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

1. IntroductionAny 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 to 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 [1-3]. 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. [4-7]. 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 [8-11]. 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.sub.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.sup.6]) respectively. The analysis is carried out in the pseudorapidity (r) and transverse momentum ([p.sub.T]) intervals: -1.0 < [eta] < 1.0 and 0.2 < [p.sub.T] < 5.0GeV/c respectively.

2. 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 were overcome 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 covariant propagation of constituent quarks and di-quarks but has been accompanied by baryonic and mesonic degrees of freedom. At low energies, [square root of [s.sub.NN]] < 5 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 [less than or equal to][square root of ([[phi].sub.tot]/[pi])] is considered, where [[phi].sub.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 zero and Thomas-Fermi momentum. The interaction term includes more than 50 baryon and 45 meson species. The model can treat the intermediate fireball both in and out of a local thermal and chemical equilibria. The URQMD model, thus, provides an ideal framework to study heavy-ion collisions. Although, the phase transition from a hadronic to partonic phase are not explicitly included in the model, thus a clear suggestion about the location of critical point can not be made. 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].

3. 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.sub.part]> using the Glauber model [7, 19]. Values of number of participating nucleons, mean charged particle multiplicities and dispersion of MDs ([sigma]) for various collision centralities at the four energies are estimated and listed in Tables 1-4. The centrality selection is made from the MDs of charged particles for the minimum bias events in the considered [eta] and [p.sub.T] ranges. This is illustrated in Figure 1, where the multiplicity distribution of minimum bias events for [E.sub.lab] = 40A GeV is displayed. The shaded regions show 10% centrality cross-section bins. Variations of <[N.sub.ch]> and [sigma] with <[N.sub.part]> for the centrality bin width = 2, 5 and 10% are presented in Figures 2 and 3. The statistical errors associated with these parameters are too small to be noticed in the figures. It may be noted from these figures that <[N.sub.ch]> and [sigma] increase smoothly with <[N.sub.part]> or collision centrality. The lines in Figure 2 are due to the best fits to the data obtained using the equation

[mathematical expression not reproducible], (1)

whereas, in Figure 3 the lines are due to the least square fits to the data of the form

[sigma] = p + q [square root of <[N.sub.part]>]. (2)

The values of coefficients, occurring in equations (1) and (2) are listed in Tables 5 and 6 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.sub.ch]> [varies] [N.sub.part] and [sigma][varies] [square root of [N.sub.part]]. However, in the present study the mean multiplicity is observed to grow with <[N.sub.part]>, as given by equation (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 o with <[N.sub.part]>, shown in Figure 3, are seen to be nicely fitted by equation (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 information regarding dynamical fluctuations arising from physical processes, fluctuations in mean number of participating nucleons are to be minimized. To achieve the same, centrality bins considered should be kept narrow because the fluctuations in the particle multiplicities are directly related to the fluctuations in the mean number of participating nucleons. The inherent fluctuations may be reduced by choosing narrow centrality bins; the inherent fluctuations are the fluctuations which arise due to the difference in the geometry even within the selected centrality bin. A very narrow centrality bin, if considered, would, therefore, minimize this effect but may cause additional fluctuations due to statistics. Centrality resolution of the detectors also demands that the chosen centrality bins should not be too narrow. Thus, our observations from Figure 3, tend to suggest that fluctuation effects dominate if the centrality bin width is somewhat larger or quite small.

Multiplicity distributions of relativistic charged particles for minimum bias events for [absolute value of [eta]] < 1.0 and [p.sub.T] = 0.2 - 5.0 GeV/c are displayed in Figure 4. It may be noted from the figure that MDs at the four beam energies considered, acquire nearly similar shapes and it is expected that the maximum values of [N.sub.ch] become higher with increasing energies. Similar trends in MDs have also been reported by Ghosh et al. [17] at the same beam energies predicted by URQMD model. MDs of relativistic charged particles for various centrality groups at the four beam energies have also been examined. Distributions for [E.sub.lab] = 40A GeV are presented in Figure 5 along-with the distribution of full sample of events (minimum bias). It is evidently clear from the figure that MD of minimum bias sample is a convolution of MDs with different centrality classes.

