Advances in High Energy Physics
Volume 2016 (2016), Article ID 1505823, 15 pages
http://dx.doi.org/10.1155/2016/1505823

Research Article

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

Li-Na Gao and Fu-Hu Liu

Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Received 13 October 2015; Accepted 9 December 2015

Academic Editor: Ming Liu

Copyright © 2016 Li-Na Gao and Fu-Hu Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

We study the transverse momentum spectra of and 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 (), proton-lead (-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.

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 of and mesons produced in proton-proton (), proton-lead (-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 and 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 () spectrum. For the th ( and 2) parton in the th component, we have the distribution

The transverse momentum () in final state is . The transverse momentum distribution in final state is the folding of and . Considering the contribution ratio of the first component, we have the (simplest) two-component Erlang distribution to be

In Monte Carlo method, let denote random numbers in . For the th component, we have

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 and denote the string tensions between the two partons in the first component and second component, respectively, , , and denotes the rest of the mass of a parton. According to [13–16], for the th ( and 2) parton in the given string in the th component, we have the distribution

where

is the normalization constant. Considering the contribution ratio of the first component, we have the two-component Schwinger mechanism

In Monte Carlo method, let denote random numbers in . For the th component, we have

3. Results

The transverse momentum spectra, , of mesons produced in different data samples in collision at center-of-mass energy  TeV are shown in Figure 1, where and 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 with no polarisation, from with no polarisation, prompt with full transverse polarisation, and prompt with full longitudinal polarisation, respectively. The values of free parameters and per degree of freedom () 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 and /dof corresponding to the curves in Figures 1–5.

Figure
Type
Two-component Erlang
Two-component Schwinger

(GeV/c)
(GeV/c)
/dof

(GeV/fm)
(GeV/fm)
/dof

Figure 1(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

Figure 1(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

Figure 1(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

Figure 1(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

Figure 2(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

Figure 2(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

Figure 3(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

Figure 3(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

Figure 3(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

Figure 4(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

Figure 4(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

Figure 5
= 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

= 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

= 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
1.274

(a)

(b)

(c)

(d)

Figure 1: Transverse momentum spectra of (a) prompt with no polarisation, (b) from with no polarisation, (c) prompt with full transverse polarisation, and (d) prompt with full longitudinal polarisation in collision at  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.

Figures 2(a) and 2(b) present the transverse momentum spectra, , of mesons produced in data samples, prompt and from , in collision at  TeV, respectively. Figures 3(a), 3(b), and 3(c) present the transverse momentum spectra, , of , , and mesons produced in the same collision, respectively, where , 2, and 3 correspond to the three mesons, respectively, and 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 -Pb collisions at center-of-mass energy per nucleon pair  TeV, and the experiment data of the LHCb Collaboration are taken from [19]. The values of free parameters and 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.

(a)

(b)

Figure 2: Transverse momentum spectra of (a) prompt and (b) from in collision at  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 3: Transverse momentum spectra of (a) , (b) , and (c) mesons produced in collision at  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 4: The same as Figure 2 but showing the results in different rapidity ranges in -Pb collisions at  TeV. The data are taken from [19].

The transverse momentum spectra, , of produced in three centrality classes and in Pb-Pb collisions at  TeV are given in Figure 5, where 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 are listed in Table 1. Similar conclusion obtained from Figure 1 can be obtained from Figure 5.

Figure 5: Transverse momentum spectra of produced in three centrality classes in Pb-Pb collisions at  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

is used, where is the temperature parameter, is the entropy index, is the chemical potential which is regarded as 0 at the LHC, and is the normalization constant. In (8), the effect of longitudinal motion is subtracted by using 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 , , and  GeV; the entropy index is taken to be , , and , with 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 of 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 Table 1 in Figures 6–11, which correspond to the relationships related to parameters , , , , , and , respectively. In these figures, the symbols and lines are parameter values and fitting lines, respectively. The intercepts, slopes, and related to the lines in Figures 6–11 are listed in Table 2. In the error range, and do not show obvious change with rapidity, centrality, energy, and size. , , , and 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, and /dof corresponding to the lines in Figures 6–11.

