# Comparing the Tsallis Distribution with and without Thermodynamical Description inp

Authors: H. Zheng; Lilin Zhu
Journal: Advances in High Energy Physics
 (2016)

---

## Abstract

We compare two types of Tsallis distribution, that is, with and without thermodynamical description, using the experimental data from the STAR, PHENIX, ALICE, and CMS Collaborations on the rapidity and energy dependence of the transverse momentum spectra inp
+
p collisions. Both of them can fit the particle spectra well. We show that the Tsallis distribution with thermodynamical description gives lower temperatures than the ones without it. The extra factor m
T (transverse mass) in the Tsallis distribution with thermodynamical description plays an important role in the discrepancies between the two types of Tsallis distribution. But for the heavy particles, the choice to use m
T or E
T (transverse energy) in the Tsallis distribution becomes more crucial.


---

## Body

## 1. Introduction

The particle spectrum is a basic quantity directly measured in the experiments and it can reveal the information of particle production mechanism in heavy-ion collisions. Many physicists have devoted themselves to studying the particle spectra produced in the heavy-ion collisions using thermodynamical approaches, phenomenological methods, transport models, and so forth [1–22]. Recently, the Tsallis distribution, which was first proposed about twenty-seven years ago as a generalization of the Boltzmann-Gibbs distribution [23], has attracted many theorists’ and experimentalists’ attention in high energy collisions [5–9, 11–17, 24–34]. The excellent ability to fit the spectra of identified hadrons and charged particles in a large range of p
T up to 200 GeV/c, which covers 15 orders of magnitude, is quite impressive. This spectacular result was first shown by Wong et al. [14–16]. In [21, 22], we have shown that Tsallis distribution can fit almost all the particle spectra produced in p
+
p, p
+
A, and A
+
A at RHIC and LHC. From the phenomenological view, there may be real physics behind the prominently phenomenological work, for example, Regge trajectory for particle classification [35]. We also note that there are different versions of Tsallis distribution in the literature and we classify them as Type A, Type B, and Type C to clarify the comparison in [21]. Type A Tsallis distribution is obtained without resorting to thermodynamical description, but it has been adopted to analyze the particle spectra by STAR [24] and PHENIX [25] Collaborations at RHIC and ALICE [26–28] and CMS [29] Collaborations at LHC. In [21], we applied it to do the systematic analysis of identified particle spectra in p
+
p collisions at RHIC and LHC and proposed a cascade particle production mechanism. On the other hand, Type B Tsallis distribution is derived by taking into account the thermodynamical consistency and is widely used by Cleymans and his collaborators to study the particle spectra in high energy p
+
p collisions [8–10]. It is also used by other authors for nucleus-nucleus interactions [11]. Type A and Type B are the most popular Tsallis distributions in the literature but they give quite different temperatures while fitting the same particle spectra; for example, for pion, Type A gives T
~
0.13 GeV while Type B gives T
~
0.075 GeV. In this paper, we would like to systematically address the question regarding the discrepancies of the temperatures for the two types of Tsallis distribution, by using particle spectra in p
+
p collisions. The data produced in p
+
p collisions with different p
T ranges and different rapidity cuts are collected from the experimental collaborations at RHIC and LHC [25–27, 30, 36–42].The paper is organized as follows. In Section2, we introduce the two types of Tsallis distribution, without and with thermodynamical description, in our comparison. In order to make our discussion clear, we also introduce another three transient distributions which are very similar to Type A and Type B Tsallis distributions. In Section 3, the results of particle spectra from the different distributions in p
+
p collisions and the comparisons are shown. A brief conclusion is given in Section 4.


