# Transverse Momentum Distributions of Final-State Particles Produced in Soft Excitation Process in High Energy Collisions

Authors: Fu-Hu Liu; Ya-Hui Chen; Hua-Rong Wei; Bao-Chun Li
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

Transverse momentum distributions of final-state particles produced in soft process in proton-proton (pp) and nucleus-nucleus (AA) collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies are studied by using a multisource thermal model. Each source in the model is treated as a relativistic and quantum ideal gas. Because the quantum effect can be neglected in investigation on the transverse momentum distribution in high energy collisions, we consider only the relativistic effect. The concerned distribution is finally described by the Boltzmann or two-component Boltzmann distribution. Our modeling results are in agreement with available experimental data.


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## Body

## 1. Introduction

High energy collisions are an important research topic in particle and nuclear physics. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) did firstly collider experiments on heavy ions [1], and the center-of-mass energy per nucleon pair (sNN) at the RHIC reached highly 200 GeV [2]. The Large Hadron Collider (LHC) at European Laboratory for Particle Physics (CERN) renovated value of sNN to TeV region [3]. It seems that a new state of matter, namely, Quark-Gluon Plasma (QGP), is possibly formed in heavy ion collisions at RHIC and LHC energies due to high temperature and density [4, 5]. At initial stage of high energy collisions, another possible new state of matter, namely, color glass condensate (CGC), is caused by strong color fields in the low-xgluon realm [6, 7], wherexdenotes the ratio of quark or gluon momentum to hadron one. A CGC is in fact a region of the nuclear wave function at low-xandQ2and exists already before the collisions, whereQ2denotes the square momentum of virtual photon. On the other hand, the CGC may not be a new state, but more like a model or calculation for initial state hadron behavior. One cannot measure the QGP and CGC directly. However, one can measure final-state particle spectra at freeze-out to extract thermal and other characteristics of interacting system and give a judgment on formation and property of the new matters.The final-state particle spectra include rapidityy (or pseudorapidity η) [8, 9], transverse momentum pT (or transverse mass) [10, 11], transverse energy [12, 13], and other distributions [14]. It is known that y and pT distributions reflect, respectively, the degrees of longitudinal extension and transverse excitation of interacting system. Especially, for transverse excitation, soft excitation and hard scattering processes can affect, respectively, distributions in low- and high-pT ranges. The soft and hard processes correspond to different physics mechanisms and distribution laws [15]. In low energy collisions, the soft process is main process, and the hard process can be neglected due to almost zero contribution. In high energy collisions, although the hard process cannot be neglected, the soft process is still main process.To understand the transverse excitation, we need firstly to study the soft excitation process. A lot of models have been introduced to describe the soft process, although some of them can be used to describe the hard process too [16, 17]. Among the models, the multisource thermal model proposed by us is a very simple one and can be used to describe pT spectra in both the soft and hard processes if source’s contribution is given by an Erlang distribution [18]. Finally, the considered distribution is described by a multicomponent Erlang distribution [19, 20]. Different from some simulation codes, our model gives directly a few statistical laws by analytical expressions in describing some quantities. In the case of being incapable of analytical expressions, we could use a Monte Carlo method to give a numerical result. Our model is easy to be used by experimental experts.Due to significances of the considered model and topic, in this paper, based on Boltzmann distribution for a single source, we describepT spectra of final-state particles produced in soft process in proton-proton (pp) and nucleus-nucleus (AA) collisions at RHIC and LHC energies. Some interesting results are obtained.


