Abstract

We present an analysis of proton and charged pion transverse momentum spectra of and reactions at 3, 8, and 15 GeV/c in the framework of a multisource thermal model. The spectra are compared closely with the experimental data of HARP-CDP at all angular intervals. The result shows that the widths of the particle distributions in both and collisions decrease with increasing the angle for the same incident momentum.

1. Introduction

The relativistic heavy ion collider (RHIC) [1, 2] in the United States and the large hadron collider (LHC) [3] in Switzerland have been built, respectively. Much higher energy collisions can lead to a new significant extension of the kinematic range in transverse momentum. The collisions bring valuable information of quark gluon plasma (QGP) at high energy due to high temperature and density [4, 5]. The state is a thermalized system consisting of strong coupled quarks and gluons in a very small region. This matter is only created for the briefest of instants, and then the fireball cools down and hadronizes into hadrons. So, we cannot observe the QGP directly in the existing laboratory conditions. However, we can extract a judgment of the creation of the quark matter by measuring and analyzing the spectra of identified particles produced after thermal freeze-out in heavy ion collisions.

In a very early stage of the collision, the energy density is expected to be sufficient to dissolve normal nuclear matter into a phase of quark matter, which exists for only a short time before the fireball cools down and the process of hadronization takes place. High-energy collisions provide an excellent probe of the quark matter. Properties of QGP will be probed further in LHC at the European Organization for Nuclear Research (CERN) [6]. The proton-nucleus collisions are important in experimental programmes performed in the LHC [7] because they not only provide baseline measurements for the nucleus-nucleus collisions but also help us better understand fundamental features of quantum chromodynamics (QCD) [8]. The transverse momentum spectra of final-state particles can give some important information of the matter created in high-energy collisions. In this paper, we use a multisource thermal model to study transverse momentum spectra of protons and charged pions produced in and collisions at 3, 8, and 15 GeV/c, recently measured in the Hadron Production Experiment (HARP) at the CERN [9].

2. Distribution of Transverse Momentum

In a framework of the multisource thermal model [10, 11], identified fragments or particles emit isotropically from different emission sources in the collisions. In order to deal conveniently with the relation between the sources and particles, we split the sources into groups in accordance with kinetic laws and geometrical positions. For the source in the group, its share of the transverse momentum spectrum of the final-state particles is given by where is the mean value of the transverse momentum which comes from the source in the group. According to statistical properties of the model, we have indicates the source number in the group. By computing convolution of the exponential functions equation (1), the total share of the group for the transverse momentum distribution is obtained as It is an Erlang distribution. Then, the transverse momentum distribution from the groups is given by where characterizes how large the contribution of the sources in the group is. Equation (5) is known as a multicomponent Erlang distribution. To simplify the calculation, the Monte Carlo method is used to calculate the transverse momentum spectra. With (1) and (4), the transverse momentum distribution is where is a random number in .

3. Comparison with HARP Results

Figures 1, 2, and 3 present the transverse momentum spectra of , , and in collisions at 3, 8, and 15 GeV/c at different angular intervals, respectively. From the first column to the third column in the figures, the momenta of incident protons are 3, 8, and 15 GeV/c, respectively. And from the first row to the third row in the figures, the angular intervals are 30°–40°, 60°–75°, and 105°–125°, separately. The symbols indicate the experimental data [9] in the HARP experiment at the CERN. The solid lines are the results of the multisource thermal model. The parameters and per degree of freedom () are given in Tables 1 and 2. In the 30°–40° angular interval, the value of for the production at 8 GeV/c incident momentum is equal to that at the 15 GeV/c. Moreover, for , the at 8 GeV/c and 15 GeV/c are the same in the three angular intervals. So is it for . In the calculation, one group with two sources is selected. We see that the model can successfully describe the experimental data. From the figures and the tables, it is also found that the width or the mean contribution of the distribution decreases with increasing the angular intervals for the same incident momentum.

Table 1 
Values of , taken in the fits of Figures 1 and 2.
Table 2 
Values of , taken in the fits of Figures 3 and 4.

Figure 1 
Transverse momentum of protons produced in collisions at 3, 8, and 15 GeV/c. The angular bins are 30°–40°, 60°–75°, and 105°–125°, respectively. The circles indicate the experimental data [9] and the curves indicate the results of the model.

Figure 2 
The same as Figure 1, but for .

Figure 3 
The same as Figure 1, but for .

Figures 4, 5, and 6 show the transverse momentum spectra of , , and in interactions at 3, 8, and 15 GeV/c with different angular bins. The symbols indicate the HARP-CDP experimental data [9] and the solid lines indicate the results of the multisource thermal model. The results of (7) are in agreement with the experimental data. The corresponding parameters and are listed in Tables 2 and 3. At forward angles θ = 30°–40°, the values of are the same for proton production at 3 GeV/c incident momentum and for the 15 GeV/c. For production, the at 8 GeV/c and 15 GeV/c are the same at the angles of 60°–75° and 105°–125°. Like the case of collisions, we still use a single group with two sources and the results agree well with the experimental data. At the same incident momentum, the decreases with the increase of the forward angles.

Table 3 
Values of , taken in the fits of Figures 5 and 6.

Figure 4 
Transverse momentum of protons produced in collisions at 3, 8, and 15 GeV/c. The angular bins are 30°–40°, 60°–75°, and 105°–125°, respectively. The circles indicate the experimental data [9] and the curves indicate the results of the model.

Figure 5 
The same as Figure 4, but for .

Figure 6 
The same as Figure 4, but for .

4. Discussion and Conclusion

In the multisource thermal model, the transverse momentum spectra of protons and charged pions produced in protons on Cu and Pb collisions at 3, 8, and 15 GeV/c in fixed angles of 30°–40°, 60°–75°, and 105°–125° are discussed. The spectra of the model are in agreement with the HARP-CDP data. The maximum value of is 1.73, and the minimum value is 0.02. From the above discussions, it is seen that the distribution widths of the concerned particles in both and reactions decrease with increasing the angular intervals for the same incident momentum.

In the work, two sources in one group are used to study the experimental data. It implies that the final-state particles emit from two sources. The model can provide an explanation not only for one group but also for more groups. In the previous work, the model was used to investigateelliptic flows [10], particle production [11, 12], longitudinal shift in (pseudo) rapidity distributions [13, 14], and so forth. For various types of collision systems, the multicomponent Erlang distribution can fit the transverse momentum spectra. The work reveals a multisource production phenomenon in the heavy ion 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 Grant nos. 11247250 and 11005071 and the National Fundamental Fund of Personnel Training (no. J1103210).