# Comparing Erlang Distribution and Schwinger Mechanism on Transverse Momentum Spectra in High Energy Collisions

Authors: Li-Na Gao; Fu-Hu Liu
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

We study the transverse momentum spectra ofJ/ψ and Υ mesons by using two methods: the two-component Erlang distribution and the two-component Schwinger mechanism. The results obtained by the two methods are compared and found to be in agreement with the experimental data of proton-proton (pp), proton-lead (p-Pb), and lead-lead (Pb-Pb) collisions measured by the LHCb and ALICE Collaborations at the large hadron collider (LHC). The related parameters such as the mean transverse momentum contributed by each parton in the first (second) component in the two-component Erlang distribution and the string tension between two partons in the first (second) component in the two-component Schwinger mechanism are extracted.


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## Body

## 1. Introduction

In the last century, scientists predicted that a new state of matter could be produced in relativistic heavy-ion (nucleus-nucleus) collisions or could exist in quark stars owing to high temperature and high density [1–3]. The new matter is named the quark-gluon plasma (QGP) or quark matter. This prediction makes the research of high energy collisions develop rapidly. A lot of physics researchers devoted themselves to researching the mechanisms of particle productions and the properties of QGP formation. Because of the fact that the reaction time of the impacting system is very short, people could not make a direct measurement for the collision process. So, only by researching the final state particles, people can presume the evolutionary process of collision system. For this reason, people proposed many models to simulate the process of high energy collisions [4].The transverse momentum (mass) spectra of particles in final state are an important observation. They play one of the major roles in high energy collisions. Other quantities which also play major roles include, but are not limited to, pseudorapidity (or rapidity) distribution, azimuthal distribution (anisotropic flow), particle ratio, and various correlations [5]. Presently, many formulas such as the standard (Fermi-Dirac, Bose-Einstein, or Boltzmann) distribution [6], the Tsallis statistics [7–11], the Tsallis form of standard distribution [11], the Erlang distribution [12], and the Schwinger mechanism [13–16] are used in describing the transverse momentum spectra. It is expected that the excitation degree (effective temperature and kinetic freeze-out temperature), radial flow velocity, and other information can be obtained by analyzing the particle transverse momentum spectra.In this paper, we use two methods, the two-component Erlang distribution and the two-component Schwinger mechanism, to describe the transverse momentum spectra ofJ/ψ and Υ mesons produced in proton-proton (pp), proton-lead (p-Pb), and lead-lead (Pb-Pb) collisions measured by the LHCb and ALICE Collaborations at the large hadron collider (LHC) [17–20]. The related parameters such as the mean transverse momentum contributed by each parton in the first (second) component in the two-component Erlang distribution and the string tension between two partons in the first (second) component in the two-component Schwinger mechanism are extracted.