## 2. Tsallis Distributions

Type A Tsallis distribution has been widely adopted by STAR [24] and PHENIX [25] Collaborations at RHIC and ALICE [26–28] and CMS [29] Collaborations at LHC:(1)
E
d
3
N
d
p
3
=
d
N
d
y
n
-
1
n
-
2
2
π
n
C
n
C
+
m
n
-
2
1
+
m
T
-
m
n
C
-
n
,where m
T
=
p
T
2
+
m
2 is the transverse mass, d
N
/
d
y, n, and C are fitting parameters, and m was used as a fitting parameter in [24], but it represents the rest mass of the particle studied in [25–29]. When p
T
≫
m, we can ignore m in the last term in (1) and obtain E
d
3
N
/
d
p
3
∝
p
T
-
n. This result is well known because high energy particles come from hard scattering and they follow a power law distribution with p
T. When p
T
≪
m which is the nonrelativistic limit, we obtain m
T
-
m
=
p
T
2
/
2
m
=
E
T
c
l
a
s
s
i
c
a
l and E
d
3
N
/
d
p
3
∝
e
-
E
T
c
l
a
s
s
i
c
a
l
/
C, that is, a thermal distribution. The parameter C in (1) plays the same role as temperature T. In [21, 22], we have obtained the simpler form of (1):(2)
E
d
3
N
d
p
3
η
<
a
=
A
1
+
E
T
n
T
-
n
,where the transverse energy E
T
=
m
T
-
m. A, n, and T are free fitting parameters in (2). We note that it has been used by CMS Collaboration [31, 32, 43] and by Wong et al. in their recent paper [16]. The STAR Collaboration also applied a formula which is very close to (2) [44]. We adopt (2) in the following study.In the framework of Tsallis statistics, the distribution function is(3)
f
E
,
q
=
1
+
q
-
1
E
-
μ
T
-
1
/
q
-
1
.Taking into account the self-consistent thermodynamical description, one has to use the effective distribution [
f
(
E
,
q
)
]
q. Therefore, the Tsallis distribution is obtained [8, 9, 11, 12]:(4)
E
d
3
N
d
p
3
=
g
V
m
T
cosh
⁡
y
2
π
3
1
+
q
-
1
m
T
cosh
⁡
y
-
μ
T
-
q
/
q
-
1
,where g is the degeneracy of the particle state, V is the volume, y is the rapidity, μ is the chemical potential, T is the temperature, and q is the entropic factor, which measures the nonadditivity of the entropy. We dubbed it as the Type B Tsallis distribution [21]. In (4), there are four parameters, namely, V, μ, T, and q. μ was assumed to be 0 in [8, 9, 11] which is a reasonable assumption because the energy is high enough and the chemical potential is much smaller than the temperature. In the midrapidity y
=
0 region, (4) is reduced to(5)
E
d
3
N
d
p
3
=
g
V
m
T
2
π
3
1
+
q
-
1
m
T
T
-
q
/
q
-
1
.It becomes very similar to (2), but there are some differences; for example, m
T replaces E
T in the bracket and there is an extra term m
T in front of the bracket. It should be pointed out that there is no direct match between n and q in (2) and (5). But we could find a connection between n and q in the limit at large p
T. When p
T
≫
m, from (5), we can obtain(6)
E
d
3
N
d
p
3
∝
p
T
-
1
/
q
-
1
.Recalling that E
d
3
N
/
d
p
3
∝
p
T
-
n when p
T
≫
m from (1), the relation between n and q is(7)
n
=
1
q
-
1
.Another treatment to find the relation between n and q can be found in [10].For the other Tsallis distributions in the literature, we refer them to [21, 22]. We noted that Type A and Type B Tsallis distributions can reproduce the particle spectra in p
+
p collisions very well, but Type B gives lower temperatures than the ones given by Type A. In this paper, we would like to address this discrepancies between the two types of Tsallis distribution. To make our discussion clear, another three transient distributions are used to bridge Type A and Type B distributions. In [45], a Tsallis-like distribution is obtained in the framework of nonextensive statistics for the particle invariant yield at midrapidity:(8)
E
d
3
N
d
p
3
=
A
m
T
1
+
q
-
1
m
T
T
-
1
/
q
-
1
,where A, q, and T are fitting parameters. Comparing (8) and (5), the only difference is the power of the distribution function, that is, q for (5) and 1 for (8). We also introduce another two forms of distribution. One is(9)
E
d
3
N
d
p
3
=
A
1
+
q
-
1
m
T
T
-
q
/
q
-
1
,where we neglect the term m
T outside of the bracket in (5) and the constants are absorbed into the parameter A. The other one is(10)
E
d
3
N
d
p
3
=
A
1
+
q
-
1
m
T
T
-
1
/
q
-
1
.Similar to (9), we neglect the term m
T in front of the bracket in (8) but keep the rest. Using the relation equation (7), we find that (10) becomes (2) in the limit of massless particle.Let us put the four distributions in order of (5), (8), (10), and (2) and (5), (9), (10), and (2); one can see that any adjacent distributions have only one term different. We successfully bridge Type A and Type B Tsallis distributions and are able to conduct our investigation. We need to point out that even though we use the same symbols for the parameters in all five distributions, they may have different values when fitting the experimental data. In the next section, we will systematically apply the five distributions to the particle spectra in p
+
p collisions, similar to our previous work [21]. But we update some experimental data which have larger p
T ranges and will focus on the temperature differences among the five distributions.


