Emission of Protons and Charged Pions in p + Cu and p + Pb Collisions at 3 , 8 , and 15 GeV / c

We present an analysis of proton and charged pion transversemomentum spectra ofp+Cu andp+Pb reactions at 3, 8, and 15GeV/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 p + Cu and p + Pb collisions decrease with increasing the angle for the same incident momentum.


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 +Cu and  + Pb collisions at 3, 8, and 15 GeV/c, recently measured in the Hadron Production Experiment (HARP) at the CERN [9].

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 th source in the th 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 th source in the th group.According to statistical properties of the model, we have indicates the source number in the th group.By computing convolution of the   exponential functions equation ( 1), the total share of the th 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 th 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 [0, 1].

Comparison with HARP Results
Figures 1, 2, and 3 present the transverse momentum spectra of ,  − , and  + in  + Cu 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  2 per degree of freedom (dof) are given in Tables 1 and 2. In the 30 ∘ -40 ∘ angular interval, the value of ⟨ Tm1 ⟩ for the  production at 8 GeV/c incident momentum is equal to that at the 15 GeV/c.Moreover, for  − , the ⟨ Tm1 ⟩ 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   [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  2 /dof are listed in Tables 2  and 3.At forward angles  = 30 ∘ -40 ∘ , the values of ⟨ Tm1 ⟩ are the same for proton production at 3 GeV/c incident momentum and for the 15 GeV/c.For  + production, the ⟨ Tm1 ⟩ at 8 GeV/c and 15 GeV/c are the same at the angles of 60 ∘ -75 ∘ and 105 ∘ -125 ∘ .Like the case of  + Cu 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 ⟨ Tm1 ⟩ decreases with the increase of the forward angles.

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  2 /dof 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  + Cu and +Pb 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.

Table 1 :
Values of ⟨ Tm1 ⟩,  1 taken in the fits of Figures1 and 2.

Table 2 :
Values of ⟨ Tm1 ⟩,  1 taken in the fits of Figures3 and 4.

Table 3 :
Values of ⟨ Tm1 ⟩,  1 taken in the fits of Figures5 and 6. the width or the mean contribution ⟨ Tm1 ⟩ of the distribution decreases with increasing the angular intervals for the same incident momentum.Figures4, 5, and 6 show the transverse momentum spectra of ,  − , and  + in  + Pb interactions at 3, 8, and 15 GeV/c with different angular bins.The symbols indicate the HARP-CDP experimental data