Contributions of Jets in Net Charge Fluctuations from the Beam Energy Scan at RHIC and LHC

Dynamical net charge fluctuations have been studied in ultra-relativistic heavy-ion collisions from the beam energy scan at RHIC and LHC energies by carrying out the hadronic model simulation. Monte Carlo model, HIJING is used to generate events in two different modes, HIJING-default with jet quenching switched off and jet/minijet production switched off. A popular variable, $\nu_{[+-,dyn]}$ is used to study the net charge fluctuations in different centrality bins and the findings are compared with the available experimental values reported earlier. Although the broad features of net charge fluctuations are reproduced by the HIJING, yet the model predicts the larger magnitude of fluctuations as compared to the one observed in experiments. The role of jets/minijets production in reducing the net charge fluctuations is, however distinctly visible from the analysis of the two types of HIJING events. Furthermore, $dN_{ch}/d\eta$ and $1/N$ scaling is partially exhibited which is due to the fact that in HIJING, nucleus-nucleus collisions are treated as multiple independent nucleon-nucleon collisions.


Introduction
The interest in the studies involving event-by-event fluctuations in hadronic (hh) and heavy-ion (AA) collisions is primarily connected to the idea that the correlations and fluctuations of dynamical origin are associated with the critical phenomena of phase transitions and leads to the local and global differences between the events produced under similar initial conditions [1,2]. Several different approaches have been made to investigate the event-by-event fluctuations in hh and AA collisions at widely different energies, e.g., multifractals [3,4,5], normalized factorial moments [6], erraticity [4,7], k-order pseudorapidity spacing [8,9] and transverse momentum(p T ) spectra, etc. Furthermore, event-by-event fluctuations in the conserved quantities, like, strangeness, baryon number, electric charge have emerged as new tools to estimate the degree of equilibration and criticality of the measured system [12]. Experiments such as RHIC and LHC are well suited for the study of these observables [12,13].
Event-by-event fluctuations of net charge of the produced relativistic charged particles serve as an important tool to investigate the composition of hot and dense matter prevailing in the 'fireball', created during the intermediate stage of AA collisions, which, in principle, be characterized in the framework of QCD [13].
It has been argued that a phase transition from QGP to normal hadronic state is an entropy conserving process [14] and therefore, the fluctuations in net electric charge will be significantly reduced in the final state in comparison to what is envisaged to be observed from a hadron gas system [15,18]. This is expected because the magnitude of charge fluctuations is proportional to the square of the number of charges present in the system which depends on the state from which charges originate. A system passing through QGP phase, quarks are the charge carriers whereas in the case of hadron gas the charge carriers are hadrons. This suggests that the charge fluctuations observed in the case of QGP with fractional charges would be smaller than those in hadron gas with integral charges [12,16,19]. A reduction in the fluctuations of net charge in Pb-Pb collision at √ s N N = 2.76 TeV in comparison to that observed at RHIC has been reported by ALICE collaboration [20]. A question arises here whether the fluctuations arising from QGP or from hadron gas would survive during the evaluation of the system [12,21,22,23,24].
The fluctuations observed at the freeze-out depend crucially on the equation of state of the system and final effects. It has been shown [25] that large charge fluctuations survive, if accompanied by large temperature fluctuations at freeze-out in context to the experiments. Measurement of charge fluctuations depends on the observation window which is so selected that the majority of the fluctuations are captured without being affected by the conservation limits [22,23,24].
An attempt is, therefore, made to carry out a systematic study of dynamical net charge fluctuations from beam energy scan at RHIC and LHC energies using the Monte-Carlo model, HIJING and the findings are compared with those obtained with the real data and other MC models. The reason for using the code HIJING is that it gives an opportunity to study the effect of jets and jet-quenching. HIJING events are generated at various beam energies corresponding to RHIC and LHC which covers an energy range from √ s N N = 62.4 GeV to 5.02 TeV. Two sets of events, i) HIJING-default with jets and minijets and ii) HIJING with no jet/minijet production are generated for each of the incident energies considered.

