# Charged-Particle Multiplicity Moments as Described by Shifted Gompertz Distribution ine+e−, pp¯, and pp Collisions at High Energies

Authors: Aayushi Singla; M. Kaur
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

In continuation of our earlier work, in which we analysed the charged particle multiplicities in leptonic and hadronic interactions at different center-of-mass energies in full phase space as well as in restricted phase space using the shifted Gompertz distribution, a detailed analysis of the normalized moments and normalized factorial moments is reported here. A two-component model in which a probability distribution function is obtained from the superposition of two shifted Gompertz distributions, as introduced in our earlier work, has also been used for the analysis. This is the first analysis of the moments with the shifted Gompertz distribution. Analysis has also been performed to predict the moments of multiplicity distribution for thee+e− collisions at s=500GeV at a future collider.


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## Body

## 1. Introduction

In one of our recent papers, we introduced a statistical distribution, the shifted Gompertz distribution to investigate the multiplicity distributions of charged particles produced ine+e− collisions at the LEP, pp¯ interactions at the SPS and pp collisions at the LHC at different center of mass energies in full phase space as well as in restricted phase space [1]. A distribution of the larger of two independent random variables, the shifted Gompertz distribution was introduced by Bemmaor [2] as a model of adoption of innovations. One of the parameters has an exponential distribution and the other has a Gumbel distribution, also known as log-Weibull distribution. The nonnegative fit parameters define the scale and shape of the distribution. Subsequently, the shifted Gompertz distribution has been widely studied in various contexts [3–5]. In our earlier work [1] by studying the charged particle multiplicities, we showed that this distribution can be successfully used to study the statistical phenomena in high energy e+e−, pp¯, and pp collisions at the LEP, SPS, and LHC colliders, respectively.A multiplicity distribution is represented by the probabilities ofn-particle events as well as by its moments or its generating function. The aim of the present work is to extend the analysis by calculating the higher moments of a multiplicity distribution. Because the moments are calculated as derivatives of the generating function, the moment analysis is a powerful tool which helps to unfold the characteristics of multiplicity distribution. The multiparticle correlations can be studied through the normalized moments and normalized factorial moments of the distribution. The dependence of moments on energy can also reveal the KNO (Koba, Nielsen, and Olesen) scaling [6–8] conservation or violation. Several analyses of moments have been done at different energies, using different probability distribution functions and different types of particles [9–12]. The higher moments also can identify the correlations amongst produced particles.InSection 2, formulae for the probability distribution function (PDF) of the shifted Gompertz distribution, normalized moments, and the normalized factorial moments used for the analysis are given. A two-component model has been used and modification of distributions carried out, in terms of these two components; one from soft events and another from semi-hard events. Superposition of distributions from these two components, using appropriate weights is done to build the full multiplicity distribution. When multiplicity distribution is fitted with the weighted superposition of two shifted Gompertz distributions, we find that the agreement between data and the model improves considerably. The details of these fits are published in [1]. The distributions have been fitted both in full phase space as well as in restricted rapidity windows for pp¯ and pp data and in five rapidity windows and in full phase space only for e+e−, in terms of soft and semi-hard components.Section3 presents the evaluations of moments from experimental data, the fitted shifted Gompertz distributions and the fitted modified shifted Gompertz distributions. Section 4 details the method of estimating uncertainties on the moments. Discussion and conclusion are presented in Section 5.


## 2. Shifted Gompertz Distribution and Moments

The particle production dynamics can be understood by analysing the charged particle multiplicity distribution as its measurements can provide relevant constraints for particle-production models. Charged particle multiplicity is defined as the average number of charged particlesn, produced in a collision at a given energy in the center of mass system.In addition, the analysis of moments of the distribution is often used to study the patterns and correlations in the multi-particle final state of high-energy collisions in the presence of statistical fluctuations. The fractal structures present in the multiplicity distributions have often been studied to search for the embedded constraints on the underlying particle production mechanism [13, 14]. The observation of fractal structures is of great interest because it imposes strong constraints on the underlying particle-production mechanism. We define different kinds of moments as follows.LetX be any nonnegative random variable having the shifted Gompertz distribution with parameters b and β, where b>0 is a scale parameter and β>0 is a shape parameter. The probability distribution function (PDF) of X is given by(1)PXx;b,β=be−bx+βe−bx1+β1−e−bx,wherex>0.The raw moments (cn) and factorial moments (fn) are defined as:(2)cn=EXnandfn=EXX−1X−2….X−n−1,whereas, the normalized moments (Cn) and normalized factorial moments (Fn) are defined as following:(3)Cn=EXnEXnandFn=EXX−1X−2….X−n−1EXn,n as a natural number ranging from 1 to ∞. The mean value (E[X]) of shifted Gompertz distribution is given by(4)EX=1bγ+lnβ+1−e−ββ+Γ0,β.and thec2 moment is given by(5)c2=EX2=2b2βγ+Γ0,β+β23F31,1,1,2,2,2,−β+lnβ.The higher order raw moments (cn) can be found by the moment generating function of the shifted Gompertz distribution [[b, β], t](6)e−β−1+tbββt/bΓ1−tb−Γ1−tb,β.Alsof2 moment is given by(7)f2=EXX−1=2b2βγ+Γ0,β+β23F31,1,1,2,2,2,−β+lnβ−1−e−β+βγ+Γ0,β+lnβbβ.The higher factorial moments (fn) can be found by the generating function of shifted Gompertz distribution [[b,β],t](8)e−β−βlnt/bΓ1−lntb−Γ1−lntb,β1+lntbβ,where(i)
γ≈0.5772156 stands for the Euler constant (also referred to as Euler–Mascheroni constant).(ii)
Γs the Euler Gamma function and Γs,x the incomplete Gamma function defined below;(9)
Γ
s
=
∫
0
∞
t
s
−
1
e
−
t
d
t
Γ
s
,
x
=
∫
x
∞
t
s
−
1
e
−
t
d
t
,(iii)
3
F
3 is a generalized hypergeometric function.(10)
3
F
3
1
,
1
,
1
,
2
,
2
,
2
,
−
β
=
∑
k
=
1
∞
−
1
k
+
1
β
k
+
1
k
!
k
2
.


