Fluctuations of identified particle yields using the {\nu} dyn variable at energies available at the BNL Relativistic Heavy Ion Collider

We study the fluctuations of net-charge, net-pion, net-kaon and net-proton using $D$-measure and $\nu_{dyn}$ variables in heavy-ion jet interaction generator (HIJING), ultra-relativistic quantum molecular dynamics (UrQMD), and hadron resonance gas (HRG) model at different collision energies \sqsn. It has been observed that, the values of $D$ strongly dependent on $\Delta \eta$ in HIJING and UrQMD models and independent in HRG model. The diffusion coefficients ($\sigma$) of identified particles are estimated at various \sqsn. It is observed that, the $\sigma$ values are independent of collision energies but emphasizes the particle species dependence of diffusion coefficient in the QGP medium. This study provides a realistic baseline for comparison with the experimental data.


I. INTRODUCTION
One of the major goal of heavy-ion experiments is to study the phase transition from hadronic matter to quark-gluon plasma (QGP). Event-by-event fluctuations of conserved quantities such as net-baryon number, netelectric charge, and net-strangeness are proposed as possible signals of the QCD phase transition [1]. It can also help to understand the nature of such phase transition. One of the observables, net charge fluctuation, has been termed to be very promising signal of such studies. The reason behind net charge fluctuation study is similar to the original study of color charge in e + e − experiment where color charge ratio is measured and depending upon the difference in fundamental degrees of freedom in quark gluon state to hadronic state, the origin of the color charge is determined [2,3]. Several experiments have measured net-charge fluctuation at SPS, RHIC and LHC energies [4][5][6][7][8][9].
Event-by-event fluctuation in high-energy heavy-ion collisions aims to study the equilibrium of thermodynamical fluctuations at freeze-out. In the QGP phase, quarks are the charge carrier with fractional unit charge of 1/3, while in hadronic phase hadrons being the charge carrier with unit charges. Hence, net-charge fluctuations in the QGP phase is a factor of 2-3 smaller as compared to that of hadron phase. These differences may be considered as indicators of the formation of quark-gluon plasma in high energy heavy-ion collisions. Thus, the net-charge fluctuations are strongly dependent on the phase of their origin. Due to the rapid expansion of the fireball created in the heavy ion collisions, the fluctuations created in the initial state may survive during the hadronization process [2]. If the relaxation time happens to be shorter than the lifetime of the hadronic stage of the collisions, then values of such fluctuations should deviate from their * Electronic address: vkr.singh@vecc.gov.in † Electronic address: dkmishra@barc.gov.in ‡ Electronic address: za@vecc.gov.in equilibrium hadron gas values towards their earlier, primordial values, typical for QGP [1,2]. The fluctuations of different length, or range, in rapidity space relax on different time scales. Since relaxation can only proceed via diffusion of the charge, the longer range of fluctuations relax slower. The relaxation time grows as a square of the range [10]. It is evident that fluctuations of total charge in a wider rapidity window relax slower. The minimal interval of rapidity we can consider must be much larger than the mean rapidity change of a charged particle in a collision, δy coll . It is observed that, the typical δy coll for the baryon and electric charge is of order 0.2 and 0.8, respectively. A large acceptance detector such as ALICE or STAR can be considered as somewhat idealized limit of rapidity windows which are much wider than δy coll . This can certainly be reasonable acceptance to study the diffusion of the baryon charge.
The Beam Energy Scan (BES) program at RHIC has drawn much attention to explore the QCD phase diagram and study the transport properties of nuclear matter at finite temperature (T ) and net-baryon (and net-electric charge) density. At lower collision energies, e.g. √ s N N = 7.7 GeV, the baryon chemical potential can reach up to µ B ∼ 400 MeV which is significant compared to the temperature achieved. At such energies, a strong gradients in the chemical potential of conserved charges are expected. Hence, lower energy beam scan program at RHIC will be useful to explore the properties of net-charge diffusion in the nuclear matter.
The conservation laws limit the dissipation of the fluctuations which suffer after the hadronization has occurred. It is observed that, due to the diffusion of particles in rapidity space, these fluctuations may also get diluted in the expanding medium [10,11]. The hadronic diffusion from time of hadronization τ 0 to a freeze-out time τ f can dissipate these fluctuations. It is argued that, the reduction of the fluctuation in the QGP phase might be observed only if the fluctuations are measured over a large rapidity range [10]. It is also discussed that, how much the fluctuations are reduced with the increase of accepted rapidity interval. The suppression of charge fluctuations observed in experimental data is consistent with the diffusion estimates. The effect of the QGP suppression is crucially different from the critical fluctuations. While the QGP suppression is the history effect, the critical fluctuations are the equilibrium fluctuations pertaining to the freeze-out point, and the study of diffusion is necessary to establish them [12]. Earlier efforts were made to estimate the fluctuation strength and diffusion parameter (σ) using transport model and HRG model for all inclusive charged particles [16]. However the contribution of different identified particles to dilution of measured fluctuation strength and diffusion parameters may be different. It would be interesting to study the diffusion coefficient of different conserved charges in heavy-ion collisions.
One can not measure the collisional volume directly in experiments. To avoid volume fluctuations, ratio of positive (+) and negative (−) charged particles normalized by total number of charged particles under consideration for a fixed centrality class of events is used to measure fluctuation strength, usually known as D-measure [13], which can be defined as: where R(= N + /N − ) is ratio of number of positive particles to number of negative particles. Q = N + − N − being the difference between number of positive and negative particles (net-charge) and N ch = N + +N − being the total number of charged particles measured in an event. The δQ 2 is the variance of the net charge Q, which is proportional to the net-charge fluctuation in the system. The value of D is found to be approximately four times smaller in the QGP phase as compared to the hadron gas phase [13]. However, D-measure has been found to be dependent on detection efficiency. Another variable, ν (±,dyn) is used to measure the fluctuation strength. The ν (±,dyn) is found to be robust and independent of detection efficiency. It is defined as The value of ν (±,dyn) gives the measure of the relative correlation strength of ("++", "−−" and "+−") charged particle pairs. The relation between D and ν (±,dyn) is given as [13] The values of ν (±,dyn) need to be corrected for global charge conservation and finite net-charge effect [14,15]. One of the important aspect of this measured fluctuation strength is their survival probability. In high multiplicity environment, these fluctuation strength gets diluted.
This dilution may occur due to diffusion of particles in the expanding medium [10,11]. It was suggested to measure the fluctuation strength over large rapidity space. In Ref. [17], all diffusion coefficient matrix for baryon, electric charge and strangeness has been calculated using kinetic theory. In the present study, we have calculated the fluctuation strength and diffusion coefficient of identified charged particles, mainly for net-pion, net-kaon and net-proton using hadron resonance gas model (HRG), HI-JING model, and transport model UrQMD which will serve as the baseline for experimental measurements.
The paper is organized as follows. In the following section, we discuss HRG model used in this paper as well as implementation of resonance decay. We also briefly discuss the HIJING and UrQMD models. In Sec. III, we discuss our estimated results on D, ν (±,dyn) and diffusion coefficients for identified particles at different ∆η and √ s N N . We finally summarize our findings in Sec. IV.

