Understanding the Negative Binomial Multiplicity Fluctuations in Relativistic

Home , Cumulant, Skellam distribution

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The main observation of this work is that: after includ- II. GENERAL DERIVATION ing the distribution of principal thermodynamic variables (PTVs) in statistical model (e.g.,distribution of volume, In this section, we derive a general expression for the the dominated effect in HIC), the sub-event A and B cor- multiplicity distribution, related to recent experiments responding to the method used in experiments are cor- at RHIC [6–12]. To avoid centrality bin width effect in related to each other in event-by-event analysis, and, as experiment, the cumulant calculations are restricted in far as we know, this feature have not been taken seri- a fine bin of centrality (a reference multiplicity bin is ously in previous studies. These correlations make the the finest centrality bin) [11, 35], the bin width depend measured multiplicity distribution becomes a conditional on the statistics. In this work, we calculate the cumu- probability distribution (Eq. (8)), instead of the tradi- lants of multiplicity distribution as function of reference tional probability distribution (Eq. (4)) discussed in pre- multiplicity, the relation between the results in reference vious studies [14–24, 27–30]. We develope an improved multiplicity bin and in centrality bin are obvious. baseline measure for multiplicity distribution under Pois- In a specific statistical ensemble (SSE), the probabil- son approximation in statistical model with the corrected ity distribution of multiplicity X is defined as PE(X; Ω), probability distributions. The improved statistical ex- where Ω represents a set of PTVs (e.g.,for GCE, Ω = pectations, albeit simple, work well in describing the (T,V,µ)). After employing the distribution of PTVs negative binomial multiplicity distribution measured in F (Ω), which was caused by the collisional geometry in experiments, e.g., HIC, we obtain the multiplicity distribution in statistical The relations among the scale variances of positive, model [27, 30] • negative and total charge distributions reported by the NA49 Collaboration [25] and the PHENIX Col- P(X)= dΩF (Ω)PE (X; Ω). (2) laboration [9]. Z 1 The variances of total charge distributions at On experimental side, P(X) stand for the multiplic- • √sNN = 27 GeV reported by the STAR Collab- ity distribution measured in a specific acceptance win- oration [11]. dows (e.g.,rapidity, pseudorapidity, transverse momentum, particle species, et.al.). It can be used for centrality The sensitivity of NBD parameters on the trans- definition or for moment analysis. Meanwhile, Eq.(2) can • verse momentum range of momentum-analysis par- be also regarded as the general formula of α-ensemble ticles reported by the PHENIX Collaboration [9]. discussed in Ref. [27]. From Eq.(2), the distribution of reference multiplicity The NBD baselines used for the cumulant products k and the distribution of multiplicity q can be written as • of net-charge distributions reported by the STAR Collaboration [34]. P (k)= dΩF (Ω)P (k; Ω), (3) A Z A The differences between the cumulants of net- • charges and net-kaons distributions reported by the PB(q)= dΩF (Ω)PB (q; Ω), (4) STAR Collaboration [10, 34]. Z

