#### Abstract

The double-differential production cross sections in interactions of charged pions on targets at high momentums are analyzed by using a multicomponent Erlang distribution which is obtained in the framework of a multisource thermal model. The calculated results are compared and found to be in agreement with the experimental data at the incident momentums of 3, 5, 8, and 12 GeV/c measured by the HARP Collaboration. It is found that the source contributions to the mean momentum of charged particles and to the distribution width of particle momentums decrease with increase of the emission angle, and the source number and temperature do not show an obvious dependence on the emission angle of the considered particle.

#### 1. Introduction

Charged particles are the main products in hadron and heavy ion induced nuclear reactions at high energies. In such collisions, the particle distributions can provide information on the properties of interacting system [1–3]. These distributions include multiplicity distribution, rapidity and pseudorapidity distributions, azimuthal and polar angular distributions, momentum and transverse momentum distributions, and double-differential cross sections.

Many models have been introduced to describe particle productions in high energy collisions. For example, the FRITIOF model [4], the VENUS model [5, 6], the RQMD model [7–9], the HIJING model [10–12], a multiphase transport model (the AMPT model) [13], the Gribov-Glauber model [14], the QGSM model [15], the color glass condensate (CGC) model [16], the perturbative QCD plus saturation plus hydrodynamics (EKRT) model [17], the ART model [18], the ZPC model [19, 20], the Hydrodynamics model [21, 22], the string percolation model [23], a running coupling nonlinear evolution [24], a consistent quantum mechanical multiple scattering approach (EPOS) [25, 26], a combination model of constituent quarks and Landau hydrodynamics [27], a two-stage gluon model or a gluon dominance model [28], the KKT model [29], a multisource thermal model [30–32], and others are used in the data analyzes of high energy collisions.

As a challenging investigation, the multisource thermal model has been proposed and developed by us to describe the multiplicity distribution of charged particles [33] and isotopic production cross section of emitted fragments [34]. It is known that the multisource thermal model is very simple in describing some experimental data. Particularly, it can result in a multicomponent Erlang distribution which describes uniformly some distributions. In most cases, the experimental data have not been presented by the momentum distribution, but the transverse momentum distribution of particles. We are interested in analyzing the momentum distribution in terms of probability or double-differential cross section for the further test of the multisource thermal model.

In this paper, we will use the multisource thermal model to describe the double-differential production cross sections in charged pion induced nuclear reactions at high momentums. The model is described in Section 2. The comparisons with the experimental data at the incident momentums of 3, 5, 8, and 12 GeV/c measured by the HARP Collaboration [35] are shown in Section 3. Finally, we give our conclusions in Section 4.

#### 2. The Model and Method

The model used in the present work can be found in our previous work [33, 34] which present related formulations in terms of multiplicity distribution of final-state products and neutron number distribution in isotopes. To give a whole presentation of the present work, we introduce briefly the model [33, 34] on the multicomponent Erlang distribution in terms of charged particle momentum and its distribution in the following. In the model, many emission sources of particles and fragments are assumed to form in collisions. According to different interaction mechanisms or event samples, the sources are divided into groups (subsamples). The source number in the th group is assumed to be .

It is assumed that each source contributes to momentum distribution to be an exponential function, that is, the momentum () distribution contributed by the th source in the th group is given by where denotes the mean momentum contributed by the th source in the th group [33, 34]. Generally, the mean momentum contributed by different sources in the same group is assumed to be the same. The particle momentum distribution contributed by the th group is then given by the folding of exponential functions [33, 34]: One can see that an Erlang distribution is obtained. The momentum distribution contributed by the groups is given by a weighted sum of Erlang distributions: where denotes the weight factor and obeys the normalization . To avoid the factorial calculation in (2) in the case of being a large value, the Monte Carlo method is used to calculate the momentum distribution [33, 34].

In the Monte Carlo calculation, let denote random variable in . We have for the th source in the th group due to (1) which obeys the exponential function. The momentum contributed by the th group is due to the fact that (2) is the folding of exponential functions. The mean momentum contributed by the groups is due to the fact that (3) is a weighted sum.