## 2. The Model

According to the multisource thermal model, many emission sub-sources of final-state particles are assumed to form in high energy collisions [19, 20]. These multiple sub-sources can be different regions in the overlap region or different mechanisms, and these particles can be created/emitted at different times in the collisions. In fact, these sub-sources can be divided into different groups (sources) due to different interacting mechanisms or event samples. Obviously, soft process corresponds to sources with low degree of excitation or to particles with low transverse momentum, and hard process corresponds to sources with high degree of excitation or to particles with high transverse momentum, where the excitation means to create particles through string breaking, direct scattering, recombination, and their hybrid.In the rest frame of a source, we consider the source as a thermodynamic system of relativistic and quantum ideal gas. The momentum (p) distribution of final-state particles in the natural unit system is given by [21]
(1)fp(p)=1NdNdp=C0p2[exp(p2+m02-μT)±1]-1,
where N is the number of particles, C0 is the normalization constant, m0 is the rest mass of a considered particle, μ is the chemical potential, T is the temperature parameter, +1 denotes fermions, and -1 denotes bosons, respectively. Our calculations show that at RHIC and LHC energies the quantum effect and chemical potential can be neglected compared to the relativistic effect [22]. Then, we have a simple expression for momentum distribution to be [23, 24]
(2)fp(p)=1NdNdp=C0p2exp(-p2+m02T),
where C0=(1/m02T)(1/K2(m0/T)) and K2(m0/T) is the modified Bessel function of order 2.ThepT distribution can be written as a Boltzmann distribution [25]:
(3)fpT(pT)=1NdNdpT=CpTexp(-pT2+m02T),
where C is the normalization constant. Because of interactions among different sources, the considered source has a deformation and/or movement in the transverse plane. Letadenote a relative deformation and letbdenote an absolute movement of the source; that is, we use apT+b instead of pT in (3). The revised pT distribution can be given by
(4)fpT(pT)=1NdNdpT=C(pT-b)a-2exp[-(pT-b)2a-2+m02T].
In the case of considering multiple sources, we have
(5)fpT(pT)=1NdNdpT=∑i=1nkiCipTexp(-pT2+m02Ti)
or
(6)fpT(pT)=1NdNdpT=∑i=1nkiCi(pT-bi)ai-2exp[-(pT-bi)2ai-2+m02Ti],
where ki, Ci, and Ti denote the contribution ratio, normalization constant, and temperature of theith source, respectively. Because the effects of deformation and movement of the source can be neglected in the calculation of transverse momentum, we take the default values of ai=1 and bi=0 in the revised pT distribution, which results from the Boltzmann distribution or a multicomponent Boltzmann distribution.We should have a few sources to describe the soft and hard processes. For the soft process, the number of sources is generally 1 or 2. For the hard process, the number of sources is also 1 or 2. The total number of sources will be from 2 to 4 for a widepT distribution. In this paper, we pay our attention on the soft process which has a narrow pT distribution. It is hard to say that what the pT distribution range is for the soft process. What we can say is that for low energy collisions the distribution range is narrower. In the present work, we regard the distribution range as 0–10 GeV/c. The difference between the single and multisource models is obvious. The former one describes a narrower distribution which corresponds to an equilibrium state with a lower degree of excitation. The latter one describes a wider distribution which corresponds to a few local equilibrium sates with different excitations.


## 3. Comparisons with Experimental Data

Figure1 presents the transverse momentum distributions of (a) π-, K- and p-, (b) π+, K+ and p, (c) π+, (d) K+, and (e) p produced in pp collision at center-of-mass energy s=200 GeV with different y ranges and magnifications shown in the figure. The symbols represent the experimental data of the STAR [26] (Figures 1(a) and 1(b)) and BRAHMS Collaborations [27] (Figures 1(c)–1(e)), and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. In the calculation, we have used a fitting method to obtain parameter values which are shown in Table 1 with values of χ2 per degree of freedom (χ2/dof). To give a short presentation, the values corresponding to “negative/positive” charged particles are given in terms of “the first value/the second value” or “value” in the case of the first value and the second value being the same. We would like to point out that the presenting styles of rapidity ranges for Figures 1(a) and 1(b) as well as for Figures 1(c)–1(e) are different due to different presentations in [26, 27]. One can see that the modeling results with 1 or 2 sources are in agreement with the experimental data. For emissions of π-/π+, K-/K+ and p-/p, the temperature parameter increases with increase of particle mass (Figures 1(a) and 1(b)), which indicates the impact of radial flow and/or the early emission of heavy hadrons. The temperature does depend nonobviously on rapidity range (Figures 1(c)–1(e)).Table 1
Parameter values corresponding to the curves in Figure1.
