# Comparing Multicomponent Erlang Distribution and Lévy Distribution of Particle Transverse Momentums

Authors: Hua-Rong Wei; Ya-Hui Chen; Li-Na Gao; Fu-Hu Liu
Journal: Advances in High Energy Physics
 (2014)

---

## Abstract

The transverse momentum spectrums of final-state products produced in nucleus-nucleus and proton-proton collisions at different center-of-mass energies are analyzed by using a multicomponent Erlang distribution and the Lévy distribution. The results calculated by the two models are found in most cases to be in agreement with experimental data from the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). The multicomponent Erlang distribution that resulted from a multisource thermal model seems to give a better description as compared with the Lévy distribution. The temperature parameters of interacting system corresponding to different types of final-state products are obtained. Light particles correspond to a low temperature emission, and heavy particles correspond to a high temperature emission. Extracted temperature from central collisions is higher than that from peripheral collisions.


---

## Body

## 1. Introduction

The Relativistic Heavy Ion Collider (RHIC) in USA and the Large Hadron Collider (LHC) in Switzerland have been built to study properties of matters formed in high-energy collisions. These collisions are helpful in understanding particles’ statistical behavior, production process, interaction mechanism, and related phenomenon in high-density and high-temperature states. Such high-energy collisions offer us opportunities to carry out investigations not only on the Higgs and dark matter [1–3], but also on particle statistical behavior at ultrahigh energy.Transverse momentum spectrums of final-state products are very important in high-energy collisions. Many models have been introduced to describe the transverse momentum spectrums of different final-state products [4]. From the spectrums, one can extract temperature parameter of interacting system. It is expected that temperature parameters extracted from different particle spectrums are different due to different emission stages and regions in collisions. Although we can compare nuclear temperature with classical temperature, they have different physical meanings.Temperature parameter in high-energy collisions is very important. Generally speaking, temperatures of interacting system at initial, intermediate, and final states are different [5]. Since these temperatures cannot be measured directly, it may, therefore, be interesting to find out an indirect method for obtaining the temperature of the interesting system. Traditionally, temperature can be extracted from measurements of spectrum slopes or double isotopic ratios at lower energies [5, 6]. In some cases, we cannot obtain absolute values of concerned temperature parameters, but relative values corresponding to different particle spectrums.Multicomponent Erlang distribution derived from multisource thermal model [7, 8] has been applied to collisions in relatively low energy region comparing to RHIC and LHC energies. Energy spectrum of nuclear fragments, multiplicity distribution of charged particles, neutron number distribution of isotope in nuclear fragments, transverse momentum (mass) spectrum of relativistic particles, and so forth were described by the multicomponent Erlang distribution. The Lévy distribution has been also applied to transverse momentum spectrums in high-energy collisions [9–11]. We can study transverse momentum spectrums by using the multicomponent Erlang distribution [7, 8] or the Lévy distribution [9–11] to extract temperature parameters.In this paper, the transverse momentum spectrums of different final-state products produced in nucleus-nucleus and proton-proton collisions at RHIC and LHC energies are studied with the two distributions mentioned above. Temperature parameters are then obtained from fitting experimental data of the STAR, CMS, and ALICE Collaborations.