To obtain the parameter values, for the purpose of convenience, we can use the idea of the least-square method. The values of and are changed from low to high step by step and the values of can be obtained. The former parameter can change continuously and the latter one is a series of integers. The best parameter values correspond to the minimum value, and the acceptable values determinate the parameter errors.

In the model, the value of denotes the number of types of emission sources. Generally, , which renders a small number of types of emission sources. The total number of emission sources can be small or large. A small number of emission sources means the hadronic sources, and a large number of emission sources means the parton degree of freedom. In the charged pion induced nuclear reactions at 3–12 GeV/c which are considered in the present work, we expect a small number of emission sources due to the saturation effect of target nuclei and not too high incident momentums.

The imbalances in mechanics and geometry render the interacting system to have kinetic, hydrodynamical, and thermaldynamical evolutions. The system or subsystems are assumed to stay in an equilibrium state or in local equilibrium states at the stage of chemical freeze-out. In the model, the inverse slope (mean transverse momentum) in the exponential distribution of transverse momentums is approximately regarded as the temperature parameter. According to the relation between the transverse momentum and the momentum , we have the temperature for a given polar angle . Although there is a in the presentation of , we do not expect that there is a dependence of on .

#### 3. Comparisons with Experimental Data

The double-differential and production cross sections measured in different emission angle ranges in -Be interactions at 3, 5, 8, and 12 GeV/c are shown in Figures 1 and 2, respectively. Different panels are scaled by multiplying different amounts to give all of them in a whole figure. The circles represent the experimental data of the HARP Collaboration [35] and the curves are our calculated results. In the calculation, we have used . The best values of and , as well as the value of per degree of freedom (/dof), are given in the figure in terms of (, , /dof), where is in the units of GeV/c. The relative errors for are about 6%, which are only statistical errors, and the systematic errors are eliminated. The error for can be neglected due to that the smallest alteration results in different distribution shape with an unexpected large . Similarly, the double-differential and production cross sections measured in different ranges in -Be interactions at 3, 5, 8, and 12 GeV/c are shown in Figures 3 and 4, respectively. From Figures 1–4 one can see that the model with describes uniformly the experimental data in most cases.

Figures 5–8 are the same as Figures 1–4, respectively, but showing the results for -C interactions. Figures 9–12 are the same as Figures 1–4, respectively, too, but showing the results for -Al interactions. We see that the model describes uniformly the experimental data in most cases. In [35], the experimental data on -Cu, Sn, Ta, and Pb interactions at high momentums have been given, too. A similar conclusion can be obtained in the case of fitting the experimental data by using the model.

From Figures 1–12 we see that the distribution width which is represented by shows a decrease with increases of . This phenomenon renders that the emission source has a forward movement along the beam direction. As a result, comparing with the situation in the source rest frame, the pion with small has a large momentum and that with large has a small momentum due to the effect of source momentum. We see also from Figures 1–12 that is only in the range from 2 to 12 and the most probability is 2. The sources are obviously incident pion and target nucleons. The number of participant target nucleons is in the range from 1 to 11 and the most probability is 1. Except the participant nucleons, other nucleons in target nucleus are the spectator nucleons.

To see clearly the changing trends of the parameters, the dependences of , , , and on for produced in -Be, C, and Al interactions are shown in Figure 13. The different symbols represent the parameter values obtained at different momentums, and for targets Be, C, and Al, the results are taken from Figures 1, 5, and 9, respectively. Similarly, the corresponding dependences for produced in the same interactions are given in Figure 14 which holds the parameter values taking from Figures 2, 6, and 10. The results for and produced in induced interactions are presented in Figures 15 and 16 which hold the parameter values taking from Figures 3, 7, and 11, as well as Figures 4, 8, and 12, respectively. We see that has a decreasing trend, and do not show an obvious change, and has a decreasing trend with increasing . This renders again that the emission source has a forward movement along the beam direction, which leads the momentum distribution width of produced pions to be small at large . The target nucleus presents a saturation effect on the source number and temperature.