## 2. Formulism

We assume that the basic impacting process in high energy collisions is binary parton-parton collision. We have two considerations on the description of violent degree of the collision. A consideration is the mean transverse momentum contributed by each parton. The other one is the string tension between two partons. The former consideration can be studied in the framework of Erlang distribution. The latter one results in the Schwinger mechanism. Considering the wide transverse momentum spectra in experiments, we use the two-component Erlang distribution and the two-component Schwinger mechanism. Generally, the first component describes the region of low transverse momentum, and the second one describes the high transverse momentum region.Firstly, we consider the two-component Erlang distribution. Let pt1 and pt2 denote the mean transverse momentums contributed by each parton in the first component and second component, respectively. Each parton is assumed to contribute an exponential transverse momentum (pt) spectrum. For the jth (j=1 and 2) parton in the ith component, we have the distribution(1)fijptj=1ptiexp⁡-ptjpti.The transverse momentum (pT) in final state is pT=pt1+pt2. The transverse momentum distribution in final state is the folding of fi1(pt1) and fi2(pt2). Considering the contribution ratio kE1 of the first component, we have the (simplest) two-component Erlang distribution to be(2)fpT=kE1∫0pTf11pt1f12pT-pt1dpt1+1-kE1∫0pTf21pt1f22pT-pt1dpt1=kE1pTpt12exp⁡-pTpt1+1-kE1pTpt22exp⁡-pTpt2.In Monte Carlo method, let R1,2 denote random numbers in [0,1]. For the ith component, we have(3)pT=pt1+pt2=-ptiln⁡R1+ln⁡R2.We would like to point out that the folding of multiple exponential distributions with the same parameter results in the ordinary Erlang distribution which is not used in the present work. Monte Carlo method performs a simpler calculation for the folding.Secondly, we consider the two-component Schwinger mechanism. Let κ1 and κ2 denote the string tensions between the two partons in the first component and second component, respectively, K1≡κ1/π, K2≡κ2/π, and m0 denotes the rest of the mass of a parton. According to [13–16], for the jth (j=1 and 2) parton in the given string in the ith component, we have the distribution(4)
f
i
j
p
t
j
=
C
0
K
i
exp
⁡
-
p
t
j
2
+
m
0
2
K
i
=
C
0
K
i
exp
⁡
-
m
0
2
K
i
exp
⁡
-
p
t
j
2
K
i
=
1
K
i
π
exp
⁡
-
p
t
j
2
K
i
,where(5)C0Ki=1∫0∞exp⁡-ptj2+m02/Kidptj=1Kiπexp⁡m02Kiis the normalization constant. Considering the contribution ratio kS1 of the first component, we have the two-component Schwinger mechanism(6)
f
p
T
=
k
S
1
∫
0
p
T
f
11
p
t
1
f
12
p
T
-
p
t
1
d
p
t
1
+
1
-
k
S
1
∫
0
p
T
f
21
p
t
1
f
22
p
T
-
p
t
1
d
p
t
1
=
k
S
1
C
0
2
K
1
∫
0
p
T
e
x
p
-
p
t
1
2
+
p
T
-
p
t
1
2
+
2
m
0
2
K
1
d
p
t
1
+
1
-
k
S
1
C
0
2
K
2
∫
0
p
T
e
x
p
-
p
t
1
2
+
p
T
-
p
t
1
2
+
2
m
0
2
K
2
d
p
t
1
=
k
S
1
K
1
π
∫
0
p
T
exp
⁡
-
p
t
1
2
+
p
T
-
p
t
1
2
K
1
d
p
t
1
+
1
-
k
S
1
K
2
π
∫
0
p
T
exp
⁡
-
p
t
1
2
+
p
T
-
p
t
1
2
K
2
d
p
t
1
