Advances in High Energy Physics
Volume 2015 (2015), Article ID 684950, 8 pages
http://dx.doi.org/10.1155/2015/684950

Research Article

Transverse Momentum Spectra of and at Midrapidity in + Au, Cu + Cu, and Collisions at  GeV

Bao-Chun Li, Guo-Xing Zhang, and Yuan-Yuan Guo

Department of Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Received 30 October 2014; Revised 31 December 2014; Accepted 31 December 2014

Academic Editor: Bhartendu K. Singh

Copyright © 2015 Bao-Chun Li et al. 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 analyze transverse momentum spectra of and at midrapidity in + Au, Cu + Cu, and collisions at  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.

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 ( or ) 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 and produced in + Au, Cu + Cu, and collisions at  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

where is the Tsallis temperature, is called the “nonequilibrium degree” of the collision system, and is the degeneracy degree. The parameters , , , and are the particle momentum, the system volume, the energy, and the chemical potential, respectively. In terms of the transverse mass and the rapidity , the transverse momentum spectra of the particles can be written as

When and , the distribution function is

Considering a fixed rapidity interval [5], the distribution function should be

where the interval of the integral represents the rapidity range observed in the experiment. is a normalization constant. The function is an improved form to satisfy the thermodynamical consistency [8]. Under the limit , the spectrum becomes a conventional form,

Next, we review the Boltzmann distribution. For Maxwell’s ideal gas, the momentum distribution function is

where is a Boltzmann constant and is the particle mass. If the relativistic effect is taken into account, the momentum distribution function is

where is the second-order modified Bessel function. For the isotropic emission in the collision, the transverse momentum distribution is

where is a normalization constant. The two-component distribution of the transverse momentum is

where indicates the contribution percentage of the first component.

3. Discussion and Conclusion

Figure 1 shows the transverse momentum spectra of meson at midrapidity in + Au, Cu + Cu, and collisions at  GeV. The experimental points measured by STAR and PHENIX collaborations [11–15] are shown with different symbols. For Cu + Cu and + Au, different centrality bins are marked by the different shapes. At the bottom of the figure, we show the 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 , , , and in the calculations are listed in Table 1 with per degree of freedom (/dof). Their values do not change obviously due to a scaling behavior. In Figure 2, we also give a comparison between the numerical results and the experimental points of (or ). The parameters , , , and are listed in Table 2 with /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 of , , , and taken in Figure 1. and units are GeV.

Figure
Centrality

/dof
Figure
Centrality

/dof

Figure 1(a)
MinBias
0.122
1.085
0.852
Figure 1(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

Figure 1(b)
MinBias
0.122
1.086
0.426
Figure 1(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
0.319

Table 2: Values of , , , and taken in Figure 2. and units are GeV.

Figure
Centrality

/dof
Figure
Centrality

/dof

Figure 2(a)
MinBias
0.120
1.084
0.372
Figure 2(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

Figure 2(b)
MinBias
0.120
1.080
0.229
Figure 2(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
0.429

(a)

(b)

(c)

(d)

Figure 1: transverse momentum spectra in + Au and Cu + Cu collisions for different centrality bins and collisions at  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: and transverse momentum spectra in + Au and Cu + Cu collisions for different centrality bins and collisions at  GeV. Experimental data [11–15] are shown with different symbols. The dashed lines and solid lines are the results of 4 and 5, respectively.

In Figures 3 and 4, we present a comparison between the two-component Boltzmann distribution and the experimental data measured in + Au, Cu + Cu, and collisions at  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 , , and used in the calculations are given in Table 3 with /dof. The values of are two to four times the values of or . The values of are about twice the values of 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 spectra. For the two-component distribution of the Boltzmann distribution, the parameter is used to denote the contribution of the soft process and is used to denote the contribution of the hard process.

Table 3: Values of , , and taken in Figures 3 and 4. and units are GeV.

Figure
Centrality

/dof
Figure
Centrality

/dof

Figure 3(a)
MinBias
0.380
0.900
0.996
1.057
Figure 4(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

Figure 3(c)
MinBias
0.450
0.900
0.936
0.858
Figure 4(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
0.320

(a)

(b)

(c)

(d)

Figure 3: The same as Figure 1, 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 Figure 2, 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.

In summary, we have compared Tsallis statistics and the Boltzmann distribution in the analysis of the transverse momentum spectra of and at midrapidity in + Au, Cu + Cu, and collisions at  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 region. Tsallis statistics is nonextensive statistics [4]. The parameter is temperature and the parameter summarily describes all features causing a departure from the Boltzmann-Gibbs statistics. In [6], 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 .

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 11247250, no. 11005071, and no. 10975095 and the Shanxi Provincial Natural Science Foundation under Grant no. 2013021006.

References

L. Adamczyk, G. Agakishiev, M. M. Aggarwal et al., “Directed flow of identified particles in Au + Au collisions at  GeV at RHIC,” Physical Review Letters, vol. 108, no. 20, Article ID 202301, 6 pages, 2012.