## 3. Results

We fit the particle spectra with differentp
T ranges and different rapidity cuts from p
+
p collisions at s
=
62.4, 200, 900, 2760, and 7000 GeV with the five distributions discussed in the last section. The fitting process is the same as that in [21, 22]. Compared with [21], the identified particle spectra data at s
=
2760 and 7000 GeV have been updated.In this work, we are interested in the differences of the parameterT from the five distributions. As we argued in [21, 22], T is one free fitting parameter and can be different for different particles even though they are produced in the same colliding system. In order to distinguish the parameters T and q in the distributions and make our discussion clear, we assign T
1
(
q
1
), T
2
(
q
2
), T
3
(
q
3
), and T
4
(
q
4
) to (5), (8), (9), and (10), respectively, while we assignT
5 and n for (2). All the five distributions can fit the particle transverse momentum spectra very well. The values of T
i (i
=
1,2
,
3,4
,
5), q
i (i
=
1,2
,
3,4), n, and corresponding χ
i
2
/
n
d
f for π, K, p, and charged particles are shown in Tables 1 and 2. Furthermore, the errors of T
i, q
i and n are also provided in the tables. Here, we only show the fitting results with the five distributions for four cases: (1) π
+ at s
=
200 GeV, (2) π
+ at s
=
900 GeV, (3) K
+
+
K
- at s
=
2760 GeV, and (4) p
+
p
¯ at s
=
7000 GeV.Table 1
The fitting parametersT
i, q
i  (
i
=
1,2
,
3
), the errors of parameters, and χ
2/ndf in (5), (8), and (9) for the particle spectra in p
+
p collisions. The unit of T is GeV.
Data source
s (GeV)
Particle
T
1
q
1
χ
1
2/ndf
T
2
q
2
χ
2
2/ndf
T
3
q
3
χ
3
2/ndf
PHENIX [37]
62.4
π
0
0.106
±
0.001
1.076
±
0.0009
2.923/11
0.101
±
0.0001
1.070
±
0.0001
3.622/11
0.140
±
0.0003
1.078
±
0.0001
1.353/11
PHENIX [25]
62.4
π
+
0.0905
±
0.0001
1.092
±
0.0002
6.981/23
0.083
±
0.0007
1.084
±
0.0006
6.781/23
0.130
±
0.0006
1.091
±
0.002
4.794/23
π
-
0.0895
±
0.0004
1.091
±
0.0009
8.617/23
0.0824
±
0.001
1.083
±
0.002
8.805/23
0.125
±
0.0001
1.093
±
0.0001
5.383/23
K
+
0.0979
±
0.0005
1.097
±
0.002
5.205/13
0.0767
±
0.0004
1.100
±
0.0008
5.253/13
0.130
±
0.0005
1.110
±
0.001
5.346/13
K
-
0.0959
±
0.0002
1.093
±
0.0001
2.718/13
0.0773
±
0.0005
1.093
±
0.001
2.432/13
0.132
±
0.001
1.096
±
0.004
2.285/13
p
0.100
±
0.0001
1.065
±
0.0002
6.755/24
0.088
±
0.0001
1.063
±
0.0001
6.855/24
0.108
±
0.0001
1.075
±
0.0002
7.651/24
p
-
0.0894
±
0.0001
1.064
±
0.0002
3.655/22
0.0901
±
0.0001
1.057
±
0.0001
3.637/22
0.0906
±
0.0001
1.079
±
0.0002
5.931/22
PHENIX [38]
200
π
0
0.0898
±
0.0005
1.107
±
0.0006
17.137/22
0.0787
±
0.0004
1.098
±
0.0004
20.184/22
0.114
±
0.0003
1.121
±
0.0002
27.743/22
PHENIX [25]
200
π
+
0.0747
±
0.0001
1.126
±
0.0005
5.638/24
0.0671
±
0.0001
1.112
±
0.0002
5.816/24
0.111
±
0.0001
1.138
±
0.0006
4.698/24
π
-
0.0799
±
0.0001
1.121
±
0.0001
4.861/24
0.0718
±
0.0001
1.107
±
0.0003
4.652/24
0.121
±
0.0001
1.126
±
0.0002
3.547/24
K
+
0.0416
±
0.0001
1.166
±
0.0005
1.886/13
0.0494
±