## 2. Formalism

The multicomponent Erlang distribution can be derived from the multisource thermal model [7, 8]. In the model, many emission sources of particles are assumed to form in high energy collisions. According to different interaction mechanisms, geometrical relations, selected conditions, or other factors, the emission sources are divided into l groups. Source number in the jth group is assumed to be m
j. Each source contributes final-state distribution to be an exponential function. We have the transverse momentum p
t
i
j spectrum contributed by the ith source in the jth group to be
(1)
f
i
j
(
p
t
i
j
)
=
1
N
d
N
d
p
t
i
j
=
1
〈
p
t
i
j
〉
exp
⁡
(
-
p
t
i
j
〈
p
t
i
j
〉
)
,
where N denotes number of final-state particles and
(2)
〈
p
t
i
j
〉
=
∫
p
t
i
j
f
i
j
(
p
t
i
j
)
d
p
t
i
j
is mean transverse momentum contributed by the sources in the jth group.The transverse momentump
T spectrum contributed by the jth group is the fold of m
j exponential functions; that is,
(3)
f
j
(
p
T
)
=
1
N
d
N
d
p
T
=
p
T
m
j
-
1
(
m
j
-
1
)
!
〈
p
t
i
j
〉
m
j
exp
⁡
(
-
p
T
〈
p
t
i
j
〉
)
.
This is an Erlang distribution. In final state, the p
T spectrum contributed by the l groups can be written as
(4)
f
(
p
T
)
=
1
N
d
N
d
p
T
=
∑
j
=
1
l
k
j
f
j
(
p
T
)
=
∑
j
=
1
l
k
j
p
T
m
j
-
1
(
m
j
-
1
)
!
〈
p
t
i
j
〉
m
j
exp
⁡
(
-
p
T
〈
p
t
i
j
〉
)
,
where k
j is the relative weight contributed by the jth group. It is a multicomponent Erlang distribution.Considering relative contribution of thejth group, we have the mean transverse momentum of final-state particles to be
(5)
〈
p
T
〉
=
∑
j
=
1
l
k
j
m
j
〈
p
t
i
j
〉
.
Generally, 〈
p
T
〉 reflects the mean excitation degree of the emission sources and can be used to describe the source temperature parameter T
E. As in the ideal gas model in which   p
T obeys Rayleigh distribution, we have
(6)
T
E
≈
2
π
〈
p
T
〉
2
m
0
γ
-
,
where m
0 denotes rest mass and γ
- is mean Lorentz factor of considered particles. Further,
(7)
m
0
γ
-
=
E
-
≈
〈
p
〉
2
+
m
0
2
=
1.5
〈
p
T
〉
2
+
m
0
2
,
where E
- and 〈
p
〉 are mean energy and mean momentum of considered particles, respectively. On other hand, as the inverse slope parameter, 〈
p
t
i
j
〉 can be used to describe excitation degree of the emission sources. We define
(8)
T
E
S
≡
∑
j
=
1
l
k
j
〈
p
t
i
j
〉
as a new temperature parameter.The Lévy distributions appear in many branches of physics, mathematics, biology, economy, computer science, and other areas, where the distribution forms may be different in different branches and the scale of fluctuations may be characterized by long tails and an asymptotic power-law-like behavior. The Lévy distributions are a generalization of the Gaussian distribution. They are similar to the Gaussian distribution and remain stable under the convolution. In fact the Lévy distributions are quite general distributions which contain Gaussian and Cauchy distributions as special cases [12].Letq be the nonextensive parameter. As a probability distribution, the Lévy distribution is commonly the following power-like distribution [9]:
(9)
G
q
(
x
)
=
C
q
[
1
-
(
1
-
q
)
x
〈
x
〉
(
3
-
2
q
)
]
1
/
(
1
-
q
)
which is just a one-parameter generation of the Boltzmann-Gibbs exponential formula with 1
≤
q
<
1.5, where C
q is the normalization constant and x is in the range from 0 to infinity. For the transverse momentum distributions in high-energy collisions, we use directly the function form of Lévy distribution [10]:
(10)
1
2
π
p
T
d
2
N
d
y
d
p
T
=
d
N
d
y
(
n
-
1
)
(
n
-
2
)
2
π
n
T
L
[
n
T
L
+
m
0
(
n
-
2
)
]
×
(
1
+
p
T
2
+
m
0
2
-
m
0
n
T
L
)
-
n
,
where T
L is the slope parameter and n represents the scale of possible fluctuation in T
L. The parameter T
L can be regarded as the temperature parameter in the Lévy distribution.