II. ESTIMATION OF D-MEASURE IN DIFFERENT MODEL
In this section, we briefly describe the models used in calculation of D-measure. We have used HRG, HIJING and UrQMD models to study the fluctuation variable (D or ν dyn ). These models have been extensively used to explain the experimental data from heavy-ion collisions.

A. Hadron resonance gas model
The partition function in HRG model has all relevant degrees of freedom of the confined, strongly interacting matter and implicitly include all the interactions that result in resonance formation [18,19]. The logarithm of the partition function is given as where i is the particle number index, V is the volume of the system, g i being the degeneracy factor for i-th particle, ±ve signs correspond to the baryon or meson, respectively. We have used the total chemical potential of individual particle µ i in our calculations as given in Ref. [18]. Using partition function one can calculate various thermodynamical quantities of the system in heavyion collisions. The susceptibilities of different orders are related to the N ch and δQ 2 representing mean and variance of the individual particle, respectively. These quantities can be calculated by taking first and second derivative of Eq. 4 with respect to µ: Equations 5 and 6 are used to calculate D-measure in HRG model. Experimentally measured stable particles (pions, kaons and protons along with their anti-particles) have contributions from production of both primordial as well as from resonance decay. Further, neutral resonances introduce positive correlations between N + and N − and hence their decayed daughters can affect the fluctuation of the final measured particles. The ensemble averaged stable particle yield will have contribution from both primordial production and the resonance decay [3,20], where N * i and N R correspond to the average primordial yield of particle species i and of the resonances R, respectively. The summation runs over all the resonances which decay to the final particle i and n being the average number of particle type i produced from the resonance R. Further, b R r is the branching ratio of the r-th decay channel of the resonance R and n R i,r is the number of particle i produced in that decay branch. The generalized n-th order susceptibility for stable particle i can be written as [21]; The first term in Eq. 8 corresponds to the contribution from primordial yield and second term corresponds the contribution from fluctuations of primordial resonances and average number of produced particle of type i, assuming the number of decay daughters are fixed.

