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

1. 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 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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