Understanding the negative binomial multiplicity fluctuations in relativistic heavy ion collisions Hao-jie Xu∗ Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China (Dated: May 31, 2016) By deriving a general expression for multiplicity distribution (a conditional probability distri- bution) in statistical model, we demonstrate the mismatches between experimental measurements and previous theoretical calculations on multiplicity fluctuations. From the corrected formula, we develop an improved baseline measure for multiplicity distribution under Poisson approximation in statistical model to replace the traditional Poisson expectations. We find that the ratio of the mean multiplicity to the corresponding reference multiplicity are crucial to systemically explain- ing the measured scale variances of total charge distributions in different experiments, as well as understanding the centrality resolution effect observed in experiment. The improved statistical ex- pectations, albeit simple, work well in describing the negative binomial multiplicity distribution measured in experiments, e.g. the cumulants (cumulant products) of total (net) electric charge distributions. PACS numbers: 25.75.-q, 25.75.Gz, 25.75.Nq I. INTRODUCTION therein), the statistical model and its variations has been regarded as one of the basic tools in studying the baseline Event-by-event multiplicity fluctuations are expected prediction for the data on multiplicity fluctuations [14– to provide us crucial informations about the hot and 24]. For the mathematical convenience, the Poisson dis- dense Quantum chromodynamics (QCD) matter created tribution, which can be obtained from grand canonical in heavy ion collision (HIC) [1–5]. In experiment [6– ensemble (GCE) with Boltzmann statistics [14, 15, 18] 12], the multiplicity distribution of total (net-conserved) have been frequently used in HIC as one basic baseline charges published by STAR and PHENIX Collaboration measure for multiplicity fluctuations [6, 7, 10]. To under- were calculated using particles with specific kinematic stand the deviations of data from Poisson distributions, cuts (denoted as sub-event B), and the centrality cuts there are many effects have been studied in statistical were made using particles with some other acceptance models, e.g. finite volume effect, quantum effect, ex- windows (denoted as sub-event A). To avoid auto cor- perimental acceptance, as well as the resonance decays relation, these two sub-events have been separated by which were once considered as one of the major contri- different pseudorapidity intervals or particle species. For butions to the deviations. Despite many improvements example, in the net-charges case [10], the kinematic cut of statistical models [14–24], however, there are still dif- for the centrality-definition particles in sub-event A is ficulties in their systemically describing the data on neg- 1.0 > η > 0.5 and for the moment-analysis particles in ative binomial multiplicity distributions. For example, sub-event| | B is η < 0.5, where η is pseudorapidity. In the measured scale variation of total charge distributions this work, we always| | use q to represent the multiplicity are very different in different centralities and different ex- in sub-event B for the study of multiplicity distribution, periments [9, 11, 25, 26]. This implies that some exter- and use k to represent the multiplicity in sub-event A nal effects [27–33], unrelated to the critical phenomenon, for the centrality definition. The latter k is also called should be included. Recently, the effect of volume fluc- reference multiplicity. It is observed in experiments [8– tuations on cumulants of multiplicity distributions have 12] that the (total, positive, negative, net) charge dis- been studied by Skokov and his collaborations [30]. tribution can be well described by the negative binomial Unfortunately, previous theoretical studies are only fo- distribution (NBD), cus on the probability distribution PB(q)(without volume P arXiv:1602.06378v2 [nucl-th] 28 May 2016 fluctuations) or B(q)(with volume fluctuations), but (q + r 1)! q r NBD(q; p, r) − p (1 p) , (1) overlook the effect of probability conditions from sub- ≡ q!(r 1)! − − event A, here we postpone the definitions of PB(q) and where p (0 <p< 1) is the success probability in each PB(q) to the next section (see Eq. 4). We will show trial, and q (r) is the number of success (failure). that neither PB (q) nor PB (q) is the correct formula of Due to its success in describing the ratios of particle probability distribution in describing the experimental multiplicities data in a broad energy range of relativis- measurements on multiplicity fluctuations. Clarifying tic heavy ion collisions (see e.g. [13] and the references the mismatches between the experiments and the previ- ous theoretical calculations on multiplicity distributions and then understanding the negative binomial multiplic- ity distributions of electric charges are the main motiva- ∗ [email protected] tion of this work. 2 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 momen- tum, 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 distribu- tions. We will also give approximate formula for higher order cumulants which can explain most of experimen- tal 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.