# Transverse Momentum Spectra ofK

Authors: Bao-Chun Li; Guo-Xing Zhang; Yuan-Yuan Guo
Journal: Advances in High Energy Physics
 (2015)

---

## Abstract

We analyze transverse momentum spectra ofK
S
0 and K
*
0 at midrapidity in d + Au, Cu + Cu, and p
+
p collisions at s
N
N
=
200 GeV in the formworks of Tsallis statistics and Boltzmann statistics, respectively. Both of them can describe the transverse momentum spectra and extract the thermodynamics parameters of matter evolution in the collisions. The parameters are helpful for us to understand the thermodynamics factors of the particle production.


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## Body

## 1. Introduction

High-energy collisions provide many final-state particles, which can be observed in experiments [1–3]. By an investigation of the particle distribution produced in different kinds of collisions, we may speculate the collision process in some ways. Among the properties of the observed particles, the transverse momentum plays a significant role in the colliding experiment. Transverse momentum spectra of hadrons produced in proton and heavy-ion collisions at RHIC and LHC energies have been described successfully through nonextensive statistical mechanics [4]. In our previous work [5], we have systematically investigated the pseudorapidity distributions of charged particles produced in high-energy nucleon-nucleon (p
p or p
p
¯) collisions and high-energy nucleus-nucleus (AA) collisions with different centralities by combining Tsallis statistics with a multisource thermal model.Recently, Tsallis statistics [6–8] and Boltzmann statistics [9] have been used to analyze the transverse momentum spectra in heavy-ion collisions at high energy. They can both extract the thermodynamics parameters of matter produced in the collisions. What is a parameter difference between the different models? What does the difference mean? In order to concretely understand the thermodynamics properties [10], we implant the Tsallis distribution and Boltzmann distribution in the multisource thermal model. In this paper, we compare the two model descriptions of the transverse momentum spectra of K
S
0 and K
*
0 produced in d + Au, Cu + Cu, and p
+
p collisions at s
N
N
=
200 GeV. Two forms of the Tsallis distribution will be taken in Tsallis statistics. One is a conventional choice and the other has been improved to satisfy the thermodynamical consistency [8].


## 2. Tsallis Distribution and Boltzmann Distribution

In the Tsallis statistics [6–8], the momentum distribution is given by
(1)
d
3
N
d
3
P
=
g
V
2
π
3
1
+
q
-
1
E
-
μ
T
-
q
/
q
-
1
,
where T is the Tsallis temperature, q is called the “nonequilibrium degree” of the collision system, and g is the degeneracy degree. The parameters p, V, E, and μ are the particle momentum, the system volume, the energy, and the chemical potential, respectively. In terms of the transverse mass m
T and the rapidity y, the transverse momentum spectra of the particles can be written as
(2)
d
2
N
d
P
T
d
y
=
g
V
P
T
m
T
cosh
⁡
y
2
π
2
1
+
q
-
1
m
T
cosh
⁡
y
-
μ
T
-
q
/
q
-
1
.
When μ
=
0 and y
=
0, the distribution function is
(3)
d
2
N
d
P
T
d
y
y
=
0
=
g
V
P
T
m
T
2
π
2
1
+
q
-
1
m
T
T
-
q
/
q
-
1
.
Considering a fixed rapidity interval [5], the distribution function should be
(4)
d
N
P
T
d
P
T
=
C
∫
y
min
⁡
y
max
⁡
cosh
⁡
y
m
T
1
+
q
-
1
m
T
cosh
⁡
y
T
-
q
/
q
-
1
,
where the interval of the integral represents the rapidity range observed in the experiment. C is a normalization constant. The function is an improved form to satisfy the thermodynamical consistency [8]. Under the limit q
→
1, the p
T spectrum becomes a conventional form,
(5)
d
N
P
T
d
P
T
=
C
∫
y
min
⁡
y
max
⁡
cosh
⁡
y
m
T
1
+
q
′
-
1
m
T
cosh
⁡
y
T
′
-
1
/
q
′
-
1
.
Next, we review the Boltzmann distribution. For Maxwell’s ideal gas, the momentum distribution function is
(6)
d
N
N
d
3
P
=
P
2
m
0
k
T
3
/
2
2
π
exp
⁡
-
P
2
2
m
0
k
T
,
where k is a Boltzmann constant and m
0 is the particle mass. If the relativistic effect is taken into account, the momentum distribution function is
(7)
d
N
N
d
3
P
=
P
2
m
0
2
k
T
K
2
m
0
/
k
T
exp
⁡
-
P
2
+
m
0
2
k
T
,
where K
2
(
m
0
/
k
T
) is the second-order modified Bessel function. For the isotropic emission in the collision, the transverse momentum distribution is
(8)
d
N
N
d
P
T
=
C
′
P
T
exp
⁡
-
P
T
2
+
m
0
2
k
T
=
g
C
′
,
T
,
where C
′ is a normalization constant. The two-component distribution of the transverse momentum is
(9)
d
N
N
d
P
T
=
w
g
C
1
′
,
T
1
+
1
-
w
g
C
2
′
,
T
2
,
where w indicates the contribution percentage of the first component.