.In Monte Carlo method, let R3–6 denote random numbers in [0,1]. For the ith component, we have(7)pT=pt1+pt2=Ki-ln⁡R3cos⁡2πR4+-ln⁡R5cos⁡2πR6.


## 3. Results

The transverse momentum spectra,d2σJ/ψ/dpTdy, of J/ψ mesons produced in different data samples in pp collision at center-of-mass energy s=7 TeV are shown in Figure 1, where σ and y denote the cross section and rapidity, respectively. The symbols represent the experimental data measured by the LHCb Collaboration [17] in different rapidity ranges and scaled by different amounts marked in the panels. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism, respectively. From Figures 1(a)–1(d), the data samples are prompt J/ψ with no polarisation, J/ψ from b with no polarisation, prompt J/ψ with full transverse polarisation, and prompt J/ψ with full longitudinal polarisation, respectively. The values of free parameters and χ2 per degree of freedom (χ2/dof) are listed in Table 1. One can see that the two methods describe the experimental data of the LHCb Collaboration.Table 1
Values of parameters andχ2/dof corresponding to the curves in Figures 1–5.
Figure
Type
Two-component Erlang
Two-component Schwinger
k
E
1
p
t
1 (GeV/c)
p
t
2 (GeV/c)
χ
2/dof
k
S
1
κ
1 (GeV/fm)
κ
2 (GeV/fm)
χ
2/dof
Figure1(a)
2.0 <y < 2.5
0.540 ± 0.062
1.008 ± 0.132
1.553 ± 0.086
0.519
0.813 ± 0.027
11.70 ± 0.88
54.30 ± 3.67
1.814
2.5 <y < 3.0
0.545 ± 0.083
1.008 ± 0.107
1.533 ± 0.082
0.846
0.810 ± 0.023
11.50 ± 1.20
50.00 ± 3.32
2.318
3.0 <y < 3.5
0.552 ± 0.102
0.978 ± 0.124
1.473 ± 0.078
0.775
0.817 ± 0.029
11.42 ± 1.24
47.82 ± 3.98
1.882
3.5 <y < 4.0
0.557 ± 0.096
0.887 ± 0.093
1.373 ± 0.070
0.927
0.830 ± 0.020
10.85 ± 1.10
44.00 ± 3.82
0.544
4.0 <y < 4.5
0.564 ± 0.093
0.878 ± 0.120
1.283 ± 0.100
0.913
0.830 ± 0.020
10.43 ± 1.12
38.00 ± 4.23
0.256
Figure1(b)
2.0 <y < 2.5
0.563 ± 0.084
1.183 ± 0.087
1.957 ± 0.094
0.226
0.713 ± 0.023
13.80 ± 3.12
65.00 ± 7.40
0.880
2.5 <y < 3.0
0.582 ± 0.063
1.120 ± 0.093
1.886 ± 0.112
0.940
0.778 ± 0.032
15.90 ± 3.78
69.50 ± 7.90
0.492
3.0 <y < 3.5
0.605 ± 0.078
1.118 ± 0.100
1.838 ± 0.098
0.922
0.786 ± 0.037
14.60 ± 3.65
66.00 ± 7.50
0.474
3.5 <y < 4.0
0.650 ± 0.068
1.300 ± 0.092
1.570 ± 0.087
0.571
0.700 ± 0.028
12.80 ± 3.07
49.80 ± 6.80
0.291
4.0 <y < 4.5
0.647 ± 0.100
1.180 ± 0.140
1.498 ± 0.153
0.105
0.657 ± 0.023
11.50 ± 3.05
40.30 ± 6.30
0.218
Figure1(c)
2.0 <y < 2.5
0.610 ± 0.077
1.180 ± 0.093
1.538 ± 0.102
0.738
0.785 ± 0.025
11.90 ± 2.75
50.00 ± 5.90
2.545
2.5 <y < 3.0
0.640 ± 0.080
1.008 ± 0.100
1.580 ± 0.087
0.705
0.860 ± 0.036
12.80 ± 2.93
55.00 ± 6.30
1.255
3.0 <y < 3.5
0.650 ± 0.065
0.953 ± 0.127
1.500 ± 0.076
0.711
0.853 ± 0.032
11.00 ± 2.67
49.30 ± 6.10
1.353
3.5 <y < 4.0
0.656 ± 0.072
0.902 ± 0.118
1.404 ± 0.092
1.049
0.855 ± 0.027
10.80 ± 2.70
45.00 ± 5.70
0.523
4.0 <y < 4.5
0.725 ± 0.082
1.010 ± 0.120
1.255 ± 0.085
0.707
0.776 ± 0.023
10.00 ± 2.63
33.20 ± 5.20
0.708
Figure1(d)
2.0 <y < 2.5
0.715 ± 0.062
1.120 ± 0.140
1.603 ± 0.117
0.842
0.793 ± 0.038
12.00 ± 3.80
50.00 ± 5.60
2.816
2.5 <y < 3.0
0.640 ± 0.074
1.008 ± 0.093
1.580 ± 0.082
1.171
0.796 ± 0.043
12.00 ± 4.20
48.00 ± 5.80
4.194
3.0 <y < 3.5
0.596 ± 0.054
0.947 ± 0.073
1.530 ± 0.078
1.108
0.847 ± 0.032
13.00 ± 3.60
52.00 ± 5.20
0.789
3.5 <y < 4.0
0.570 ± 0.067
0.930 ± 0.056
1.413 ± 0.060
1.147
0.835 ± 0.034
