Centrality Dependence of Multiplicity Fluctuations in Ion-Ion Collisions from the Beam Energy Scan at FAIR

Multiplicity distributions and event-by-event multiplicity fluctuations in AuAu collisions at energies in future heavy-ion experiment at the Facility for Anti-proton and Ion Research (FAIR) are investigated. Events corresponding to FAIR energies are simulated in the frame work of Ultra Relativistic Quantum Molecular Dynamics (URQMD) model. It is observed that the mean and the width of multiplicity distributions monotonically increase with beam energy. The trend of variations of dispersion with mean number of participating nucleons for the centrality-bin width of 5\% are in accord with the Central Limit Theorem. The multiplicity distributions in various centrality bins as well as for full event samples are observed to obey Koba, Nielsen and Olesen (KNO) scaling. The trends of variations of scaled variance with beam energy are also found to support the KNO scaling predictions for larger collision centrality. The findings also reveal that the statistical fluctuations in 5\% centrality-bin width appear to be under control.


Introduction
Any physical quantity measured in an experiment is subject to fluctuations. These fluctuations depend on the property of the system and are expected to provide important information about the nature of the system under study [1,2]. As regards relativistic heavy-ion (AA) collisions, the system so created is a dense and hot fireball consisting of partonic and (or) hadronic matter [1,2].
To investigate the existence of partonic matter in the early life of fireball is one of the main goals of AA collisions. Study of fluctuations in AA collisions would help check the idea that fluctuations of a thermal system are directly related to various susceptibilities and could be an indicator for the possible phase transitions [1,2,3]. Fluctuations in experimental observables, such as charged particle multiplicity, particle ratios, mean transverse momentum and other global observables are related to the thermodynamic properties of the system, like, entropy, specific heat, chemical potential, etc. [4,5,6,7]. Event-by-event (ebe) fluctuations of these quantities are regarded as an important mean to understand the particle production dynamics which, in turn, would lead to understand the nature of phase transition and the critical fluctuations at the QCD phase boundary. A non-monotonic behavior of the fluctuations as a function of collision centrality and energy of the colliding beam may signal the onset of confinement and may be used to probe the critical point in the QCD phase diagram [7]. The multiplicity of charged particles produced in heavy-ion collisions is the simplest and day-one observable which provides a mean to investigate the dynamics of highly excited multi-hadron system. Studies involving multiplicity distributions (MDs) of the relativistic charged particles produced would allow finding the deviations from a simple superposition of multiple independent nucleon-nucleon (nn) collisions. Such studies, if carried out in limited rapidity space are envisaged to provide useful information on dynamical fluctuations [8,9,10,11]. It has been stressed that moments of MDs in full and limited rapidity bins would lead to make some interesting remarks about the production mechanisms involved.
Dependence of MDs and their moments on collision centrality is also expected to lead to some interesting conclusions because of the fact that in narrow centrality windows the geometrical fluctuations may be treated as under control, whereas, such windows, if correspond to most central collisions, may be of additional importance because of the extreme conditions of temperature and excitation energy [7]. An attempt is, therefore, made to study the multiplicity fluctuations in the narrow centrality windows in AuAu collisions for the Beam Energy Scan (BES) at FAIR energies (for E lab = 10, 20, 30 and 40A GeV) in the frame work of URQMD model, using the code, urqmd-v3.4 [12,13]. The number of events simulated at these energies are 2.3, 2.3, 2.1 and 2.2M (M = 10 6 ) respectively. The analysis is carried out in the pseudorapidity and p T intervals with −1.0 < η < 1.0 and 0.2 < p T < 5.0 GeV/c respectively.