## 3. Discussion and Conclusion

Figure1 shows the transverse momentum spectra of K
S
0 meson at midrapidity in d + Au, Cu + Cu, and p
+
p collisions at s
N
N
=
200 GeV. The experimental points measured by STAR and PHENIX collaborations [11–15] are shown with different symbols. For Cu + Cu and d + Au, different centrality bins are marked by the different shapes. At the bottom of the figure, we show the p
+
p data as a reference. The dashed lines and solid lines are numerical results from the thermodynamically consistent Tsallis distribution equation 4 and the conventional Tsallis distribution equation 5, respectively. It is seen that the two forms of Tsallis distribution can both agree with the data. The difference of the numerical results is very small. The parameters T, q, T
′, and q
′ in the calculations are listed in Table 1 with χ
2 per degree of freedom (χ
2/dof). Their values do not change obviously due to a P
T scaling behavior. In Figure 2, we also give a comparison between the numerical results and the experimental points of K
*
0 (or K
*
0
¯). The parameters T, q, T
′, and q
′ are listed in Table 2 with χ
2/dof. Similarly, the values have no significant or no regular changes. With Tsallis statistics’ success in dealing with nonequilibrated complex systems in condensed matter research, it is used to study the particle production in high-energy physics. The Tsallis statistics is widely applied in the description of the experimental data in RHIC [12, 16, 17] and LHC [18–20]. It is an advantage that the Tsallis statistics is connected to thermodynamics by the entropy; for example, see [21] for more detailed discussions and its references.Table 1
Values ofT, q, T
′, and q
′ taken in Figure 1. T and T
′ units are GeV.
Figure
Centrality
T
q
χ
2/dof
Figure
Centrality
T
′
q
′
χ
2/dof
Figure1(a)
MinBias
0.122
1.085
0.852
Figure1(c)
MinBias
0.120
1.077
0.755
0–20%
0.124
1.084
0.621
0–20%
0.120
1.077
0.782
20–40%
0.124
1.084
0.566
20–40%
0.121
1.077
0.528
40–60%
0.126
1.084
0.589
40–60%
0.122
1.077
0.495
60–88%
0.129
1.084
0.631
60–88%
0.122
1.077
0.670
pp data
0.134
1.083
0.952
pp data
0.122
1.077
1.026
Figure1(b)
MinBias
0.122
1.086
0.426
Figure1(d)
MinBias
0.120
1.079
0.383
0–20%
0.124
1.086
0.215
0–20%
0.122
1.079
0.402
20–60%
0.124
1.086
0.270
20–60%
0.122
1.079
0.240
60–94%
0.125
1.085
0.354
60–94%
0.122
1.079
0.319Table 2
Values ofT, q, T
′, and q
′ taken in Figure 2. T and T
′ units are GeV.
Figure
Centrality
T
q
χ
2/dof
Figure
Centrality
T
′
q
′
χ
2/dof
Figure2(a)
MinBias
0.120
1.084
0.372
Figure2(c)
MinBias
0.122
1.075
0.391
0–20%
0.120
1.084
0.446
0–20%
0.122
1.075
0.426
20–40%
0.130
1.082
0.515
20–40%
0.124
1.075
0.510
40–60%
0.130
1.082
0.658
40–60%
0.124
1.073
0.624
60–88%
0.140
1.081
0.675
60–88%
0.125
1.073
0.759
pp
data
0.150
1.077
0.788
pp data
0.127
1.073
0.806
Figure2(b)
MinBias
0.120
1.080
0.229
Figure2(d)
MinBias
0.118
1.073
0.255
0–20%
0.120
1.080
0.275
0–20%
0.118
1.073
0.279
20–40%
0.122
1.080
0.210
20–40%
0.118
1.073
0.246
40–60%
0.124
1.080
0.462
40–60%
0.122
1.074
0.503
60–94%
0.126
1.080
0.404
60–94%
0.122
1.074
0.429Figure 1
K
S
0 transverse momentum spectra in d + Au and Cu + Cu collisions for different centrality bins and p
+
p collisions at s
N
N
=
200 GeV. Experimental data [11–15] are shown with different symbols. The dashed lines and solid lines are the results of 4 and 5, respectively.
(a)
(b)
(c)
(d)Figure 2
K
*
0 and K
*
0
¯ transverse momentum spectra in d + Au and Cu + Cu collisions for different centrality bins and p
+
p collisions at s
N
N
=
200 GeV. Experimental data [11–15] are shown with different symbols. The dashed lines and solid lines are the results of 4 and 5, respectively.
(a)
(b)
(c)