### 2.1. Modified Shifted Gompertz Distribution

It is well established that at high energies, charged particle multiplicity distribution in full phase space becomes broader than a Poisson distribution. The most widely adopted, negative binomial distribution [15] to describe the multiplicity spectra, fails to explain the experimental data. As a corrective measure to explain the failure, a two-component approach, was introduced by Giovannini et al. [15]. The details are included in our earlier publication [1] on the shifted Gompertz distribution.To better explain the data at high energies, a superposition of two shifted Gompertz distributions, which are interpreted as soft and hard components, is used. The multiplicity distribution is produced by adding weighted superposition of multiplicity in soft events and multiplicity distribution in semi-hard events. This approach combines merely two classes of events and not two different particle–production mechanisms. Therefore, no interference terms need to be introduced. The final distribution is the superposition of the two independent distributions. We call it “modified shifted Gompertz distribution”.(11)Pn=αPsoftshGompn+1−αPsemi-hardshGompn.Adopting this approach for the multiplicity distributions ine+e−, pp, and pp¯ collisions at high energies, the data at different energies are fitted with the distribution which involves five parameters as given below;(12)Pnα:b1,β1;b2,β2=αPnsoft+1−αPnsemi-hard,whereα is the fraction of soft events, (b1, β1) and (b2, β2) are respectively the scale and shape parameters of the two distributions.Values ofα evaluated for different interactions, were published in the Tables in our previous paper [1]. In Figure 1, we show the variation of α as a function of collision energy and pseudorapidity for pp interactions at LHC energies. The values show that the alpha decreases with collision energy as well as with increasing pseudo-rapidity window.Figure 1
Dependence ofα on c.m. energy and pseudprapidity for pp interaction.


## 2.1. Modified Shifted Gompertz Distribution

It is well established that at high energies, charged particle multiplicity distribution in full phase space becomes broader than a Poisson distribution. The most widely adopted, negative binomial distribution [15] to describe the multiplicity spectra, fails to explain the experimental data. As a corrective measure to explain the failure, a two-component approach, was introduced by Giovannini et al. [15]. The details are included in our earlier publication [1] on the shifted Gompertz distribution.To better explain the data at high energies, a superposition of two shifted Gompertz distributions, which are interpreted as soft and hard components, is used. The multiplicity distribution is produced by adding weighted superposition of multiplicity in soft events and multiplicity distribution in semi-hard events. This approach combines merely two classes of events and not two different particle–production mechanisms. Therefore, no interference terms need to be introduced. The final distribution is the superposition of the two independent distributions. We call it “modified shifted Gompertz distribution”.(11)Pn=αPsoftshGompn+1−αPsemi-hardshGompn.Adopting this approach for the multiplicity distributions ine+e−, pp, and pp¯ collisions at high energies, the data at different energies are fitted with the distribution which involves five parameters as given below;(12)Pnα:b1,β1;b2,β2=αPnsoft+1−αPnsemi-hard,whereα is the fraction of soft events, (b1, β1) and (b2, β2) are respectively the scale and shape parameters of the two distributions.Values ofα evaluated for different interactions, were published in the Tables in our previous paper [1]. In Figure 1, we show the variation of α as a function of collision energy and pseudorapidity for pp interactions at LHC energies. The values show that the alpha decreases with collision energy as well as with increasing pseudo-rapidity window.Figure 1
Dependence ofα on c.m. energy and pseudprapidity for pp interaction.