The URQMD Model
Multiparticle production in relativistic collisions have been described earlier within the hydrodynamic approach [14]. At a later stage the Regge theory [15] and multiperipheral models were developed [15,16]. Although the difficulties attributed to the statistical models wereover come in these models yet the inconvenience of this approach is the large number of free parameters which are to be fixed by comparison with the experiments. Subsequently various quark-parton models motivated by QCD were introduced and as a consequence a large variety of models for hadronic and heavy-ion collisions were proposed. These models may be classified into macroscopic (statistical and thermodynamic) models [17] and microscopic (string, transport, cascade, etc.) models, like URQMD, VENUS, RQMD, etc. The microscopic models describe the individual hadron-hadron collisions.
URQMD model is based on the co-variant propagation of constituent quarks and di-quarks but  GeV, the collisions are described in terms of interactions between hadrons and their excited states [17], whereas at higher energies ( > 5 GeV), the quark and gluon degrees of freedom are considered and the concept of color string excitation is introduced with their subsequent fragmentation into hadrons [13]. In a transport model, AA collisions are considered as the superposition of all possible binary nn collisions. Every nn collision corresponding to the impact parameter, b ≤ σ tot /π is considered, where σ tot represents the total cross section. The two colliding nuclei are described by Fermi gas model [17] and hence the initial momentum of each nucleon is taken at random between The study, however, might help in the interpretation of the experimental data since it will permit subtraction of simple dynamical and geometrical effects from the expected Quark Gluon Plasma (QGP) signals [18].

Results and discussion
The URQMD model gives the value of impact parameter, b on ebe basis which allows to determine the collision centrality and mean number of participating nucleons, N part using the Glauber model [7,19]. Values of number of participating nucleons, mean charged particle multiplicities and dispersion of MDs (σ) for various collision centralities at the four energies are estimated and listed in Tables I -IV The values of coefficients, occurring in Eqs.1 and 2 are listed in Tables V and VI respectively. As described in ref.7, the centrality dependence of the moments may be understood by the Central Limit Theorem (CLT), according to which, N ch ∝ N part and σ ∝ N part . However, in the present study the mean multiplicity is observed to grow with N part , as given by Eq.1., i.e. a slight deviation from linearity is exhibited by the data irrespective of the fact that how large or small the centrality bins are chosen. The variations of σ with N part , shown in FIG.3, is seen to be nicely fitted by Eq.2 for 5% centrality bin width, while for the centrality bin widths of 2% and 10% the data are seen to be fitted only for centrality > 20%, as indicated by the lines in this figure; the lines are drawn for the range of centrality for which the fits of the data have been performed.
Similar deviations from CLT have also been observed in AuAu collisions at RHIC and lower energies [7]. In order to extract dynamical fluctuations arising from physical processes, fluctuations in mean number of participating nucleons are to be minimized. To achieve the same, centrality bins    Yet another way to examine and predict the MDs, is to plot MDs in terms of KNO scaling variable Z (= N ch / N ch ). It has been observed that MDs in hadron-hadron collisions exhibit a universal behavior in a wide range of incident energies if plotted as N ch P (N ch ) against the variable Z [20,21,22,23,24,25]. It was shown that MDs corresponding to pp collisions in the energy range ∼ (50 -303) GeV are nicely reproduced by the functional form given by Slattery [22]. MDs in pp collisions, for non single diffractive events at ISR energies have also been observed to exhibit KNO scaling [26]. Since the width of MDs for a given centrality gives the extent of fluctuations, the here ω is regarded as a quantitative measure of the particle number fluctuations [7,18,27,28,29].
The scaled variance, ω is an intensive quantity which does not depend on the volume It may be noted from the figure that ω increases with beam energy as well as in centrality bin widths. It may also be noted that increase of ω with c.m. energy becomes linear for the centrality classes 35% and above. If the data obey the KNO scaling [21], it is predicted that ω should increase linearly with mean charge multiplicity [29]. It may also be noticed in FIG.8 that increase of ω with beam energy is somewhat weaker for the central collisions. Similar trends of variations of ω with energy have also been reported in pp collisions by NA61 collaboration [29].
Centrality dependence of scaled variance at the four incident energies are exhibited in FIG.9. It is observed that for 10% centrality bins ω increases with centrality bin widths, whereas for 5%    considering a bin as wide as 5%, would help arrive at some meaningful conclusions on dynamical fluctuations, if present.