## 3. Comparisons with Experimental Data

The transverse momentum spectrums of final-state particles produced in Cu-Cu and Au-Au collisions at RHIC energy (s
N
N
=
0.2 TeV) are shown in Figure 1. The symbols represent experimental data of the STAR Collaboration [11]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution with l
=
1 or 2 and the Lévy distribution, respectively. The results for different centralities (0–10%, 20–30%, and 40–60% in Cu + Cu, as well as 0–5%, 20–40%, and 60–80% in Au + Au) and also for different particles (K
s
0, Λ, Ξ, and Ω
+
Ω
- in Cu + Cu, as well as K
s
0 and Λ in Au + Au) in central rapidity range (|
y
|
<
0.5) are displayed in different panels. For the sake of convenience, the spectrums are for various centrality bins, with each being scaled by the amount indicated in the legend. The parameter values used in the calculations are shown in Table 1 along with values of χ
2 per degree of freedom (χ
2
/
dof) and extracted temperatures. One can see that the concerned experimental data are described approximately by the two distributions. Light particles correspond to a lower temperature comparing with the heavy particles. The multicomponent Erlang distribution seems to give a better description than the Lévy distribution. We can use the new distribution, the multicomponent Erlang distribution, to describe the transverse momentum spectrums.Table 1
Parameter values for the two kinds of curves in Figure1. The values of χ
2
/
dof and extracted temperatures are given. The errors for m
1, m
2, and n can be neglected, and the relative errors for other parameters are less than 10%.
Figures
Centralities
m
1
〈
p
t
i
1
〉 (GeV/c)
k
1
m
2
〈
p
t
i
2
〉 (GeV/c)
T
E (GeV)
χ
2
/
dof
n
T
L (GeV)
χ
2
/
dof
Figure1(a)
0–10%
2
0.310
0.990
2
0.680
0.274
1.162
13
0.200
2.823
20–30%
2
0.320
0.987
2
0.680
0.286
1.004
12
0.190
3.991
40–60%
2
0.310
0.982
2
0.680
0.277
1.160
11
0.180
1.260
Figure1(b)
0–10%
4
0.275
0.950
2
0.550
0.440
0.928
23
0.260
7.272
20–30%
4
0.260
0.920
2
0.540
0.408
0.209
21
0.250
4.588
40–60%
4
0.240
0.900
2
0.530
0.367
0.988
18
0.240
4.000
Figure1(c)
0–10%
4
0.290
0.950
4
0.420
0.456
0.322
31
0.310
0.890
20–30%
4
0.290
0.950
4
0.450
0.460
0.341
26
0.310
1.160
40–60%
4
0.290
0.950
4
0.430
0.458
0.594
33
0.310
0.877
Figure1(d)
0–10%
3
0.420
1.000
—
—
0.444
1.130
30
0.360
0.940
20–30%
3
0.410
1.000
—
—
0.428
0.932
25
0.330
0.354
40–60%
3
0.410
1.000
—
—
0.428
0.474
25
0.340
0.147
Figure1(e)
0–5%
2
0.320
0.990
2
0.640
0.284
1.260
16
0.230
2.840
20–40%
2
0.320
0.984
2
0.650
0.287
0.797
14
0.230
4.481
60–80%
2
0.310
0.970
2
0.610
0.280
0.812
14
0.220
2.540
Figure1(f)
0–5%
5
0.240
0.900
2
0.510
0.486
0.939
42
0.300
9.609
20–40%
5
0.230
0.800
2
0.490
0.450
1. 537
35
0.310
4.180
60–80%
5
0.200
0.620
2
0.455
0.365
0.834
22
0.270
1.670Transverse momentum spectrums of final-state particles produced in Cu-Cu and Au-Au collisions ats
N
N
=
0.2 TeV. The symbols represent experimental data of the STAR Collaboration [11]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively, (a), (b), (c), (d), (e), and (f) correspond to different final-state particles and collisions.
(a)
(b)
(c)
(d)
(e)
(f)In Figure2, we give the transverse momentum spectrums of leading and subleading jets produced in Pb-Pb and p-p collisions at the LHC energy (s
N
N or s
=
2.76 TeV), where the selections of leading and subleading jets can be found in experimental material [13]. The symbols represent experimental data of the CMS Collaboration [13]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively. Figures 2(a), 2(b), and 2(c) correspond to different selected conditions shown in the panels, where ∫
L
d
t, ϕ,anti-
k
T, R, and PFlow denote the integral luminosity, azimuth, sequential recombination algorithm for high-p
T particle, resolution parameter, and particle flow, respectively. The parameter values used in the calculations are shown in Table 2 with values of χ
2
/
dof and extracted temperatures. It is again observed that the two distributions describe approximately the concerned experimental data.Table 2