The centrality resolution effect observed in experi- where PA(k; Ω) and PB(q; Ω) stand for multiplicity dis- • ment [35]. tribution in a SSE with specific acceptance cuts for sub- event A and sub-event B, respectively. The results indicate that the probability conditions from It is worth noting that, although PA(k) can been re- sub-event A play crucial roles to explain the negative bi- garded as distribution of reference multiplicity measured nomial multiplicity distributions of (net) electric charges in experiment, neither PB (k; Ω) nor PB (k) can be used measured in sub-event B. to represent the experiment measurements [6, 7, 10–12]. The paper is organized as follows. In Sec. II, we will This is because the multiplicity distribution of moment- demonstrate the mismatches between experimental mea- analysis particles measured in experiment is a condi- surements and previous theoretical calculations, by de- tional probability distribution. Briefly stated, con- riving a general formula for the multiplicity fluctuation dition refers to the notion that the calculations of cumu- corresponding to the method used in experiment [6–12]. lants are restricted in a specific centrality (reference mul- In Sec. III, under Poisson approximation, we will show tiplicity) bin. We note that PB (k; Ω) and PB (k) are in- how to calculate the improved statistical baseline mea- dependent of the definition of reference multiplicity, and sure for higher order cumulants of multiplicity distributions. We will also give approximate formula for higher order cumulants which can explain most of experimental observables related to multiplicity fluctuations such 1 We always use P to represent the probability distribution in a as the scale variance, the centrality resolution effect, et. SSE, and use P to represent the probability distribution mea- al. We will give a summary in the final section. sured in experiment. 3 they have been widely discussed in previous studies [14– The outline of the present section is as follows. In 24, 27–30]. Unfortunately, both of them are not the cor- Sec. IIIA, we calculate the cumulants of PA(k) and rect formula for the multiplicity distributions measured P (q k) under Poisson approximation. With the help B|A | in experiment. of the data of reference multiplicity PA(k) and mean The conditional probability distribution for multiplic- value distribution M (k) measured in experiment, we ity q in given reference multiplicity bin k reads, demonstrate how to obtain the higher order cumulants of multiplicity distribution in the improved statistical P A∩B(q, k) model. The calculations are directly applied to the net- PB|A(q k)= . (5) | PA(k) conserved charges case in Sec. IIIB. In Sec. IIIC, we calculate the approximate solutions of these higher or- where der cumulants which can explain most of the experiment observables. Finally, in Sec. IIID, a short discussion is P (q, k)= dΩF (Ω)P (q, k; Ω) (6) A∩B Z A∩B given to highlight some of the difficulties in the improved statistical baseline measure. and PA∩B(q, k; Ω) is a joint probability distribution for sub-event A and B in a SSE. With some experimental techniques, the two sub-events are expected to be inde- A. Improved statistical baseline measure pendent of each other. In this case, we have In this section, we consider the discussion of one PTV, PA∩B(q, k; Ω)= PB(q; Ω)PA(k; Ω). (7) e.g., the system volume as the dominated effect in HIC. With Poisson approximation In this work, we focus on such independent approximation. We note that, due to dynamic evolution and the λke−λ P (k; λ)= (9) correlation between different particle species in HIC, the A k! independent approximation might be contaminated. for sub-event A, where the Poisson parameter λ λ(Ω) With independent approximation, Eq.(5) can be writ- ≡ ten as is determined by Ω and acceptance cuts, the distribution of reference multiplicity PA(k) in Eq.(3) can be written Ω Ω Ω Ω as, P d F ( )PB (q; )PA(k; ) B|A(q k)= R P . (8) | A(k) λke−λ PA(k)= dΩF (Ω) Consequently, we derive a general expression in statisti- Z k! cal model for arbitrary statistical ensemble and arbitrary λk e−λ = dλf(λ) , (10) distribution of PTVs, related to recent data [6, 7, 10, 11] Z k! on multiplicity distributions. For a specific calculation, the informations of P (k; Ω), P (q; Ω), as well as F (Ω) where f(λ) is the normalized distribution of Poisson pa- A B P are required. rameter. The scale variance of A(k) reads Due to PA(k; Ω) and F (Ω) appeared in both Eq.