Multiplicity fluctuations in the Glauber Monte Carlo approach

Maksym Deliyergiyev∗ and Maciej Rybczyński† Institute of Physics, Jan Kochanowski University, 25-406 Kielce, Poland Abstract We discuss multiplicity fluctuations of charged particles produced in nuclear collisions measured event-by-event by the NA49 experiment at CERN SPS within the Glauber Monte Carlo approach. We use the concepts of wounded nucleons and wounded quarks in the mechanism of multiparticle production to characterize multiplicity fluctuations expressed by the scaled variance of multiplicity distribution. Although Wounded Nucleon Model correctly reproduce the centrality dependence of the average multiplicity in Pb+Pb collisions, it completely fails in description of corresponding centrality dependence of scaled variance of multiplicity distribution. Using subnucleonic degrees of freedom, i.e. wounded quarks within Wounded Quark Model, it is possible to describe quite well the multiplicity distribution of charged particles produced in proton+proton interactions. However, the Wounded Quark Model with parameters describing multiplicity distribution of particles produced in proton+proton interactions substantially exceeds the average multiplicity of charged particles produced in Pb+Pb collisions. To obtain values of average multiplicities close to those experimentally measured in Pb+Pb collisions, the concept of shadowed quark sources is implemented. Wounded Quark Model with implemented shadowing source scenario reproduces the centrality dependence of scaled variance of multiplicity distribution of charged particles produced in Pb+Pb collisions in the range from the most central to mid-peripheral interactions.


I. INTRODUCTION
Particle interactions in collisions of relativistic ions usually lead to production of secondary particles whose number rises with increasing collision energy. The multiplicity of produced charged particles, N is one of the fundamental observables being an important characteristic of the global properties of the system. can investigate fluctuations of various observables which may be sensitive to the transitions between hadronic and partonic phases [1,2] with the use the event-by-event method [3].
Fluctuations in the initial conditions are essential to the full understanding of the dynamics of collisions of relativistic ions. The simplest way of modeling these fluctuations is based on random selection of positions of nucleons in each nucleus before the collision and deterministic designation, obtained after a collision, energy density according to the made assumptions. That is what happens, for example, in the standard Glauber Monte Carlo (GMC) model [4] or Kharzeev-Levin-Nardi (KLN) Monte Carlo [5], which have been used for many years.
Recently, the PHENIX Collaboration suggested [6] that the Wounded Quark Model (WQM) [7][8][9] works better than the popular Wounded Nucleon Model (WNM) [10,11], in particular in description of average multiplicities. In Ref. [7] it was shown that data on multiplicities obtained at RHIC may be reproduced within a WQM. Larger number of constituents and a decreased quark-quark cross section with respect to nucleons, allow to obtain an approximately linearly increased particle production at midrapidity as a function of wounded quarks number, dN/dη ∝ Q W . The quark scaling for the SPS energies was discussed in Ref. [12]. The agreement with the data may be achieved without the introduction of the binary-collision component [13,14], which introduces non-linearity between the number of nucleons participating in collision and the multiplicity of produced particles.
In this paper we are interested to examine an impact of subnucleonic components of matter on the dynamics of the early stage of the collision of relativistic nuclei. We extend and generalize the approach proposed in [15] to describe the centrality dependence of multiplicity fluctuations of charged particles registered by the NA49 experiment at CERN SPS [16].
Quite unexpectedly, the measured scaled variance of the multiplicity distribution in Pb+Pb collisions show quite non-trivial dependence on collision centrality [16]. It is close to unity at very central collisions, however it shows a substantial difference from unity at peripheral interactions. This effect is not present in a commonly used models of nuclear collisions. In Ref. [16] the NA49 data on centrality dependence of scaled variance of the multiplicity distribution was compared to HIJING [17], HSD [19], UrQMD [20], and VENUS [21] simulations.
A detailed discussion of HSD and UrQMD predictions for centrality dependence of scaled variance was also presented in [22]. The models produce approximately Poissonian 1 multiplicity distributions independent of centrality. Although there are some models trying to describe the non-monotonic behavior of the scaled variance of multiplicity distribution as a function of collision centrality [23][24][25] up to now there is no commonly accepted explanation of this phenomenon.
The article is organized as follows. In Sec. II we discuss the experimental data used in this paper, in Sec. III we briefly depict the idea of GMC approach of description of relativistic nuclear collisions. The resultant multiplicity fluctuations obtained in WNM are discussed in Sec. IV. The next section is devoted to analysis of the multiplicity fluctuations within WQM.
Finally, Sec. VI contains our summary, with the conclusion that WQM works much better in description of charged particles multiplicity fluctuations than WNM. In the Appendix we use a simple example do demonstrate the influence of fluctuating number of sources for the final multiplicity distribution.

