1 Introduction

Multiparticle production in proton-carbon collisions

at GeV/

Pei-Pin Yang, Mai-Ying Duan***E-mail: duanmaiying@sxu.edu.cn, Fu-Hu LiuE-mail: fuhuliu@163.com; fuhuliu@sxu.edu.cn, Raghunath SahooE-mail: Raghunath.Sahoo@cern.ch; raghunath.phy@gmail.com

Institute of Theoretical Physics and Department of Physics and

State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan, Shanxi 030006, China

Discipline of Physics, School of Basic Sciences, Indian Institute of Technology Indore, Simrol, Indore 453552, India


Abstract: The momentum spectra of charged pions ( and ) and kaons ( and ), as well as protons (), produced in the beam protons induced collisions in a 90-cm-long graphite target [proton-carbon (-C) collisions] at the beam momentum GeV/ are studied in the framework of multisource thermal model by using the Boltzmann distribution and Monte Carlo method. The theoretical model results are approximately in agreement with the experimental data measured by the NA61/SHINE Collaboration. The related free parameters (effective temperature, rapidity shifts, and fraction of leading protons) and derived quantities (average transverse momentum and initial temperature) under given experimental conditions are obtained. It is shown that the considered free parameters and derived quantities to be strongly dependent on emission angle over a range from 0 to 380 mrad and weakly dependent on longitudinal position (graphite target thickness) over a range from 0 to 90 cm.

Keywords: Momentum spectra, effective temperature, rapidity shift, average transverse momentum, initial temperature

PACS: 14.40.-n, 14.20.-c, 24.10.Pa


1 Introduction

High energy (relativistic) nucleus-nucleus (heavy ion) collisions with nearly zero impact parameter or centrality percentage are believed to form Quark-Gluon Plasma (QGP) or quark matter [1–3]. High energy nucleus-nucleus collisions with large impact parameter are not expected to form QGP due to low particle multiplicity yielding lower energy density and temperature [4]. Small collision systems such as proton-nucleus and proton-proton collisions at high energy produce usually low multiplicity, which are not expected to form QGP, but are useful to study the multiparticle production processes. However, a few of proton-nucleus and proton-proton collisions at high energy can produce high multiplicity due to nearly zero “impact parameter”, which are possibly expected to form QGP, where the concept “impact parameter” or “centrality” used in nuclear collisions are used in proton-proton collisions [5] due to the degree of collectivity, strangeness enhancement etc., which are considered as QGP-like signatures, achieved in these high multiplicity events [6–8].

Assuming nucleus-nucleus as a mere superposition of proton-proton collisions in the absence of any nuclear effects, usually one considers proton-proton collisions as the baseline measurements. On the other hand, proton-nucleus collisions serve as studying the initial state effects and making a bridge between proton-proton to nucleus-nucleus collisions while studying the multiparticle production processes. In addition, proton-nucleus collisions are relatively simple in the studies of theories and models comparing with nucleus-nucleus collisions.

There are different types of models or theories being introduced in the studies of high energy collisions [9, 10]. Among these models or theories, the different versions of thermal and statistical models [11–14] are the most understandable and easy-to-use due to their similarity to thermodynamics and statistical mechanics. In particular, as a basic concept, temperature is ineluctable to be used in the analyses. In fact, not only the “temperature is surely one of the central concepts in thermodynamics and statistical mechanics” [15], but also it is very important due to its extremely wide applications in experimental measurements and theoretical studies in subatomic physics, especially in high energy and nuclear physics.

Seeing that the importance and simplicity, in this paper, we are interested in the study of proton-nucleus collisions at high energy by using the Boltzmann distribution and Monte Carlo Method in the framework of the multisource thermal model [16]. The theoretical model results are compared with the experimental data of the beam protons induced collisions in a 90-cm-long graphite target [proton-carbon (-C) collisions] at the beam momentum GeV/ measured by the NA61/SHINE Collaboration [17] at the Super Proton Synchrotron (SPS), the European Organisation for Nuclear Research or the European Laboratory for Particle Physics (CERN).