(3) 2 2 σA dλf(λ)(λ MA) and Eq.(8), the connection between the distribution of ωA =1+ − , (11) ≡ MA R MA reference multiplicity PA(k) and multiplicity distribution of moment-analysis particles PB|A(q k) has been 2 where MA = dλf(λ)λ and σ are the mean value and established. In the next section, we will show| that this A variance of PRA(k) . The most significant feature of Eq. connection is crucial to explain the centrality resolution (11) is that we obtain ωA > 1 except one special case effect measured in experiment [35]. f(λ)= δ(M) 2. Using Poisson approximation for both sub-event A and sub-event B, we obtain the conditional probability dis- III. APPLICATIONS: STATISTICAL EXPECTATIONS UNDER POISSON tribution from Eq.(8) as APPROXIMATION 1 λke−λ µqe−µ PB|A(q k)= dΩF (Ω) | PA(k) Z k! q! In this section, we calculate the improved baseline mea- λk e−λ µqe−µ sure of cumulants of multiplicity fluctuations under a = N (k) dλf(λ) , (12) Z k! q! simple approximation: PA(k; Ω) and PB(q; Ω), the distributions in a SSE, can be regarded as Poisson distributions. In a SSE [1, 14, 21, 36, 37], there are many other effects that make the distribution deviates from Poisson 2 This feature might be interesting in elementary nucleon-nucleon distribution, e.g., finite volume effect, quantum effect, collisions. Because we notice that in this case, P(k) have been resonance decays, experimental acceptance, et.al, which solely used to calculate the corresponding cumulants, and the can be a topic for our future study. results show a typical NBD feature: ω > 1 [38–42]. 4 where λ, µ = µ(Ω) = µ(λ) are the Poisson parameters B. Net-conserved charges for sub-event A and B respectively. N (k)=1/PA(k) is the normalization factor. Here we have assumed the in- If we assume the independent production of positive dependent production of A and B in each event (thermal and negative conserved charges in each event, under the system). Poisson approximation, the conditional probability dis- In Statistics, it is convenient to characterize a distribu- tribution of net-conserved charges can be obtained from tion with its moments or cumulants (see Appendix A for Eq.(8) as the definitions). The first four cumulants of PB|A(q k) | λk e−λ read P (n k)= N (k) dλf(λ)( )Sk(n; q, λ). (21) B|A | Z k! c1 = µ M (k), (13) h 2i≡ 2 n/2 Here Sk(n; q, λ) = (µ+/µ−) In(2√µ+µ−) exp[ (µ+ + c2 = µ + µ µ , (14) − h 3i h i − h i 2 2 µ−)] is the Skellam distribution [6, 22] with Poisson pa- c3 = µ + (1 µ ) 3 µ 2 µ + µ , (15) h i − h i h i− h i h i rameters µ+ = µ+(λ) and µ− = µ−(λ) of positive and 4 3 2 3 c4 = µ + µ 3 µ µ +2 µ (6 4 µ ) negative-conserved charges, respectively. n is the mul- h i h i− h ih i h i − h i + µ2 7 3 µ2 + µ 7 µ 2+2 µ 4, (16) tiplicity of net-conserved charges in sub-event B. The h i − h i h i− h i h i corresponding cumulants read k −λ N λ e N µ+ µ− where (...) (k) dλf(λ) k! (...). The scale vari- c = c + c 2( µ+µ µ+ µ ), (22) h Pi ≡ 2 2 2 − h −i − h ih −i ance of B|A(q k) isR n−1 | n! 2 cN = mN mN cN , (23) (µ µ ) n+1 n+1 − s!(n s)! n−s s+1 ω =1+ h − h i i 1. (17) Xs=0 B µ ≥ − µ+ µ− h i where cn , cn are the cumulants of positive and In generally, if we have the distribution of f(λ) and negative-conserved charges respectively. mN are the raw u(λ), the cumulants in Eq.(13,14,15,16) can be obtained n moments of PB|A(n k). Here we give the first four mo- accordingly. Here we introduce a new approach to calcu- ments which will be| used in the following discussions, late the higher order cumulants of P (q k) using the B|A N | 3 m1 = µ+ µ− , (24) distributions PA(k) and M (k) measured in experiment . N h i − h 2i Using series expansion, we have m2 = (µ+ µ−) + µ+ + µ− , (25) N h − 3i h 2i h i2 N m = (µ+ µ ) +3 µ 3 µ + m , (26) N 3 h − − i h +i− h −i 1 m N 4 2 µ = amλ . (18) m4 = (µ+ µ−) +6 (µ+ µ−) (µ+ + µ−) X=0 h −2 2 i h − i m +6 µ + µ + mN , (27) h + −i 2 Therefore, and N N N N N N N n m n µ = .. am1 am2 ...amn µ µ = .. .. a ..a m h i h + −i s1 s mX1=0 mX2=0 mXn=0 sX1=0 sXm=0 rX1=0 rXn=0 n P n m n (k + i=1 mi)! A(k + i=1 mi) (k + i=1 si + i=1 ri)! . (19) a¯r1 ..a¯rn × Pk! PAP(k) × P k! P P (k + m s + n r ) The coefficients am can be extracted by fitting the data A i=1 i i=1 i . (28) of M (k) × PPA(k) P