10.80 ± 3.20
45.00 ± 5.10
0.486
4.0 <y < 4.5
0.600 ± 0.085
0.990 ± 0.065
1.253 ± 0.067
1.051
0.805 ± 0.027
10.00 ± 3.10
37.00 ± 4.70
0.268
Figure2(a)
2.0 <y < 2.5
0.870 ± 0.064
1.200 ± 0.058
1.950 ± 0.117
1.196
0.823 ± 0.043
13.00 ± 3.57
57.80 ± 6.40
1.478
2.5 <y < 3.0
0.837 ± 0.053
1.137 ± 0.053
1.852 ± 0.130
1.114
0.830 ± 0.057
12.50 ± 3.05
56.00 ± 6.10
1.880
3.0 <y < 3.5
0.883 ± 0.057
1.145 ± 0.055
1.868 ± 0.132
0.982
0.858 ± 0.068
12.80 ± 3.12
54.80 ± 6.10
1.930
3.5 <y < 4.0
0.825 ± 0.045
1.068 ± 0.062
1.693 ± 0.127
1.118
0.873 ± 0.057
12.80 ± 3.07
55.30 ± 6.20
2.041
4.0 <y < 4.5
0.838 ± 0.065
1.066 ± 0.057
1.478 ± 0.142
1.192
0.853 ± 0.047
11.80 ± 2.98
43.00 ± 5.90
3.430
Figure2(b)
2.0 <y < 2.5
0.788 ± 0.063
1.372 ± 0.087
2.420 ± 0.190
1.545
0.757 ± 0.047
15.60 ± 3.87
75.50 ± 8.20
0.763
2.5 <y < 3.0
0.644 ± 0.056
1.218 ± 0.092
2.080 ± 0.217
1.787
0.725 ± 0.045
14.70 ± 3.73
70.00 ± 7.50
1.226
3.0 <y < 3.5
0.795 ± 0.078
1.300 ± 0.085
2.183 ± 0.204
1.279
0.782 ± 0.053
15.00 ± 3.75
70.60 ± 7.70
0.831
3.5 <y < 4.0
0.827 ± 0.053
1.295 ± 0.075
2.120 ± 0.185
1.363
0.803 ± 0.049
14.60 ± 3.65
66.00 ± 7.30
1.065
4.0 <y < 4.5
0.810 ± 0.057
1.220 ± 0.083
1.853 ± 0.177
1.733
0.822 ± 0.052
13.80 ± 3.58
58.30 ± 7.10
1.342
Figure3(a)
2.0 <y < 2.5
0.802 ± 0.188
2.560 ± 0.087
3.030 ± 0.190
0.945
0.510 ± 0.047
30.00 ± 4.80
132.00 ± 12.00
0.230
2.5 <y < 3.0
0.830 ± 0.155
2.490 ± 0.073
2.740 ± 0.178
1.523
0.524 ± 0.053
32.00 ± 4.80
120.00 ± 11.70
0.268
3.0 <y < 3.5
0.705 ± 0.165
2.360 ± 0.077
2.687 ± 0.173
1.653
0.553 ± 0.057
33.00 ± 5.20
115.00 ± 11.30
0.303
3.5 <y < 4.0
0.635 ± 0.157
2.276 ± 0.084
2.652 ± 0.168
1.099
0.527 ± 0.050
28.00 ± 4.00
107.00 ± 10.80
0.307
4.0 <y < 4.5
0.771 ± 0.169
1.987 ± 0.078
2.292 ± 0.153
8.640
0.550 ± 0.064
26.00 ± 3.80
87.00 ± 10.10
4.617
Figure3(b)
2.0 <y < 2.5
0.513 ± 0.137
2.836 ± 0.134
2.900 ± 0.157
2.844
0.530 ± 0.057
42.00 ± 7.60
140.00 ± 22.00
1.192
2.5 <y < 3.0
0.614 ± 0.156
2.463 ± 0.129
3.537 ± 0.123
1.791
0.517 ± 0.052
36.00 ± 7.30
145.00 ± 25.00
0.732
3.0 <y < 3.5
0.570 ± 0.135
2.675 ± 0.135
3.238 ± 0.137
1.715
0.503 ± 0.049
33.00 ± 7.30
150.00 ± 25.00
0.726
3.5 <y < 4.0
0.587 ± 0.143
2.587 ± 0.138
2.620 ± 0.145
3.942
0.552 ± 0.062
37.00 ± 7.5
117.00 ± 20.00
2.540
4.0 <y < 4.5
0.535 ± 0.195
2.782 ± 0.128
3.583 ± 0.237
1.970
0.510 ± 0.054
48.00 ± 7.80
175.00 ± 29.00
2.163
Figure3(c)
2.0 <y < 2.5
0.555 ± 0.148
3.342 ± 0.145
3.583 ± 0.207
4.141
0.447 ± 0.078
43.00 ± 9.00
185.00 ± 28.00
3.373
2.5 <y < 3.0
0.783 ± 0.187
3.107 ± 0.137
3.506 ± 0.198
2.554
0.493 ± 0.059
45.00 ± 7.00
180.00 ± 21.00
1.594
3.0 <y < 3.5
0.603 ± 0.152
2.886 ± 0.132
3.583 ± 0.212
3.949
0.402 ± 0.062
34.00 ± 8.00
140.00 ± 19.00
2.585
3.5 <y < 4.0
0.647 ± 0.160
2.285 ± 0.107
3.350 ± 0.205
13.242
0.516 ± 0.057
30.00 ± 7.50
150.00 ± 20.00
8.291
4.0 <y < 4.5
0.533 ± 0.133
3.012 ± 0.153
3.627 ± 0.223
11.216
0.423 ± 0.083
40.00 ± 10.70
170.00 ± 25.00
11.114
Figure4(a)
1.5 <y < 2.0
0.797 ± 0.133
1.285 ± 0.105
1.768 ± 0.242
3.624
0.687 ± 0.076
12.60 ± 3.25
52.00 ± 5.70
0.472
2.0 <y < 2.5
0.802 ± 0.148
1.356 ± 0.094
1.752 ± 0.216
2.201
0.692 ± 0.052
13.10 ± 3.30
53.10 ± 5.80
0.614
2.5 <y < 3.0
0.873 ± 0.097
1.322 ± 0.108
1.687 ± 0.198
2.842
0.708 ± 0.067
12.40 ± 3.23
52.10 ± 5.80
0.243
3.0 <y < 3.5
