In Tsallis statistics, we investigate charged pion and proton production for pCu and pPb interactions at 3, 8, and 15 GeV/c. Two versions of Tsallis distribution are implemented in a multisource thermal model. A comparison with experimental data of the HARP-CDP group shows that they both can reproduce the transverse momentum spectra, but the improved form gives a better description. It is also found that the difference between q and q′ is small when the temperature T = T′ for the same incident momentum and angular interval, and the value of q is greater than q′ in most cases.
1. Introduction
Heavy ion collisions at Large Hadron Collider (LHC) are essential for the investigation of strongly interacting matter at high-energy density [1]. Proton-nucleus (pA) program is used as a baseline measurement for nucleus-nucleus (AA) collisions and is also crucial to discuss the various domains of quantum chromodynamics (QCD). So, the pA collision has been considered as an important component of the heavy ion collisions [2]. Measurements of the transverse momentum pT spectra of identified particles, to some extent, can provide an insight into the dynamics of the colliding systems.
In recent years, many models have been proposed in the interpretation of the pT spectra in high-energy collisions, such as multisource thermal model [3–5], diffusion model [6], and Tsallis statistics [7–11]. In particular, Tsallis distribution has successfully reproduced the pT spectra and has aroused interest of scientists recently. The statistics can extract two parameters, Tsallis temperature T and q, which is used to characterize a degree of nonequilibrium in the system. For example, a Tsallis-like distribution has given excellent descriptions to the experimental data, which have been measured by the STAR [7] and PHENIX [8] collaborations at the RHIC and by the ALICE [9], ATLAS [10], and CMS [11] collaborations at the LHC. Generally, the Tsallis parameter q goes to 1. A thermodynamically consistent form of Tsallis statistics has also been proposed to fit the transverse momentum spectra [12, 13]. In our previous work [14], we have consistently embedded the improved form of the Tsallis distribution into a multisource thermal model to describe systematically pseudorapidity distributions in pp (pp-), AuAu, CuCu, and PbPb collisions at RHIC and LHC energies. The result shows that a rapidity shift of longitudinal sources needs to be considered. In this paper, we will use the Tsallis distributions with the rapidity shift to analyze proton and charged pion distributions in pCu and pPb interactions at 3, 8, and 15 GeV/c in the hadron production (HARP) experiment at CERN [15]. The results obtained from the two forms of Tsallis distribution are compared in detail.
The paper is organized as follows: in Section 2, the improved Tsallis distribution is introduced and the results are compared with the experimental data; at the end, we give discussions and conclusions in Section 3.
2. Tsallis Statistics Description of the Transverse Momentum Spectra
According to Tsallis statistics, the particle number is(1)N=gV∫d3p2π31+q-1E-μT-q/q-1,where g, V, p, E, and μ are the degeneracy factor, the volume, the momentum, the energy, and the chemical potential, respectively. The parameters T and q are temperature and nonequilibrium factor, respectively. The distribution of the corresponding momentum is given by(2)d3Nd3p=gV2π31+q-1E-μT-q/q-1.In terms of the transverse momentum pT and the rapidity y, the distribution function is(3)d2NdpTdy=gVpTmTcoshy2π21+q-1mTcoshy-μT-q/q-1.
For μ=0, at midrapidity y=0, the transverse momentum pT distribution is(4)d2NdpTdyy=0=gVpTmT2π21+q-1mTT-q/q-1.The Tsallis distribution is a quantum form, which can meet the thermodynamic consistency [10, 11]. Approximately, the q can equal 1 and the pT distribution is given by (5)d2NdpTdyy=0=gVpTmT2π21+q′-1mTT′-1/q′-1.The two distribution functions both represent a single pT spectrum of one source at y=0. Therefore, we need to consider the distribution width of the rapidity of final-state particles [14]. Equations (4) and (5) will be used in the following analysis.
Figures 1, 2, and 3 show p, π-, and π+ double-differential cross-sections as a function of the transverse momentum in pCu collisions, respectively. From left to right, the incident proton momenta are 3, 8, and 15 GeV/c, respectively. And from top to bottom, the angular intervals are 30°–40°, 60°–75°, and 105°–125°, respectively. The symbols denote the experimental data measured in the hadron production (HARP) experiment at CERN [15]. The solid lines are results fitted by (4) and the dashed lines are results fitted by (5). The Tsallis parameters q, T, q′, and T′ are given in Tables 1 and 2. From the figures, one can see that the results of (4) and (5) are in agreement with the experimental data in the whole observed pT region, but (4) can give a better fit. The values of q and q′ show slight difference when T=T′ for the fixed incident momentum and angular interval. And, in most cases, q>q′. The minimum difference is 0.002 and the maximum difference is 0.02. For proton production, the pT scaling properties behave well at 60°≤θ≤75°. The case is similar for π- production at 60°–75° and 105°–125°.
Values of T, q, T′, and q′ taken in Figures 1 and 2.