N The coefficients as anda ¯r are determined by Eq.(20) (k + m)! P (k + m) M (k)= a A . (20) with the mean value distribution of positive and negative- m k! P (k) mX=0 A conserved charges measured in experiment. Although they were assumed to be produced independently in each with a finite truncation order N. event, the relations cN = cµ+ + ( 1)ncµ− are broken in P n n n Consequently, with the help of the data of A(k) and event-by-event analysis (see e.g. Eq.(− 22)), due to the cor- M (k), Eq.(19,20) and Eq.(14,15,16) provide a new ap- relations of positive and negative-conserved charges from proach to calculate the second, third and fourth order cu- the distribution of PTVs. mulants of P (q k). Here we have assumed the contri- B|A | Obviously, the statistical expectations of multiplicity bution from critical fluctuations, if any, can be neglected distribution depend on the multiplicity of reference par- P M for the measured A(k) and (k). The higher order ticles. However, this feature has not been taken seriously cumulants can be calculated analogously. in previous studies, and only few observations have been reported. In the following subsection, with the insufficient data, we calculate the approximate solutions of 3 In principle, the distributions f(λ) and µ(λ) can be solved from these high cumulants. We will show that these solutions Eq.(10) and Eq.(13) if we known the informations of PA(k) and can qualitatively or quantitatively describe most of the M (k). observables related to multiplicity fluctuations. 5