0.835 ± 0.085
1.280 ± 0.107
1.352 ± 0.255
3.825
0.737 ± 0.054
12.00 ± 3.05
47.00 ± 5.40
0.989
3.5 <y < 4.0
0.870 ± 0.090
1.195 ± 0.095
1.342 ± 0.188
3.814
0.726 ± 0.054
10.80 ± 3.10
43.00 ± 5.20
1.383
Figure4(b)
1.5 <y < 2.0
0.812 ± 0.158
1.656 ± 0.084
1.830 ± 0.190
0.918
0.518 ± 0.065
12.60 ± 3.70
57.50 ± 5.50
1.921
2.0 <y < 2.5
0.684 ± 0.136
1.363 ± 0.082
2.497 ± 0.203
1.309
0.543 ± 0.060
12.60 ± 3.57
57.60 ± 5.80
1.807
2.5 <y < 3.0
0.850 ± 0.107
1.405 ± 0.078
2.293 ± 0.238
4.805
0.634 ± 0.067
12.30 ± 3.42
56.40 ± 5.30
1.401
3.0 <y < 3.5
0.847 ± 0.123
1.285 ± 0.080
2.050 ± 0.255
9.194
0.620 ± 0.070
12.20 ± 3.37
50.60 ± 5.00
2.658
3.5 <y < 4.0
0.835 ± 0.130
1.282 ± 0.065
2.261 ± 0.273
9.018
0.657 ± 0.065
11.70 ± 3.15
50.30 ± 5.10
4.329
Figure5
C = 0–20%
0.702 ± 0.158
0.937 ± 0.060
0.960 ± 0.195
0.778
0.567 ± 0.063
5.00 ± 0.170
17.00 ± 2.30
0.012
C = 20–40%
0.982 ± 0.113
1.000 ± 0.102
1.200 ± 0.227
1.038
0.512 ± 0.055
5.70 ± 0.183
18.60 ± 3.10
1.428
C = 40–60%
0.823 ± 0.083
1.020 ± 0.087
1.200 ± 0.253
1.163
0.518 ± 0.047
5.80 ± 0.180
19.40 ± 3.42
1.274Figure 1
Transverse momentum spectra of (a) promptJ/ψ with no polarisation, (b) J/ψ from b with no polarisation, (c) prompt J/ψ with full transverse polarisation, and (d) prompt J/ψ with full longitudinal polarisation in pp collision at s=7 TeV. The symbols represent the experimental data measured by the LHCb Collaboration [17] in different rapidity ranges and scaled by different amounts marked in the panels. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism, respectively.
(a)
(b)
(c)
(d)Figures2(a) and 2(b) present the transverse momentum spectra, d2σJ/ψ/dpTdy, of J/ψ mesons produced in data samples, prompt J/ψ and J/ψ from b, in pp collision at s=8 TeV, respectively. Figures 3(a), 3(b), and 3(c) present the transverse momentum spectra, BnSd2σnS/dpTdy, of Υ(1S), Υ(2S), and Υ(3S) mesons produced in the same collision, respectively, where n=1, 2, and 3 correspond to the three mesons, respectively, and B denotes the branching ratio. The symbols represent the experimental data of the LHCb Collaboration [18] and the curves are our results. Figure 4 is similar to Figure 2, but it presents the results in p-Pb collisions at center-of-mass energy per nucleon pair sNN=5 TeV, and the experiment data of the LHCb Collaboration are taken from [19]. The values of free parameters and χ2/dof corresponding to Figures 2–4 are listed in Table 1. Similar conclusion obtained from Figure 1 can be obtained from Figures 2–4, though some of fitting results are approximately in agreement with the data.Figure 2
Transverse momentum spectra of (a) promptJ/ψ and (b) J/ψ from b in pp collision at s=8 TeV. The symbols represent the experimental data of the LHCb Collaboration [18]. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism, respectively.
(a)
(b)Figure 3
Transverse momentum spectra of (a)Υ(1S), (b) Υ(2S), and (c) Υ(3S) mesons produced in pp collision at s=8 TeV. The symbols represent the experimental data of the LHCb Collaboration [18]. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism, respectively.
(a)
(b)
(c)Figure 4
The same as Figure2 but showing the results in different rapidity ranges in p-Pb collisions at sNN=5 TeV. The data are taken from [19].
