On Antiproton Production in 158 GeV / c Proton-Carbon Collisions and Nuclear Temperature of Interacting System

Themultisource thermalmodel is used in this paper to analyze the antiproton (p) production process in high-energy proton-carbon (p-C) collisions. The transverse momentum, Feynman variable, and rapidity distributions of antiprotons in the nucleon-nucleon center-of-mass system are calculated by using themodel.Themodeling results are compared and found to be in agreement with the experimental data measured by the NA49 Collaboration at 158GeV/c beam momentum. As a parameter, the nuclear temperature of interacting system extracted from the antiproton spectrum is estimated to be about 150MeV.


Introduction
High-energy collisions are an important experimental phenomenon in modern physics.From fixed target experiments at accelerators to collider experiments, a lot of experimental results have been reported.In the collisions, an incident projectile and a fixed target (or another incident target at colliders) can be particles, ions, or nuclei.Generally, the products in nucleus-nucleus collisions are more abundant than those in particle-particle collision.The analysis of the former one is also more complex.As a transition stage from particle-particle collision to nucleus-nucleus collisions, particle-nucleus collisions have not only abundant experimental results but also simpler physics process.In fact, in proton induced nuclear collisions, the projectile is simple and has no spectator's contribution, and the target is complex and has spectator's contribution to final state.
Many models have been proposed in the field of highenergy collisions, for example, the equivalent quark-gluon string model [1], the hadron resonance gas model [2,3], the statistical multifragmentation model [4], the expanding and emitting source model [5] or the expanding-evaporating source model [6], the nonequilibrium-statistical relativistic diffusion model [7], the dual parton model [8,9], the relativistic or ultrarelativistic quantum molecular dynamics model [10][11][12][13], and so forth.In a workshop [14] held a few years ago at the CERN Theory Institute, more models have reported their predictions for the collision program at the LHC energies.Most of the mentioned models are microscopic models based on quantum chromodynamics (QCD) and concern the system evolution and dynamical process.Parts of them are thermal and statistical models and focus on the global properties of interacting system and finalstate products.
In the past years, we have proposed a multisource thermal model [15,16] and extended it to relativistic situation [17] for descriptions of particle production in high-energy collisions.Some experimental results are described by the model.Recently, the NA49 Collaboration reported inclusive productions in proton-carbon (-C) collisions at 158 GeV/c beam momentum [18].We are interested in the description of antiproton () production and will give a description in this paper.Because there is no effect of leading particles, the distribution law of antiprotons in analysis is simpler than that of protons.
Advances in High Energy Physics homogeneously in a projectile cylinder and a target cylinder due to the penetration of the projectile and the target.In the rapidity space, the projectile and target cylinders are located in the rapidity intervals [  min ,   max ] and [  min ,   max ], respectively.For antiproton production in proton-carbon collisions, the leading nucleons have no contribution, but the target spectator contributes a cylinder in the rapidity interval [ TS min ,  TS max ] due to the produced particles causing the cascade collisions in the spectator.Let  TS denote the weight of the target spectator cylinder; the weights of the projectile and target cylinders are the same: We define the beam direction to be the  axis and the reaction plane to be the  plane.In the source rest frame, let   and   denote the source temperature and particle momentum, respectively.Considering the relativistic effect [19,20], we have the   distribution in the relativistic ideal gas model to be where  denotes the Boltzmann constant,  0 denotes the rest mass of the considered particle,  = 1/( 2 0   ) ⋅ 1/ ( 2 ( 0 /  )) is the normalization constant, and  2 ( 0 /  ) is the modified Bessel function of order 2. In the previous equation, we have taken the natural system of units in which the speed  of light in vacuum is 1.
In the Monte Carlo method,   can be obtained by solving the inequality | ∫  0    (  )  −  1 ≤ |, where  1 and  are a random number distributed evenly in the range from 0 to 1 where 0 and 1 are included (i.e., in [0, 1]) and a small quantity, respectively.Let   denote the emission angle of the considered particle.An isotropic emission gives   = arccos(1−2 respectively, where  1 and  2 are the normalization constants.We would like to point out that ( 2) is in fact the Boltzmann distribution which is used in the literature (e.g., [21]).Both ( 2) and ( 3) are valid because they are in agreement with the Monte Carlo calculations based on the definitions.The distributions of   and    are respectively.The distribution of velocity V  is In the nucleon-nucleon center-of-mass system, let   denote the source rapidity.According to different weights, we have   = (  max −   min ) 3 +   min for the projectile cylinder,   = (  max −   min ) 4 +   min for the target cylinder, or   = ( TS max −  TS min ) 5 +  TS min for the target spectator cylinders, where  3 ,  4 , and  5 are random variables in [0, 1].Then, the particle rapidity  =   +   , transverse momentum   =    , longitudinal momentum   =    cosh  +   sinh , momentum  = √ 2  +  2  , energy  = √ 2 +  2 0 , transverse mass   =    , and Feynman variable   = 2  /√ can be obtained, where √ is the energy in the center-of-mass system.Then, we transform all of the concerned kinetic quantities from the rest frame of a single source to the nucleon-nucleon center-ofmass system.The effects of the multiple sources are obtained by different   .On the other hand, from the definition of Feynman variable, we have are the functions of   .Particularly, the relationship between  and   at a given   is built.