C. Approximate solutions 600 M (STAR Preliminary) To give the analytic solutions, we consider only the ef- σ2 (STAR Preliminary) fect from distribution of volume. Due to µ and λ are Approx. both proportional to volume in statistical model, the 400 Poisson parameter µ can be written as µ = bλ and b is s =27GeV independent of λ. This consideration is also inspired by NN the near-linear feature of mean value distribution M (k) measured in experiments (see e.g. Fig. 1). Secondly, cumulants 200 except the rapid decreasing of PA(k) in most-central and most-peripheral collision range, the assumption of P P A(k + m)/ A(k) 1 is comfortable when m is not 0 too large [43]. ≃ 0 100 200 300 In general, the high order cumulants of P (q k) and k (refrence multiplicity) B|A | PB|A(n k) in semi-central and semi-peripheral collision range can| be well described by the approximate solutions. 2 But for the central and peripheral collision range, the FIG. 1. (Color online). Approximate solutions of σ (c2) approximate solutions are questionable due to the fact of the total charge multiplicity distribution in Au+Au colli- that the assumption of P (k + m)/P (k) 1 becomes sions at √sNN = 27GeV. The approximate solutions are ob- A A tained from Eq.(29). The input distribution M (k) are taken invalid [44]. ≃ from [11]. The approximate solutions of higher order cumulants of P (q k)) from Eq.(14,15,16) read B|A | M 2 From the approximate solutions, we obtain the rela- c2 = + M, (29) tionship among the scale variance of total charge hadrons k +1 ωch, positive hadrons ω+ and negative hadrons ω− 2M 3 3M 2 c3 = 2 + + M, (30) (k + 1) k +1 ω = ω+ + ω 1. (34) ch − − 6M 4 12M 3 7M 2 c4 = + + + M. (31) (k + 1)3 (k + 1)2 k +1 Within the accuracy errors this relations can be used to explain the experiment measurements from NA49 collab- where M M (k). We find that these approximate solu- orations [25] and PHENIX collaborations [9] surprisingly tions obey≡ the standard NBD expectations and the NBD well, even the effect of resonance decays have not been parameters r and p (Eq.(1)) are included in the present study. Moreover, the M/k ratios help to explain the differences on scale variance of to- r = k +1, (32) tal charge distributions measured in different centralities M and different experiments [9, 11, 25, 26]. p = . (33) M + k +1