(a)
(b)The transverse momentum spectra,d2Y/dpTdy, of J/ψ produced in three centrality classes and 2.5<y<4 in Pb-Pb collisions at sNN=2.76 TeV are given in Figure 5, where Y denotes the yield. The symbols represent the experimental data of the ALICE Collaboration [20]. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism, respectively. The values of free parameters and χ2/dof are listed in Table 1. Similar conclusion obtained from Figure 1 can be obtained from Figure 5.Figure 5
Transverse momentum spectra ofJ/ψ produced in three centrality classes in Pb-Pb collisions at sNN=2.76 TeV. The symbols represent the experimental data of the ALICE Collaboration [20]. The dashed, solid, and dotted curves are our results calculated by using the two-component Erlang distribution, the two-component Schwinger mechanism, and the Tsallis statistics, respectively.To give a comparison of fit quality with some other approach, as an example, we show the result of the Tsallis statistics [7–11] in Figure 5 by the dotted curves. A simplified form of the Tsallis transverse momentum distribution(8)fpT=CTpTpT2+m021+q-1TpT2+m02-μ-q/q-1is used, where T is the temperature parameter, q is the entropy index, μ is the chemical potential which is regarded as 0 at the LHC, and CT is the normalization constant. In (8), the effect of longitudinal motion is subtracted by using y=0 to obtain the temperature parameter as accurately as possible. For the centrality from 0–20% to 40–90%, the temperature parameter is taken to be 0.285±0.008, 0.273±0.006, and 0.264±0.009 GeV; the entropy index is taken to be 1.067±0.003, 1.080±0.004, and 1.085±0.007, with χ2/dof to be 0.969, 1.290, and 1.781, respectively. One can see the compatibility of the three approaches, which shows the multiformity of fit functions.In the above fit to the experimental data of LHCb and ALICE Collaborations, the uncorrelated and correlated uncertainties in experimental data are together included in the calculation ofχ2/dof by using the quadratic sums. No matter for the part of uncorrelated or correlated uncertainty, even for the part of correlated, especially multiplicative, common for all bins uncertainties, we just use the experimental values directly.To see clearly the relationships between parameters and rapidity, parameters and centrality, and parameters and others, we plot the values listed in Table1 in Figures 6–11, which correspond to the relationships related to parameters kE1, pt1, pt2, kS1, κ1, and κ2, respectively. In these figures, the symbols and lines are parameter values and fitting lines, respectively. The intercepts, slopes, and χ2/dof related to the lines in Figures 6–11 are listed in Table 2. In the error range, kE1 and kS1 do not show obvious change with rapidity, centrality, energy, and size. pt1, pt2, κ1, and κ2 do not show obvious change with rapidity and centrality, or they decrease slightly with increases of rapidity and centrality.Table 2
Values of intercepts, slopes, andχ2/dof corresponding to the lines in Figures 6–11.
Figure
Type
Intercept
Slope
χ
2/dof
Figure6(a)
PromptJ/ψ (no polarisation)
0.513 ± 0.001
0.012 ± 0.001
0.001
J
/
ψ from b (no polarisation)
0.456 ± 0.021
0.047 ± 0.006
0.003
PromptJ/ψ (transverse polarisation)
0.496 ± 0.035
0.049 ± 0.011
0.007
PromptJ/ψ (longitudinal polarisation)
0.819 ± 0.065
−0.060 ± 0.020
0.030
Figure6(b)
PromptJ/ψ
0.900 ± 0.045
−0.015 ± 0.014
0.013
J
/
ψ from b
0.625 ± 0.135
0.045 ± 0.041
0.139
Figure6(c)
Υ
1
S
0.916 ± 0.142
−0.051 ± 0.043
0.053
Υ
2
S
0.553 ± 0.085
0.003 ± 0.025
0.025
Υ
3
S
0.741 ± 0.200
−0.036 ± 0.060
0.111
Figure6(d)
PromptJ/ψ
0.737 ± 0.040
0.036 ± 0.014
0.008
J
/
ψ from b
0.691 ± 0.110
0.042 ± 0.039
0.051
Figure6(e)
Inclusive
0.785 ± 0.143
0.001 ± 0.003
0.344
Figure7(a)