Figure
Collision
Particle
Rapidity
T
1 (GeV)
k
1
T
2 (GeV)
χ
2
/
dof
Figure1(a)/Figure 1(b)
pp 200 GeV
π
-
/
π
+
|
y
|
<
0.1
0.165
1.000
—
1.128/1.212
K
-
/
K
+
|
y
|
<
0.1
0.187
1.000
—
0.211/0.243
p
-/p
|
y
|
<
0.1
0.197/0.212
1.000
—
0.340/0.274
Figure1(c)
pp 200 GeV
π
+
0.00
±
0.10
0.175
0.896
0.360
0.849
0.90
±
0.10
0.175
0.930
0.360
0.958
1.20
±
0.10
0.175
0.912
0.360
0.968
2.95
±
0.05
0.175
0.823
0.270
1.415
3.00
±
0.10
0.175
0.856
0.270
1.170
3.30
±
0.10
0.166
0.888
0.270
1.669
3.50
±
0.10
0.205
1.000
—
1.273
Figure1(d)
pp 200 GeV
K
+
0.00
±
0.10
0.160
0.700
0.360
0.499
0.90
±
0.10
0.160
0.600
0.360
0.786
1.20
±
0.10
0.160
0.600
0.360
0.437
2.95
±
0.05
0.198
0.800
0.360
1.048
3.00
±
0.10
0.176
0.788
0.300
0.758
3.30
±
0.10
0.160
0.780
0.273
1.689
3.50
±
0.10
0.160
0.667
0.253
0.714
Figure1(e)
pp 200 GeV
p
0.00
±
0.10
0.160
0.400
0.265
1.028
0.90
±
0.10
0.160
0.540
0.290
1.240
1.20
±
0.10
0.160
0.540
0.290
0.821
2.95
±
0.05
0.156
0.875
0.280
1.457
3.00
±
0.10
0.156
0.872
0.280
0.750
3.30
±
0.10
0.190
0.859
0.280
1.561
3.50
±
0.10
0.176
1.000
—
0.907The transverse momentum distributions of (a)π-, K-, and p-, (b) π+, K+, and p, (c) π+, (d) K+, and (e) p produced in pp collision at s=200 GeV with differentyranges and magnifications are shown in the figure. The symbols represent the experimental data of the STAR [26] and BRAHMS Collaborations [27] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.
(a)
(b)
(c)
(d)
(e)Figure2 shows the pT distributions of J/ψ produced in (a) and (b) pp collision at s=200 GeV, (c) d-Au collisions at sNN=200 GeV, and (d) p-Pb collisions at beam energy being 400 GeV with different y ranges, global scale uncertainty (GSU) (or invariant mass (Mμ+μ-)) ranges, and magnifications shown in the figure, where σ andBdenote the cross section and dilepton branching ratio, respectively. The symbols represent the experimental data of the PHENIX [28] (Figures 2(a) and 2(c)) [29, 30] (Figure 2(b)) and NA50 Collaborations [31] (Figure 2(d)), and the curves are our results calculated by the Boltzmann or two-component Boltzmann distributions. The values of parameters and χ2/dof are given in Table 2. We see again that the model with 1 or 2 sources describes the experimental data. For emission of J/ψ the temperature parameter does depend nonobviously on rapidity range (Figures 2(a) and 2(c)).Table 2
Parameter values corresponding to the curves in Figure2.
Figure
Collision
Particle
Rapidity
T
1 (GeV)
k
1
T
2 (GeV)
χ
2
/
dof
Figure2(a)
pp 200 GeV
J
/
ψ in pp
-
2.2
<
y
<
-
1.2
0.462
1.000
—
0.663
|
y
|
<
0.35
0.462
1.000
—
1.157
1.2
<
y
<
2.2
0.462
1.000
—
1.383
Figure2(b)
pp 200 GeV
J
/
ψ in pp
1.2
<
|
y
|
<
2.2
0.160
0.175
0.541
1.371
|
y
|
<
0.35
0.160
0.175
0.654
1.638
Figure2(c)
d-Au 200 GeV
J
/
ψ in d-Au
-
2.2
<
y
<
-
1.2
0.525
1.000
—
1.332
|
y
|
<
0.35
0.525
1.000
—
1.125
1.2
<
y
<
2.2
0.525
1.000
—
1.845
Figure2(d)
400 GeVp-Pb
J
/
ψ in p-Pb
-
0.425
<
y
<
0.575
0.205
0.650
0.368
1.041The transverse momentum distributions ofJ/ψ produced in (a) and (b) pp collision at s=200 GeV, (c) d-Au collisions at sNN=200 GeV, and (d) p-Pb collisions at beam energy being 400 GeV with differentyranges, GSU (or Mμ+μ-) ranges, and magnifications are shown in the figure. The symbols represent the experimental data of the PHENIX [28–30] and NA50 Collaborations [31] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.