## 3. Analysis and Results

Calculations of the normalized moments and the normalized factorial moments are presented by using the data from different experiments and following three collision types;(i)
e+e− annihilations at different collision energies, from 91 GeV up to the highest energy of 206 GeV at LEP2, from two experiments L3 [16] and OPAL [17–20] are analysed.(ii)
pp collisions at LHC energies from 900 GeV, 2360 GeV, and 7000 GeV [21] are analysed for five intervals of increasing extent in pseudorapidity η<0.5 up to η<2.4.(iii)
pp¯ collisions at energies from 200 GeV, 540 GeV, and 900 GeV [7, 8] are analysed in full phase space as well as in pseudorapidity intervals from η<0.5 up to η<5.0, where η is defined as −lntanθ/2, and θ is the polar angle of the particle with respect to the counter-clockwise beam direction.The PDF defined by Equation (1) is used to fit the experimental data on charged particle multiplicity distributions, for the shifted Gompertz function and the modified (two-component) function. Results from these fits to the above mentioned data were published in our earlier work [1]. As an example, results for L3 data are shown in Figure 2. To avoid repetition, the details of other Figures are not given here. It was shown that the data are very well explained by the modified shifted Gompertz distribution and the χ2 values for the fits in almost all cases reduce substantially. In the present analysis we calculate the normalized moments and the normalized factorial moments defined in Equations (2)–(8).Figure 2
Charged multiplicity distributions from L3 experiment. Solid lines represent the shifted Gompertz distribution and the modified shifted Gompertz distributions.Figures3–6 show the normalized moments (Cq) and the normalized factorial moments (Fq) calculated from the data of two experiments at LEP, at different energies for e+e− collisions and also from the fitted shifted Gompertz and modified shifted Gompertz distributions. The values of the moments are documented in Table 1.Figure 3
Normalized moments of shifted Gompertz distribution for thee+e− collisions recorded by the L3 and OPAL experiments at different energies.Figure 4
Normalized factorial moments of shifted Gompertz distribution for thee+e− collisions recorded by the L3 and OPAL experiments at different energies.Figure 5
Normalized moments of modified shifted Gompertz distribution for thee+e− collisions recorded by the L3 and OPAL experiments at different energies.Figure 6
Normalized factorial moments of modified shifted Gompertz distribution for thee+e− collisions recorded by the L3 and OPAL experiments at different energies.Table 1
Moments of experimental, shifted Gompertz and modified shifted Gompertz distributions fore+e− collisions.
Energy (GeV) (e+e−)
〈
n
〉
Normalized moments (experiment)
Normalized factorial moments (experiment)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
L3
91
20.464 ± 0.076
1.093 ± 0.004
1.296 ± 0.008
1.656 ± 0.016
2.268 ± 0.030
1.044 ± 0.003
1.141 ± 0.007
1.304 ± 0.013
1.560 ± 0.021
131
23.301 ± 1.002
1.080 ± 0.044
1.252 ± 0.104
1.546 ± 0.195
2.016 ± 0.343
1.038 ± 0.043
1.117 ± 0.093
1.245 ± 0.157
1.430 ± 0.245
136
24.012 ± 1.116
1.094 ± 0.051
1.295 ± 0.124
1.638 ± 0.237
2.194 ± 0.428
1.052 ± 0.050
1.162 ± 0.111
1.335 ± 0.195
1.586 ± 0.312
172
27.031 ± 1.821
1.095 ± 0.059
1.299 ± 0.142
1.657 ± 0.275
2.245 ± 0.504
1.058 ± 0.057
1.181 ± 0.128
1.384 ± 0.228
1.692 ± 0.378
182
26.795 ± 0.863
1.091 ± 0.030
1.287 ± 0.070
1.626 ± 0.135
2.180 ± 0.245
1.054 ± 0.028
1.168 ± 0.063
1.355 ± 0.112
1.633 ± 0.184
188
26.653 ± 0.609
1.084 ± 0.020
1.260 ± 0.045
1.557 ± 0.085
2.026 ± 0.149
1.046 ± 0.019
1.141 ± 0.041
1.290 ± 0.070
1.502 ± 0.110
194
27.178 ± 0.847
1.093 ± 0.027
1.295 ± 0.064
1.644 ± 0.123
2.216 ± 0.224
1.057 ± 0.026
1.177 ± 0.058
1.375 ± 0.102
1.670 ± 0.168
200
27.575 ± 0.852
1.093 ± 0.027
1.294 ± 0.064
1.644 ± 0.124
2.215 ± 0.227
1.057 ± 0.026
1.178 ± 0.058
1.378 ± 0.103
1.676 ± 0.171
206
27.917 ± 0.720
1.092 ± 0.021
1.290 ± 0.051
1.634 ± 0.098
2.195 ± 0.178
1.056 ± 0.021
1.176 ± 0.046
1.372 ± 0.082
1.665 ± 0.135
OPAL
91
21.394 ± 0.219
1.092 ± 0.011
1.293 ± 0.027
1.648 ± 0.051
2.249 ± 0.095
1.045 ± 0.011
1.144 ± 0.024
1.311 ± 0.041
1.572 ± 0.067
161
24.463 ± 0.859
1.095 ± 0.039
1.304 ± 0.093
1.673 ± 0.180
2.295 ± 0.333
1.054 ± 0.038
1.173 ± 0.084
1.373 ± 0.149
1.684 ± 0.247
183
26.856 ± 0.754
1.097 ± 0.028
1.312 ± 0.068
1.696 ± 0.133
2.348 ± 0.252
1.060 ± 0.027
1.192 ± 0.062
1.419 ± 0.112
1.777 ± 0.195
189
26.886 ± 0.505
1.096 ± 0.018
1.304 ± 0.043
1.669 ± 0.083
2.280 ± 0.153
1.059 ± 0.018
1.184 ± 0.039
1.395 ± 0.070
1.719 ± 0.117
Energy (GeV) (e+e−)
〈
n
〉
Normalized moments (shifted Gompertz)
Normalized factorial moments (shifted Gompertz)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
L3
91
20.467 ± 0.029
1.084 ± 0.002
1.278 ± 0.004
1.643 ± 0.008
2.307 ± 0.015
1.035 ± 0.002
1.124 ± 0.004
1.296 ± 0.006
1.604 ± 0.010
131
23.101 ± 0.325
1.084 ± 0.017
1.268 ± 0.039
1.591 ± 0.074
2.126 ± 0.131
1.041 ± 0.016
1.131 ± 0.035
1.284 ± 0.060
1.515 ± 0.094
136
23.530 ± 0.348
1.096 ± 0.018
1.308 ± 0.043
1.684 ± 0.083
2.317 ± 0.152
1.054 ± 0.017
1.172 ± 0.039
1.372 ± 0.068
1.680 ± 0.111
172
26.706 ± 0.359
1.097 ± 0.016
1.309 ± 0.038
1.686 ± 0.073
2.317 ± 0.134
1.059 ± 0.015