Parameter values for the two kinds of curves in Figures2, 4, and 5. The values of χ
2
/
dof and extracted temperatures are given. The abbreviations LJ and SJ represent leading and subleading jets, respectively. The errors for m
1, m
2, and n can be neglected, and the relative errors for other parameters are less than 10%.
Figures
Types
m
1
〈
p
t
i
1
〉 (GeV/c)
k
1
m
2
〈
p
t
i
2
〉 (GeV/c)
T
E (GeV)
χ
2
/
dof
n
m
0 (GeV/c
2)
T
L (GeV)
χ
2
/
dof
Figure2(a)
LJ
5
14.600
0.9800
6
22.000
25.925
0.865
14
100
13.000
1.022
SJ
4
16.000
0.9000
5
18.500
13.163
1.648
17
200
10.500
2.442
Figure2(b)
LJ
5
12.600
0.9900
6
26.500
20.499
1.207
10
100
9.500
2.322
SJ
7
10.500
0.9300
6
23.000
17.479
0.939
8
200
8.500
3.841
Figure2(c)
LJ
5
12.200
0.9600
6
17.800
19.915
1.006
13
100
10.500
1.327
SJ
8
10.000
0.9500
9
15.500
19.535
1.449
12
200
10.000
4.304
Figure4(a)
LJ
5
11.200
0.9000
6
17.800
19.016
0.754
9
100
11.000
1.019
SJ
8
9.500
0.9200
9
14.200
18.352
22.441
12
200
10.000
32.055
Figure4(b)
LJ
5
11.500
0.9200
6
16.800
18.956
2.904
10
100
10.300
2.154
SJ
6
8.450
0.9400
7
13.000
15.055
2.024
12
100
9.000
3.266
Figure4(c)
LJ
5
12.000
0.9000
6
17.000
20.624
1.446
13
100
12.000
1.788
SJ
4
9.500
0.9700
5
15.500
8.812
33.377
10
100
6.000
50.246
Figure4(d)
LJ
5
11.500
0.8900
6
16.900
11.067
0.907
10
210
7.800
1.320
SJ
4
6.900
0.9900
5
12.500
4.704
0.439
8
100
3.000
0.996
Figure5
0–10%
2
5.210
0.9993
1
30.000
4.270
0.449
18
10
3.800
6.910
10–30%
2
4.800
0.9992
1
25.000
3.808
0.245
15
10
3.200
7.534
30–50%
2
3.500
0.9986
1
17.000
1.004
1.216
6
30
0.800
6.223
50–80%
1
3.000
0.9980
1
12.200
0.180
2.070
4
32
0.190
1.630Transverse momentum spectrums of leading and subleading jets produced in Pb-Pb and p-p collisions ats
N
N or s
=
2.76 TeV. The symbols represent experimental data of the CMS Collaboration [13]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively, (a), (b), and (c) correspond to different selected conditions.
(a)
(b)
(c)In the Lévy distribution, we need to know the rest mass of final-state product. However, the rest mass of jet is uncertain. In fact, we regardedm
0 as a parameter in Figure 2. To see dependence of jet p
T spectrum on m
0 in the Lévy distribution, we redraw the Lévy distribution curves for different m
0 values in Figure 3, where the same experimental data [13] as those cited in Figure 2 are used. Different values of m
0 correspond to different results shown in the figure by different types of curves. All the parameter values with values of χ
2
/
dof are given in Table 3. We see that the temperature extracted from a given jet spectrum decreases with increase of the jet mass and is greater than that extracted from particle spectrums. It should be noticed that the jet mass is the total mass of particles in the jet. For a jet with a given total transverse momentum, a larger mass corresponds to more particle number. Then, the transverse momentum per particle will be smaller, which renders a lower temperature.Table 3
Parameter values for different curves of the Lévy distributions in Figure3. The values of χ
2
/
dof and extracted temperatures are given. The little marks LJ and SJ represent leading and subleading jets, respectively. The relative errors for the parameters are less than 10%.
Figures
n
LJ
m
0 (GeV/c
2)
T
LJ (GeV)
χ
2
/
dof
n
SJ
m
0 (GeV/c
2)
T
SJ (GeV)
χ
2
/
dof
Figure3(a)
7
200
6.0
0.813
17
200
10.5
2.442
9.3
150
9.0
0.911
40
150
14.2
3.170
14
100
13.0
1.022
55
100
18.0
2.444
14.5
50
14.0
0.902
300
50
25.0
3.914
18
1
14.0
0.919
500
1
30.0
7.714
Figure3(b)
9
200
7.0
2.883
8
200
8.5
3.841
10
100
9.5
2.322
9
100
12.0
4.684
14
10
12.0
2.135
12
10
15.0
8.050
Figure3(c)
9
200
6.5
1.279
12
200
10.0
4.304
13
100
10.5
1.327
22
100
15.0
5.891
14.5
10
11.0
1.282
42
10
20.0
9.446Dependence of jetp
T spectrum on m
0 in the Lévy distribution. The same experimental data [13] as those cited in Figure 2 are used. Different values of m
0 correspond to different results shown in the figure by different types of curves. The unit of m
0 is GeV/c2.
(a)
(b)