(a)
(b)
(c)
(d)ThepT distributions of identified charged particles produced ind-Au collisions at sNN=200 GeV, Cu-Cu collisions at sNN=22.5 GeV, Au-Au collisions at sNN=62.4, 130, and 200 GeV with different centrality classes are displayed in Figures 3–7, respectively. The symbols represent the experimental data of the STAR [26] (Figures 3, 5, 6, and 7) and PHENIX Collaboration [32] (Figure 4), and the curves are our results calculated by the Boltzmann or two-component Boltzmann distributions. Correspondingly, the values of parameters and χ2/dof are given in Tables 3–7, respectively. Once more, the model with 1 or 2 sources describes the experimental data. From Tables 5, 6, and 7 we see clearly that for emissions of π-, K-, and p- (Figures 5(a), 6(a), and 7(a)), as well as π+, K+, and p (Figures 5(b), 6(b), and 7(b)), the temperature parameter increases with increases of particle mass, impact centrality, and sNN, where we would like to point out that a large centrality (a small percentage) corresponds to a small impact parameter. The similar conclusions can be obtained from Tables 3 and 4.Table 3
Parameter values corresponding to the curves in Figure3(a)/Figure 3(b). The collisions are d + Au at 200 GeV.
Particle
Centrality
T
1 (GeV)
χ
2
/
dof
π
-
/
π
+
0–20%
0.178
0.564/0.704
20–40%
0.170/0.175
1.053/0.779
40–100%
0.170
0.879/0.905
0–100%
0.173
0.796/0.771
K
-
/
K
+
0–20%
0.206/0.228
0.119/0.052
20–40%
0.206/0.218
0.112/0.114
40–100%
0.206/0.213
0.216/0.074
0–100%
0.206/0.217
0.137/0.131
p
-
/
p
0–20%
0.243/0.239
0.036/0.032
20–40%
0.232/0.229
0.047/0.036
40–100%
0.215/0.212
0.091/0.082
0–100%
0.229
0.032/0.013Table 4
Parameter values corresponding to the curves in Figure4.
Figure
Collision
Particle
Centrality
T
1 (GeV)
k
1
T
2 (GeV)
χ
2
/
dof
Figure4(a)/Figure 4(b)
Cu-Cu22.5 GeV
π
-
/
π
+
0–10%
0.183
0.934/0.912
0.310
1.079/1.008
10–30%
0.183
0.934/0.917
0.310
1.195/0.848
30–60%
0.173/0.183
0.934/0.931
0.300/0.310
0.992/1.810
60–100%
0.162/0.160
0.934/0.900
0.293/0.270
1.394/0.959
0–100%
0.183
0.934/0.920
0.310
1.229/1.084
Figure4(c)/Figure 4(d)
Cu-Cu22.5 GeV
K
-
/
K
+
0–10%
0.183
0.860/0.747
0.310
0.636/0.593
10–30%
0.183
0.840/0.732
0.310
0.632/0.818
30–60%
0.183
0.900/0.755
0.310
1.087/1.440
60–100%
0.183
0.975/0.747
0.310
1.677/1.311
0–100%
0.183
0.860/0.747
0.310
0.632/0.515
Figure4(e)/Figure 4(f)
Cu-Cu22.5 GeV
p
-
/
p
0–10%
0.183/0.200
0.760/0.510
0.310
1.898/0.904
10–30%
0.183/0.200
0.860/0.567
0.310
0.825/1.159
30–60%
0.183/0.200
0.883/0.580
0.310
1.849/1.747
60–100%
0.160/0.200
1.000/0.888
—/0.310
1.953/1.855
0–100%
0.183/0.200
0.862/0.566
0.310
1.090/1.466Table 5
Parameter values corresponding to the curves in Figure5(a)/Figure 5(b). The collisions are Au + Au at 62.4 GeV.