Formalism
The charge fluctuations are usually studied in terms of two types of measures [26].
The first one is the D which is the direct measure of the variance of event-by-event net charge δQ 2 = Q 2 − Q 2 , where Q = N + − N − ; N + and N − respectively denote the multiplicities of positively and negatively charged particles produced in an event in the considered phase space. Since the net charge fluctuations may get affected by the uncertainties arising out of volume fluctuations, the fluctuations in the ratio, R = N + N − is taken as the other suitable parameter. R is related to the net charge fluctuations via the parameter D as [15,16,19,20]: which gives a measure of charge fluctuations per unit entropy. It has been shown that D acquires a value ∼ 4 for an uncorrelated pion gas which decreases to ∼ 3 after taking into account the resonance yields [16]. For QGP, the value of D has been reduced to ∼ 1 − 1.5, where the uncertainty arises due to the uncertainties involved in relating the entropy to the multiplicity of the charged hadrons in the final state [27]. The parameter D, thus, may be taken as an efficient probe for distinguishing between the hadron gas and QGP phases. These fluctuations are however, envisaged to be diluted in the rapidly expanding medium due to the diffusion of particles in rapidity space [22,23]. Resonance decays, collision dynamics, radial flow and final state interactions may also affect the amount of fluctuations measured [16,28,29,30]. The first results on net charge fluctuations at RHIC were presented by PHENIX [31] in terms of reduced variance ω d = δQ 2 N ch while STAR [30] results were based on a dynamical net charge fluctuations measure, ν [+−,dyn] and were treated as a rather reliable measure of the net charge fluctuations as ν [+−,dyn] was found to be robust against detection efficiency.
Furthermore, the contributions from statistical fluctuations would also be present if net charge fluctuations are studied in terms of parameter D and it will be difficult to extract the contribution due to fluctuations of dynamical origin. The novel method of estimating the net charge fluctuations takes into account the correlation strength between + +, --, and + -charge particle pairs [12,32]. The difference between the relative multiplicities of positively and negatively charged particles is given as, where the angular brackets represent the mean value over the entire sample of events. The Poisson limit of this quantity is expressed as [30]: The dynamical net charge fluctuations may, therefore, be written as the difference of these two quantities: From the theoretical point of view, ν [+−,dyn] can be expressed in terms of two particle integral correlation functions as where the term R αβ gives the ratio of integrals of two-and single-particle pseudorapidity density function, defined as, The variable, ν [+−,dyn] is, thus, basically a measure of relative correlation strength of + +, --, and + -charged hadron pairs. For independent emission of particles, these correlations should be ideally zero. However, in practice, a partial correlations is observed due to string and jet-fragmentation, resonance decays, etc. The strength of R ++ , R −− and R +− are expected to vary with system size and beam energy. Moreover, as the charge conservation, + -pair are expected to be rather strongly correlated as compared to like sign charge pairs and hence 2R +− in Eq.6 is envisaged to be larger than the sum of the other two terms [30] giving ν [+−,dyn] values less than zero, which is evident from the results based on pp andpp collisions at CERN ISR, FNAL and later on in heavy-ion collisions at RHIC [30,32,34,35,36] and LHC energies [20,30].

Results and discussion
Several sets of MC events corresponding to different collision systems in a wide range of beam energies are generated using the code HIJING-1.37 [37] for the present analysis. The details of the events simulated are listed in Table-1. Two sets of events for each beam energy and colliding nuclei, HIJING-default with jet-quenching off and with jet/minijet production switched off are simulated and analysed. It has been argued [38,39] that the minijets (semi-hard parton scattering with few GeV/c momentum transfer) are copiously produced in the early state of AA collisions at RHIC and higher energies. In a QGP medium, if present, the jets/minijets will lose energy through induced gluon radiation [40], a process jets/minijets off are somewhat larger. The jets-off multiplicities reflects the soft processes, whereas, the jets-on multiplicities includes the contributions from the jets and minijets [36]. This may cause the reduction in the contributions coming from the third term of Eq.5 which represents the correlations between + -pairs. This is expected to occur at these energies, as the events have high multiplicities and are dominated by multiple minijet production which might cause the reduction in the strengths of correlations and fluctuations [41].