Yet another way to examine and predict the MDs, is to plot MDs in terms of KNO scaling variable Z (=[N.sub.ch]/<[N.sub.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.sub.ch]>P([N.sub.ch]) against the variable Z [20-25]. 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 origin of the fluctuations are, thus, inherent in the width of MDs. To understand this behavior, MDs should be plotted for different centrality bins in terms of KNO scaling variable. MDs for 10, 30, and 50% centrality are plotted in terms of KNO scaling variable in Figure 6. For clarity sake, each next distribution is shifted upwards on y-scale by a factor of 10. It is observed that the distributions become wider with increasing collision centrality, but exhibits a perfect scaling behavior. MDs, plotted in terms of KNO variable for full event sample in Figure 7, are also noticed to show a perfect KNO scaling.

The scaled variance, [omega] of the MDs defined as,

[omega] = [[sigma].sup.2]/<[N.sub.ch]>, (3)

here u> is regarded as a quantitative measure of the particle number fluctuations [7, 18, 27-29]. The scaled variance, u> is an intensive quantity which does not depend on the volume of the system within the grand canonical ensemble (GCE) of statistical mechanics or on the number of sources within models of independent source, like wounded nucleon model. The value of scaled variance will be zero in the absence of fluctuations in MDs and unity for Poisson MDs. Since the volume of the system created in AA collisions fluctuates from event to event, and [omega] would depend on volume fluctuations, it becomes important to reduce the fluctuation effects in fluctuation studies [28]. As mentioned earlier, one way to reduce the fluctuation effects is to reduce the number of participating nucleons by selecting the narrow centrality bins. However, the choices of centrality should be such that it does not introduce additional fluctuations due to finite multiplicity and detector resolutions. Once the statistical fluctuation part is under control, the fluctuation effects present will be mostly of dynamical origin, which may contain interesting physics associated with the collisions, like hydrodynamic expansion, hadronization at freeze-out, etc.

Variation of scaled variance with center of mass (c.m.) energy for different centrality bins are plotted in Figure 8. It may be noted from the figure that o increases with beam energy as well as in centrality bin widths. It may also be noted that increase of [omega] 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 [omega] should increase linearly with mean charge multiplicity [29]. It may also be noticed in Figure 8 that increase of [omega] with beam energy is somewhat weaker for the central collisions. Similar trends of variations of [omega] with energy have also been reported in pp collisions by NA61 collaboration [29].

Centrality dependence of scaled variance at the four incident energies is exhibited in Figure 9. It is observed that for 10% centrality bins [omega] increases with centrality bin widths, whereas for 5% and 2% this parameter slowly decreases with increasing centrality and thereafter tends to acquire nearly constant values. This observation, thus, supports that statistical fluctuations arising due to fluctuations in [N.sub.part] becomes visible if the centrality bin width is 10% or more and hence considering a bin as wide as 5%, would help to arrive at some meaningful conclusions on dynamical fluctuations, if present.

4. Conclusions

MDs and ebe multiplicity fluctuations in AuAu collisions from the beam energy scan in future heavy-ion experiment at the Facility for Antiproton and Ion Research (FAIR) are examined in the frame work of Ultra-Relativistic Quantum Molecular Dynamics model, URQMD. The mean values of MDs are observed to shift towards the higher multiplicity and the width of the distributions are found to become wider from central to peripheral collisions. The MDs are also observed to obey KNO scaling in various centrality windows as well as for full event (minimum bias) samples. Centrality-bin width dependence of the 2nd moments and scaled variance gives the idea of bin width effect and centrality window-width selection, where the statistical fluctuations may be treated as under control.

https://doi.org/10.1155/2019/3905376

Data availability

The data used to support the findings of this study are available

with the corresponding author and may be provided on request.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Anuj Chandra [ID], Bushra Ali, and Shakeel Ahmad [ID]

Department of Physics, Aligarh Muslim University, Aligarh 202002, India

Correspondence should be addressed to Shakeel Ahmad; shakeel.ahmad@cern.ch

Received 12 June 2019; Accepted 30 July 2019; Published 2 December 2019

Academic Editor: Fu-Hu Liu

Caption: Figure 1: An example of centrality selection from the multiplicity distribution of minimum bias simulated events at [E.sub.lab] = 40A GeV.

Caption: Figure 2: Variations of mean multiplicity with mean number of participating nucleons. The lines are due to fits obtained using equation (1).

Caption: Figure 3: Variations of dispersion with mean number of participating nucleons. The lines are due to fits obtained using equation (2).

Caption: Figure 4: Multiplicity distributions of charged particles for AuAu collisions at 10A, 20A, 30A, and 40A GeV in the range [p.sub.T] = 0.2 - 5.0 GeV/c and [absolute value of [eta]] = 1.0.