II. MULTIPLICITY FLUCTUATIONS DATA
In this paper the charged particle multiplicity distribution P (N ) and its scaled variance ω is used to describe multiplicity fluctuations. Here P (N ) denote the probability to detect a charged particle multiplicity N in the single event of high energy nuclear collision. P (N ) is normalized to unity by definition, N P (N ) = 1. The scaled variance of multiplicity distribution (the so-called Fano-factor), ω (N ) provides a suitable index for the degree of deviation from a Poisson distribution 2 and is defined as: where V ar (N ) = N (N − N ) 2 · P (N ) is the variance of the distribution and N = N N · P (N ) is the average multiplicity.
In some models of nuclear collisions the scaled variance of multiplicity distribution do not depend on the number of sources of particle production. Commonly used models of nuclear interactions, the models of superposition, are constructed using the concept of particle production by the independent sources. In these models the scaled variance has two contributions. The first is due to the fluctuations of the number of particles emitted by a single source ω s , the second is due to the fluctuations in the number of sources ω k : where N s is the mean multiplicity of hadrons from a single source. The participant nucleons of a collision are considered to be proportional to the sources of particle production. In 2 If ω > 1, the distribution is called to be over-dispersed or super-Poissonian, namely the existence of clusters of occurrences may happen; if ω < 1, it is said the distribution to be under-dispersed or sub-Poissonian, namely this situation relates to arrangements of occurrences that are more ordinary than the randomness connected with a Poisson process. For the Poisson distribution, ω = 1.
The NA49 experiment located at CERN SPS published data on system-size and centrality dependence of fluctuations of the number of charged particles produced in proton+proton and Pb+Pb collisions at √ s N N = 17.3 GeV [16]. The NA49 detector registered multiplicity distributions of particles produced in the restricted rapidity interval 1.1 < y π < 2.6 3 in the centre of mass frame. The azimuthal acceptance of the NA49 detector was also limited.
Such restrictions correspond to a fraction of about 17% of accepted charged particles [16].
The particles produced in both proton+proton and Pb+Pb collisions were measured at exactly the same experimental acceptance. This makes a unique possibility to describe both proton+proton and Pb+Pb data in exactly the same way without introduction any additional biases.
The NA49 is a fixed-target experiment and is equipped in the forward calorimeter allow- However, in [25] it was observed that even with a fixed number of participants from projectile nucleus, the number of target participants fluctuates and it was suggested that the observed sizable multiplicity fluctuations in the forward rapidity domain of Pb+Pb collisions are due to fluctuations of the number of participants from the target. In [25] it was assumed that the nucleons participants in the target (projectile) nucleus contribute to the projectile (target) nucleus fragmentation region, which means that these areas overlap.
The authors of Ref. [25] called it a mixing model, in contrast to the transparency model in which the projectile participants contribute only to the projectile fragmentation region, and the target participants into the target fragmentation region. The transparency model is compatible with the limiting fragmentation hypothesis 5 [26] while the mixing model 3 y π means rapidity calculated under assumption of mass of π meson. 4 In the present work by the nucleons-spectators we call those nucleons which did not interact with other nucleons during the collision, and nucleons-participants are those nucleons which suffer at least one inelastic collision. 5 Hypothesis of limiting fragmentation states that for a sufficiently high collision energy particle production becomes target and energy independent in the projectile (target) fragmentation domain corresponding to the rapidities close to that of the projectile (target).
contradicts it. Both models should obviously be treated as idealizations. The analysis of d+Au collisions at RHIC [27] shows that in reality we have both mixing and transparency.
In [28] it was shown that the target nucleus does not affect the multiplicity distributions of particles produced in proton+Pb minimum bias collisions in the forward rapidity region and the effect of the increased multiplicity fluctuations due to influence of target participants is not observed. The results were compared with the corresponding results obtained in the proton+proton interactions and the predictions of models. Fig. 5 of Ref. [28] clearly shows that the transparency model describes the data quite well, but the mixing model seems to be excluded. Of course, the mechanism of the particle production in Pb+Pb collisions may be different than in proton+Pb interactions. Nevertheless, the results of [28] indicate that there is no strong mixing of the projectile and target nucleus fragmentation regions, at least in proton+nucleus collisions.