Particle
Centrality
T
1 (GeV)
χ
2
/
dof
π
-
/
π
+
0–5%
0.185
0.184/0.184
5–10%
0.185
0.192/0.156
10–20%
0.182
0.224/0.246
20–30%
0.180
0.230/0.228
30–40%
0.180
0.276/0.302
40–50%
0.175
0.112/0.128
50–60%
0.170
0.118/0.146
60–70%
0.170
0.138/0.234
70–80%
0.165
0.254/0.292
K
-
/
K
+
0–5%
0.275
0.056/0.010
5–10%
0.275
0.024/0.018
10–20%
0.275
0.040/0.018
20–30%
0.253
0.064/0.046
30–40%
0.243
0.102/0.154
40–50%
0.235
0.082/0.132
50–60%
0.215
0.166/0.084
60–70%
0.205
0.100/0.080
70–80%
0.205
0.362/0.130
p
-
/
p
0–5%
0.512
0.114/0.166
5–10%
0.475
0.068/0.120
10–20%
0.442
0.048/0.096
20–30%
0.393
0.040/0.136
30–40%
0.355
0.076/0.082
40–50%
0.305
0.104/0.080
50–60%
0.270
0.070/0.086
60–70%
0.250
0.038/0.058
70–80%
0.220
0.020/0.084Table 6
Parameter values corresponding to the curves in Figure6(a)/Figure 6(b). The collisions are Au + Au at 130 GeV.
Particle
Centrality
T
1 (GeV)
χ
2
/
dof
π
-
/
π
+
0–6%
0.185
0.210/0.268
6–11%
0.182
0.290/0.232
11–18%
0.180
0.430/0.328
18–26%
0.180
0.394/0.310
26–34%
0.175
0.222/0.172
34–45%
0.170
0.126/0.144
45–58%
0.170
0.192/0.234
58–85%
0.165
0.334/0.354
K
-/K+
0–6%
0.275
0.088/0.146
6–11%
0.275
0.268/0.087
11–18%
0.273/0.263
0.151/0.584
18–26%
0.283
0.189/0.584
26–34%
0.255/0.235
0.101/0.598
34–45%
0.260/0.245
0.846/0.293
45–58%
0.245/0.205
0.639/0.448
58–85%
0.205
0.311/0.543
p
-/p
0–6%
0.580
0.177/0.177
6–11%
0.530
0.157/0.128
11–18%
0.450
0.123/0.104
18–26%
0.410
0.105/0.118
26–34%
0.410
0.103/0.068
34–45%
0.363
0.071/0.121
45–58%
0.303
0.076/0.065
58–85%
0.265
0.029/0.055Table 7
Parameter values corresponding to the curves in Figure7(a)/Figure 7(b). The collisions are Au + Au at 200GeV.
Particle
Centrality
T
1 (GeV)
χ
2
/
dof
π
-
/
π
+
0–5%
0.193
0.940/0.746
5–10%
0.193
0.865/0.907
10–20%
0.193/0.190
0.780/0.444
20–30%
0.190/0.188
0.628/0.661
30–40%
0.188
1.432/1.449
40–50%
0.188
0.570/0.577
50–60%
0.180
0.974/0.914
60–70%
0.175
1.269/1.295
70–80%
0.170
1.269/1.311
K
-
/
K
+
0–5%
0.327
0.019/0.020
5–10%
0.327
0.019/0.032
10–20%
0.285
0.044/0.034
20–30%
0.285
0.029/0.022
30–40%
0.243
0.112/0.107
40–50%
0.243
0.082/0.079
50–60%
0.243
0.007/0.021
60–70%
0.215
0.081/0.071
70–80%
0.200
0.235/0.143
p
-
/
p
0–5%
0.570
0.046/0.031
5–10%
0.516
0.071/0.013
10–20%
0.516
0.037/0.045
20–30%
0.435
0.047/0.035
30–40%
0.396
0.039/0.036
40–50%
0.338/0.345
0.035/0.032
50–60%
0.312/0.320
0.025/0.031
60–70%
0.270
0.086/0.057
70-80%
0.246
0.045/0.022The transverse momentum distributions of (a)π-, K-, and p- as well as (b) π+, K+, and p produced in d-Au collisions at sNN=200 GeV with |y|<0.1 and different centrality classes and magnifications are shown in the figure. The symbols represent the experimental data of the STAR Collaboration [26] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.
