency corrected cumulants, which will be implemented in the future data analysis.

It should be noted that one would need to consider how to treat the momentum resolution in real experiments. Since we use $p_{T}$ of individual particles for the efficiency correction, the momentum resolution might directly affect the cumulants. This effect can be studied by smearing $p_{T}$ for each particle with the known value of momentum resolution. Finally, we note that bin-by-bin method (even single efficiency bin) using the averaged efficiency should also work in this toy model. This is because only one probability distribution function is considered, which indicates we assume that the underlying physics is identical for whole $p_{T}$ region for each electric charge Nonaka et al. (2017); Kitazawa and Luo (2017).

V Summary

In this paper, we pointed out three difficulties arising in the experimental application of the current efficiency correction method based on the arbitrary binning with respect to track-wise variables. It was shown that those difficulties can be addressed by using the track-by-track efficiency in the efficiency correction formula based on factorial cumulants. We don’t need to worry about how many efficiency bins are enough to calculate the efficiency corrected cumulants by using the analytical parameterization of the efficiency directly. Thus, the averaged efficiency doesn’t need to be estimated and the cumulant analysis can be proceeded without the corrected spectra. Furthermore, the calculation cost for the statistical error estimation have been significantly reduced. Experimentally, single particle efficiency of the detectors would depend on transverse momentum, rapidity and azimuthal angle in one event. We assume that efficiencies for individual particles are independent, which leads to the binomial response function of the single particle efficiency. One thing we have to work hard is to parametrize the single particle efficiency as a function of track-wise variables with the best precision as long as the computing source allows. On the other hand, one should also study the possible non-binomial efficiency effects in the experimental condition. Finally, we emphasize that the method shown in this paper could serve as one of the most precise and efficient way for the cumulant efficiency correction with binomial response function. It will play an important role for the QCD critical point search and can be applied for the cumulant analysis in the future heavy-ion collision 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 China 973-Project No. 2015CB856901, the National Natural Science Foundation of China under Grants (No. 11575069, 11828501, 11890711 and 11861131009), Fundamental Research Funds for the Central Universities No. CCNU19QN054 and China Postdoctoral Science Foundation funded project 2018M642878.