1.189 ± 0.035
1.408 ± 0.062
1.747 ± 0.102
182
26.779 ± 0.259
1.096 ± 0.010
1.307 ± 0.025
1.679 ± 0.048
2.303 ± 0.089
1.059 ± 0.010
1.187 ± 0.023
1.403 ± 0.040
1.737 ± 0.067
188
26.624 ± 0.172
1.088 ± 0.007
1.276 ± 0.016
1.599 ± 0.030
2.121 ± 0.053
1.050 ± 0.006
1.156 ± 0.014
1.328 ± 0.025
1.581 ± 0.040
194
27.007 ± 0.232
1.095 ± 0.009
1.305 ± 0.022
1.674 ± 0.043
2.288 ± 0.078
1.058 ± 0.009
1.186 ± 0.020
1.400 ± 0.036
1.729 ± 0.060
200
27.599 ± 0.251
1.095 ± 0.009
1.300 ± 0.022
1.660 ± 0.043
2.253 ± 0.078
1.058 ± 0.009
1.184 ± 0.020
1.392 ± 0.036
1.709 ± 0.060
206
27.940 ± 0.209
1.093 ± 0.007
1.295 ± 0.018
1.646 ± 0.034
2.222 ± 0.062
1.057 ± 0.007
1.180 ± 0.016
1.383 ± 0.029
1.688 ± 0.047
OPAL
91
21.547 ± 0.086
1.093 ± 0.005
1.306 ± 0.012
1.706 ± 0.023
2.425 ± 0.043
1.046 ± 0.005
1.159 ± 0.010
1.368 ± 0.018
1.726 ± 0.031
161
24.322 ± 0.325
1.099 ± 0.017
1.325 ± 0.040
1.742 ± 0.078
2.479 ± 0.148
1.058 ± 0.016
1.193 ± 0.036
1.436 ± 0.065
1.838 ± 0.110
183
26.980 ± 0.244
1.102 ± 0.011
1.333 ± 0.026
1.761 ± 0.051
2.517 ± 0.097
1.065 ± 0.010
1.213 ± 0.023
1.481 ± 0.043
1.926 ± 0.075
189
27.176 ± 0.164
1.103 ± 0.007
1.334 ± 0.017
1.750 ± 0.033
2.466 ± 0.062
1.066 ± 0.007
1.214 ± 0.016
1.471 ± 0.028
1.883 ± 0.048
Energy (GeV) (e+e−)
〈
n
〉
Normalized moments (modified shifted Gompertz)
Normalized factorial moments (modified shifted Gompertz)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
L3
91
20.209 ± 0.031
1.092 ± 0.002
1.297 ± 0.005
1.670 ± 0.009
2.329 ± 0.016
1.043 ± 0.002
1.139 ± 0.004
1.313 ± 0.007
1.607 ± 0.011
131
23.293 ± 0.427
1.081 ± 0.022
1.255 ± 0.050
1.555 ± 0.093
2.042 ± 0.164
1.038 ± 0.021
1.120 ± 0.045
1.253 ± 0.076
1.451 ± 0.117
136
23.867 ± 0.499
1.092 ± 0.025
1.291 ± 0.059
1.635 ± 0.113
2.198 ± 0.203
1.050 ± 0.024
1.157 ± 0.054
1.331 ± 0.093
1.589 ± 0.148
172
26.852 ± 0.458
1.092 ± 0.020
1.293 ± 0.048
1.645 ± 0.091
2.229 ± 0.165
1.055 ± 0.019
1.174 ± 0.044
1.372 ± 0.077
1.677 ± 0.125
182
26.771 ± 0.317
1.093 ± 0.013
1.295 ± 0.030
1.650 ± 0.058
2.239 ± 0.106
1.055 ± 0.012
1.175 ± 0.028
1.376 ± 0.049
1.683 ± 0.081
188
26.509 ± 0.212
1.084 ± 0.008
1.264 ± 0.020
1.570 ± 0.036
2.061 ± 0.064
1.047 ± 0.008
1.144 ± 0.018
1.301 ± 0.030
1.528 ± 0.048
194
26.943 ± 0.316
1.092 ± 0.013
1.291 ± 0.030
1.638 ± 0.057
2.210 ± 0.104
1.055 ± 0.012
1.172 ± 0.027
1.367 ± 0.048
1.662 ± 0.079
200
27.548 ± 0.306
1.093 ± 0.011
1.296 ± 0.026
1.651 ± 0.050
2.235 ± 0.092
1.057 ± 0.011
1.180 ± 0.024
1.384 ± 0.042
1.693 ± 0.070
206
27.928 ± 0.259
1.093 ± 0.009
1.295 ± 0.021
1.646 ± 0.040
2.223 ± 0.072
1.057 ± 0.008
1.180 ± 0.019
1.383 ± 0.033
1.689 ± 0.055
OPAL
91
21.461 ± 0.107
1.092 ± 0.006
1.297 ± 0.015
1.670 ± 0.028
2.322 ± 0.051
1.045 ± 0.006
1.149 ± 0.013
1.333 ± 0.022
1.637 ± 0.037
161
24.378 ± 0.396
1.096 ± 0.020
1.310 ± 0.048
1.696 ± 0.093
2.361 ± 0.171
1.055 ± 0.020
1.179 ± 0.044
1.393 ± 0.077
1.739 ± 0.127
183
26.957 ± 0.272
1.101 ± 0.012
1.329 ± 0.028
1.750 ± 0.056
2.492 ± 0.107
1.064 ± 0.011
1.209 ± 0.026
1.470 ± 0.047
1.904 ± 0.082
189
27.007 ± 0.202
1.100 ± 0.009
1.319 ± 0.021
1.710 ± 0.040
2.377 ± 0.075
1.063 ± 0.009
1.199 ± 0.019
1.434 ± 0.034
1.804 ± 0.057Figures7–10 show the normalized moments (Cq) and the normalized factorial moments (Fq) calculated from the data and from the modified shifted Gompertz distribution for the pp¯ data at energies from 200, 540, and 900 GeV, in four rapidity bins and full phase space. The values of the moments are documented in Table 2.Figure 7
Normalized momentsCn of modified shifted Gompertz distribution for the pp¯ collisions recorded by the UA5 experiment at different energies, in full phase space and pseudo-rapidity bin, η<0.5.Figure 8
Normalized momentsCn of modified shifted Gompertz distribution for the pp¯ collisions recorded by the UA5 experiment at different energies and in three pseudo-rapidity bins, η<1.5, 3.0 and 5.0.Figure 9
Normalized factorial momentsFn of modified shifted Gompertz distribution for the pp¯ collisions recorded by the UA5 experiment at different energies, in full phase space and pseudo-rapidity bin, η<0.5.Figure 10
Normalized factorial momentsFn of modified shifted Gompertz distribution for the pp¯ collisions recorded by the UA5 experiment at different energies and in three pseudo-rapidity bins, η<1.5, 3.0, and 5.0.Table 2
Moments of experimental, shifted Gompertz and modified shifted Gompertz distributions forpp¯ collisions.
Energy (GeV) (pp¯)
|
η
|
〈
n
〉
Normalized moments (experiment)
Normalized factorial moments (experiment)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
200
0.5
2.501 ± 0.074
1.911 ± 0.053
4.895 ± 0.322
15.711 ± 1.876
60.718 ± 11.403
1.511 ± 0.046
2.923 ± 0.238
6.945 ± 1.147
19.796 ± 5.362
1.5
7.940 ± 0.206
1.556 ± 0.037
3.159 ± 0.167
7.792 ± 0.695
22.349 ± 3.004
1.430 ± 0.035
2.603 ± 0.144
5.664 ± 0.548
14.140 ± 2.136
3
15.540 ± 0.294
1.368 ± 0.024
2.329 ± 0.082
4.630 ± 0.249
10.275 ± 0.754
1.304 ± 0.024
2.073 ± 0.075
3.791 ± 0.209
7.615 ± 0.577
5
20.285 ± 0.479
1.278 ± 0.028
1.948 ± 0.087
3.379 ± 0.233
6.442 ± 0.605
1.229 ± 0.027
1.764 ± 0.080
2.836 ± 0.198
4.935 ± 0.470
Full
21.175 ± 0.500
1.244 ± 0.028
1.823 ± 0.082
3.023 ± 0.207
5.496 ± 0.510
1.197 ± 0.027
1.651 ± 0.075
2.536 ± 0.176
4.204 ± 0.395
540
0.5
3.004 ± 0.047
1.915 ± 0.029
5.037 ± 0.175
16.684 ± 1.004
66.035 ± 6.085
1.582 ± 0.026