(a)
(b)The transverse momentum distributions of (a)π-, (b) π+, (c) K-, (d) K+, (e) p-, and (f) p produced in Cu-Cu collisions at sNN=22.5 GeV with different centrality classes and magnifications are shown in the figure. The symbols represent the experimental data of the PHENIX Collaboration [32] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.
(a)
(b)
(c)
(d)
(e)
(f)The transverse momentum distributions of (a)π-, K-, and p-, and (b) π+, K+, and p produced in Au-Au collisions at sNN=62.4 GeV with different centrality classes and magnifications are shown in the figure. The symbols represent the experimental data of the STAR Collaboration [26] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.
(a)
(b)The same as Figure5, but showing the results at sNN=130 GeV.
(a)
(b)The same as for Figure5, but showing the results at sNN=200 GeV.
(a)
(b)Figure8 gives the pT distributions of (a)–(c) identified particles and (d) charged particles in range of |η|<0.8 in nonsingle diffraction (NSD) produced in pp collision at s=900 GeV, where NEV and Nch denote numbers of events and charged particles, respectively. The symbols represent the experimental data of the ALICE Collaboration [33, 34] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. The values of parameters and χ2/dof are given in Table 8. We see that the model with 1 or 2 sources describes the experimental data. For emissions of charged hadrons, the temperature parameter increases with increase of particle mass.Table 8
Parameter values corresponding to the curves in Figure8.
Figure
Collision
Particle
T
1 (GeV)
k
1
T
2 (GeV)
χ
2
/
dof
Figure8(a)/Figure 8(b)
pp 900 GeV
π
-
/
π
+
0.148/0.142
0.803/0.765
0.349/0.341
0.153/0.369
K
-/K+
0.185
0.670
0.427
0.158/0.375
p
-/p
0.185
0.700/0.670
0.390/0.427
0.103/0.156
Figure8(c)
pp 900 GeV
K
S
0
0.132
0.505
0.369
0.375
Λ
0.323
1.000
—
1.026
Λ
-
0.311
1.000
—
1.375
ϕ
0.362
1.000
—
0.890
Ξ
-
+
+
Ξ
-
0.325
1.000
—
0.664
Figure8(d)
pp 900 GeV
charged
0.259
0.985
0.754
1.903The transverse momentum distributions of (a)π-, K-, and p-, (b) π+, K+, and p, (c) KS0,Λ, Λ-, ϕ, and Ξ-++Ξ-, and (d) charged particles in range of |η|<0.8 in NSD produced in pp collisions at s=900 GeV with different magnifications are shown in the figure. The symbols represent the experimental data of the ALICE Collaboration [33, 34] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.
(a)
(b)
(c)
(d)ThepT distributions of identified particles produced in (a) central Pb-Pb collisions at sNN=2.76 TeV, (b) Pb-Pb collisions with different centralities at sNN=2.76 TeV and inelastic pp collision at s=2.76 TeV, (c) central rapidity region in Pb-Pb collisions with different centralities at sNN=2.76 TeV, and (d) central rapidity region in pp collision at s=7 TeV are presented in Figure 9. The symbols represent the experimental data of the ALICE Collaboration [35–38] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. The values of parameters and χ2/dof are given in Table 9. We see that in most cases the model with 1 or 2 sources describes the experimental data. Especially, for emissions of π-, K-, and p- (Figure 9(a)), the temperature parameter increases with increase of particle mass; for emission of π0 (Figure 9(b)), the temperature parameter does not depend on impact centrality; and for emission of p- (Figure 9(c)), the temperature parameter increases with increase of impact centrality (or with decrease of impact parameter).Table 9
Parameter values corresponding to the curves in Figure9.