The remainder of this paper is structured as follows. The formalism and method are shortly described in Section 2. Results and discussion are given in Section 3. In Section 4, we summarize our main observations and conclusions.

2 Formalism and method

According to the multisource thermal model [16], it is assumed that there are many local emission sources to be formed in high energy collisions due to different excitation degrees, rapidity shifts, reaction mechanisms, impact parameters (or centralities). In the transverse plane, the local emission sources have the same excitation degree form a (large) emission source. In the rapidity space, the local emission sources have the same rapidity shift form a (large) emission source. In the rest frame of an emission source with a determined excitation degree, the particles are assumed to emit isotropically.

In the rest frame of a given emission source, let denote the temperature parameter. The particles with rest mass produced in the rest frame of the emission source are assumed to have the simplest Boltzmann distribution of momenta [18]. That is

(1)

where is the normalization constant which is related to . As a probability density function, Eq. (1) is naturally normalized to 1. If we need to consider multiple sources, we can use a superposition of different equations with different temperatures and fractions. We have

(2)

where , , and are the fraction, normalization constant, and temperature for the -th source or component. The average temperature obtained from Eq. (2) is .

It should be noted that or is not the “real” temperature of the emission source, but the effective temperature due to the fact that the flow effect is not excluded in the momentum spectrum. The “real” temperature is generally smaller than the effective temperature which contains the contribution of flow effect. The effect of chemical potential is not included in Eq. (1) as well, due to the fact that the chemical potential affects only the normalization, but not the trend, of the spectrum. The contribution of spin being small, is not included in Eq. (1).

In the Monte Carlo method [19, 20], let denote random numbers distributed evenly in . To obtain a concrete value of which satisfies Eq. (1) or one of the components in Eq. (2), we can perform the solution of

(3)

where denote a small shift relative to .

Under the assumption of isotropic emission in the rest frame of emission source, the emission angle of the considered particle has the probability density function:

(4)

In the Monte Carlo method, satisfies

(5)

Considering and obtained from Eqs. (3) and (5), we have the transverse momentum to be

(6)

the longitudinal momentum to be

(7)

the energy to be

(8)

and the rapidity to be

(9)

In the center-of-mass reference frame or the laboratory reference frame, the rapidity of the considered emission source is assumed to be in the rapidity space. Then, the rapidity of the considered particle in the center-of-mass or laboratory reference frame is

(10)

due to the additivity of rapidity. Multiple emission sources are assumed to distribute evenly in the rapidity range , where and are the minimum and maximum rapidity shifts of the multiple sources. In the Monte Carlo method,

(11)

In particular, protons exhibit the effect of leading particles which are assumed to distribute evenly in the rapidity range , where and are the minimum and maximum rapidity shifts of the leading protons. We have

(12)

The fraction of the leading protons in total protons is assumed to be .

In the center-of-mass or laboratory reference frame, the transverse momentum is

(13)

the longitudinal momentum is

(14)

the momentum is

(15)

and the emission angle is

(16)

The whole calculation is performed by the Monte Carlo method. To compare the theoretical model results with the experimental momentum spectra in a given range, we analyze the momentum distribution of particles which are in the given range. It should be noted that another experimental selection, i.e. the longitudinal position [17], is not regarded as the selected condition in the theoretical model work due to the fact that is only a reflection of target thickness in a 90-cm-long graphite target. From to cm, the beam momentum is slightly decreasing, which is neglected in the present work.

3 Results and discussion

Figures 1 and 2 present the momentum spectra, , of charged pions ( and ) produced in -C collisions at 31 GeV/ in the laboratory reference frame respectively, where denotes the number of protons on target and denotes the number of particles. Panels (a)–(c), (d)–(f), (g)–(i), (j)–(l), (m)–(o), and (p)(q) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The spectra, in different ranges, scaled by different amounts shown in the panels are represented by different symbols which are the experimental data measured by the NA61/SHINE Collaboration [17]. The curves are our results fitted by the multisource thermal model due to Eq. (1) and Monte Carlo method. The values of free parameters (, and ), normalization constant (), , and number of degree of freedom (dof) corresponding to the fits for the spectra of and are listed in Tables 1 and 2 respectively. In two cases, the degrees of freedom (dof) in the fittings are negative which appear in the tables in terms of “” and the corresponding curves are for eye guiding only. One can see that the theoretical model results are approximately in agreement with the experimental data of and measured by the NA61/SHINE Collaboration.