3.347 ± 0.134
8.737 ± 0.654
26.799 ± 3.240
1.5
9.471 ± 0.137
1.610 ± 0.022
3.480 ± 0.107
9.345 ± 0.482
29.750 ± 2.289
1.504 ± 0.021
2.992 ± 0.096
7.331 ± 0.403
21.149 ± 1.773
3
18.968 ± 0.252
1.474 ± 0.017
2.857 ± 0.073
6.815 ± 0.284
19.083 ± 1.127
1.422 ± 0.017
2.630 ± 0.068
5.955 ± 0.254
15.758 ± 0.956
5
26.330 ± 0.342
1.353 ± 0.016
2.271 ± 0.056
4.480 ± 0.176
10.016 ± 0.556
1.315 ± 0.015
2.120 ± 0.053
3.984 ± 0.159
8.425 ± 0.479
Full
28.375 ± 0.374
1.305 ± 0.016
2.079 ± 0.054
3.868 ± 0.159
8.128 ± 0.465
1.269 ± 0.016
1.943 ± 0.051
3.446 ± 0.143
6.853 ± 0.400
900
0.5
3.614 ± 0.087
1.940 ± 0.046
5.410 ± 0.318
19.789 ± 2.173
89.311 ± 15.604
1.664 ± 0.042
3.952 ± 0.269
12.314 ± 1.648
47.134 ± 10.245
1.5
11.503 ± 0.247
1.582 ± 0.031
3.368 ± 0.144
8.736 ± 0.614
26.012 ± 2.634
1.496 ± 0.029
2.970 ± 0.131
7.107 ± 0.522
19.258 ± 2.066
3
21.925 ± 0.428
1.456 ± 0.027
2.695 ± 0.103
5.828 ± 0.341
13.956 ± 1.108
1.410 ± 0.026
2.500 ± 0.096
5.124 ± 0.303
11.487 ± 0.920
5
32.263 ± 0.642
1.354 ± 0.025
2.278 ± 0.088
4.464 ± 0.266
9.736 ± 0.797
1.323 ± 0.025
2.154 ± 0.083
4.054 ± 0.244
8.427 ± 0.698
Full
35.632 ± 0.686
1.284 ± 0.023
2.003 ± 0.073
3.606 ± 0.203
7.185 ± 0.554
1.256 ± 0.022
1.897 ± 0.069
3.279 ± 0.186
6.227 ± 0.485
Energy (GeV) (pp¯)
|
η
|
〈
n
〉
Normalized moments (shifted Gompertz)
Normalized factorial moments (shifted Gompertz)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
0.5
2.475 ± 0.029
1.912 ± 0.024
4.864 ± 0.122
15.015 ± 0.573
53.249 ± 2.762
1.508 ± 0.020
2.873 ± 0.078
6.261 ± 0.264
14.708 ± 0.873
1.5
7.917 ± 0.055
1.536 ± 0.011
3.018 ± 0.045
7.021 ± 0.158
18.455 ± 0.560
1.410 ± 0.011
2.468 ± 0.037
4.992 ± 0.115
11.123 ± 0.349
200
3
15.173 ± 0.061
1.381 ± 0.007
2.348 ± 0.022
4.626 ± 0.063
10.137 ± 0.181
1.315 ± 0.007
2.083 ± 0.020
3.762 ± 0.052
7.425 ± 0.134
5
20.152 ± 0.094
1.297 ± 0.006
2.014 ± 0.020
3.560 ± 0.053
6.915 ± 0.137
1.247 ± 0.006
1.826 ± 0.018
2.995 ± 0.045
5.314 ± 0.106
Full
21.064 ± 0.152
1.245 ± 0.010
1.827 ± 0.028
3.039 ± 0.070
5.560 ± 0.171
1.198 ± 0.009
1.654 ± 0.026
2.549 ± 0.059
4.254 ± 0.132
0.5
2.976 ± 0.016
1.917 ± 0.012
5.082 ± 0.064
17.035 ± 0.324
67.968 ± 1.738
1.581 ± 0.010
3.376 ± 0.045
8.940 ± 0.182
27.480 ± 0.756
1.5
9.428 ± 0.044
1.599 ± 0.008
3.421 ± 0.035
9.095 ± 0.143
28.618 ± 0.611
1.493 ± 0.008
2.935 ± 0.031
7.109 ± 0.115
20.226 ± 0.444
540
3
18.774 ± 0.076
1.448 ± 0.006
2.705 ± 0.023
6.155 ± 0.081
16.359 ± 0.293
1.394 ± 0.006
2.479 ± 0.021
5.335 ± 0.071
13.338 ± 0.243
5
26.154 ± 0.093
1.345 ± 0.005
2.258 ± 0.017
4.523 ± 0.053
10.440 ± 0.166
1.307 ± 0.005
2.107 ± 0.016
4.026 ± 0.048
8.822 ± 0.141
Full
28.112 ± 0.084
1.307 ± 0.004
2.099 ± 0.014
3.982 ± 0.040
8.625 ± 0.115
1.271 ± 0.004
1.962 ± 0.013
3.551 ± 0.036
7.299 ± 0.099
0.5
3.555 ± 0.028
1.894 ± 0.016
4.905 ± 0.087
15.756 ± 0.424
59.066 ± 2.159
1.613 ± 0.015
3.465 ± 0.065
8.991 ± 0.259
26.371 ± 1.043
1.5
11.352 ± 0.054
1.557 ± 0.008
3.187 ± 0.032
7.850 ± 0.121
22.057 ± 0.460
1.469 ± 0.007
2.791 ± 0.029
6.294 ± 0.099
15.956 ± 0.342
900
3
21.835 ± 0.075
1.441 ± 0.005
2.604 ± 0.019
5.486 ± 0.060
12.867 ± 0.189
1.395 ± 0.005
2.410 ± 0.018
4.803 ± 0.053
10.539 ± 0.156
5
32.073 ± 0.101
1.324 ± 0.004
2.134 ± 0.014
3.980 ± 0.040
8.275 ± 0.111
1.293 ± 0.004
2.013 ± 0.013
3.595 ± 0.036
7.105 ± 0.096
Full
35.567 ± 0.157
1.268 ± 0.006
1.935 ± 0.018
3.406 ± 0.048
6.680 ± 0.125
1.240 ± 0.006
1.830 ± 0.017
3.091 ± 0.043
5.774 ± 0.109
Energy (GeV) (pp¯)
|
η
|
〈
n
〉
Normalized moments (modified shifted Gompertz)
Normalized factorial moments (modified shifted Gompertz)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
200
0.5
2.477 ± 0.041
1.881 ± 0.033
4.604 ± 0.166
13.493 ± 0.742
45.166 ± 3.376
1.478 ± 0.028
2.651 ± 0.104
5.320 ± 0.326
11.404 ± 0.993
1.5
7.895 ± 0.071
1.547 ± 0.016
3.085 ± 0.063
7.340 ± 0.226
19.818 ± 0.819
1.420 ± 0.015
2.529 ± 0.053
5.257 ± 0.167
12.101 ± 0.520
3
15.537 ± 0.107
1.367 ± 0.010
2.321 ± 0.034
4.593 ± 0.101
10.130 ± 0.300
1.302 ± 0.009
2.065 ± 0.030
3.758 ± 0.084
7.493 ± 0.226
5
20.484 ± 0.134
1.270 ± 0.009
1.923 ± 0.026
3.317 ± 0.068
6.293 ± 0.172
1.221 ± 0.008
1.742 ± 0.024
2.786 ± 0.057
4.827 ± 0.134
Full
21.124 ± 0.185
1.245 ± 0.012
1.830 ± 0.036
3.054 ± 0.089
5.610 ± 0.216
1.197 ± 0.012
1.657 ± 0.033
2.565 ± 0.075
4.301 ± 0.167
540
0.5
2.981 ± 0.022
1.901 ± 0.016
4.929 ± 0.085
15.986 ± 0.419
61.449 ± 2.190
1.566 ± 0.014
3.241 ± 0.059
8.192 ± 0.234
23.951 ± 0.945
1.5
9.463 ± 0.055
1.606 ± 0.011
3.468 ± 0.047
9.313 ± 0.192
29.587 ± 0.828
1.501 ± 0.010
2.981 ± 0.041
7.305 ± 0.154
21.009 ± 0.607
3
18.863 ± 0.092
1.470 ± 0.008
2.836 ± 0.031
6.714 ± 0.112
18.569 ± 0.421
1.417 ± 0.007
2.608 ± 0.028
5.857 ± 0.099
15.278 ± 0.353
5
26.179 ± 0.113
1.352 ± 0.006
2.282 ± 0.022
4.582 ± 0.067
10.568 ± 0.207
1.314 ± 0.006
2.130 ± 0.021
4.080 ± 0.060
8.931 ± 0.177
Full
28.144 ± 0.100
1.306 ± 0.005
2.096 ± 0.017
3.964 ± 0.048
8.553 ± 0.138
1.271 ± 0.005
1.959 ± 0.016
3.535 ± 0.043
7.234 ± 0.118
900
0.5
3.578 ± 0.038
1.907 ± 0.022
5.041 ± 0.120
16.619 ± 0.631
63.995 ± 3.464
1.627 ± 0.019
3.599 ± 0.093
9.671 ± 0.412
29.394 ± 1.868
1.5
11.428 ± 0.071
1.575 ± 0.010
3.303 ± 0.044
8.345 ± 0.172
23.917 ± 0.675
1.487 ± 0.010
2.905 ± 0.039
6.740 ± 0.142
17.448 ± 0.511
3
22.026 ± 0.098
1.449 ± 0.007
2.672 ± 0.026
5.755 ± 0.084
13.741 ± 0.266
1.404 ± 0.007
2.479 ± 0.024
5.060 ± 0.074