Figure
Collision
Particle
Type
T
1 (GeV)
k
1
T
2 (GeV)
χ
2
/
dof
Figure9(a)
Pb-Pb 2.76 TeV
π
-
0–5%
0.169
0.595
0.364
0.858
K
S
0
+
K
-
0–5%
0.355
0.900
0.460
0.141
p
-
0–5%
0.697
1.000
—
0.832
Figure9(b)
Pb-Pb 2.76 TeV
π
0
0–20%
0.332
0.985
0.900
1.321
20–40%
0.332
0.985
0.900
1.760
40–60%
0.332
0.985
0.900
1.324
60–80%
0.332
0.975
0.900
0.583
pp 2.76 TeV
Inelastic
0.332
0.965
0.900
1.816
Figure9(c)
Pb-Pb 2.76 TeV
p
-
0–5%,|y|<0.5
0.632
1.000
—
0.517
5–10%,|y|<0.5
0.600
1.000
—
0.324
10–20%,|y|<0.5
0.580
1.000
—
0.275
20–30%,|y|<0.5
0.540
1.000
—
0.200
30–40%,|y|<0.5
0.532
1.000
—
0.275
40–50%,|y|<0.5
0.461
1.000
—
0.432
50–60%,|y|<0.5
0.450
1.000
—
0.296
60–70%,|y|<0.5
0.385
1.000
—
0.371
Figure9(d)
pp 7 TeV
Ξ
|
y
|
<
0.5
0.350
0.700
0.730
0.510
Ω
|
y
|
<
0.5
0.565
1.000
—
0.666The transverse momentum distributions of (a)π-, KS0+K-, and p-, (b) π0, (c) p-, and (d) Ξ and Ω produced in Pb-Pb collisions at sNN=2.76 TeV ((a), (b), and (c)) and pp collision at s=2.76 and 7 TeV ((b) and (d)) with differentyranges, centrality classes, and magnifications are shown in the figure. The symbols represent the experimental data of the ALICE Collaboration [35–38] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.
(a)
(b)
(c)
(d)From Tables1–9 we see that some values of χ2/dof are too low pointing to overestimated errors of the experimental points. In fact, in the case of errors being not available in related references, we have used a half size of the experimental points to give the errors. This treatment may cause larger errors in some cases.


## 4. Discussions and Conclusions

From the above discussions we see that the model used in the present work is just a simple phenomenology which does not contain other processes such as parton-hadron string dynamics, hydrodynamic flows, and resonances. The successful description renders that the mentioned processes should contribute a higher transverse momentum at multi-GeV energy or a refined structure in distribution curve. In the concerned transverse momentum region and for the concerned distribution curves, we just need to consider the Maxwell-Boltzmann thermal law.The present work is justified to compare fits in a low transverse momentum region (<10 GeV/c) for different particles by the same thermal law. Although the difference for charged and neutral particles is unlikely due to Coulomb effects which are important for very soft charged particles only, both the charged and neutral particles obey the same thermal law. The transverse momentum can extend to more than 100 GeV/c at multi-GeV energy. The distribution in the low transverse momentum region is mainly contributed by the soft processes. The hard processes which contribute high transverse momentums can be partly described by the thermal law.To conclude, we have used the multisource thermal model to describe the transverse momentum distributions of particles produced in the soft process inpp and AA collisions at RHIC and LHC energies. For single source, the relativistic ideal gas model is applied in description of particle behavior. The concerned distribution is finally described by single source or two sources which result from a Boltzmann or two-component Boltzmann distributions. The modeling results are in agreement with available experimental data, which renders that an equilibrium or two local equilibriums are reached in high energy collisions. Because of the evolvement time of interesting system in collisions being very short, the particles should reach rapidly to the state of equilibrium.The present work can be used to extract nuclear temperature for soft process. For emissions of charged hadrons, the temperature parameter increases with increases of particle mass, impact centrality, and center-of-mass energy and does depend nonobviously on rapidity range. That the temperature increases with particle mass indicates the impact of radial flow and/or the early emission of heavy hadrons. The temperature parameter for emission ofJ/ψ does depend nonobviously on rapidity range too, which is consistent with charged hadrons. However, for emission of π0 the temperature parameter does not depend on impact centrality, which is inconsistent with charged hadrons. Different behaviors for π0 and π-/π+ render different production mechanisms. Especially, there are Coulomb corrections for emissions of charged particles, which affects the extraction of temperature [39].The values of temperature parameter for emissions ofπ-/π+ are about 160–190 MeV which reaches the temperature (166–172 MeV) of creating QGP at zero baryon-chemical potential, where 172 MeV is the equilibrium phase transition temperature and 166 MeV is due to finite hadron size [40]. In most cases the temperature for emission of heavy hadrons is greater than that for pions, which renders the impact of radial flow and/or the early emission of heavy hadrons in collisions.



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*Source: 965735-2013-11-03.xml*