0.0004
1.131
±
0.002
2.157/13
0.0819
±
0.0001
1.176
±
0.003
2.062/13
K
-
0.0672
±
0.0001
1.140
±
0.0003
3.819/13
0.0613
±
0.0001
1.121
±
0.0003
4.039/13
0.0960
±
0.0001
1.162
±
0.0004
3.355/13
p
0.0444
±
0.0001
1.111
±
0.0002
19.868/31
0.0395
±
0.0001
1.100
±
0.0001
19.986/31
0.0386
±
0.0001
1.135
±
0.0001
24.485/31
p
-
0.0414
±
0.0001
1.110
±
0.0001
15.918/31
0.0321
±
0.001
1.102
±
0.005
20.450/31
0.0541
±
0.0001
1.122
±
0.0002
15.892/31
STAR [39]
200
π
+
0.0860
±
0.0001
1.108
±
0.0003
6.953/20
0.0735
±
0.0001
1.099
±
0.0002
12.536/20
0.124
±
0.0001
1.115
±
0.0002
5.979/20
π
-
0.0884
±
0.0003
1.107
±
0.0003
6.240/20
0.0791
±
0.003
1.097
±
0.001
5.977/20
0.128
±
0.0005
1.113
±
0.0002
5.099/20
p
0.0774
±
0.0006
1.093
±
0.0006
12.112/17
0.0576
±
0.0002
1.091
±
0.0001
11.137/17
0.074
±
0.0001
1.115
±
0.0002
12.495/17
p
-
0.0676
±
0.0001
1.098
±
0.0002
9.595/17
0.0620
±
0.0001
1.090
±
0.0001
9.017/17
0.0829
±
0.002
1.110
±
0.004
9.572/17
ALICE [26]
900
π
0
0.0915
±
0.0001
1.131
±
0.0002
7.588/10
0.0951
±
0.0006
1.108
±
0.0008
9.523/10
0.122
±
0.0002
1.154
±
0.0001
7.715/10
ALICE [27]
900
π
+
0.0712
±
0.0002
1.147
±
0.0006
21.599/30
0.0623
±
0.0004
1.128
±
0.002
21.912/30
0.123
±
0.0002
1.148
±
0.0005
13.382/30
π
-
0.0723
±
0.0002
1.144
±
0.0001
15.642/30
0.0636
±
0.0001
1.126
±
0.0001
14.984/30
0.126
±
0.0002
1.141
±
0.003
11.206/30
K
+
0.0656
±
0.0001
1.168
±
0.0008
19.692/24
0.0516
±
0.0001
1.148
±
0.0002
16.661/24
0.0661
±
0.0001
1.240
±
0.0005
16.831/24
K
-
0.0696
±
0.0005
1.157
±
0.0001
7.919/24
0.0578
±
0.0009
1.138
±
0.002
7.240/24
0.0985
±
0.0003
1.191
±
0.0001
6.804/24
p
0.0513
±
0.0001
1.133
±
0.0002
14.723/21
0.0466
±
0.0001
1.117
±
0.0002
14.808/21
0.0359
±
0.0001
1.179
±
0.0009
14.901/21
p
-
0.0446
±
0.0001
1.137
±
0.0006
14.094/21
0.0457
±
0.0003
1.115
±
0.0007
14.348/21
0.0416
±
0.0001
1.178
±
0.0003
16.739/21
ALICE [41]
2760
π
0
0.0945
±
0.0001
1.141
±
0.0002
6.164/15
0.0826
±
0.0001
1.123
±
0.0002
6.162/15
0.143
±
0.0004
1.159
±
0.0004
5.981/15
ALICE [36]
2760
π
+
+
π
-
0.0846
±
0.0002
1.144
±
0.0001
167.103/60
0.0743
±
0.0001
1.126
±
0.0002
172.168/60
0.128
±
0.0005
1.166
±
0.0002
31.492/60
K
+
+
K
-
0.0966
±
0.0002
1.143
±
0.0003
10.339/55
0.0845
±
0.0005
1.125
±
0.0008
10.347/55
0.144
±
0.0001
1.163
±
0.0001
13.534/55
p
+
p
-
0.0812
±
0.0001
1.122
±
0.0001
27.683/46
0.0789
±
0.001
1.106
±
0.0007
20.581/46
0.120
±
0.002
1.133
±
0.0009
20.062/46
ALICE [26]
7000
π
0
0.0951
±
0.0002
1.146
±
0.0002
11.071/30
0.0828
±
0.0002
1.128
±
0.0002
11.112/30
0.139
±
0.0001
1.170
±
0.0002
13.443/30
ALICE [42]
7000
π
+
+
π
-
0.0712
±
0.0005
1.172
±
0.0007
68.408/38
0.0608
±
0.0005
1.147
±
0.001
68.193/38
0.125
±
0.0002
1.191
±
0.0004
26.041/38
K
+
+
K
-
0.0986
±
0.0001
1.158
±
0.0008
7.807/45
0.0840
±
0.0002
1.137
±
0.0001
7.910/45
0.152
±
0.0003
1.184
±
0.0004
5.335/45
p
+
p
-
0.113
±
0.0001
1.117
±
0.0003
16.534/43
0.105
±
0.0002
1.103
±
0.0002
18.608/43
0.159
±
0.0001
1.129
±
0.002
13.778/43
CMS [30]
900
Charged
0.0875
±
0.0002
1.132
±