PromptJ/ψ (no polarisation)
1.200 ± 0.049
−0.076 ± 0.015
0.007
J
/
ψ from b (no polarisation)
1.067 ± 0.144
0.035 ± 0.043
0.054
PromptJ/ψ (transverse polarisation)
1.301 ± 0.163
−0.089 ± 0.049
0.070
PromptJ/ψ (longitudinal polarisation)
1.219 ± 0.110
−0.068 ± 0.033
0.047
Figure7(b)
PromptJ/ψ
1.342 ± 0.041
−0.068 ± 0.012
0.008
J
/
ψ from b
1.429 ± 0.112
−0.045 ± 0.034
0.034
Figure7(c)
Υ
1
S
3.219 ± 0.128
−0.272 ± 0.039
0.028
Υ
2
S
2.658 ± 0.315
0.003 ± 0.095
0.087
Υ
3
S
3.890 ± 0.670
−0.296 ± 0.202
0.449
Figure7(d)
PromptJ/ψ
1.428 ± 0.080
−0.051 ± 0.028
0.021
J
/
ψ from b
1.853 ± 0.144
−0.165 ± 0.051
0.075
Figure7(e)
Inclusive
0.937 ± 0.020
0.001 ± 0.001
0.008
Figure8(a)
PromptJ/ψ (no polarisation)
1.898 ± 0.054
−0.140 ± 0.016
0.007
J
/
ψ from b (no polarisation)
2.552 ± 0.119
−0.247 ± 0.036
0.025
PromptJ/ψ (transverse polarisation)
1.938 ± 0.116
−0.148 ± 0.035
0.031
PromptJ/ψ (longitudinal polarisation)
2.039 ± 0.096
−0.173 ± 0.029
0.025
Figure8(b)
PromptJ/ψ
2.485 ± 0.142
−0.221 ± 0.043
0.026
J
/
ψ from b
2.842 ± 0.227
−0.219 ± 0.068
0.036
Figure8(c)
Υ
1
S
3.697 ± 0.191
−0.313 ± 0.057
0.025
Υ
2
S
2.884 ± 0.858
0.090 ± 0.258
0.511
Υ
3
S
3.574 ± 0.229
−0.014 ± 0.069
0.022
Figure8(d)
PromptJ/ψ
2.269 ± 0.151
−0.250 ± 0.053
0.027
J
/
ψ from b
1.958 ± 0.442
0.083 ± 0.156
0.180
Figure8(e)
Inclusive
0.985 ± 0.092
0.004 ± 0.002
0.062
Figure9(a)
PromptJ/ψ (no polarisation)
0.785 ± 0.009
0.011 ± 0.003
0.001
J
/
ψ from b (no polarisation)
0.850 ± 0.096
−0.038 ± 0.029
0.132
PromptJ/ψ (transverse polarisation)
0.841 ± 0.087
−0.005 ± 0.026
0.108
PromptJ/ψ (longitudinal polarisation)
0.774 ± 0.047
0.013 ± 0.014
0.025
Figure9(b)
PromptJ/ψ
0.780 ± 0.027
0.021 ± 0.008
0.005
J
/
ψ from b
0.643 ± 0.041
0.042 ± 0.012
0.015
Figure9(c)
Υ
1
S
0.479 ± 0.027
0.017 ± 0.008
0.007
Υ
2
S
0.526 ± 0.041
−0.001 ± 0.012
0.016
Υ
3
S
0.472 ± 0.100
−0.005 ± 0.030
0.106
Figure9(d)
PromptJ/ψ
0.642 ± 0.016
0.025 ± 0.006
0.003
J
/
ψ from b
0.399 ± 0.040
0.071 ± 0.014
0.017
Figure9(e)
Inclusive
0.559 ± 0.023
−0.001 ± 0.001
0.031
Figure10(a)
PromptJ/ψ (no polarisation)
13.254 ± 0.305
−0.638 ± 0.092
0.002
J
/
ψ from b (no polarisation)
18.725 ± 2.440
−1.540 ± 0.733
0.037
PromptJ/ψ (transverse polarisation)
15.070 ± 1.188
−1.160 ± 0.357
0.012
PromptJ/ψ (longitudinal polarisation)
14.940 ± 1.750
−1.040 ± 0.526
0.021
Figure10(b)
PromptJ/ψ
13.945 ± 0.703
−0.420 ± 0.211
0.004
J
/
ψ from b
17.145 ± 0.616
−0.740 ± 0.185
0.002
Figure10(c)
Υ
1
S
37.600 ± 4.512
−2.400 ± 1.357
0.042
Υ
2
S
30.750 ± 11.613
2.600 ± 3.491
0.137
Υ
3
S
52.050 ± 11.186
−4.200 ± 3.363
0.130
Figure10(d)
PromptJ/ψ
14.765 ± 0.803
−0.840 ± 0.283
0.007
J
/
ψ from b
13.490 ± 0.227
−0.440 ± 0.080
0.001
Figure10(e)
Inclusive
5.037 ± 0.247
0.013 ± 0.006
0.111
Figure11(a)
PromptJ/ψ (no polarisation)
71.714 ± 2.053
−7.720 ± 0.617
0.007
J
/
ψ from b (no polarisation)
103.035 ± 12.805
−13.820 ± 3.850
0.110
PromptJ/ψ (transverse polarisation)
74.840 ± 9.489
−8.720 ± 2.853
0.107
PromptJ/ψ (longitudinal polarisation)
65.250 ± 7.664
−5.800 ± 2.304
0.075
Figure11(b)
PromptJ/ψ
73.075 ± 7.290
−6.060 ± 2.192
0.053
J
/
ψ from b
93.040 ± 4.414
−7.680 ± 1.327
0.012
Figure11(c)
Υ
1
S
179.150 ± 8.275
−20.600 ± 2.488
0.019
Υ
2
S
118.100 ± 41.500
8.400 ± 12.477
0.185
Υ
3
S
204.000 ± 35.512
−12.000 ± 10.677
0.121
Figure11(d)
PromptJ/ψ
62.695 ± 3.633
−4.820 ± 1.280
0.019
J
/
ψ from b
66.250 ± 2.734
−4.280 ± 0.963
0.011
Figure11(e)
Inclusive
16.785 ± 0.374
0.047 ± 0.009
0.005Figure 6
Dependence of contribution ratiokE1 for (a) J/ψ on y in pp collision at s=7 TeV, (b) J/ψ on y in pp collision at s=8 TeV, (c) Υ on y in pp collision at s=8 TeV, (d) J/ψ on y in p-Pb collisions at sNN=5 TeV, and (e) J/ψ on C in Pb-Pb collisions at sNN=2.76 TeV. The symbols are quoted in Table 1 and the lines are our fitting results.