Figure
Type
Intercept
Slope
/dof

Figure 6(a)
Prompt (no polarisation)
0.513 ± 0.001
0.012 ± 0.001
0.001

from (no polarisation)
0.456 ± 0.021
0.047 ± 0.006
0.003

Prompt (transverse polarisation)
0.496 ± 0.035
0.049 ± 0.011
0.007

Prompt (longitudinal polarisation)
0.819 ± 0.065
−0.060 ± 0.020
0.030

Figure 6(b)
Prompt
0.900 ± 0.045
−0.015 ± 0.014
0.013

from
0.625 ± 0.135
0.045 ± 0.041
0.139

Figure 6(c)

0.916 ± 0.142
−0.051 ± 0.043
0.053

0.553 ± 0.085
0.003 ± 0.025
0.025

0.741 ± 0.200
−0.036 ± 0.060
0.111

Figure 6(d)
Prompt
0.737 ± 0.040
0.036 ± 0.014
0.008

from
0.691 ± 0.110
0.042 ± 0.039
0.051

Figure 6(e)
Inclusive
0.785 ± 0.143
0.001 ± 0.003
0.344

Figure 7(a)
Prompt (no polarisation)
1.200 ± 0.049
−0.076 ± 0.015
0.007

from (no polarisation)
1.067 ± 0.144
0.035 ± 0.043
0.054

Prompt (transverse polarisation)
1.301 ± 0.163
−0.089 ± 0.049
0.070

Prompt (longitudinal polarisation)
1.219 ± 0.110
−0.068 ± 0.033
0.047

Figure 7(b)
Prompt
1.342 ± 0.041
−0.068 ± 0.012
0.008

from
1.429 ± 0.112
−0.045 ± 0.034
0.034

Figure 7(c)

3.219 ± 0.128
−0.272 ± 0.039
0.028

2.658 ± 0.315
0.003 ± 0.095
0.087

3.890 ± 0.670
−0.296 ± 0.202
0.449

Figure 7(d)
Prompt
1.428 ± 0.080
−0.051 ± 0.028
0.021

from
1.853 ± 0.144
−0.165 ± 0.051
0.075

Figure 7(e)
Inclusive
0.937 ± 0.020
0.001 ± 0.001
0.008

Figure 8(a)
Prompt (no polarisation)
1.898 ± 0.054
−0.140 ± 0.016
0.007

from (no polarisation)
2.552 ± 0.119
−0.247 ± 0.036
0.025

Prompt (transverse polarisation)
1.938 ± 0.116
−0.148 ± 0.035
0.031

Prompt (longitudinal polarisation)
2.039 ± 0.096
−0.173 ± 0.029
0.025

Figure 8(b)
Prompt
2.485 ± 0.142
−0.221 ± 0.043
0.026

from
2.842 ± 0.227
−0.219 ± 0.068
0.036

Figure 8(c)

3.697 ± 0.191
−0.313 ± 0.057
0.025

2.884 ± 0.858
0.090 ± 0.258
0.511

3.574 ± 0.229
−0.014 ± 0.069
0.022

Figure 8(d)
Prompt
2.269 ± 0.151
−0.250 ± 0.053
0.027

from
1.958 ± 0.442
0.083 ± 0.156
0.180

Figure 8(e)
Inclusive
0.985 ± 0.092
0.004 ± 0.002
0.062

Figure 9(a)
Prompt (no polarisation)
0.785 ± 0.009
0.011 ± 0.003
0.001

from (no polarisation)
0.850 ± 0.096
−0.038 ± 0.029
0.132

Prompt (transverse polarisation)
0.841 ± 0.087
−0.005 ± 0.026
0.108

Prompt (longitudinal polarisation)
0.774 ± 0.047
0.013 ± 0.014
0.025

Figure 9(b)
Prompt
0.780 ± 0.027
0.021 ± 0.008
0.005

from
0.643 ± 0.041
0.042 ± 0.012
0.015

Figure 9(c)