0.0002
63.863/17
0.0780
±
0.0001
1.117
±
0.0001
63.940/17
0.127
±
0.0002
1.150
±
0.0002
85.844/17
CMS [40]
2760
Charged
0.0972
±
0.0006
1.145
±
0.0002
83.860/19
0.0856
±
0.0005
1.126
±
0.0003
84.026/19
0.136
±
0.002
1.169
±
0.0001
110.614/19
CMS [30]
7000
Charged
0.101
±
0.0009
1.152
±
0.0004
64.009/24
0.0879
±
0.0005
1.132
±
0.0001
63.955/24
0.147
±
0.001
1.179
±
0.0002
80.316/24Table 2
The fitting parametersT
4 and  q
4 in (10), T
5 and n  in (2), the errors of parameters, and χ
2/ndf  for the particle spectra in p  + p collisions. The unit of T is GeV.
Data source
s (GeV)
Particle
T
4
q
4
χ
4
2/ndf
T
5
n
χ
5
2/ndf
PHENIX [37]
62.4
π
0
0.130
±
0.0003
1.072
±
0.0001
1.340/11
0.139
±
0.0003
13.729
±
0.121
1.302/11
PHENIX [25]
62.4
π
+
0.121
±
0.0007
1.081
±
0.002
5.386/23
0.131
±
0.0002
11.867
±
0.017
4.779/23
π
-
0.114
±
0.0002
1.085
±
0.0001
5.402/23
0.125
±
0.0003
11.661
±
0.349
5.194/23
K
+
0.111
±
0.0004
1.104
±
0.004
5.423/13
0.160
±
0.0004
9.384
±
0.011
5.121/13
K
-
0.119
±
0.001
1.089
±
0.0001
2.271/13
0.161
±
0.0001
10.95
±
0.007
2.186/13
p
0.099
±
0.0002
1.071
±
0.0003
7.861/24
0.176
±
0.0001
16.099
±
0.281
6.966/24
p
-
0.119
±
0.0004
1.055
±
0.0003
4.136/22
0.149
±
0.0008
12.854
±
0.005
7.178/22
PHENIX [38]
200
π
0
0.108
±
0.0001
1.106
±
0.0001
20.705/22
0.115
±
0.0004
9.180
±
0.018
23.025/22
PHENIX [25]
200
π
+
0.0971
±
0.001
1.122
±
0.002
4.740/24
0.115
±
0.0008
8.291
±
0.025
4.485/24
π
-
0.107
±
0.0002
1.113
±
0.0006
3.352/24
0.122
±
0.0001
8.848
±
0.020
3.354/24
K
+
0.0531
±
0.0003
1.165
±
0.009
1.665/13
0.136
±
0.0009
6.222
±
0.315
1.587/13
K
-
0.0915
±
0.0006
1.133
±
0.009
4.321/13
0.148
±
0.0009
6.941
±
0.081
2.999/13
p
0.0372
±
0.0003
1.117
±
0.002
22.734/31
0.142
±
0.0002
8.164
±
0.006
24.581/31
p
-
0.0481
±
0.0001
1.109
±
0.0001
15.932/31
0.151
±
0.0009
9.210
±
0.006
13.535/31
STAR [39]
200
π
+
0.113
±
0.003
1.103
±
0.002
5.743/20
0.127
±
0.0001
9.725
±
0.001
5.972/20
π
-
0.115
±
0.0002
1.101
±
0.0001
5.144/20
0.129
±
0.0002
9.912
±
0.011
4.705/20
p
0.0703
±
0.0001
1.101
±
0.0001
11.452/17
0.174
±
0.0001
10.736
±
0.089
10.359/17
p
-
0.0709
±
0.0001
1.101
±
0.0002
9.769/17
0.177
±
0.0004
10.751
±
0.006
9.991/17
ALICE [26]
900
π
0
0.107
±
0.0004
1.133
±
0.0006
7.687/10
0.137
±
0.0002
7.947
±
0.002
7.537/10
ALICE [27]
900
π
+
0.107
±
0.0006
1.130
±
0.0001
13.871/30
0.125
±
0.0005
7.703
±
0.058
13.460/30
π
-
0.111
±
0.0001
1.123
±
0.0003
11.199/30
0.128
±
0.002
8.090
±
0.601
12.483/30
K
+
0.0689
±
0.0035
1.178
±
0.002
14.271/24
0.159
±
0.0001
5.733
±
0.017
12.980/24
K
-
0.0875
±
0.0001
1.157
±
0.0003
7.515/24
0.168
±
0.0007
6.588
±
0.022
6.609/24
p
0.0407
±
0.0001
1.145
±
0.0002
15.148/21
0.195
±
0.0006
8.537
±
0.008
13.974/21
p
-
0.0404
±
0.0001
1.147
±
0.0002
15.637/21
0.190
±
0.0008
8.660
±
0.379
13.675/21
ALICE [41]
2760
π
0
0.123
±
0.0002
1.137
±
0.0002
5.979/15
0.142
±
0.002
7.300
±
0.112
5.987/15
ALICE [36]
2760
π
+
+
π
-
0.110
±
0.0002
1.142
±
0.0003
30.885/60
0.130
±
0.0002
7.034
±
0.001
31.583/60
K
+
+
K
-
0.123