B. The HIJING and UrQMD models
We have used HIJING (V.1.37) and UrQMD (V.1.30) to study the fluctuation variables (D or ν dyn ). Both HI-JING and UrQMD models are Monte Carlo event generators used for nucleon-nucleon and nucleus-nucleus collisions in high energy physics simulations. These models provide the proper baseline to compare with the experimental data.
The HIJING model is based on perturbative QCD (pQCD) considering the multiple mini-jet partons produced in collisions are transformed into string fragments and later fragments into hadrons. It uses the PYTHIA model to generate kinetic variables for each hard scattering and JETSET model for jet fragmentation. In pQCD, the cross sections for hard parton scattering is determined using the leading order to account for the higherorder corrections. While the soft contributions are determined using the diquark-quark strings with gluon kinks induced by soft gluon radiation. The HIJING model considers the nucleus-nucleus collisions as a superposition of proton-proton collisions, it also takes into account the other physics processes like multiple scattering, jet quenching and nuclear shadowing to study the nuclear effects [22].
The UrQMD model considers the microscopic transport of quarks and diquarks with mesonic and baryonic degrees of freedom. The model preserves the conservation of baryon number, electric charge, and strangeness number. In this model, the space-time evolution of the fireball is studied in terms of excitation and fragmentation of color strings, and the formation and decay of hadronic resonances [15]. Interaction of the produced particles, which may influence the acceptance of certain windows is included in the model. The formation of hadrons are explained by color string fragmentation, it also considers the resonance decays, multiple scattering between hadrons during the evolution including baryon stopping phenomena, which is one of the feature of heavyion collisions specially at lower collision energies [23]. The UrQMD model has been applied successfully to study the thermalization [24], particle yields [25,26], leptonic and photonic probes [27], and event-by-event fluctuations [28][29][30][31][32][33].