11.315 ± 0.221
5
32.341 ± 0.129
1.350 ± 0.006
2.259 ± 0.019
4.393 ± 0.056
9.482 ± 0.163
1.319 ± 0.006
2.136 ± 0.018
3.988 ± 0.052
8.197 ± 0.142
Full
35.734 ± 0.199
1.285 ± 0.008
2.015 ± 0.024
3.653 ± 0.066
7.354 ± 0.177
1.257 ± 0.008
1.908 ± 0.023
3.326 ± 0.060
6.386 ± 0.155Figures11–14 show the normalized moments (Cq) and the normalized factorial moments (Fq) calculated from the data and from the modified shifted Gompertz distribution for the pp data at energies from 900, 2360, and 7000 GeV in five rapidity bins. The values of the moments are documented in Table 3.Figure 11
Normalized moments (Cn) of modified shifted Gompertz distribution for the pp collisions recorded by the CMS experiment at different energies and in two pseudo-rapidity bins, η<0.5 and 1.0.Figure 12
Normalized moments (Cn) of modified shifted Gompertz distribution for the pp collisions recorded by the CMS experiment at different energies and in three pseudo-rapidity bins, η<1.5, 2.0, and 2.4Figure 13
Normalized factorial moments (Fn) of modified shifted Gompertz distribution for the pp collisions recorded by the CMS experiment at different energies and in two pseudo-rapidity bins, η<0.5 and 1.0.Figure 14
Normalized factorial moments (Fn) of modified shifted Gompertz distribution for the pp collisions recorded by the CMS experiment at different energies and in three pseudo-rapidity bins, η<1.5, 2.0, and 2.4.Table 3
Moments of experimental, shifted Gompertz and modified shifted Gompertz distributions forpp collisions.
Energy (GeV) (pp)
|
η
|
〈
n
〉
Normalized moments (experiment)
Normalized factorial moments (experiment)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
900
0.5
4.426 ± 0.084
1.582 ± 0.036
3.386 ± 0.152
8.893 ± 0.592
27.013 ± 2.387
1.356 ± 0.032
2.416 ± 0.113
5.122 ± 0.356
12.119 ± 1.125
1.0
8.059 ± 0.121
1.570 ± 0.029
3.272 ± 0.117
8.229 ± 0.432
23.642 ± 1.636
1.446 ± 0.027
2.719 ± 0.099
6.047 ± 0.324
15.050 ± 1.064
1.5
11.744 ± 0.151
1.538 ± 0.024
3.092 ± 0.094
7.413 ± 0.331
20.116 ± 1.188
1.453 ± 0.023
2.714 ± 0.083
5.952 ± 0.269
14.542 ± 0.874
2.0
15.507 ± 0.183
1.504 ± 0.022
2.912 ± 0.080
6.659 ± 0.270
17.095 ± 0.914
1.440 ± 0.021
2.630 ± 0.073
5.599 ± 0.229
13.205 ± 0.714
2.4
18.432 ± 0.211
1.474 ± 0.020
2.759 ± 0.073
6.042 ± 0.235
14.748 ± 0.755
1.420 ± 0.020
2.525 ± 0.067
5.190 ± 0.203
11.742 ± 0.607
2360
0.5
5.289 ± 0.095
1.585 ± 0.034
3.334 ± 0.138
8.443 ± 0.521
24.490 ± 2.041
1.396 ± 0.031
2.507 ± 0.107
5.243 ± 0.337
12.194 ± 1.086
1.0
9.722 ± 0.144
1.565 ± 0.027
3.185 ± 0.106
7.689 ± 0.383
21.099 ± 1.426
1.463 ± 0.026
2.723 ± 0.092
5.900 ± 0.300
14.287 ± 0.998
1.5
13.968 ± 0.196
1.523 ± 0.025
2.956 ± 0.093
6.738 ± 0.316
17.392 ± 1.114
1.451 ± 0.024
2.640 ± 0.084
5.552 ± 0.264
13.072 ± 0.858
2.0
18.323 ± 0.251
1.490 ± 0.023
2.806 ± 0.085
6.177 ± 0.282
15.347 ± 0.955
1.435 ± 0.023
2.568 ± 0.079
5.306 ± 0.245
12.256 ± 0.776
2.4
21.696 ± 0.298
1.471 ± 0.023
2.736 ± 0.084
6.008 ± 0.280
15.092 ± 0.973
1.425 ± 0.022
2.537 ± 0.078
5.285 ± 0.249
12.518 ± 0.823
7000
0.5
6.924 ± 0.077
1.732 ± 0.026
4.159 ± 0.114
12.245 ± 0.486
41.444 ± 2.156
1.588 ± 0.024
3.450 ± 0.097
9.020 ± 0.366
26.546 ± 1.413
1.0
13.122 ± 0.118
1.705 ± 0.021
3.945 ± 0.088
11.020 ± 0.353
35.044 ± 1.461
1.629 ± 0.020
3.567 ± 0.081
9.323 ± 0.301
27.410 ± 1.154
1.5
19.421 ± 0.154
1.669 ± 0.018
3.725 ± 0.073
9.946 ± 0.279
30.027 ± 1.094
1.617 ± 0.018
3.472 ± 0.069
8.843 ± 0.249
25.240 ± 0.926
2.0
25.870 ± 0.188
1.640 ± 0.016
3.562 ± 0.064
9.187 ± 0.234
26.573 ± 0.879
1.601 ± 0.016
3.375 ± 0.061
8.388 ± 0.215
23.203 ± 0.771
2.4
30.790 ± 0.213
1.616 ± 0.015
3.433 ± 0.059
8.613 ± 0.209
24.169 ± 0.760
1.584 ± 0.015
3.277 ± 0.056
7.963 ± 0.194
21.495 ± 0.678
Energy (GeV) (pp)
|
η
|
〈
n
〉
Normalized moments (shifted Gompertz)
Normalized factorial moments (shifted Gompertz)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
900
0.5
4.433 ± 0.033
1.571 ± 0.017
3.348 ± 0.065
8.804 ± 0.245
26.927 ± 0.973
1.345 ± 0.015
2.387 ± 0.048
5.083 ± 0.146
12.190 ± 0.453
1.0
8.099 ± 0.038
1.522 ± 0.010
3.056 ± 0.038
7.474 ± 0.132
21.215 ± 0.482
1.399 ± 0.010
2.522 ± 0.032
5.455 ± 0.097
13.480 ± 0.310
1.5
11.768 ± 0.038
1.521 ± 0.007
3.019 ± 0.025
7.213 ± 0.086
19.783 ± 0.307
1.437 ± 0.007
2.645 ± 0.022
5.791 ± 0.070
14.371 ± 0.226
2.0
15.536 ± 0.040
1.499 ± 0.005
2.887 ± 0.019
6.604 ± 0.062
17.122 ± 0.209
1.435 ± 0.005
2.606 ± 0.017
5.556 ± 0.052
13.271 ± 0.163
2.4
18.445 ± 0.043
1.470 ± 0.004
2.737 ± 0.016
5.996 ± 0.049
14.790 ± 0.159
1.416 ± 0.004
2.504 ± 0.014
5.152 ± 0.043
11.810 ± 0.128
2360
0.5
5.275 ± 0.035
1.580 ± 0.015
3.318 ± 0.056
8.439 ± 0.206
24.846 ± 0.798
1.390 ± 0.013
2.491 ± 0.043
5.249 ± 0.132
12.514 ± 0.422
1.0
9.608 ± 0.040
1.570 ± 0.009
3.206 ± 0.034
7.845 ± 0.119
22.148 ± 0.447
1.466 ± 0.008
2.738 ± 0.029
6.023 ± 0.093
15.112 ± 0.314
1.5
13.762 ± 0.045
1.536 ± 0.007
3.013 ± 0.024
7.000 ± 0.082
18.718 ± 0.297
1.464 ± 0.006
2.688 ± 0.022
5.774 ± 0.069
14.160 ± 0.231
2.0
17.994 ± 0.051
1.512 ± 0.005
2.884 ± 0.020
6.472 ± 0.065
16.550 ± 0.224
1.456 ± 0.005
2.639 ± 0.018
5.560 ± 0.056
13.253 ± 0.183
2.4
21.374 ± 0.055
1.486 ± 0.005
2.795 ± 0.017
6.276 ± 0.058
16.423 ± 0.207
1.439 ± 0.005
2.591 ± 0.016
5.527 ± 0.052
13.694 ± 0.176
7000
0.5
7.178 ± 0.028
1.652 ± 0.010
3.749 ± 0.040
10.568 ± 0.158
34.861 ± 0.668
1.513 ± 0.009