Figure 1
T
q
T′
q′
Figure 2
T
q
T′
q′
Figure 1(a)
0.060
1.025
0.060
1.025
Figure 2(a)
0.010
1.070
0.010
1.065
Figure 1(b)
0.020
1.065
0.020
1.060
Figure 2(b)
0.010
1.065
0.010
1.060
Figure 1(c)
0.020
1.065
0.020
1.060
Figure 2(c)
0.005
1.075
0.005
1.070
Figure 1(d)
0.060
1.020
0.060
1.025
Figure 2(d)
0.060
1.020
0.060
1.020
Figure 1(e)
0.060
1.020
0.060
1.025
Figure 2(e)
0.060
1.020
0.060
1.020
Figure 1(f)
0.060
1.020
0.060
1.025
Figure 2(f)
0.060
1.020
0.060
1.020
Figure 1(g)
0.010
1.030
0.010
1.027
Figure 2(g)
0.080
0.970
0.080
0.972
Figure 1(h)
0.010
1.035
0.010
1.032
Figure 2(h)
0.080
0.970
0.080
0.972
Figure 1(i)
0.010
1.040
0.010
1.037
Figure 2(i)
0.080
0.970
0.080
0.972
Values of T, q, T′, and q′ taken in Figures 3 and 4.
Figure 3
T
q
T′
q′
Figure 4
T
q
T′
q′
Figure 3(a)
0.080
0.980
0.080
0.980
Figure 4(a)
0.080
1.020
0.080
1.020
Figure 3(b)
0.060
1.120
0.060
1.100
Figure 4(b)
0.080
1.080
0.080
1.070
Figure 3(c)
0.060
1.120
0.060
1.100
Figure 4(c)
0.080
1.090
0.080
1.080
Figure 3(d)
0.080
0.970
0.080
0.970
Figure 4(d)
0.050
1.090
0.050
1.080
Figure 3(e)
0.060
1.090
0.060
1.070
Figure 4(e)
0.080
1.040
0.080
1.040
Figure 3(f)
0.060
1.080
0.060
1.075
Figure 4(f)
0.080
1.040
0.080
1.040
Figure 3(g)
0.070
0.880
0.070
0.880
Figure 4(g)
0.070
0.870
0.070
0.880
Figure 3(h)
0.050
1.030
0.050
1.030
Figure 4(h)
0.050
1.040
0.050
1.030
Figure 3(i)
0.050
1.030
0.050
1.030
Figure 4(i)
0.050
1.050
0.050
1.040
Transverse momentum spectra of protons produced in p+Cu collisions at 3, 8, and 15 GeV/c at different angular intervals. The symbols represent the HARP-CDP experimental data [15]. The solid curves and dashed lines are the results calculated by (4) and (5), respectively.
Same as Figure 1, but for π- production.
Same as Figure 1, but for π+ production.
The transverse momentum spectra of p, π-, and π+ produced in pPb interactions are displayed in Figures 4, 5, and 6, respectively. The symbols denote the HARP-CDP experimental data [15]. The solid lines and dashed lines are fitting results from (4) and (5), respectively. Both the solid lines and dashed lines pass through the experimental data points, but the results of (4) are in better agreement with the data. The values of T, q, T′, and q′ taken in the calculations are listed in Tables 2 and 3. The parameters q and T correspond to (4) and q′ and T′ correspond to (5). Similar to Figures 1–3, the difference between q and q′ is small when T=T′ for the same incident momentum and angular interval, and q>q′ in most cases. The maximum value of the difference is 0.02.
Values of T, q, T′, and q′ taken in Figures 5 and 6.
Figure 5
T
q
T′
q′
Figure 6
T
q
T′
q′
Figure 5(a)
0.050
1.060
0.050
1.050
Figure 6(a)
0.090
1.010
0.090
1.010
Figure 5(b)
0.080
1.060
0.080
1.040
Figure 6(b)
0.090
1.070
0.090
1.050
Figure 5(c)
0.080
1.060
0.080
1.040
Figure 6(c)
0.090
1.070
0.090
1.050
Figure 5(d)
0.050
1.090
0.030
1.110
Figure 6(d)
0.060
1.070
0.060
1.060
Figure 5(e)
0.050
1.100
0.050
1.100
Figure 6(e)
0.080
1.050
0.080
1.040
Figure 5(f)
0.050
1.110
0.050
1.100
Figure 6(f)
0.080
1.060
0.080
1.050
Figure 5(g)
0.040
1.010
0.040
1.010
Figure 6(g)
0.030
1.050
0.030
1.050
Figure 5(h)
0.040
1.050
0.040
1.050
Figure 6(h)
0.040
1.070
0.040
1.060
Figure 5(i)
0.040
1.050
0.040
1.050
Figure 6(i)
0.040
1.080
0.040
1.070
Same as Figure 1, but for p+Pb collisions.
Same as Figure 4, but for π- production.
Same as Figure 4, but for π+ production.