III. GLAUBER-LIKE MODELS AND GLAUBER MONTE CARLO APPROACH
The Glauber model [29] was released almost seventy years ago to find description of high-energy collision of atomic nuclei treated as composite structures. Until then there was no systematic calculations regarding nuclear systems as projectile or target. Glauber model containing quantum theory of collisions of composite particles allows to describe experimental results on collisions of protons with deuterons and heavier nuclei. In the mid 1970s Bialas et al [10,30] used Glauber model to describe inelastic nuclear collisions in their WNM. Bialas et al [10] formulation allows to treat a collision between nuclei as a superposition of incoherent collisions between their nucleons. A review of Glauber modelling of high-energy collisions was given in [4].
In the last years the popular GMC approach became an important tool in the analysis of collisions of relativistic ions [4]. One of the most important application of the GMC simulation is the estimate of the number of participants dependence on the centrality, especially in the collider experiments [4,[31][32][33][34]. The presence of the event-by-event fluctuations in the initial Glauber phase is a very important aspect of the approach. These fluctuations are transferred to the distributions of the experimentally registered hadrons. GMC initial state is often used as an starting point for event-by-event hydrodynamics [35].
The multiplicity of particles produced in nuclear collisions fluctuates event-by-event. In the GMC approach using wounded nucleons or wounded partons only a part of these fluctuations can be described by the fluctuations of the number of sources emitting particles (nucleons or partons). In order to describe the experimentally observed charged hadrons multiplicity distributions the model multiplicity distribution should be expressed as a convolution of the distribution of the number of emitting sources, P S with the distribution, P H , of hadrons emitted from a singe source. So, the number of charged particles is given by: where n i follows from P H with the generating function H (z) and N p comes from P S with the generating function S (z). Thus, the measured multiplicity distribution P (N ) is given by the compound generating function and finally To parameterize P H we use Negative Binomial (NB) distribution which is a statistical tool frequently used to describe multiplicity distributions of particles produced in nuclear collisions: NB has two free parameters: N describing mean multiplicity and, not necessarily integer parameter k (k ≥ 1) affecting shape of the distribution.
In this article we use GLISSANDO [32][33][34] which is a versatile GMC generator for early-