III. RESULTS AND DISCUSSION
Due to the diffusion of charge particles in the hadronic phase, the measured fluctuations may get diluted during evolution of the system and keep on approaching the equilibrated values in the hadronic medium until their kinetic freeze-out. Hence, the experimental measurements of not only the magnitudes of fluctuation strength at a fixed ∆η but also their dependence on ∆η enable to explore various aspect of the time evolution of the hot medium and the hadronization mechanism. It is proposed to study the fluctuations of identified particle species and estimate the rate of diffusion in different rapidity interval.  model are performed for different cases by considering all the charged hadrons, individual identified particles (π, K, p) and contribution of resonance decays. The estimated values of D-measure are found to be independent of ∆η in case of HRG model, while in case of HIJING and UrQMD models, the value of D decreases with increase in ∆η interval. However, there is strong dependence of resonance decay effects on the identified particles. The HRG calculations for net-charge, net-pion, net-kaon and net-proton fluctuations are performed within the same kinematic acceptance as done for HIJING and UrQMD. The calculation of D from the HRG model will provide a pure thermal baseline contribution as a function ∆η. There is strong dependence of D values on ∆η observed for net-charge as well as identified particles in HIJING and UrQMD models. The higher D value at smaller ∆η interval suggests that the correlation is maximum for the smaller ∆η interval which gets diluted with increasing ∆η window [11]. It is also suggested that opening a larger ∆η allows to see deeper back into the history of the collision [10]. The curvature of D shows a decreasing slope up to higher ∆η intervals. This is in contrast to observation made by ALICE experiment at higher collision energies √ s N N = 2.76 TeV, which shows a flattening trend by extrapolating the fitted curve to higher ∆η range [34].  The D values for net-charge, net-pion, and net-kaon are independent of √ s N N in HI-JING and UrQMD models. The D values for net-proton case shows small energy dependence in both the models. There is clear particle dependence of D values in both the models. It is to be seen whether the particle dependence is because of mass of the particles or due to the type of the particle (meson, baryon or strange). to kinetic freeze-out because of the diffusion of charged hadrons in rapidity space [11]. Figures 3 and 4 show the N ch ν corr (±,dyn) and D as a function of ∆η intervals for (0-5%) centrality in Au+Au collisions at different √ s N N using HIJING and UrQMD models, respectively. Following the Refs. [10,11], the simulated data points are fitted with the error function, Erf(∆η/ √ 8σ) representing the diffusion in rapidity space. The fitted functions are shown as solid lines in Figs. 3 and 4. The data points are fitted within ∆η range 0.35 to 5.0 for net-charge, net-pion, net-kaon and net-proton. The fit parameter, σ in Erf(∆η/ √ 8σ), characterizes the diffusion at freezeout that accounts for the broadening of the rapidity distributions due to interactions and particle production. We have calculated the diffusion coefficients at √ s N N = 7.7, 19.6, 27, 39, 62.4 and 200 GeV for all charge and the identified particles which represent the proxy for conserved quantities (net-baryon, net-electric charge and net-strangeness) using both HIJING and UrQMD models. The slope of the fit function decreases with increasing particle mass. The ∆η dependence of N ch ν corr (±,dyn) and D for net-proton is qualitatively different in both HIJING and UrQMD models, whereas net-charge, netpion and net-kaon have similar behavior in both the models. In case of UrQMD model, the D values flattened at higher ∆η with increasing √ s N N . Hence, the N ch ν corr (±,dyn) as a function of ∆η for net-proton not able to fit with Erf(∆η/ √ 8σ) function unlike pions and kaons. Figure 5 shows the diffusion coefficient as a function of ∆η window for net-charge, net-pion, net-kaon, and netproton in Au+Au collisions at √ s N N = 200 GeV. The σ values are obtained by fitting the N ch ν corr (±,dyn) up to different ∆η range with the error function. In both the HIJING and UrQMD models, the diffusion coefficient for net-charge and net-pion are independent of ∆η window and match with each other. In HIJING model, the diffusion coefficients of net-kaon and net-proton show small ∆η dependence. The σ values of net-proton are systematically above, whereas σ values of net-kaon are systematically below the net-charge and net-pion values. Due to the qualitatively different nature of curvature of D as a function ∆η for net-proton in UrQMD model, it was not possible to extract the σ values.
The extracted values of diffusion coefficient of identified particles as a function of √ s N N are shown in Fig. 6 from both HIJING and UrQMD models. The resulting values of σ are obtained by fitting the N ch ν corr (±,dyn) values up to ∆η = 5.0 with the error function. The σ values for net-charge and net-pion are close to each other at all the studied energies. The σ of net-proton and net-kaon are closer to each other and systematically higher than net-pion at all energies in both the models. The diffusion coefficients are constant as a function of studied collision energy range √ s N N = 7.7 to 200 GeV. However slight variation is observed in case of HIJING model at lower energies particularly for net-kaon and and net-proton.

IV. SUMMARY
In summary, we have studied the fluctuations of netcharge, net-pion, net-kaon, and net-proton using the Dmeasure observable within the ambit of HRG, HIJING, and UrQMD models at different collision energies. The D values are estimated up to higher ∆η window. A stronger dependence of D value observed for lower ∆η and the decreasing trend continues up to higher ∆η with lower slope in both the models, except net-proton in UrQMD model. In case of net-proton in UrQMD model, the curvature of D values as a function of ∆η shows different behavior. The D values obtained from different model calculations are independent of collision energies but show particle species dependent. The diffusion coefficient (σ) has been estimated by fitting the D-measure as a function of ∆η with the error function. The σ values are independent of collision energies. The σ values of net-kaon and net-proton are systematically higher than net-pion at all the studied energies. This study emphasizes the particle species dependence of diffusion coefficient and provides a realistic baseline for comparison with the experimental data.