The scale variance ω =1+ M/(k + 1) increases with M 1. Net-conserved charges while r is independent of M, these features have been observed in Ref. [9]. In that paper, the authors found Analogously, we obtain the approximate solutions of that ω increases with transverse momentum (pT ) range first four cumulants of P (n k) as of moment-analysis particles (see Fig.6 and Fig.7 in that B|A | paper), but r (denoted as kNBD in the reference) show N c1 = M+ M−, (35) no significant pT -dependence (see Fig.8 and Fig.9 in that − 2 paper). This is because in Ref. [9] a narrower pT range N (M+ M−) c = − + M+ + M , (36) correspond to a smaller M. 2 k +1 − 2 3 2 2 In Fig. 1, we show the approximate solutions of σ of 2(M+ M ) 3(M+ M ) cN = − − + − − + cN , (37) the total charge multiplicity distribution in Au+Au col- 3 (k + 1)2 k +1 1 lisions at √sNN = 27GeV as function of reference mul- 4 2 tiplicity k. The input distribution M (k) (open triangle N 6(M+ M−) 12(M+ M−) (M+ + M−) c4 = − + − symbol) are taken from [11]. We find that the approxi- (k + 1)3 (k + 1)2 mate solution (black-dashed line) can reproduce the ex- 6(M 2 + M 2 ) + + − + cN , (38) perimental results(open star symbol) expect the central k +1 2 collision range. The deviations in most central collision are due to the non-trivial features of PA(k) in this range, where M+ and M− are the mean values of positive and that make the second assumption PA(k+m)/PA(k) 1 negative conserved charges in a given reference multiplic- becomes invalid. ≃ ity bin k. 6

10 (a) Sσ (b) σκ 2 0.8 1 0.8

0.6 0.6 α α 0.4 10-1 0.4

0.2 0.2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 β β

FIG. 2. (Color online). The β α plane of Sσ(upper panel) and κσ2(lower panel) of multiplicity distribution of net-conserved charges. β is the multiplicity ratio− between positive-conserved charges and the reference multiplicity, α is the multiplicity ratio between negative and positive-conserved charges. For the details, see Eq.(39) and Eq.(40).

Due to less sensitive to the interaction volume and ex- with β and decrease with α, as it was shown in perimental efficiency [6, 7, 10, 18–20], the moment prod- Fig. 2(a). Meanwhile, due to the smaller β in N N 2 N N ucts Sσ c3 /c2 and κσ c4 /c2 have been frequently net-kaons case, its cumulants will be more closer discussed≡ in both theory and≡ experiment. From the above to the Skellam baseline measure than in the net- approximate solutions, we have charges case. These features are in consist with data [10, 34]. β(1 α2)+1 α Sσ =2β(1 α)+ − − , (39) − β(1 α)2 +1+ α 3. Independent production approximation. As − 2α we have mentioned before, the independent pro- κσ2 =6β(γ )+1, (40) duction relations of positive and negative-conserved − γ charges has been violated in event-by-event anal- where α = M−/M+, β = M+/(k + 1) and γ = β(1 ysis. Moreover, the NBD baselines obtained by N µ+ n µ− 2 − c = cn + ( 1) cn overestimate the higher or- α) +1+ α. If β 0, Eq.(39) and Eq.(40) will back n − to the Skellam expectations:→ Sσ = (1 α)/(1 + α) and der cumulants of net-conserved charges distribu- 2 κσ2 = 1. − tions [10]. However, the corrections for Sσ and κσ In Fig. 2 we show the β α plane of Sσ and κσ2 from depended on the parameters β and α. Eq.(39) and Eq.(40). The− approximate solutions can ex- 4. Quantitative estimation. Using (M+ + M ) plain many observations on multiplicity fluctuations ex- − ≃ k (M+ M ) in the net-charge case [10], we cept the most-central and most-peripheral centralities: − have≫ β 1/−(1 + α), α 1 and ≃ ≃ Centrality resolution effect 1. . The moments and 4(1 α) its products Sσ and κσ2 not only dependent on Sσ − , (41) the multiplicity ratio between negative and positive ≃ 1+ α κσ2 4, (42) conserved charges, but also depend on the multi- ≃ plicity used for centrality definition. This property has been found in both experimental measurements which are about four times of the Skellam expecta- and some model calculations [35], which was con- tions. The results are shown in Fig. 3 and Fig. 4. sidered as centrality resolution effect. More specif- We find that the approximate solutions of Sσ are ically, a larger pseudorapidity range of reference closer to the experiment data/NBD baselines than multiplicity contribute to a smaller values of Sσ the Skellam baselines given in [10]. The approxi- 2 and κσ2 due to its smaller β, and vice versa. mate solutions of κσ are colser to the NBD baselines, but fail to quantitatively reproduce the data. 2. Net-charge versus net-kaon. Comparison with This indicate the existence of correlations of posi- the cumulants of net-charges and net-kaons distri- tive and negative charges [10] and/or the correla- butions, the κσ2 of net-charges distributions will tions between the moment-analysis parameters and be larger than the net-kaons one due to its larger the reference particles. Notice that, though it have β and α, see Fig. 2(b). But for Sσ, there is a been shown in the figures, the approximate solu- competition between β and α, because Sσ increase tions in 0 5% and 60 80% centrality bins are ques- − − 7