Comparison with Experimental Data
Figure 1 shows the invariant cross-section, (  ,   ), in the units of mb/(GeV 2 / 3 ) for  produced in -C collisions at 158 GeV/c beam momentum.The distributions of   , (  ,   ) ∝ (1/)(/    ), at different   are given.The symbols represent the experimental data of the NA49 Collaboration [18], and the curves are our results calculated by the multisource thermal model [15][16][17].To show a clear representation, the values of (  ,   ) at different   are scaled by multiplying constants shown in the figure.In the calculation, we have used the method of   2 and 3, respectively.The symbols represent the experimental data of the NA49 Collaboration [18] and the curves are our calculated results.The value of   is the same as that for Figure 1 and  TS = 0.15 ± 0.01.The values of other parameters (rapidity shifts) which have a relative error of 6% are given in Table 1 with the values of  2 /dof.Once again the model describes well the experimental data.
To see the dependences of different rapidity shifts on   , the relations of   max −   ,   min −   ,   max −   ,   min −   ,  TS max −   , and  TS min −   are given in Figure 4.The symbols represent the parameter values used in Figures 2 and 3.The lines are our fitted results described by the equations   =   + , where   denotes different rapidity shifts and the values of  and  are given in Table 2 with the values of  2 /dof.One can see that   max decreases and   min and  TS min increase with the increases of   .A very slight decrease in   max and very slight increases in   min and  TS min are observed with the increase of   .If we define   max −   min as the total length of the projectile and target cylinders, this length shows a decrease with the increase of   .The condition   max −   min = 0 results in a zero length cylinder which gives approximately the maximum   to be 3.82 GeV/c.From   = 2 GeV/c to 3.82 GeV/c, the relations of   min −  ,   max −  , and  TS max −   will not be the same linear changes as those in the region of   < 2 GeV/c.The projectile and target cylinders will be from partly overlapping to totally overlapping.The length will become in fact shorter and shorter and finally will be a single source.   2 and 3 due to the experimental data of the NA49 Collaboration [18], and the lines are our fitted results.

Discussions
In the previous discussions, we have assumed in fact that a thermal equilibrium (or a local thermal equilibrium) is achieved in the collisions when the thermal source freezes out final-state particles.Thus, the concept of temperature can be used in our calculation.In the calculation, we have not given special attention to the possible presence of early emitting source before an achievement of equilibrium.In fact, when studying antiprotons instead of protons with leading particle contributions, the preequilibrium emissions are naturally excluded.This is the reason that we have not studied protons but antiprotons.
In a previous work [21], the transverse momentum and rapidity distributions of  mesons produced in Pb-Pb collisions at 20A, 30A, 40A, 80A, and 158A GeV were studied by using the multisource thermal model (or multisource ideal gas model).In another previous work [22], the transverse mass distributions of protons produced in Au-Au collisions at 8A GeV and Pb-Pb collisions at 158A GeV were studied by using the same or similar model.Besides, the transverse mass spectra of protons, pions, kaons, Lambdas, and Antilambdas produced in Au-Au collisions at 2A, 4A, 6A, and 8A GeV and Pb-Pb collisions at 20A, 30A, 40A, 80A, and 158A GeV were studied [23].The pseudorapidity distributions of charged particles produced in - collisions at 0.053, 0.2, 0.54, 0.546, 0.63, 0.9, and 1.8 TeV; the pseudorapidity and multiplicity distributions of charged particles produced in - collisions at 0.9, 2.36, and 7 GeV; and the pseudorapidity distributions of charged particles produced in Pb-Pb collisions at 2.76A TeV were studied too [24][25][26][27].We would like to say that the multisource thermal model used in the present work describes different particles produced in different collisions at different energies.However, the present work is the first one to describe the antiproton production by the model.
We would like to point out that the temperature extracted in the present work reflects excitation degree of thermal source when it freezes out antiprotons.According to production process, the source temperature extracted from proton spectrum is on average less than 150 MeV due to some protons being leading particles and that from meson spectrum is higher than 150 MeV due to violent collisions.If we exclude the contributions of leading particles, the temperature extracted from proton spectrum is approximately equal to 150 MeV.These results are consistent with other measurements in the field [18] and with other model expectations [14].
The target spectator weighting factor is found to be about 15%.This reflects the contribution of cascade collisions in the target spectator caused by the produced particles.In highenergy collisions, the spectator effect cannot be neglected in most cases.Particularly, in asymmetric collisions such as proton-carbon collisions, one can obtain some asymmetric rapidity spectra.The contribution of target spectators has to be considered.In nucleus-nucleus collisions, both the contributions of projectile and target spectators have to be considered.