0.479 ± 0.027
0.017 ± 0.008
0.007

0.526 ± 0.041
−0.001 ± 0.012
0.016

0.472 ± 0.100
−0.005 ± 0.030
0.106

Figure 9(d)
Prompt
0.642 ± 0.016
0.025 ± 0.006
0.003

from
0.399 ± 0.040
0.071 ± 0.014
0.017

Figure 9(e)
Inclusive
0.559 ± 0.023
−0.001 ± 0.001
0.031

Figure 10(a)
Prompt (no polarisation)
13.254 ± 0.305
−0.638 ± 0.092
0.002

from (no polarisation)
18.725 ± 2.440
−1.540 ± 0.733
0.037

Prompt (transverse polarisation)
15.070 ± 1.188
−1.160 ± 0.357
0.012

Prompt (longitudinal polarisation)
14.940 ± 1.750
−1.040 ± 0.526
0.021

Figure 10(b)
Prompt
13.945 ± 0.703
−0.420 ± 0.211
0.004

from
17.145 ± 0.616
−0.740 ± 0.185
0.002

Figure 10(c)

37.600 ± 4.512
−2.400 ± 1.357
0.042

30.750 ± 11.613
2.600 ± 3.491
0.137

52.050 ± 11.186
−4.200 ± 3.363
0.130

Figure 10(d)
Prompt
14.765 ± 0.803
−0.840 ± 0.283
0.007

from
13.490 ± 0.227
−0.440 ± 0.080
0.001

Figure 10(e)
Inclusive
5.037 ± 0.247
0.013 ± 0.006
0.111

Figure 11(a)
Prompt (no polarisation)
71.714 ± 2.053
−7.720 ± 0.617
0.007

from (no polarisation)
103.035 ± 12.805
−13.820 ± 3.850
0.110

Prompt (transverse polarisation)
74.840 ± 9.489
−8.720 ± 2.853
0.107

Prompt (longitudinal polarisation)
65.250 ± 7.664
−5.800 ± 2.304
0.075

Figure 11(b)
Prompt
73.075 ± 7.290
−6.060 ± 2.192
0.053

from
93.040 ± 4.414
−7.680 ± 1.327
0.012

Figure 11(c)

179.150 ± 8.275
−20.600 ± 2.488
0.019

118.100 ± 41.500
8.400 ± 12.477
0.185

204.000 ± 35.512
−12.000 ± 10.677
0.121

Figure 11(d)
Prompt
62.695 ± 3.633
−4.820 ± 1.280
0.019

from
66.250 ± 2.734
−4.280 ± 0.963
0.011

Figure 11(e)
Inclusive
16.785 ± 0.374
0.047 ± 0.009
0.005

(a)

(b)

(c)

(d)

(e)

Figure 6: Dependence of contribution ratio for (a) on in collision at  TeV, (b) on in collision at  TeV, (c) on in collision at  TeV, (d) on in -Pb collisions at  TeV, and (e) on in Pb-Pb collisions at  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 Figure 6 but showing the results for .

(a)

(b)

(c)

(d)

(e)

Figure 8: The same as Figure 6 but showing the results for .

(a)

(b)

(c)

(d)

(e)

Figure 9: The same as Figure 6 but showing the results for .

(a)

(b)

(c)

(d)

(e)

Figure 10: The same as Figure 6 but showing the results for .

(a)

(b)

(c)

(d)

(e)

Figure 11: The same as Figure 6 but showing the results for .

4. Discussions

To discuss the Schwinger mechanism, if the charmonium () or bottomonium () can be produced in the collision, the minimum distance between the two partons, for the th component, is [16]

where is the mass of produced charmed or bottom quark but not that of the participant parton. The mean minimum distance is as follows:

The minimal minimum distance is as follows:

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 , where 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

The mean maximum potential energy is as follows:

The maximal maximum potential energy is as follows:

Correspondingly, the distribution of the maximum potential energy can be obtained by .

From Table 1, we see that the values of , , , and 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 , 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 of and 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 , -Pb, and Pb-Pb collisions at the LHC.(b)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. At the same time, the contribution ratios and of the two first components in the two methods are obtained.(c)In the error range, and do not show obvious change with rapidity, centrality, energy, and size. , , , and 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 11575103.

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