IV. MULTIPLICITY FLUCTUATIONS IN WOUNDED NUCLEON MODEL
The NA49 data on charged hadrons multiplicity distributions in proton+proton and centrality selected Pb+Pb collisions were obtained using the same experimental conditions.
The basic concept of description these results within WNM is then as follows. First, we fit the experimental proton+proton multiplicity distribution by the NB distribution. Then, we overlay the distribution of emitting sources, which are wounded nucleons with the NB distribution with the parameters obtained from the fit to proton+proton data. The NB fit to proton+proton data is based on the χ 2 method with the errors taken from [16,28]. The  slowly when going to peripheral. However it is always higher than the corresponding number in proton+proton interactions.
In order to decrease the average charged multiplicity we implement the source shadowing procedure as presented by Chatterjee et al in [36]. In short: the contribution to particle production by quark sources located inside the nucleus is shadowed by those being in front.
The particle production suppression factor S (n, λ) for the contribution from a quark source shadowed by the n other quark sources from the same nucleus ahead is S (n, λ) = exp (−n · λ) with λ being a phenomenological parameter. Such idea of shadowing was discussed first in Ref. [37]. In the left panel of Fig. 6  There are also possible other variants of shadowing. Namely, instead of decreased production from the shadowed source as discussed above, such source may emit particles with a certain probability, proportional to suppression factor, given by Eq. (7). We also checked such mechanism but we did not find differences in comparison to the discussed above shadowing scenario.

VI. CONCLUDING REMARKS
The main purpose of the study presented in this article is to check the influence of nucleonic and subnucleonic degrees of freedom for the dynamics of the early stage of the collision of relativistic nuclei. We use the concept of wounded nucleons and wounded quarks in the mechanism of multiparticle production to describe charged particles multiplicity fluctuations expressed by the scaled variance of multiplicity distribution and observed in collisions of relativistic ions by the NA49 experiment at CERN SPS. We take the opportunity that NA49 data on multiplicity fluctuations for proton+proton and centrality selected Pb+Pb collisions were obtained at the same experimental acceptance. Wounded nuleons and wounded quarks are implemented using Glauber Monte Carlo approach. Our results are as follows.
• WNM describe reasonably well the centrality dependence of the average multiplicity of charged particles produced in Pb+Pb collisions. It is possible when distribution of wounded nucleons is overlaid with the NB distribution with parameters describing multiplicity distribution in proton+proton interactions.
• WNM does not describe the centrality dependence of the scaled variance of multiplicity distribution of charged particles produced in Pb+Pb collisions at any centrality.
• Using subnucleonic degrees of freedom, i.e. wounded quarks within WQM, it is possible to describe quite well the multiplicity distribution of charged particles produced in proton+proton interactions.
• However, the WQM with parameters describing multiplicity distribution of particles produced in proton+proton interactions substantially exceeds the average multiplicity of charged particles produced in Pb+Pb collisions. This is due to higher average number of wounded quarks per nucleon in centrality selected Pb+Pb collisions with respect to proton+proton interactions.
• To obtain values of average multiplicities close to those experimentally measured in Pb+Pb collisions, the concept of shadowed quark sources was implemented. In this scenario each quark source which is behind other source in the same nucleus emit less number of particles, proportionally to the number of shadowing sources. • WQM shows that implementation of sources, like quarks, helps to describe heavy ion data in the range 80 < N proj p < 200. This leads to idea that there should be a triggering process that cause an additional increase in number of sources (sea quarks/gluons) in the range of 0 < N proj p < 80, which is smoothly switching off beyond this threshold.

Acknowledgements
The where V ar (n) means variance of the multiplicity distribution from a single source and V ar (N p ) is the variance of the number of sources distribution.
In the first scenario we generate the sum of the N p = 2 numbers generated from P 1 (n) distribution. In the second scenario we sum the N p numbers given by P 1 (n), but N p is generated from the set of N p ∈ {1, 2, 3} uniformly generated numbers. Obviously N p = 2 in both scenarios. Also average multiplicities N have the same values in both scenarios.
However variances of the resultant multiplicity distributions are substantially different since in the first scenario V ar (N p ) = 0 and in the second one V ar (N p ) = 0.667.  In both cases distribution from the single source is the same, given by NB distribution with the average n = 5 and shape parameter k = 5. See text for details.
As an example we provide results from Monte Carlo simulations of the two discussed scenarios. P 1 (n) distribution is given by NB distribution with average value n = 5 and variance V ar (n) = 10, what corresponds to shape parameter k = 5, see