(c)In Figure4, another data sample on p
T spectrums of leading and subleading jets produced in Pb-Pb collisions at s
N
N
=
2.76 TeV is analyzed. The symbols represent experimental data of the CMS Collaboration [13]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively. The values of all the parameters along with the values of χ
2
/
dof are given in Table 2. We see that except for a few points the two distributions describe approximately the experimental data. Different spectrums corresponding to different A
J (dijet imbalance parameter) ranges can be described by the same distribution which reflects a common law in the spectrums.The same as that for Figure2, but showing another data sample in which the dijet imbalance parameter A
J is regarded as the selected condition. (a), (b), (c), and (d) correspond to different A
J ranges.
(a)
(b)
(c)
(d)Thep
T spectrums of charged jets produced in Pb-Pb collisions at s
N
N
=
2.76 TeV is given in Figure 5. The symbols represent experimental data of the ALICE Collaboration [14]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively. All the parameter values along with values of χ
2
/
dof and extracted temperatures are given in Table 2. One can see that both the distributions describe approximately the experimental data, and the former one gives a better description than the latter one.Figure 5
Thep
T spectrums of charged jets produced in Pb-Pb collisions at s
N
N
=
2.76 TeV. The symbols represent experimental data of the ALICE Collaboration [14]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.Thep
T spectrums of charged particles (which can be approximately regarded as π
±) produced in s
N
N
=
2.76 TeV Pb-Pb collisions in different centrality bins with different multiplications are shown in Figure 6(a). Meanwhile, the p
T spectrums of π
-, K
s
0, K
-, and p
- produced in central (0–5%) Pb-Pb collisions at the same energy are shown in Figure 6(b). The symbols represent experimental data of the ALICE Collaboration [14, 15] measured in the pseudorapidity range of |
η
|
<
0.8. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively. Corresponding to Figures 6(a) and 6(b), the parameter values with values of χ
2
/
dof and extracted temperatures are given in Tables 4 and 5, respectively. One can see that the multicomponent Erlang distribution describes well the p
T spectrums in all the cases. The Lévy distribution describes well the spectrums in some cases, and in other cases it describes approximately the mean trends of the spectrums.Table 4
Parameter values for the two kinds of curves in Figures6(a), 8, and 10. The values of χ
2
/
dof and extracted temperatures are given. The errors for m
1,2
,
3,4 and n can be neglected, and the relative errors for other parameters are less than 10%.
Figures
Figure6(a)
Figure6(a)
Figure6(a)
Figure8(a)
Figure8(a)
Figure8(b)
Figure10
Types
0–5%
20–40%
40–80%
Ξ
Ω
Inclusive electron
Charged particle
m
1
1
1
1
3
3
1
1
〈
p
t
i
1
〉 (GeV/c)
0.500
0.500
0.500
0.390
0.390
0.108
0.370
k
1
0.799184
0.928772
0.918786
0.91000
0.70000
0.88822
0.93906
m
2
1
1
1
4
4
1
1
〈
p
t
i
2
〉 (GeV/c)
0.600
0.800
0.820
0.625
0.530
0.380
0.800
k
2
0.199796
0.069908
0.079894
0.09000
0.30000
0.10978
0.05994
m
3
1
1
1
—
—
5
1
〈
p
t
i
3
〉 (GeV/c)
2.300
2.300
2.300
—
—
0.480
1.800
k
3
0.001000
0.001300
0.001300
—
—
0.00200
0.00100
m
4
1
1
1
—
—
—
—
〈
p
t
i
4
〉 (GeV/c)
6.700
6.700
6.700
—
—
—
—
T
E (GeV)
0.265
0.266
0.268
0.515
0.551
0.074
0.198
χ
2
/
dof
0.303
0.299
0.512
0.535
0.023
0.070
0.563
n
7.5
7
7
9
10
5
8
T
L (GeV)
0.200
0.190
0.160
0.320
0.390
0.040
0.150
χ
2
/
dof
66.600
46.825
4.944
0.288
0.099
1.184
0.503Table 5
Parameter values for the two kinds of curves in Figures6(b), 7(b), 7(c), and 9. The values of χ
2
/
dof and extracted temperatures are given. The errors for m
1, m
2, and n can be neglected, and the relative errors for other parameters are less than 10%.
Figures
Types
m
1
〈
p
t
i
1
〉 (GeV/c)
k
1
m
2
〈
p
t
i
2
〉 (GeV/c)
T
E (GeV)
χ
2
/
dof
n
T
L (GeV)
χ
2
/
dof
Figure6(b)
π
-
2
0.195
0.785
3
0.320