±
0.0001
1.140
±
0.0001
13.608/55
0.193
±
0.0003
7.131
±
0.0008
14.481/55
p
+
p
-
0.104
±
0.0002
1.119
±
0.0004
21.531/46
0.216
±
0.0003
8.484
±
0.0005
32.877/46
ALICE [26]
7000
π
0
0.120
±
0.0005
1.145
±
0.001
13.240/30
0.139
±
0.0003
6.885
±
0.006
16.750/30
ALICE [42]
7000
π
+
+
π
-
0.105
±
0.0002
1.160
±
0.0004
26.539/38
0.128
±
0.0004
6.257
±
0.0002
26.216/38
K
+
+
K
-
0.131
±
0.0002
1.154
±
0.0003
5.373/45
0.206
±
0.0003
6.497
±
0.009
5.333/45
p
+
p
-
0.140
±
0.0002
1.114
±
0.0002
13.768/43
0.247
±
0.0006
8.750
±
0.006
12.700/43
CMS [30]
900
charged
0.115
±
0.0001
1.129
±
0.0005
85.776/17
0.126
±
0.0002
7.627
±
0.018
84.867/17
CMS [40]
2760
charged
0.117
±
0.0004
1.144
±
0.0001
110.770/19
0.137
±
0.005
6.915
±
0.035
110.4/19
CMS [30]
7000
charged
0.124
±
0.004
1.152
±
0.0001
80.353/24
0.145
±
0.0003
6.583
±
0.014
80.25/24In Figures1 and 2, we show the fitting results for π
+ at s
=
200 GeV from STAR Collaboration and s
=
900 GeV from ALICE Collaboration using the five distributions: (5) (solid line), (8) (dashed line), (9) (dotted line), (10) (dash-dotted line), and (2) (bold dash-dotted line), respectively. Since the lines are so close to each other, they are almost indistinguishable in the figures. To visualize the fitting quality better, we also plot the ratios of experimental data and fitting results at the bottom of the figures. We can see that the five distributions can describe the experimentally measured π
+ transverse momentum spectra very well. The errors are within 20%.Figure 1
Fitting results using the distributions equations (5), (8), (9), (10), and (2) for π
+ in p
+
p collisions at s
=
200 GeV. The solid line, dashed line, dotted line, dash-dotted line, and bold dash-dotted line refer to (5), (8), (9), (10), and (2), respectively, but they are hardly to distinguish. The ratios of data/fit are shown at the bottom. Data are taken from STAR [39].Figure 2
Same as in Figure1 for π
+ in p
+
p collisions at s
=
900 GeV. Data are taken from ALICE [27].To show the fitting results other than pions, we selectK
+
+
K
- at s
=
2760 GeV and p
+
p
¯ at s
=
7000 GeV at LHC in Figures 3 and 4, respectively. For the five distributions, we can not distinguish them at all in the two cases. Similar to pions, the fitting qualities are also very good.Figure 3
Same as in Figure1 for K
+
+
K
- in p
+
p collisions at s
=
2760 GeV. Data are taken from ALICE [36].Figure 4
Same as in Figure1 for p
+
p
¯ in p
+
p collisions at s
=
7000 GeV. Data are taken from ALICE [42].As we can see from Tables1 and 2, the values of parameter T
i are different. It is the main purpose of this work to target what causes the temperature discrepancies among different Tsallis distributions, with and without thermodynamical description. In order to avoid confusion, we emphasis the meaning of each T
i. Using the Tsallis distribution classification in [21], T
1 refers to Type B Tsallis distribution and T
5 refers to Type A Tsallis distribution. T
2, T
3, and T