3. Discussions and Conclusions
We combine the picture of the multisource thermal model and Tsallis statistics to investigate the transverse momentum spectra of protons and charged pions produced in the collisions of 3, 8, and 15 GeV/c protons on Cu and Pb at fixed angles of 30°–40°, 60°–75°, and 105°–125°. In practice, we choose (5), where q tends to 1. Equation (4) has been improved from (5) to satisfy the thermodynamic consistency [12, 13]. By comparing their results to the HARP-CDP data, it is found that they both agree with the experimental data and (4) can better reproduce the transverse momentum spectra. In addition, we notice that the values of q and q′ show slight difference when T=T′ for the same incident momentum and angular interval, and the value of q is greater than q′ in most cases.
In the present work, we focus on the two versions of Tsallis distribution in the picture of the multisource production for the description of the transverse momentum spectra of produced particles. According to the multisource thermal model [16, 17], many emission sources of produced particles and nuclear fragments are formed in pCu and pPb interactions. Every source was regarded as a thermal equilibrium system, which is comprised approximately of ideal gases. Understandably, the Maxwell distribution was selected in the discussion of the particle production of high-energy collisions [18, 19]. So, it is only an approximate classical method. If the relativity effect and quantum effect are considered, the improved Tsallis distribution [20–22] is a better choice. The observed particles are emitted isotropically in the rest frame of emission sources with the different excitation degree. The Tsallis distributions are embedded consistently into the model. The rapidity location in the framework of the Tsallis description is tightly linked to the rapidity (pseudorapidity) shift of the emission sources. And, the rapidity width is taken into account in the analysis of final-state particles. By the multisource-production discussion, the Tsallis statistics can not only describe the transverse momentum spectra but also obtain the underlying physical picture of the particle production in high-energy collisions.
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 Grants no. 11247250, 11005071, and 10975095; the National Fundamental Fund of Personnel Training under Grant no. J1103210; the Shanxi Provincial Natural Science Foundation under Grants no. 2013021006 and 2011011001.
ArmestoN.JeonS.BorghiniN.Heavy-ion collisions at the LHC—last call for predictionsSalgadoC. A.Alvarez-MuñizJ.ArleoF.Proton–nucleus collisions at the LHC: scientific opportunities and requirementsLiuF. H.ChenY. H.WeiH. R.LiB. C.Transverse momentum distributions of final-state particles produced in soft excitation process in high energy collisionsLiB. C.FuY. Y.WangL. L.LiuF. H.Dependence of elliptic flows on transverse momentum and number of participants in Au + Au collisions at SNN = 200 GeVLiB. C.WangY. Z.LiuF. H.Formulation of transverse mass distributions in Au-Au collisions SNN = 200 GeV/nucleonSuzukiN.BiyajimaM.Transverse momentum distribution with radial flow in relativistic diffusion modelAbelevB. I.AdamsJ.AggarwalM. M.Strange particle production in p + p collisions at s = 200 GeVAdareA.AfanasievS.AidalaC.Measurement of neutral mesons in p + p collisions at s = 200 GeV and scaling properties of hadron productionAamodtK.AbelN.AbeysekaraU.Production of pions, kaons and protons in pp collisions at S=900GeV with ALICE at the LHCAadG.AbbottB.AbdallahJ.Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHCKhachatryanV.SirunyanA. M.TumasyanA.Strange particle production in pp collisions at s = 0.9 and 7 TeVWilkG.WlodarczykZ.Interpretation of the nonextensivity parameter in some applications of Tsallis statistics and Lévy distributionsRybczyńskiM.WłodarczykZ.Tsallis statistics approach to the transverse momentum distributions in p-p collisionsLiB. C.WangY. Z.LiuF. H.WenX. J.DongY. E.Particle production in relativistic PP(P-) and AA collisions at RHIC and LHC energies with Tsallis statistics using the two-cylindrical multisource thermal modelAbdel-WagedK.FelembanN.UzhinskiiV. V.GEANT4 hadronic cascade models analysis of proton and charged pion transverse momentum spectra from p + Cu and Pb collisions at 3, 8, and 15 GeV/cLiuF. H.TianC. X.DuanM. Y.LiB. C.Relativistic and quantum revisions of the multisource thermal model in high-energy collisionsLiB. C.FuY. Y.WangL. L.WangE. Q.LiuF. H.Transverse momentum distributions of strange hadrons produced in nucleus–nucleus collisions at SNN=62.4 and 200 GeVLiB. C.WangY. Z.WangE. Q.Meson production in high energy p+p collisions at the RHIC energiesLiB. C.FuY. Y.WangE. Q.WangL. L.LiuF. H.Transverse momentum dependence of charged and strange hadron elliptic flows in Cu-Cu collisionsWongC. Y.WilkG.Tsallis fits to pT spectra and multiple hard scattering in pp collisions at the LHCAzmiM. D.CleymansJ.Transverse momentum distributions in proton-proton collisions at LHC energies and Tsallis thermodynamicsWilkG.WlodarczykZ.Uncertainty relations in terms of the Tsallis entropy