0.260
0.045
10
0.185
0.435
K
-
,
K
s
0
8
0.230
0.060
3
0.270
0.411
0.155
68
0.350
0.335
p
-
11
0.131
0.130
3
0.463
0.636
0.193
22
0.450
12.390
Figure7(b)
0–20%
1
0.450
0.990
1
1.250
0.231
0.406
9
0.180
1.171
20–40%
1
0.400
0.980
1
1.000
0.207
0.283
9
0.175
0.599
40–60%
1
0.400
0.975
1
1.100
0.210
0.122
8
0.170
0.362
60–80%
1
0.420
0.975
1
1.220
0.222
0.152
7
0.145
0.361
p-p
1
0.390
0.970
1
1.150
0.207
0.258
7
0.140
0.097
Figure7(c)
0–5%
6
0.272
0.500
3
0.370
0.622
0.199
300
0.450
21.836
5–10%
6
0.262
0.400
3
0.410
0.620
0.212
300
0.450
21.136
10–20%
6
0.269
0.380
3
0.390
0.604
0.113
300
0.450
16.885
20–30%
6
0.267
0.330
3
0.395
0.595
0.065
300
0.470
6.772
30–40%
6
0.260
0.220
3
0.395
0.564
0.161
500
0.470
2.645
40–50%
6
0.260
0.230
3
0.365
0.527
0.187
200
0.450
1.074
50–60%
6
0.280
0.110
3
0.355
0.488
0.145
100
0.420
0.509
60–70%
6
0.320
0.070
3
0.330
0.444
0.627
22
0.346
1.117
Figure9(a)
π
+
2
0.159
0.850
3
0.280
0.198
0.834
8
0.129
0.270
K
+
3
0.180
0.870
7
0.210
0.293
0.310
7
0.180
0.999
p
8
0.230
0.060
3
0.240
0.293
0.304
8
0.195
0.486
Figure9(b)
π
-
2
0.170
0.895
3
0.305
0.200
0.302
8
0.129
0.176
K
-
3
0.180
0.130
7
0.192
0.284
0.684
8
0.185
1.025
p
-
8
0.210
0.050
3
0.230
0.267
0.347
9
0.190
0.606
Figure9(c)
K
s
0
2
0.250
0.860
3
0.410
0.259
0.136
6.8
0.168
0.426
Λ
2
0.290
0.780
3
0.400
0.230
0.484
9
0.220
0.953
Λ
¯
2
0.300
0.800
3
0.400
0.232
0.053
8
0.198
0.126
ϕ
2
0.448
1.000
—
—
0.341
0.396
9
0.270
0.224
Ξ
-
+
Ξ
-
+
2
0.430
1.000
—
—
0.279
0.147
8
0.230
0.076Thep
T spectrums of (a) charged and (b) identified particles produced in Pb-Pb collisions at s
N
N
=
2.76 TeV. The symbols represent experimental data of the ALICE Collaboration [14, 15]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.
(a)
(b)Figures7(a), 7(b), and 7(c) show, respectively, p
T spectrums of final-state particles π
+
+
π
-, π
0, and p
- produced in s
N
N
=
2.76 TeV Pb-Pb collisions in different centrality bins with different multiplications. Selected condition for p
- is rapidity being in the range of |
y
|
<
0.5. For the sake of comparison, the results for π
+
+
π
- and π
0 produced in 2.76 TeV p-p collisions are also given in Figures 7(a) and 7(b), respectively. The symbols represent experimental data of the ALICE Collaboration [16, 17]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively. All the parameter values with values of χ
2
/
dof and extracted temperatures are given in Tables 5 (for Figures 7(b) and 7(c)) and 6 (for Figure 7(a)), respectively. One can see that the multicomponent Erlang distribution describes well the p
T spectrums in all the cases. The Lévy distribution describes well the spectrums in some cases, and in other cases it describes approximately mean trends of the spectrums.Table 6
Parameter values for the two kinds of curves in Figure7(a). The values of χ
2
/
dof and extracted temperatures are given. The errors for m
1,2
,
3 and n can be neglected, and the relative errors for other parameters are less than 10%.
Types
0–5%
5–10%
10–20%
20–40%
40–60%
60–80%
p-p
m
1
1
1
1
1
1
1
1
〈
p
t
i
1
〉 (GeV/c)
0.550
0.500
0.600
0.650
0.650
0.6400
0.660
k
1
0.994980
0.995480
0.990941
0.984941
0.981941
0.979951
0.974942
m
2
1
1
1
1
1
1
1
〈
p
t
i
2
〉 (GeV/c)
1.85
1.700
1.800
1.800
1.750
1.720
1.680
k
2
0.005000
0.004500
0.008999
0.014999
0.017999
0.019999
0.024998
m
3
5
5
5
5
5
5
5
〈
p
t
i
3
〉 (GeV/c)
2.800
2.200
2.000
2.200
2.300
2.200
2.300
T
E (GeV)
0.283
0.256
0.312
0.342
0.343
0.339
0.352
χ
2
/
dof
0.166
0.208
0.139
0.320
0.105
0.074
0.077
n
6
6
6
6
6
7
7
T
L (GeV)
0.00001
0.00010
0.00080
0.00700
0.03000
0.15000
0.14000
χ
2
/
dof
2.656
1.544
1.373
0.814
0.480
0.531
1.220Thep
T spectrums of (a) π
+
+
π
-, (b) π
0, and (c) p
- produced in s
N
N
=
2.76 TeV Pb-Pb collisions in different centrality bins. For the sake of comparison, the results for π
+
+
π
- and π
0 produced in 2.76 TeV p-p collisions are also given. The symbols represent experimental data of the ALICE Collaboration [16, 17]. The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.
(a)
(b)