Conclusions
We have analyzed the transverse momentum, Feynman variable, and rapidity distributions of antiprotons produced in proton-carbon collisions at high-energy by using the multisource thermal model.This model assumes that many sources are formed in high energy collisions.Each source is treated as a relativistic ideal gas.The distributions of momenta, transverse momenta, longitudinal momenta, energies, kinetic energies, transverse mass, velocities, and other related quantities for a given kind of particles at a given temperature in the source rest frame can be obtained.
In the comparisons with the experimental data, the transverse momentum distribution is not related to the source positions and arrangements but to the temperature of the rest source.A single-temperature distribution describes the  transverse momentum distributions of antiprotons produced in proton-carbon collisions at 158 GeV/c measured by the NA49 Collaboration [18].From the transverse momentum distributions of antiprotons, we determine that the source temperature is approximately 150 MeV.In longitudinal distributions, the distributions of Feynman variables and rapidities are mainly related to the source positions and arrangements.In rapidity space, these sources form a projectile cylinder and a target cylinder.Particularly, in proton-carbon collisions, a target spectator cylinder consisting of a series of sources has a weight for the production of antiprotons.By using the three cylinders, the model describes the distributions of Feynman variables and rapidities for antiprotons in proton-carbon collisions at high energy.Our analyses show that the weight of the target spectator cylinder is 15%.Both the weights of the projectile cylinder and the target cylinder are the same.
With the increase of the transverse momentum, the maximum rapidity shift of the projectile cylinder has a decrease, the minimum rapidity shifts of the target and target spectator cylinders have increases, the maximum rapidity shift of the target cylinder has a very slight decrease, and the minimum rapidity shift of the projectile cylinder and the maximum rapidity shift of the target spectator cylinder have very slight increases.The length of the total projectile and target cylinders decreases with the increase of the transverse momentum.
Combined with the previous works [15,16,[21][22][23][24][25][26][27] analyzed by the multisource thermal model, we guess the energy behavior of some parameters of the model here.For a given projectile-target impacting system, the rapidity shifts of the cylinders increase slowly with the logarithmic center-of-mass energy and do not depend obviously on the centrality.The temperatures extracted from the hadron spectra increase slowly with the energy and centrality.The relative contributions of the cylinders, leading nucleons, and spectators have no obvious relation to the energy but the centrality.In central collisions, the leading nucleons and spectators have very limited contributions or no contribution to the finalstate distributions; and in peripheral collisions, they have large contributions.

3 /GeV 2 )Figure 1 :
Figure 1: Transverse momentum distributions of antiprotons in -C collisions at 158 GeV/c.The symbols represent the experimental data of the NA49 Collaboration [18] at different   , and the curves are our results calculated by the multisource thermal model.

Figure 2 :
Figure 2: Feynman variable distributions of antiprotons in -C collisions at 158 GeV/c.The symbols represent the experimental data of the NA49 Collaboration [18] at different   , and the curves are our results calculated by the multisource thermal model.

Figure 3 :
Figure 3: Rapidity distributions of antiprotons in -C collisions at 158 GeV/c.The symbols represent the experimental data of the NA49 Collaboration [18] at different   , and the curves are our results calculated by the multisource thermal model.
y T min , y TS min

Figure 4 :
Figure 4: Correlations between the rapidity shifts and   .The symbols represent the parameter values used in Figures2 and 3due to the experimental data of the NA49 Collaboration[18], and the lines are our fitted results.
MeV for different   values and with the values of  2 per degree of freedom ( 2 /dof) is in the order 0.759, 1.042, 0.564, 0.888, 0.699, 0.994, 0.436, 1.348, 0.895, 0.552, 0.298, 0.424, and 0.450, respectively.One can see that the model describes well the experimental data.The distributions of   and  at different   for  produced in -C collisions at 158 GeV/c are presented in Figures 2testing to determine the parameter values.The value of the only free parameter is   = (150± 10)

Table 1 :
Values of the rapidity shifts and  2 /dof for the fits in Figures2 and 3.The relative errors of the rapidity shifts are 6%. Te value of  2 is given instead of  2 /dof.

Table 2 :
Values of  and  in equations   =   + .The unit of   is GeV/c.