The parameter D and ν [+−,dyn] are related to each other as per the relation: The magnitude of net charge fluctuations is limited by the global charge conservation of the produced particles [32]. Considering the effect of global charge conservation, the dynamical fluctuations need to be corrected by a factor of −4/ N total , where N total denotes the total charged particle multiplicity of an event in full phase space. Taking into account the global charge conservation and finite acceptance, the corrected value of ν [+−,dyn] is given by, Values of ν corr [+−,dyn] for various data sets are presented in the last column of Tables  2 -4 Moreover, for a given N part the values of product N part ν [+−,dyn] decrease with the beam energy. It is interesting to note that the difference in the values observed at RHIC and LHC energies, after applying the corrections to ν [+−,dyn] values almost vanishes. It is also interesting to note that the HIJING simulated data points lie closer to the corresponding ones reported earlier using the ALICE data [20]. The production. This may lead to the conclusion that as one moves from RHIC to LHC energies, contributions to the particle multiplicity coming from the jet/minijet production causes the reduction in the magnitude of charge fluctuations.
As mentioned earlier, if AA collisions are the superpositions of m number of nn collisions the single particle density for nn and AA collisions would be written as: ρ n 1 n(η) = dN ch /dη and ρ A 1 A(η) = mρ nn 1 η. In such a scenario, the invariant cross section is proportional to the number of nn collisions, m, and the quantity (dN ch /dη)ν [+−,dyn] is independent of centrality of collision and the system size [14]. STAR results, however, give ∼ 40% increase in (dN ch /dη)ν It has been suggested [43] that any multiplicity scaling should be based on the mean multiplicities of charged particles. In the model independent sources [44], mean particle multiplicity is taken to be proportional to the number of sources, N s , which changes from event to event. The multiplicity of positively and negatively charged particles may be expressed as where, α i and β i represent the contributions from i th source. The first and second moments of multiplicity distributions are written as here α , β and α 1 , β 1 , αβ are the first and second moments of the probability distributions P (α, β) for a single source.
Following the details as given in ref [44] and using the equation: the following form of ν dyn may be obtained [45] where, ν * [α, β] is the quantity of the multiplicities of types a and b for each source. This gives ν a,b to be inversely proportional to the size of the colliding nuclei. On the other hand, as the term N 2 s − N s cancels out by construction, ν dyn is independent of the system size but requires an additional scaling due to the remaining term, 1/ N s . If 1/( 1 Na + 1 N b ) type of scaling is used, then substituting Eqs.12 and 13 in Eq.17, the term 1/ N s vanishes and the following form of the scaling is obtained: The scaling of this type has been tested and the results for the various data sets are shown in Figs.10 and 11. It may be seen in these figures that the scaled ν [+−,dyn] values for a given energy are nearly independent of charged particle density. It is, further, observed that the magnitude of scaled ν [+−,dyn] values increases as one moves from RHIC to LHC energies. The magnitude of ν [+−,dyn] is observed to be inversely proportional to the number of sub collisions leading to the particle production. If number of particles produced in each sub collisions is independent of collision centrality, ν [+−,dyn] would exhibit 1/N scaling [11]. It has been reported [11] that in Au-Au collisions at 130 GeV 1/N scaling is clearly noted by the data. HIJING simulated data, however, supports such scaling. In contrast to this, findings from URQMD simulations do not support 1/N scaling which maybe because in URQMD re-scattering effects are included which would reduce the magnitude of Nν [+−,dyn] for central collisions [11]. On the basis of various types of scaling of ν [+−,dyn] tested in the present study and also the ones by other workers it may be concluded here that 1/( 1 Na + 1 N b ) scaling of ν [+−,dyn] is relatively a better scaling as compared to other scalings.