Caption: Figure 5: Multiplicity distributions of relativistic charged particles for various centrality classes at 40A GeV in the range [p.sub.T] = 0.2 - 5.0 GeV/c and [absolute value of [eta]] = 1.0.

Caption: Figure 6: Scaled Multiplicity distributions of relativistic charged particles for the centrality bins 0-10%, 30-40% and 50-60%. Distributions corresponding to 30-40% and 50-60% are shifted upwards on the y-scale by factors 10 and 100 for clarity sake only.

Caption: Figure 7: Scaled Multiplicity distributions of relativistic charged particles for the minimum bias events at the four beam energies considered.

Caption: Figure 8: Dependence of scaled variance, [omega] on beam energy.

Caption: Figure 9: Dependence of scaled variance, [omega] on <[N.sub.part]> in different centrality-bin widths.

Table 1: Values of <[N.sub.part]>, <[[N.sub.ch]>, dispersion ([sigma]) and scaled variance ([omega]) in various centrality bins at [E.sub.lab] = 10A GeV/c. Centrality <[N.sub.part]> <[N.sub.ch]> (%) 5 348.00 [+ or -] 0.0020 231.03 [+ or -] 0.07 10 289.90 [+ or -] 0.0020 182.37 [+ or -] 0.06 15 238.45 [+ or -] 0.0020 144.36 [+ or -] 0.05 20 195.27 [+ or -] 0.0020 113.62 [+ or -] 0.04 25 159.21 [+ or -] 0.0020 89.70 [+ or -] 0.04 30 127.17 [+ or -] 0.0020 70.10 [+ or -] 0.04 35 100.08 [+ or -] 0.0020 53.49 [+ or -] 0.03 40 77.97 [+ or -] 0.0010 39.72 [+ or -] 0.03 45 58.89 [+ or -] 0.0010 28.99 [+ or -] 0.02 50 44.44 [+ or -] 0.0010 20.62 [+ or -] 0.02 55 31.99 [+ or -] 0.0010 13.86 [+ or -] 0.02 60 22.27 [+ or -] 0.0010 9.13 [+ or -] 0.01 65 14.83 [+ or -] 0.0010 5.95 [+ or -] 0.01 70 10.15 [+ or -] 0.0010 3.67 [+ or -] 0.01 75 7.06 [+ or -] 0.0020 2.17 [+ or -] 0.01 80 5.53 [+ or -] 0.0050 1.20 [+ or -] 0.00 Centrality [sigma] [omega] (%) 5 25.07 [+ or -] 0.05 2.7198 [+ or -] 0.0009 10 21.63 [+ or -] 0.05 2.5650 [+ or -] 0.0009 15 19.34 [+ or -] 0.05 2.5898 [+ or -] 0.0010 20 17.26 [+ or -] 0.05 2.6214 [+ or -] 0.0012 25 15.19 [+ or -] 0.05 2.5727 [+ or -] 0.0013 30 13.85 [+ or -] 0.05 2.7352 [+ or -] 0.0016 35 12.18 [+ or -] 0.05 2.7734 [+ or -] 0.0019 40 10.76 [+ or -] 0.05 2.9171 [+ or -] 0.0022 45 9.22 [+ or -] 0.05 2.9297 [+ or -] 0.0027 50 7.84 [+ or -] 0.05 2.9803 [+ or -] 0.0032 55 6.46 [+ or -] 0.05 3.0073 [+ or -] 0.0039 60 5.17 [+ or -] 0.05 2.9287 [+ or -] 0.0049 65 4.14 [+ or -] 0.05 2.8798 [+ or -] 0.0059 70 3.21 [+ or -] 0.05 2.8056 [+ or -] 0.0071 75 2.42 [+ or -] 0.05 2.6991 [+ or -] 0.0089 80 1.77 [+ or -] 0.05 2.6062 [+ or -] 0.0111 Table 2: Values of the same variables, as in Table 1, but for [E.sub.lab] = 20A GeV/c. Centrality (%) <[N.sub.part]> <[N.sub.ch]> 5 348.00 [+ or -] 0.0020 288.81 [+ or -] 0.07 10 289.90 [+ or -] 0.0020 227.70 [+ or -] 0.06 15 238.45 [+ or -] 0.0020 179.76 [+ or -] 0.06 20 195.27 [+ or -] 0.0020 141.24 [+ or -] 0.05 25 159.21 [+ or -] 0.0020 111.38 [+ or -] 0.05 30 127.17 [+ or -] 0.0020 86.96 [+ or -] 0.04 35 100.08 [+ or -] 0.0020 66.34 [+ or -] 0.04 40 77.97 [+ or -] 0.0010 49.44 [+ or -] 0.03 45 58.89 [+ or -] 0.0010 36.03 [+ or -] 0.03 50 44.44 [+ or -] 0.0010 25.75 [+ or -] 0.02 55 31.99 [+ or -] 0.0010 17.36 [+ or -] 0.02 60 22.27 [+ or -] 0.0010 11.53 [+ or -] 0.02 65 14.83 [+ or -] 0.0010 7.55 [+ or -] 0.01 70 10.15 [+ or -] 0.0010 4.71 [+ or -] 0.01 75 7.06 [+ or -] 0.0020 2.79 [+ or -] 0.01 80 5.53 [+ or -] 0.0050 2.25 [+ or -] 0.21 Centrality (%) [sigma] [omega] 5 27.91 [+ or -] 0.05 2.6980 [+ or -] 0.0008 10 24.58 [+ or -] 0.05 2.6535 [+ or -] 0.0008 15 22.37 [+ or -] 0.04 2.7839 [+ or -] 0.0010 20 20.27 [+ or -] 0.04 2.9100 [+ or -] 0.0012 25 18.10 [+ or -] 0.03 2.9414 [+ or -] 0.0014 30 16.52 [+ or -] 0.03 3.1384 [+ or -] 0.0017 35 14.62 [+ or -] 0.03 3.2211 [+ or -] 0.0020 40 12.90 [+ or -] 0.02 3.3637 [+ or -] 0.0024 45 11.05 [+ or -] 0.02 3.3876 [+ or -] 0.0030 50 9.49 [+ or -] 0.02 3.4943 [+ or -] 0.0035 55 7.80 [+ or -] 0.01 3.5043 [+ or -] 0.0042 60 6.32 [+ or -] 0.01 3.4638 [+ or -] 0.0055 65 5.11 [+ or -] 0.01 3.4627 [+ or -] 0.0066 70 4.00 [+ or -] 0.01 3.4008 [+ or -] 0.0080 75 3.06 [+ or -] 0.01 3.3583 [+ or -] 0.0100 80 2.62 [+ or -] 0.15 3.0484 [+ or -] 0.3079 Table 3: Values of the same variables, as in Table 1, but for [E.sub.lab] = 30A GeV/c. Centrality (%) <[N.sub.part]> <[N.sub.ch]> 5 348.00 [+ or -] 0.0020 327.23 [+ or -] 0.09 10 289.90 [+ or -] 0.0020 257.60 [+ or -] 0.08 15 238.45 [+ or -] 0.0020 203.53 [+ or -] 0.07 20 195.27 [+ or -] 0.0020 159.94 [+ or -] 0.06 25 159.21 [+ or -] 0.0020 126.11 [+ or -] 0.06 30 127.17 [+ or -] 0.0020 98.62 [+ or -] 0.05 35 100.08 [+ or -] 0.0020 75.28 [+ or -] 0.04 40 77.97 [+ or -] 0.0010 56.15 [+ or -] 0.04 45 58.89 [+ or -] 0.0010 40.95 [+ or -] 0.03 50 44.44 [+ or -] 0.0010 29.36 [+ or -] 0.03 55 31.99 [+ or -] 0.0010 19.87 [+ or -] 0.02 60 22.27 [+ or -] 0.0010 13.20 [+ or -] 0.02 65 14.83 [+ or -] 0.0010 8.68 [+ or -] 0.02 70 10.15 [+ or -] 0.0010 5.44 [+ or -] 0.01 75 7.06 [+ or -] 0.0020 3.23 [+ or -] 0.01 80 5.53 [+ or -] 0.0050 3.01 [+ or -] 0.25 Centrality (%) [sigma] [omega] 5 31.63 [+ or -] 0.06 3.0566 [+ or -] 0.0009 10 27.73 [+ or -] 0.05 2.9857 [+ or -] 0.0010 15 25.22 [+ or -] 0.05 3.1245 [+ or -] 0.0011 20 22.70 [+ or -] 0.04 3.2205 [+ or -] 0.0013 25 20.26 [+ or -] 0.04 3.2551 [+ or -] 0.0016 30 18.56 [+ or -] 0.03 3.4930 [+ or -] 0.0019 35 16.47 [+ or -] 0.03 3.6032 [+ or -] 0.0023 40 14.48 [+ or -] 0.03 3.7316 [+ or -] 0.0027 45 12.40 [+ or -] 0.02 3.7530 [+ or -] 0.0033 50 10.63 [+ or -] 0.02 3.8484 [+ or -] 0.0040 55 8.83 [+ or -] 0.02 3.9211 [+ or -] 0.0048 60 7.19 [+ or -] 0.01 3.9103 [+ or -] 0.0063 65 5.81 [+ or -] 0.01 3.8935 [+ or -] 0.0076 70 4.60 [+ or -] 0.01 3.8949 [+ or -] 0.0094 75 3.51 [+ or -] 0.01 3.8103 [+ or -] 0.0116 80 3.13 [+ or -] 0.18 3.2661 [+ or -] 0.2933 Table 4: Values of the same variables, as in Table 1, but for [E.sub.lab] = 40A GeV/c. Centrality (%) <[N.sub.part]> <[N.sub.ch]> 5 348.00 [+ or -] 0.0020 353.86 [+ or -] 0.10 10 289.90 [+ or -] 0.0020 278.50 [+ or -] 0.08 15 238.45 [+ or -] 0.0020 219.93 [+ or -] 0.07 20 195.27 [+ or -] 0.0020 172.92 [+ or -] 0.07 25 159.21 [+ or -] 0.0020 136.43 [+ or -] 0.06 30 127.17 [+ or -] 0.0020 106.50 [+ or -] 0.05 35 100.08 [+ or -] 0.0020 81.49 [+ or -] 0.05 40 77.97 [+ or -] 0.0010 60.80 [+ or -] 0.04 45 58.89 [+ or -] 0.0010 44.31 [+ or -] 0.04 50 44.44 [+ or -] 0.0010 31.76 [+ or -] 0.03 55 31.99 [+ or -] 0.0010 21.58 [+ or -] 0.02 60 22.27 [+ or -] 0.0010 14.31 [+ or -] 0.02 65 14.83 [+ or -] 0.0010 9.43 [+ or -] 0.02 70 10.15 [+ or -] 0.0010 5.90 [+ or -] 0.01 75 7.06 [+ or -] 0.0020 3.50 [+ or -] 0.01 80 5.53 [+ or -] 0.0050 2.04 [+ or -] 0.03 Centrality (%) [sigma] [omega] 5 35.23 [+ or -] 0.07 3.5065 [+ or -] 0.0010 10 30.55 [+ or -] 0.06 3.3503 [+ or -] 0.0011 15 27.86 [+ or -] 0.05 3.5289 [+ or -] 0.0013 20 24.98 [+ or -] 0.05 3.6082 [+ or -] 0.0015 25 22.09 [+ or -] 0.04 3.5766 [+ or -] 0.0017 30 20.16 [+ or -] 0.04 3.8159 [+ or -] 0.0020 35 17.75 [+ or -] 0.03 3.8673 [+ or -] 0.0024 40 15.78 [+ or -] 0.03 4.0966 [+ or -] 0.0029 45 13.50 [+ or -] 0.03 4.1148 [+ or -] 0.0036 50 11.55 [+ or -] 0.02 4.1974 [+ or -] 0.0042 55 9.61 [+ or -] 0.02 4.2835 [+ or -] 0.0051 60 7.78 [+ or -] 0.01 4.2327 [+ or -] 0.0066 65 6.28 [+ or -] 0.01 4.1880 [+ or -] 0.0079 70 4.95 [+ or -] 0.01 4.1634 [+ or -] 0.0097 75 3.77 [+ or -] 0.01 4.0707 [+ or -] 0.0120 80 2.97 [+ or -] 0.02 4.3207 [+ or -] 0.0620 Table 5: Values of parameters, a, b, and c, occurring in equation (1) at different energies. [E.sub.lab] Fit Par. Centrality 10% a x [10.sup.-1] -17.425 [+ or -] 0.032 10A GeV b x [10.sup.-2] 49.595 [+ or -] 0.018 c x [10.sup.-4] 4.895 [+ or -] 0.008 a x [10.sup.-1] -15.317 [+ or -] 0.051 20A GeV b x [10.sup.-2] 60.103 [+ or -] 0.024 c x [10.sup.-4] 6.668 [+ or -] 0.010 a x [10.sup.-1] -16.633 [+ or -] 0.061 30A GeV b x [10.sup.-2] 68.297 [+ or -] 0.029 c x [10.sup.-4] 7.458 [+ or -] 0.012 a x [10.sup.-1] -18.499 [+ or -] 0.063 40A GeV b x [10.sup.