3.098 ± 0.034
7.771 ± 0.118
22.472 ± 0.435
1.0
13.744 ± 0.032
1.615 ± 0.006
3.488 ± 0.022
9.202 ± 0.082
28.179 ± 0.321
1.542 ± 0.006
3.146 ± 0.020
7.771 ± 0.070
22.099 ± 0.253
1.5
20.066 ± 0.034
1.609 ± 0.004
3.426 ± 0.016
8.811 ± 0.056
26.028 ± 0.213
1.560 ± 0.004
3.191 ± 0.015
7.829 ± 0.050
21.925 ± 0.180
2.0
25.909 ± 0.034
1.642 ± 0.003
3.559 ± 0.012
9.216 ± 0.045
27.029 ± 0.169
1.604 ± 0.003
3.371 ± 0.012
8.419 ± 0.041
23.653 ± 0.148
2.4
30.658 ± 0.034
1.628 ± 0.003
3.471 ± 0.010
8.794 ± 0.037
25.165 ± 0.136
1.596 ± 0.003
3.313 ± 0.010
8.134 ± 0.034
22.423 ± 0.122
Energy (GeV) (pp)
|
η
|
〈
n
〉
Normalized moments (modified shifted Gompertz)
Normalized factorial moments (modified shifted Gompertz)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
900
0.5
4.423 ± 0.046
1.579 ± 0.024
3.376 ± 0.094
8.878 ± 0.352
27.104 ± 1.379
1.353 ± 0.022
2.407 ± 0.069
5.117 ± 0.207
12.219 ± 0.634
1.0
8.097 ± 0.048
1.534 ± 0.014
3.101 ± 0.049
7.601 ± 0.171
21.522 ± 0.618
1.411 ± 0.013
2.563 ± 0.042
5.549 ± 0.126
13.651 ± 0.395
1.5
11.784 ± 0.051
1.521 ± 0.010
3.020 ± 0.035
7.199 ± 0.117
19.628 ± 0.410
1.436 ± 0.009
2.647 ± 0.031
5.778 ± 0.095
14.234 ± 0.300
2.0
15.510 ± 0.054
1.497 ± 0.008
2.891 ± 0.027
6.624 ± 0.087
17.169 ± 0.290
1.433 ± 0.007
2.609 ± 0.024
5.573 ± 0.074
13.299 ± 0.226
2.4
18.433 ± 0.059
1.465 ± 0.007
2.732 ± 0.023
5.995 ± 0.072
14.792 ± 0.228
1.411 ± 0.007
2.500 ± 0.021
5.152 ± 0.062
11.809 ± 0.183
2360
0.5
5.309 ± 0.050
1.575 ± 0.021
3.290 ± 0.080
8.292 ± 0.292
24.062 ± 1.111
1.386 ± 0.019
2.471 ± 0.062
5.148 ± 0.186
12.032 ± 0.579
1.0
9.746 ± 0.052
1.554 ± 0.012
3.141 ± 0.044
7.570 ± 0.151
20.928 ± 0.546
1.452 ± 0.011
2.684 ± 0.038
5.810 ± 0.117
14.237 ± 0.380
1.5
13.978 ± 0.062
1.516 ± 0.009
2.937 ± 0.033
6.711 ± 0.108
17.510 ± 0.374
1.445 ± 0.009
2.622 ± 0.030
5.533 ± 0.090
13.208 ± 0.288
2.0
18.284 ± 0.069
1.486 ± 0.008
2.797 ± 0.027
6.183 ± 0.085
15.515 ± 0.283
1.431 ± 0.007
2.559 ± 0.025
5.313 ± 0.074
12.414 ± 0.230
2.4
21.760 ± 0.079
1.463 ± 0.007
2.710 ± 0.024
5.949 ± 0.078
15.039 ± 0.263
1.417 ± 0.007
2.513 ± 0.022
5.235 ± 0.069
12.498 ± 0.221
7000
0.5
6.979 ± 0.035
1.707 ± 0.013
4.024 ± 0.056
11.645 ± 0.226
38.944 ± 0.966
1.564 ± 0.013
3.331 ± 0.047
8.554 ± 0.168
24.908 ± 0.623
1.0
13.220 ± 0.041
1.677 ± 0.008
3.800 ± 0.033
10.424 ± 0.125
32.771 ± 0.502
1.601 ± 0.008
3.431 ± 0.030
8.802 ± 0.106
25.612 ± 0.394
1.5
19.665 ± 0.047
1.634 ± 0.006
3.561 ± 0.024
9.316 ± 0.086
27.750 ± 0.324
1.583 ± 0.006
3.317 ± 0.022
8.275 ± 0.076
23.324 ± 0.273
2.0
25.817 ± 0.048
1.636 ± 0.005
3.547 ± 0.018
9.152 ± 0.065
26.588 ± 0.240
1.597 ± 0.005
3.360 ± 0.017
8.354 ± 0.059
23.225 ± 0.210
2.4
30.731 ± 0.050
1.612 ± 0.004
3.417 ± 0.015
8.582 ± 0.054
24.209 ± 0.193
1.579 ± 0.004
3.262 ± 0.015
7.934 ± 0.050
21.540 ± 0.172It is observed that the moments decrease with the increase in rapidity window, at all energies. It is also interesting to compare the results frompp to pp¯ collisions at the same c.m. energy. From the comparison at 900 GeV, we find that the moments have higher values in case of pp¯ than pp. For example, we include Figure 15 to show the dependence for η<0.5. However values of multiplicity for pp collisions at 900 GeV in full phase space are not available. In addition, for pp and for pp¯ the rapidity intervals are also different. So a direct comparison is not feasible.Figure 15
Comparison of normalised factorial moments forpp and pp¯ interactions at the same c.m. energy in the same rapidity window.Figures5–14, described above, are shown only for the modified shifted Gompertz distributions. To avoid repetition and cluttering of figures, the figures for shifted Gompertz distributions are only included for e+e− collisions and are not included for pp and pp¯ data. However the values of the moments are given in Tables 1–3.The predictions for normalized moments and normalized factorial moments are also made fore+e− collisions at 500 GeV at a future collider. Using the shifted Gompertz distribution, the prediction for probability distribution is made, as shown in Figure 16. Using this predicted distribution, 1σ confidence interval band, moments have been calculated, as given in Table 4.Figure 16
Probability distribution predicted from shifted Gompertz distribution, in 1σ confidence interval band, for the e+e− collisions at 500 GeV.Table 4
Predicted moments ats=500GeV from shifted Gompertz distribution for e+e− collisions.
Energy (GeV) (e+e−)
〈
n
〉
Normalized moments (predicted)
Normalized factorial moments (predicted)
C
2
C
3
C
4
C
5
F
2
F
3
F
4
F
5
500
37.081 ± 0.087
1.145 ± 0.003
1.487 ± 0.008
2.167 ± 0.018
3.496 ± 0.038
1.118 ± 0.003
1.396 ± 0.008
1.936 ± 0.016
2.948 ± 0.032It is observed that in case ofpp¯ and pp interactions, C2, C3 and F2, F3 remain roughly constant with energy while higher moments C4, C5, F4, F5 show an increase with increasing energy. The increase becomes more evident for larger rapidity windows. This leads to the depiction of violation of KNO scaling at high energies. Same conclusions have been reported in Ref. [21] for pp collisions at the LHC. However for the e+e− collisions, the moments are roughly independent of energy. This is expected as the collision energy is low, nearly at the onset of energy range, which marks the start of KNO scaling violation.