(c)The transverse momentum spectrums ofΞ and Ω as well as inclusive electrons produced in inelastic p-p collision at 7 TeV are given in Figures 8(a) and 8(b), respectively. Experimental data measured by the ALICE Collaboration [15, 18] are shown by the symbols. Results calculated by using the multicomponent Erlang distributions and the Lévy distributions are shown by the solid and dashed curves, respectively. The parameter values used in the calculation are listed in Table 4. We see that both distributions describe approximately the experimental data.Thep
T spectrums of (a) Ξ and Ω as well as (b) inclusive electrons produced in inelastic p-p collision at 7 TeV. Experimental data measured by the ALICE Collaboration [15, 18] are shown by the symbols. Results calculated by using the multicomponent Erlang distributions and the Lévy distributions are shown by the solid and dashed curves, respectively.
(a)
(b)The transverse momentum spectrums ofπ
+, K
+, and  p; π
-, K
-, and   p
-; K
s
0, Λ, Λ
-, ϕ, and Ξ
-
+
Ξ
-
+ produced in p-p collisions at 0.9 TeV are displayed in Figures 9(a), 9(b), and 9(c), respectively. The symbols represent experimental data of the ALICE Collaboration [19, 20]. The solid and dashed curves represent results calculated by using the multicomponent Erlang distribution and the Lévy distribution, respectively. The related parameter values are given in Table 5. One can see that both the two distributions describe approximately the experimental data.Thep
T spectrums of (a) π
+, K
+,   and p; (b) π
-, K
-, and p
-; and (c) K
s
0, Λ, Λ
-,   ϕ, and Ξ
-
+
Ξ
-
+ produced in p-p collisions at 0.9 TeV. The symbols represent experimental data of the ALICE Collaboration [19, 20]. The solid and dashed curves represent results calculated by using the multicomponent Erlang distribution and the Lévy distribution, respectively.
(a)
(b)
(c)In Figure10, the transverse momentum spectrum of charged particles (which can be approximately regarded as π
±) produced in nonsingle diffractive (NSD) p-p collisions at 0.9 TeV is presented. The symbols represent experimental data measured in the pseudorapidity range of |
η
|
<
0.8 by the ALICE Collaboration [19]. The solid and dashed curves represent results of the multicomponent Erlang distribution and the Lévy distribution, respectively. The related parameter values are given in Table 4. One can see that both the two distributions describe approximately the experimental data.Figure 10
Thep
T spectrum of charged particles produced in NSD p-p collisions at 0.9 TeV. The symbols represent experimental data measured by the ALICE Collaboration [19]. The solid and dashed curves represent results of the multicomponent Erlang distribution and the Lévy distribution, respectively.To see dependences of temperatureT (T
E and T
L) on centrality and s
N
N, in Figures 11 and 12, we plot different values of T
E and T
L taken from Tables 1–6. The related impacting types, s
N
N, centralities, and final-state products are shown in the figures. Figures 11(a), 11(b), 11(c) and 11(d) as well as 11(e) and 11(f) correspond to dependence on centrality for particle productions at 0.2 and 2.76 TeV and jet production at 2.76 TeV, respectively. Figure 12 corresponds to dependence on s
N
N for particle productions at RHIC and LHC energies. One can see that the extracted temperature for light particles is less than that for heavy particles. Central collisions or high s
N
N correspond to a relative high temperature. The multicomponent Erlang distribution extracts a relatively high temperature comparing to the Lévy distribution. Besides, from the parameter tables (Tables 1, 2, and 4–6) and (8), one can easily obtain values of T
E
S which show similar behaviors as those of T
E.Dependences of temperaturesT
E and T
L on centrality. (a), (b), (c), and (d) as well as (e) and (f) correspond to dependence on centrality for particle productions at 0.2 and 2.76 TeV and jet production at 2.76 TeV, respectively.
(a)
(b)
(c)
(d)
(e)
(f)Dependences of temperaturesT
E and T
L on s
N
N.
(a)
(b)