K. Adcox, S. S. Adler, N. N. Ajitanand et al., “Centrality dependence of , , p, and production from  GeV Au + Au collisions at RHIC,” Physical Review Letters, vol. 88, Article ID 242301, 2002.

P. S. B. Dev, A. Pilaftsis, and U. K. Yang, “New production mechanism for heavy neutrinos at the LHC,” Physical Review Letters, vol. 112, no. 8, Article ID 081801, 5 pages, 2014.

C. Tsallis, “Possible generalization of Boltzmann-Gibbs statistics,” Journal of Statistical Physics, vol. 52, no. 1-2, pp. 479–487, 1988.

B. C. Li, Y. Z. Wang, F. H. Liu, X. J. Wen, and Y. E. Dong, “Particle production in relativistic and collisions at RHIC and LHC energies with Tsallis statistics using the two-cylindrical multisource thermal model,” Physical Review D, vol. 89, Article ID 054014, 2014.

C. Y. Wong and G. Wilk, “Tsallis fits to spectra and multiple hard scattering in collisions at the LHC,” Physical Review D, vol. 87, Article ID 114007, 2013.

B. C. Li, Y. Z. Wang, and F. H. Liu, “Formulation of transverse mass distributions in Au–Au collisions at GeV/nucleon,” Physics Letters B, vol. 725, no. 4-5, pp. 352–356, 2013.

M. Rybczyński and Z. Włodarczyk, “Tsallis statistics approach to the transverse momentum distributions in p–p collisions,” The European Physical Journal C, vol. 74, no. 2, p. 2785, 2014.

F.-H. Liu, Y.-H. Chen, H.-R. Wei, and B.-C. Li, “Transverse momentum distributions of final-state particles produced in soft excitation process in high energy collisions,” Advances in High Energy Physics, vol. 2013, Article ID 965735, 15 pages, 2013.

B.-C. Li, Y.-Y. Fu, E.-Q. Wang, L.-L. Wang, and F.-H. Liu, “Transverse momentum dependence of charged and strange hadron elliptic flows in Cu–Cu collisions,” Journal of Physics G: Nuclear and Particle Physics, vol. 39, no. 8, Article ID 085109, 2012.

A. Adare, S. Afanasiev, C. Aidala et al., “Measurement of and in p+p, d+Au, and Cu + Cu collisions at  GeV,” Physical Review C, vol. 90, Article ID 054905, 2014.

A. Adare, S. Afanasiev, C. Aidala et al., “Measurement of neutral mesons in p+p collisions at  GeV and scaling properties of hadron production,” Physical Review D, vol. 83, Article ID 052004, 2011.

J. Adams, V. Eckardt, J. Putschke et al., “ resonance production in Au+Au and collisions at GeV,” Physical Review C, vol. 71, Article ID 064902, 2005.

B. I. Abelev, M. M. Aggarwal, Z. Ahammed et al., “Hadronic resonance production in collisions at GeV measured at the BNL relativistic heavy ion collider,” Physical Review C, vol. 78, no. 4, Article ID 044906, 20 pages, 2008.

M. M. Aggarwal, Z. Ahammed, and A. V. Alakhverdyants, “K *0 production in Cu + Cu and Au + Au collisions at  GeV and 200 GeV,” Physical Review C, vol. 84, no. 3, Article ID 034909, 2011.

A. Adare, S. Afanasiev, C. Aidala et al., “Identified charged hadron production in collisions at and 62.4 GeV,” Physical Review C, vol. 83, Article ID 064903, 2011.

B. I. Abelev, J. Adams, M. M. Aggarwal et al., “Strange particle production in collisions at  GeV,” Physical Review C, vol. 75, no. 6, Article ID 064901, 21 pages, 2007.

G. Aad, B. Abbott, J. Abdallah et al., “Charged-particle multiplicities in interactions measured with the ATLAS detector at the LHC,” New Journal of Physics, vol. 13, Article ID 053033, 2011.

K. Aamodt, N. Abel, U. Abeysekara et al., “Transverse momentum spectra of charged particles in proton-proton collisions at  GeV with ALICE at the LHC,” Physics Letters B, vol. 693, no. 2, pp. 53–68, 2010.

V. Khachatryan, A. M. Sirunyan, A. Tumasyan et al., “Transverse-momentum and pseudorapidity distributions of charged hadrons in pp collisions at   TeV,” Physical Review Letters, vol. 105, Article ID 022002, 2010.

G. Wilk and Z. Włodarczyk, “Interpretation of the nonextensivity parameter q in some applications of Tsallis statistics and Lévy distributions,” Physical Review Letters, vol. 84, no. 13, pp. 2770–2773, 2000.

G. Wilk and Z. Wlodarczyk, “Consequences of temperature fluctuations in observables measured in high-energy collisions,” The European Physical Journal A, vol. 48, article 161, 2012.