(d)In Figures3 and 4, we present a comparison between the two-component Boltzmann distribution and the experimental data measured in d + Au, Cu + Cu, and p
+
p collisions at s
N
N
=
200 GeV. The solid lines denote the results of the two-component Boltzmann distribution equation 9. The two-component Boltzmann distribution also can agree with the experimental points. The dashed lines and the dotted lines denote the contributions of the first component and the second component, respectively. It is seen clearly that the soft and hard interactions behave in the low and high transverse momentum of the identified particles. The parameters T
1, T
2, and w used in the calculations are given in Table 3 with χ
2/dof. The values of T
1 are two to four times the values of T or T
′. The values of T
2 are about twice the values of T
1 because of the hard interaction. In 9, the first component is the contribution of soft process and the second component is the contribution of hard process. The distribution in the low transverse momentum region is mainly contributed by the soft processes. The hard processes contribute high transverse momentums in the p
T spectra. For the two-component distribution of the Boltzmann distribution, the parameter w is used to denote the contribution of the soft process and 1
-
w is used to denote the contribution of the hard process.Table 3
Values ofT
1, T
2, and w taken in Figures 3 and 4. T
1 and T
2 units are GeV.
Figure
Centrality
T
1
T
2
w
χ
2/dof
Figure
Centrality
T
1
T
2
w
χ
2/dof
Figure3(a)
MinBias
0.380
0.900
0.996
1.057
Figure4(a)
MinBias
0.300
0.600
0.977
1.248
0–20%
0.380
0.900
0.998
0.932
0–20%
0.310
0.580
0.974
0.890
20–40%
0.390
0.900
0.996
0.811
20–40%
0.310
0.580
0.950
0.780
40–60%
0.400
0.910
0.996
0.670
40–60%
0.310
0.580
0.974
0.605
60–88%
0.410
0.920
0.996
0.715
60–88%
0.320
0.580
0.940
0.643
pp data
0.420
0.930
0.995
0.600
pp data
0.300
0.600
0.979
0.580
Figure3(c)
MinBias
0.450
0.900
0.936
0.858
Figure4(c)
MinBias
0.280
0.580
0.979
1.150
0–20%
0.470
0.920
0.989
0.705
0–20%
0.280
0.580
0.984
0.700
20–60%
0.470
0.920
0.990
0.426
20–40%
0.280
0.580
0.982
0.570
60–94%
0.470
0.920
0.987
0.395
40–60%
0.280
0.600
0.979
0.352
—
—
—
—
—
60–94%
0.280
0.600
0.980
0.320Figure 3
The same as Figure1, but the solid lines are the results of 9. The dashed lines and the dotted lines denote the contributions of the first component and the second component, respectively.
(a)
(b)
(c)
(d)Figure 4
The same as Figure2, but the solid lines are the results of 9. The dashed lines and the dotted lines denote the contributions of the first component and the second component, respectively.
(a)
(b)
(c)
(d)In summary, we have compared Tsallis statistics and the Boltzmann distribution in the analysis of the transverse momentum spectra ofK
S
0 and K
*
0 at midrapidity in d + Au, Cu + Cu, and p
+
p collisions at s
N
N
=
200 GeV. The two methods can both describe the distribution of the final-state particles. They have their own advantage and proper scope. The two forms of Tsallis distribution can consistently agree with the experimental points in the low and high p
T region. Tsallis statistics is nonextensive statistics [4]. The parameter T is temperature and the parameter q summarily describes all features causing a departure from the Boltzmann-Gibbs statistics. In [6], Var
⁡
(
T
)
/
T
2
=
q
-
1 directly reflects intrinsic fluctuations of the temperature. However, the Tsallis distribution also emerges from a number of other dynamical mechanisms [22]. The two-component Boltzmann distribution can directly show the contribution of the soft interaction and the hard interaction in the observed spectra by the weight parameter w.



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