From Figures 1–16, we see that the changing trends of the parameters on incident momentum and target size are not obvious. We have not seen an obvious difference of parameters for and produced in reactions induced by incident and . To see clearly the dependences of parameters on incident momentum () and target size, we combine the results of produced in the four ranges in reactions induced by and give the mean values of , , , and (i.e., , , , and ) for different incident momentums and target nuclei in Figures 17(a)–17(d), respectively, where the error bars for , , and are obtained according to the error transformation formula, and those for are standard deviations. One can see that the concerned mean values have no obvious dependence on incident momentum and target size. We would like to point out that the nondependence on incident momentum may be a result of the too narrow range of incident momentums, and the nondependence on target size renders the saturation effect of target nuclei in charged pion induced reactions at the considered incident momentums. Particularly, the extracted mean temperatures are in the range from MeV to MeV which are lower than the critical temperature (156 MeV) for phase transition from hadron matter to quark matter obtained by the thermaldynamic model [36].

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#### 4. Conclusions and Discussions

To conclude, the multisource thermal model is used to give a new and simple description of the double-differential production cross sections measured in different emission angle ranges in interactions of on different targets at the considered incident momentums. In most cases, the model describes well the experimental data of the HARP Collaboration. In the calculation we have used . This means that there is only one type of emission sources in the considered emission angle ranges. The parameter is small, which renders that the emission sources are incident pion and target nucleons. The structures of pion and nucleon do not play an important role at the considered incident momentums.

The parameter does not show an obvious change with increases of . For a given interaction, it is a natural result that the source number has no relation with the emission angle of produced particle. The parameter and the product decrease obviously with increases of . Considering the noneffect of , has the same behavior as . As the distribution width of particle momentum, has a small value at large because the emission source has a forward movement along the beam direction and the particle has a momentum transformation from the source rest frame to the laboratory reference frame.

As the source number, is not related to the concerned incident momentum and target size. The source contribution to the mean momentum of charged particles and the distribution width of the particle momentums are not related to the concerned incident momentum and target size, too. The nonrelation to the concerned incident momentum is caused by the narrow momentum range, and the nonrelation to the target size is the result of saturation effect of target nuclei in the concerned reactions. It is expected that the concerned parameters will increase at a high enough incident momentum.

The extracted temperatures do not show an obvious dependence on the incident momentum, target size, and polar angle in the considered collisions. The target nucleus presents a saturation effect on the source temperature. The extracted mean temperatures over polar angles are in the range from MeV to MeV which are lower than the critical temperature (156 MeV) for phase transition from hadron matter to quark matter obtained by the thermaldynamic model [36]. It is expected that higher temperatures will be obtained at higher incident momentums.

Because the transverse momentum , the distribution of is similar to that of at a given . The present modelling description can be used for the transverse momentum distribution if we use instead of in (1)–(3). In fact, in our previous work [37], the transverse momentum distributions of identified particles (, , , , , and ) produced in proton-proton, proton- (deuteron-) nucleus, and nucleus-nucleus collisions at the Relativistic Heavy Ion Collider and related energies are studied by using the multisource thermal model. This means that the present model can be used in different interacting systems in a wider energy region.

#### Conflict of Interests

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

#### Acknowledgments

This work was partly finished at the State University of New York at Stony Brook, USA. One of the authors Fu-Hu Liu thanks Professor Dr. Roy A. Lacey and the members of the Nuclear Chemistry Group of Stony Brook University for their hospitality. The authors acknowledge the supports of the National Natural Science Foundation of China (under Grant no. 10975095, no. 11005071, and no. 11247250), the China National Fundamental Fund of Personnel Training (under Grant no. J1103210), the Open Research Subject of the Chinese Academy of Sciences Large-Scale Scientific Facility (under Grant no. 2060205), the Shanxi Scholarship Council of China, and the Overseas Training Project for Teachers at Shanxi University.