## 4. Uncertainties on Moments

Given a distributionPn which is normalized to unity with an uncertainty ∈n, and assuming that the errors on the individual bins are uncorrelated, the moment errors can be calculated using the method described in [22], using the partial derivatives:(13)∂Cq∂Pn=nqn−nqqnnq+1.(14)∂Fq∂Pn=nn−1…n−q+1n−nn−1…n−q+1qnnq+1.The total error is then(15)Eq2=∑n∂Xq∂Pnϵn2,whereXq is Cq or Fq.In the published data, multiplicities are given either as (value + statistical error + systematic error) or as (value + error). The errors on the multiplicities have been taken as the total error, by adding the statistical and systematic errors in quadrature.


## 5. Conclusion

An analysis of moments of multiplicity distributions described within a newly proposed statistical distribution, the shifted Gompertz distribution and its modified form has been done. We had proposed and shown that the use of this statistical distribution for studying the multiplicity distributions in high energy collisions reproduces the results ine+e−,pp¯, and pp collisions very well.A good agreement between the normalized moments as well as normalized factorial moments obtained from the shifted Gompertz distribution and its modified form with the experimental values, serves as a good test of the validity of the proposed distribution. The results have reproduced the violation of KNO scaling as observed for higher moments in the measured data. At higher energies, particularly at LHC energies, the moments strongly are dependent on the energy. The information dissemination from such an analysis is often used to study the patterns and correlations in the multiparticle final state of high-energy collisions in the presence of statistical fluctuations. In this connection, we find that the factorial moments are large indicating correlations amongst the produced particles. The predictions for normalized moments and normalized factorial moments are also made fore+e− collisions at 500 GeV at a future collider.



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*Source: 5192193-2020-01-25.xml*