4 are from the transient distributions to bridge Type A and Type B Tsallis distributions. To have a clear picture for the parameter T
i from the five distributions, we plot T
i versus the colliding energy s for π, K, and p, as shown in Figure 5. We can clearly see that Type B Tsallis distribution gives lower T than Type A Tsallis distribution, as we mentioned in [21]. For the light particles, that is, pions (Figure 5(a)),  T
1 and T
2 which are from the distributions ((5) and (8)) with extra m
T term are lower than T
3, T
4, and T
5 which are from the distributions ((9), (10), and (2)) without it. As we see that T
1 is larger than T
2, since (5) and (8) are similar except the power in the distributions, we conclude that the power q in (5) causes larger T. This can be verified by comparing T
3 with T
4. With the same argument, since T
4 is smaller than T
5, m
T in (10) causes smaller T. To see the effects of q and m
T in the distribution, we can compare T
3 with T
5, which are similar. The effects of the power q causing larger T and m
T causing smaller T cancel each other. For the heavier particles, that is, kaons and protons (Figures 5(b) and 5(c)), the effect of m
T in (9) wins and T
3 is smaller than T
5. We note that the effect of the extra m
T term is crucial for the temperature difference between Type A and Type B Tsallis distributions, especially for the light particles, while for the heavier particles, the effect of the choice of m
T or E
T in the Tsallis distribution becomes more important as we can see that T
5 is larger than the other T
i for kaons and protons in Figure 5.Figure 5
(Color online) temperatureT in the five distributions versus s for (a) π, (b) K, and (c) p in p
+
p collisions.
(a)
(b)
(c)


## 4. Summary

In this paper, we have presented a detailed investigation of two types of Tsallis distribution, with and without the thermodynamical description, by thep
T spectra measured from STAR and PHENIX Collaborations at RHIC and ALICE and CMS Collaborations at LHC. The power q in the Tsallis distribution with thermodynamical description is responsible for the thermodynamical consistency. To show a clear and complete comparison, another three transient distributions to bridge the two types of Tsallis distribution are also given. Good agreements are obtained, but they give different temperatures. Agreed with our previous work [21], the Tsallis distribution with thermodynamical description gives lower T than the ones given by the distribution without it. The extra term m
T in the Tsallis distribution with thermodynamical description is responsible for the discrepancies of the temperatures. But for the heavier particles, the effect of the choice of m
T or E
T in the Tsallis distribution beats the effect of the extra term m
T. The data for p
+
p collisions at 8 and 13 TeV are expected to provide further support for the conclusions presented here.



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*Source: 9632126-2016-04-10.xml*