Figure 3 presents the momentum spectra of (a)(b) and (c)(d) in (a)(c) –40 mrad and (b)(d) –140 mrad in six ranges with different scaled amounts shown in the panels. The symbols represent the experimental data measured by the NA61/SHINE Collaboration in -C collisions at 31 GeV/ [17]. The curves are our results fitted by the multisource thermal model. The values of , , , , , and dof corresponding to the fits for the spectra of and are listed in Table 3. One can see again that the theoretical model results are approximately in agreement with the experimental data of and measured by the NA61/SHINE Collaboration.

Similar to Figs. 1 and 2, Figs. 4 and 5 show the momentum spectra of positively and negatively charged kaons ( and ) produced in -C collisions at 31 GeV/ respectively. Panels (a), (b), (c), (d), (e), and (f) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The values of , , , , , and dof corresponding to the fits for the spectra of and are listed in Tables 4 and 5 respectively. One can see that the theoretical model results are approximately in agreement with the experimental data of and measured by the NA61/SHINE Collaboration.

Fig. 1. Momentum spectra of produced in -C collisions at 31 GeV/. Panels (a)–(c), (d)–(f), (g)–(i), (j)–(l), (m)–(o), and (p)(q) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The symbols represent the experimental data measured by the NA61/SHINE Collaboration [17]. The curves are our results fitted by the multisource thermal model due to Eq. (1) and Monte Carlo method.

Table 1. Values of , , , , , and dof corresponding to the curves in Fig. 1 in which different data are measured in different and ranges. In the table, is in the units of cm and is not listed. In one case, dof is negative which appears in terms of “” and the corresponding curve is just for eye guiding purpose.

Figure (GeV) /dof
22/2
Fig. 1(a) 42/12
018 93/12
87/9
85/9
60/9
Fig. 1(b) 15/9
018 16/6
37/6
57/6
70/6
Fig. 1(c) 35/3
018 28/2
12/1
2/
49/3
Fig. 1(d) 48/12
1836 76/12
90/9
67/9
70/9
Fig. 1(e) 29/9
1836 3/6
19/6
50/6
64/6
Fig. 1(f) 44/3
1836 22/2
6/1
7/0
38/3
Fig. 1(g) 108/12
3654 85/12
60/9
69/9
42/9
Fig. 1(h) 29/9
3654 13/6
40/6
44/6
97/6
Fig. 1(i) 37/3
3654 5/2
6/1
22/1
66/3
Fig. 1(j) 145/12
5472 81/12
60/9
72/9
76/9
Fig. 1(k) 36/9
5472 13/6
87/6
35/6
94/6
Fig. 1(l) 22/3
5472 5/2
8/1
16/1
49/3
Fig. 1(m) 56/12
7290 67/12
52/9
66/9
64/9
Fig. 1(n) 73/9
7290 9/6
14/6
36/6
66/6
Fig. 1(o) 35/3
7290 13/2
3/1
19/1
46/3
Fig. 1(p) 97/12
90 88/12
50/9
104/9
83/9
Fig. 1(q) 64/6
90 118/6
71/3
29/2

Fig. 2. Same as Fig. 1, but showing the spectra of .

Table 2. Values of , , , , , and dof corresponding to the curves in Fig. 2 in which different data are measured in different and ranges. In the table, is in the units of cm and is not listed. In one case, dof is negative which appears in terms of “” and the corresponding curve is just for eye guiding only.