0.8 (a) 7.7 GeV Sσ 10 (a) 7.7 GeV σκ 2 0.6 Approx. Approx. 0 0.4 STAR STAR Skellam(STAR) Skellam(STAR) 0.2 NBD(STAR) -10 NBD(STAR) 0 0 100 200 300 0 100 200 300 (b) 11.5 GeV 0.4 (c) 19.6 GeV (b) 11.5 GeV 0.6 (c) 19.6 GeV 5 5 0.4 0.2 0.2 0 0 0 0 -5 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 (d) 27 GeV (e) 39 GeV 0.3 (d) 27 GeV (e) 39 GeV 0.2 0.2 5 5 0.1 0.1 0 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 (f) 62.4 GeV (g) 200 GeV (f) 62.4 GeV (g) 200 GeV 0.2 0.1 5 5 0.1 0 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300

< Npart> < Npart > < Npart> < Npart >

FIG. 3. (Color online). Approximate solutions of Sσ of the FIG. 4. (Color online). Similar to Fig. 3, but for κσ2. The net-charge multiplicity distribution in Au+Au collisions at approximate solutions are four times of the Skellam measures, √sNN = 7.7 to 200 GeV. The data, Skellam and NBD base- see Eq.(42). lines are taken from [10]. The approximate solutions are four times of the Skellam measures, see Eq.(41). These correlations might be one of the reason why the Binomial distribution instead of NBD have be observed tionable due to the non-trivial features of PA(k) in in experiment [7] for the protons and anti-protons distri- these ranges. butions. We notice that the two sub-events used for centrality definition and for moment analysis share a com- mon pseudorapidity range. Using a transport dynamic model [45], the author found that the high order cumu- D. Comments and discussion lants of net-proton distributions are sensitive to the definition of reference multiplicity. Meanwhile, due to the In this section we have calculated the improved base- small β in proton and anti-proton cases, some other cor- line measure of higher order cumulants of multiplicity rections might overcome the correction discussed in this distribution. We found that, even uner Poisson approx- work, and alter the classifications of proton and anti- imation, the statistical baseline measure deviates from proton distributions. the Poisson measure. However, as we have mentioned, even in a SSE there are some other effects that make the multiplicity fluctuation deviates from Poisson distri- IV. CONCLUSION bution. These corrections should be taken into account especially in the case of β 0 when the former devia- The traditional calculations of higher order cumulants tions are small. → of multiplicity distributions are incomplete due to lack In general, the two sub-events used for centrality defi- of the distribution of principal thermodynamic variables nition and for moment-analysis are expected to be totally and the probability condition from reference multiplic- independent of each event. However, the unexpected ity. After including the distribution of principal thermo- correlations between them, as well as the correlations dynamic variables, we have derived a general expression between the positive and negative-conserved charges in for the multiplicity distribution in terms of a conditional net-conserved charges case, might contaminate our dis- probability with arbitrary statistical ensembles and discussions. tribution of thermodynamic variables. As an application, 8 we have used the general formula to calculate higher or- The cumulant-generating function is defined as der cumulants under the Poisson approximation. ∞ tn We found that the improved baseline measure for mul- K(t) = ln M(t)= c , (A4) tiplicity distribution mimics the negative binomial distri- n n! nX=1 bution instead of Poisson one, though the Poisson distri- where cn is the nth-order cumulant of f(x). Then we bution was used as input in a specific statistical ensemble. have The deviation of the new baseline measure from the Pois- ∞ ∞ son one increases with the ratio of the mean multiplicity tn tn M(t)= m = exp( c ). (A5) M to the corresponding reference multiplicity (k + 1). n n X=0 n! X=1 n! The basic statistical expectations work well in describing n n the negative binomial multiplicity distribution measured By taking nth order derivatives at t = 0, we have in experiments, e.g. the cumulants (cumulant products) n for multiplicity distribution of total (net) charges. n! m +1 = m c +1, (A6) Similar to the trivial Poisson expectations, the basic n p!(n p)! n−p p Xp=0 statistical expectations can be directly constructed from − n−1 experiment, but with the data of mean multiplicity M (k) n! cn+1 = mn+1 mn−pcp+1, (A7) and distribution of reference multiplicity PA(k) . How- − X=0 p!(n p)! ever, we note that currently the exact statistical measure p − cannot be fully determined because of insufficient data. The first four order explicit relation, which was fre- These data are crucial for calculation of the new baseline quently used in this paper, reads, measure especially in most central collision due to non- trivial feature of PA(k) in this range. The measurements m1 = c1, (A8) 2 of these distributions are highly expected in the future m2 = c2 + c1, (A9) to pin down the exact statistical measure. 3 m3 = c3+3c1c2 + c1, (A10) 2 2 4 m4 = c4 +4c3c1 +3c2 +6c2c1 + c1, (A11) and ACKNOWLEDGMENTS

c1 = m1 µ, (A12) ≡ 2 2 The author would like to thanks Qun Wang for a care- c2 = m2 m1 σ , (A13) ful reading of the manuscript and useful comments. The − ≡ 3 3 c3 = m3 3m2m1 +2m1 Sσ , (A14) author also acknowledges fruitful discussions with T. S. − 2 ≡ 2 4 Biro, L. J. Jiang, J. X. Li, H. C. Song, N. R. Sahoo, and c4 = m4 4m3c1 3m2 + 12m2m1 6m1 4− − − A. H. Tang. This work is supported by China Postdoc- κσ , (A15) ≡ toral Science Foundation with grant No. 2015M580908. where µ, σ2, S and κ are mean value, variance, skew- ness and kurtosis of probability distribution f(x), respectively. Appendix A: Moments and cumulants For the Poisson distribution, we have

c1 = c2 = c3 = c4 = λ, (A16) For a probability distribution f(x), the moment- generating function can be written as, where λ is the Poisson parameter shown in Eq.(9). The scale variance for Poisson distribution is ω = c2/c1 = 1. ∞ For the NBD, we have M(t)= f(x)etxdx. (A1) Z−∞ rp c1 = , (A17) 1 p We obtain the sereies expansion, −rp c2 = , (A18) (1 p)2 ∞ − tn rp(1 + p) M(t)= mn , (A2) c3 = , (A19) n! (1 p)3 nX=0 − 6rp2 rp c4 = 4 + 2 , (A20) where mn is the nth-order raw moment for f(x) (1 p) (1 p) − − ∞ where r and p are NBD parameters shown in Eq.(1). The n mn = dxx f(x). (A3) scale variance for NBD is ω = c2/c1 =1/(1 p) > 1. Z−∞ − 9

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