-2] 74.059 [+ or -] 0.030 c x [10.sup.-4] 7.986 [+ or -] 0.013 [E.sub.lab] Fit Par. Centrality 5% a x [10.sup.-1] -19.211 [+ or -] 0.071 10A GeV b x [10.sup.-2] 49.581 [+ or -] 0.020 c x [10.sup.-4] 4.452 [+ or -] 0.007 a x [10.sup.-1] -18.215 [+ or -] 0.063 20A GeV b x [10.sup.-2] 60.250 [+ or -] 0.021 c x [10.sup.-4] 6.670 [+ or -] 0.007 a x [10.sup.-1] -18.670 [+ or -] 0.066 30A GeV b x [10.sup.-2] 68.483 [+ or -] 0.024 c x [10.sup.-4] 7.428 [+ or -] 0.008 a x [10.sup.-1] -19.733 [+ or -] 0.065 40A GeV b x [10.sup.-2] 73.900 [+ or -] 0.026 c x [10.sup.-4] 8.080 [+ or -] 0.010 [E.sub.lab] Fit Par. Centrality 2% a x [10.sup.-1] -13.718 [+ or -] 0.004 10A GeV b x [10.sup.-2] 43.791 [+ or -] 0.015 c x [10.sup.-4] 5.350 [+ or -] 0.005 a x [10.sup.-1] -17.031 [+ or -] 0.045 20A GeV b x [10.sup.-2] 53.923 [+ or -] 0.017 c x [10.sup.-4] 6.900 [+ or -] 0.006 a x [10.sup.-1] -17.852 [+ or -] 0.050 30A GeV b x [10.sup.-2] 60.889 [+ or -] 0.019 c x [10.sup.-4] 7.872 [+ or -] 0.007 a x [10.sup.-1] -18.947 [+ or -] 0.049 40A GeV b x [10.sup.-2] 65.885 [+ or -] 0.020 c x [10.sup.-4] 8.468 [+ or -] 0.007 Table 6: Values of parameters, p and q, occurring in equation (2) at different energies. [E.sub.lab] Fit Par. Centrality 10% 10A GeV p x [10.sup.-1] -21.171 [+ or -] 0.061 q x [10.sup.-1] 16.436 [+ or -] 0.017 20A GeV p x [10.sup.-1] -19.790 [+ or -] 0.082 q x [10.sup.-1] -19.050 [+ or -] 0.019 30A GeV p x [10.sup.-1] -21.280 [+ or -] 0.097 q x [10.sup.-1] 21.334 [+ or -] 0.023 40A GeV p x [10.sup.-1] -22.988 [+ or -] 0.101 q x [10.sup.-1] 23.292 [+ or -] 0.026 [E.sub.lab] Fit Par. Centrality 5% 10A GeV p x [10.sup.-1] -10.181 [+ or -] 0.087 q x [10.sup.-1] 13.244 [+ or -] 0.001 20A GeV p x [10.sup.-1] -8.694 [+ or -] 0.008 q x [10.sup.-1] 15.308 [+ or -] 0.001 30A GeV p x [10.sup.-1] -8.795 [+ or -] 0.008 q x [10.sup.-1] 17.131 [+ or -] 0.002 40A GeV p x [10.sup.-1] -11.275 [+ or -] 0.008 q x [10.sup.-1] 18.901 [+ or -] 0.002 [E.sub.lab] Fit Par. Centrality 2% 10A GeV p x [10.sup.-1] 8.087 [+ or -] 0.055 q x [10.sup.-1] 11.704 [+ or -] 0.011 20A GeV p x [10.sup.-1] -6.706 [+ or -] 0.067 q x [10.sup.-1] 13.634 [+ or -] 0.013 30A GeV p x [10.sup.-1] -6.936 [+ or -] 0.073 q x [10.sup.-1] 15.295 [+ or -] 0.015 40A GeV p x [10.sup.-1] -8.658 [+ or -] 0.075 q x [10.sup.-1] 16.814 [+ or -] 0.016

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Title Annotation: | Research Article |
---|---|

Author: | Chandra, Anuj; Ali, Bushra; Ahmad, Shakeel |

Publication: | Advances in High Energy Physics |

Date: | Dec 1, 2019 |

Words: | 5866 |

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