Figure (GeV) /dof
33/2
Fig. 2(a) 116/12
018 71/12
88/9
18/9
33/9
Fig. 2(b) 18/9
018 27/6
50/6
37/6
39/6
Fig. 2(c) 28/3
018 38/2
12/1
2/
57/3
Fig. 2(d) 73/12
1836 84/12
69/9
9/9
57/9
Fig. 2(e) 64/9
1836 21/6
30/6
55/6
84/6
Fig. 2(f) 28/3
1836 52/2
7/1
176/0
36/3
Fig. 2(g) 95/12
3654 79/12
58/9
33/9
48/9
Fig. 2(h) 63/9
3654 25/6
9/6
41/6
74/6
Fig. 2(i) 30/3
3654 47/2
9/1
8/0
41/3
Fig. 2(j) 65/12
5472 61/12
60/9
13/9
50/9
Fig. 2(k) 70/9
5472 28/6
48/6
25/6
66/6
Fig. 2(l) 31/3
5472 44/2
16/1
26/1
43/3
Fig. 2(m) 73/12
7290 88/12
86/9
23/9
51/9
Fig. 2(n) 33/9
7290 7/6
45/6
41/6
50/6
Fig. 2(o) 23/3
7290 30/2
15/1
19/1
47/3
Fig. 2(p) 18/12
90 70/12
66/9
64/9
91/9
Fig. 2(q) 30/6
90 37/6
13/3
34/2

Fig. 3. Same as Fig. 1, but showing the spectra of (a)(b) and (c)(d) in (a)(c) 20–40 mrad and (b)(d) 100–140 mrad in six ranges.

Table 3. Values of , , , , , and dof corresponding to the curves in Fig. 3 in which different data are measured in different and ranges. In the table, is in the units of mrad and is not listed.

Figure (GeV) /dof
48/12
57/12
Fig. 3(a) 95/12
2040 55/12
68/12
231/12
30/9
19/9
Fig. 3(b) 11/9
100140 17/9
20/9
37/9
57/12
16/12
Fig. 3(c) 52/12
2040 23/12
26/12
40/12
17/9
15/9
Fig. 3(d) 36/9
100140 11/9
27/9
36/9

Fig. 4. Same as Fig. 1, but showing the spectra of . Panels (a), (b), (c), (d), (e), and (f) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively.

Table 4. Values of , , , , , and dof corresponding to the curves in Fig. 4 in which different data are measured in different and ranges. In the table, is in the units of cm and is not listed.

Figure (GeV) /dof
18/2
Fig. 4(a) 30/2
018 24/2
31/2
8/2
Fig. 4(b) 26/2
1836 27/2
25/2
47/4
Fig. 4(c) 39/2
3654 21/2
14/2
38/4
Fig. 4(d) 18/2
5472 13/2
38/2
14/4
Fig. 4(e) 32/2
7290 17/2
22/2
Fig. 4(f) 10/5
90 36/2

Fig. 5. Same as Fig. 1, but showing the spectra of . Panels (a), (b), (c), (d), (e), and (f) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively.

Table 5. Values of , , , , , and dof corresponding to the curves in Fig. 5 in which different data are measured in different and ranges. In the table, is in the units of cm and is not listed.

Figure (GeV) /dof
27/2
Fig. 5(a) 18/2
018 67/2
19/2
39/2
Fig. 5(b) 72/2
1836 18/2
25/2
33/4
Fig. 5(c) 18/2
3654 18/2
29/2
67/4
Fig. 5(d) 21/2
5472 17/2
55/2
39/4
Fig. 5(e) 10/2
7290 5/2
23/2
Fig. 5(f) 36/5
90 39/2

Fig. 6. Same as Fig. 1, but showing the spectra of . Panels (a)(b), (c)(d), (e)(f), (g)(h), (i)(j), and (k)(l) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively.

Table 6. Values of , , , , , , , , and dof corresponding to the curves in Fig. 6 in which different data are measured in different and ranges. In the table, is in the units of cm and is not listed. In a few cases, dof are negative which appear in terms of “” and the corresponding curves are just for eye guiding only.

Figure (GeV) /dof
10/
Fig. 6(a) 118/9
018 86/7
202/6
128/6