(a)
(b)
(c)
(d)
(e)Figure 7
The same as Figure6 but showing the results for pt1.
(a)
(b)
(c)
(d)
(e)Figure 8
The same as Figure6 but showing the results for pt2.
(a)
(b)
(c)
(d)
(e)Figure 9
The same as Figure6 but showing the results for kS1.
(a)
(b)
(c)
(d)
(e)Figure 10
The same as Figure6 but showing the results for κ1.
(a)
(b)
(c)
(d)
(e)Figure 11
The same as Figure6 but showing the results for κ2.
(a)
(b)
(c)
(d)
(e)


## 4. Discussions

To discuss the Schwinger mechanism, if the charmonium (cc-) or bottomonium (bb-) can be produced in the collision, the minimum distance between the two partons, for the ith component, is [16](9)Lmin,i=2mTκi=2pT2+m02κi,where m0 is the mass of produced charmed or bottom quark but not that of the participant parton. The mean minimum distance is as follows:(10)Lmin,i=2mTκi=2pT2+m02κi.The minimal minimum distance is as follows:(11)Lmin,imin=2m0κi.We see that the string tension is an important parameter which is related to the minimum distance, the mean minimum distance, and the minimal minimum distance. Correspondingly, the distribution of the minimum distance can be obtained by 1/NdN/dLmin,i, where N denotes the number of charmonia.Further, if the produced charmed or bottom quark stays at haphazard at the middle between the two participant partons, the maximum potential energy of the charmed quark staying in the colour field of the two partons is(12)Vmax,i=-κiLmin,i2=-mT=-pT2+m02.The mean maximum potential energy is as follows:(13)Vmax,i=-κiLmin,i2=-mT=-pT2+m02.The maximal maximum potential energy is as follows:(14)Vmax,imax=-κiLmin,i2max=-m0.Correspondingly, the distribution of the maximum potential energy can be obtained by 1/NdN/dVmax,i.From Table1, we see that the values of pt1, pt2, κ1, and κ2 in the collisions at the LHC are large, which render that the interactions between partons are violent and the minimal minimum distance between the two interacting partons is small. According to κ2, the minimal minimum distance is ~0.03–0.06 fm which is a few percent of nucleon size. We believe that, in the collisions at the LHC, nucleons penetrate through each other totally. Projectile partons are exceedingly close to target partons with large probability. At the same time, both the contribution ratios of the two second components in the two calculation methods are large and cannot be neglected.


## 5. Conclusions

From the above discussions, we obtain the following conclusions:(a)
The transverse momentum spectra ofJ/ψ and Υ mesons produced in high energy collisions are described by using both the two-component Erlang distribution and the two-component Schwinger mechanism. The results obtained by the two methods are compared and found to be in agreement with the experimental data of pp, p-Pb, and Pb-Pb collisions at the LHC.(b)
The related parameters such as the mean transverse momentumpt1 (pt2) contributed by each parton in the first (second) component in the two-component Erlang distribution and the string tension κ1 (κ2) between two partons in the first (second) component in the two-component Schwinger mechanism are extracted. At the same time, the contribution ratios kE1 and kS1 of the two first components in the two methods are obtained.(c)
In the error range,kE1 and kS1 do not show obvious change with rapidity, centrality, energy, and size. pt1, pt2, κ1, and κ2 do not show obvious change with rapidity and centrality, or they decrease slightly with increases of rapidity and centrality. Both the contribution ratios of the two second components cannot be neglected. The minimal minimum distance between interacting partons is ~0.03–0.06 fm which is a few percent of nucleon size.



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*Source: 1505823-2016-01-04.xml*