## 4. Conclusions and Discussions

The transverse momentum spectrums of final-state products produced in high-energy collisions are analysed by using the multicomponent Erlang distribution and the Lévy distribution. In most cases, both the distributions are approximately in agreement with experimental data at RHIC and LHC energies. The multicomponent Erlang distribution seems to give a better description as compared to the Lévy distribution. Although the Lévy distribution is well known to give the transverse momentum spectrums, the multicomponent Erlang distribution gives a new method to describe the transverse momentum spectrums.The temperature parameters of interacting system corresponding to different types of final-state products are extracted from transverse momentum spectrums. Light particles correspond to a low temperature emission, and heavy particles correspond to a high temperature emission. For a jet with a given transverse momentum, larger mass corresponds to larger particle number and lesser transverse momentum per particle, which renders a lower temperature. Central collisions or highs
N
N correspond to a relative high temperature. The multicomponent Erlang distribution extracts a relatively high temperature comparing with the Lévy distribution.System size dependence of the hadronic spectrums is well described by the two modeling distributions in the present work. We see some correlations between the parameter values and system size. Particularly, the extracted temperature increases with increase of the system size from p-p collision to Cu-Cu and Au-Au (Pb-Pb) collisions at the sames
N
N. This renders that the excitation degree of the interacting system increases with increase of the system size. Comparing with light nuclear collisions, a participant nucleon in heavy nuclear collisions takes part in more binary collisions, and more energy per nucleon deposits in heavy nuclear collisions.It is well known that most of the hadrons in low transverse momentum region are produced in the process dominated by soft interaction, whereas the hadrons with high transverse momentums are produced in the process dominated by hard parton-parton scattering. According to the discussions in the present work, the first group of sources in the multicomponent Erlang distribution corresponds generally to the soft interaction, and the second or third group of sources corresponds to the hard scattering. The Lévy distribution does not distinguish the transverse momentum regions of soft interaction and hard scattering.Although there are more or less differences in both the modeling distributions for the observed transverse momentum spectrums, the multicomponent Erlang distribution and the Lévy distribution describing approximately the transverse momentum spectrums in different systems render that there are some common laws or universality in multihadron production [21, 22], even in general probability distributions. For example, the multicomponent Erlang distribution is also used to describe the probability distributions of some plant seed masses and sizes [23], and the Lévy distribution has more other applications [12, 24]. We are interested in searching new applications of the two distributions.



---

*Source: 782631-2014-04-10.xml*