Chemical Potentials of Quarks Extracted from Particle Transverse Momentum Distributions in Heavy Ion Collisions at RHIC Energies

In the framework of amultisource thermalmodel, the transversemomentumdistributions of charged particles produced in nucleusnucleus (A-A) and deuteron-nucleus (d-A) collisions at relativistic heavy ion collider (RHIC) energies are investigated by a twocomponent revised Boltzmann distribution. The calculated results are in agreement with the PHENIX experimental data. It is found that the source temperature increases obviously with increase of the particle mass and incident energy, but it does not show an obvious change with the collision centrality. Then, the values of chemical potentials for up, down, and strange quarks can be obtained from the antiparticle to particle yield ratios in a wide transverse momentum range.The relationship between the chemical potentials of quarks and the transverse momentum with different centralities is investigated, too.


Introduction
As an interesting research field, high energy heavy ion collisions have a very important significance for understanding of particle physics, nuclear physics, and astrophysics in both theoretical and experimental aspects [1][2][3].Investigating the multiparticle production process at high energies provides a unique opportunity for us to comprehend the nuclear reaction mechanism and rare phenomena at high density and high temperature [4][5][6].A large number of experimental and theoretical studies have been carried out over an energy range from a few tens of MeV to a few TeV per nucleon in the past decades, and many theoretical models [7][8][9][10][11][12] have been proposed in the field of multiparticle production to explain the different features of the experimental results, such as angular distributions, multiplicity distributions, transverse momentum (or mass) distributions, (pseudo) rapidity distributions, the longitudinal flow, transverse flow, production cross-sections, and others [13][14][15][16][17][18][19].
The relativistic heavy ion collider (RHIC) and the large hadron collider (LHC) [20][21][22] have been opening new energy regions for the research of multiparticle production in nuclear collisions.One of the major objectives of them is searching for the evidence of a new state of matter, namely, Quark-Gluon plasma (QGP) [23][24][25][26].Traditionally, QGP is considered crucial to determine whether the thermal and dynamical equilibrations do exist in the interacting region in relativistic collisions [27].Although this idea was put forward many years ago, up to now there is no unambiguous test to probe the degree of equilibration in the system.For any system, we can get the important thermodynamic information by the chemical potential.The possibility of combining different substances for forming optimal mixtures is strictly related to knowledge of the chemical potential of each component in the mixture environment [28].In previous literature [27][28][29][30][31], chemical potentials of different particles have been researched with different models and formulas, simple scaling relations connecting chemical potentials are found.However, the study concerning chemical potentials of quarks is correspondingly lacking.In this paper, we will use a practical method to extract chemical potentials of quarks from the antiparticle to particle yield ratios in transverse momentum distributions.

Advances in High Energy Physics
To describe the multifragment emission and multiparticle production processes, we have proposed a multisource thermal model [15][16][17] and explained some distributions of fragments and particles produced in nuclear collisions in a wider energy range.To give a further test to the multisource thermal model, in this paper, we will describe the transverse momentum distributions of charged particles produced in nucleus-nucleus collisions at RHIC energies by using the model in which the Boltzmann distribution is revised to fit the data at high transverse momentum.Then, the chemical potentials of quarks from the antiparticle to particle yield ratios are extracted in a wide transverse momentum range.

The Model
According to the multisource thermal model [15][16][17] which is used in the descriptions of particle (fragment) multiplicity, emission angles, azimuths, projected angles, (pseudo) rapidity, and transverse flows in high energy collisions, many emission sources are assumed to form in high energy collisions.These sources emit final-state particles and nuclear fragments [18,18,32].Each source is considered to emit particles isotropically in its rest frame and is treated as a thermodynamic system of relativistic and quantum ideal gas [33][34][35][36].We have   distribution to be the Boltzmann distribution [18]: where  denotes the number of particles,  is the normalization constant,  is the temperature parameter, and  0 is the rest mass of the considered particle.Considering multiple temperatures, the   distribution can be described by a multicomponent Boltzmann distribution: where   ,   , and   are the contribution ratio, normalization constant, and temperature parameter of the th source, respectively.
In the case of using the parameters as less as possible, we found that (1) and ( 2) underestimate the tail part of the transverse momentum distribution.To revise the two equations, we multiply experientially the right side by   and make a renormalization.Then, we have a two-component revised Boltzmann distribution to be where  1 is the contribution ratio of the first temperature,  1,2 and  1,2 denote the new normalization constant and new temperature parameter of the first and second components, respectively.
According to the statistical arguments based on chemical and thermal equilibrium at the quark level, we have the relations between antiparticle to particle yield ratios and quark chemical potentials to be [37] where  is the number density of considered particles;   ,   , and   are the chemical potentials of up, down, and strange quarks, respectively;   ,   , and   are the temperature parameters extracted from proton (antiproton), negatively (positively) pion, and negatively (positively) kaon spectrums, respectively.We would like to point out that the chemical potentials of up and down quarks have been differentiated in the above discussions, although the two chemical potentials were not differentiated in previous literature [37].From (4), the relations between the parameter values and the chemical potentials of quarks can be obtained.Then, the chemical potentials of quarks can be extracted from the antiparticle to particle yield ratios in transverse momentum distributions.

Comparisons with Experimental Data
The transverse momentum distributions of  + / − ,  + / − , and / produced in Au-Au collisions at center-of-mass energy per nucleon pair √  = 130 and 200 GeV are presented in Figures 1 and 2, respectively.The symbols represent the experimental data with different centralities measured by the PHENIX collaboration [38][39][40], and the curves are our results calculated by the two-component revised Boltzmann distribution.In the calculation, the values of the fitted parameters are obtained by fitting the experimental data and shown in Table 1 with values of  2 /dof ( 2 per degree of freedom).It seems that in most cases the model describes the experimental data.From the parameter values, one can see that the temperature parameter increases with increase of particle mass for emissions of the six types of particles.For emission of  + / − , the temperature does depend nonobviously on impact centrality, and for emission of  + / − and /, the temperature parameter increases with increases of impact centrality.
Figure 3 presents the transverse momentum distributions of positively and negatively charged particles produced in Au-Au and d-Au collisions at √  = 200 GeV with different centrality classes and magnifications.The symbols represent the experimental data measured by the PHENIX collaboration [41], and the curves are our results calculated by the two-component revised Boltzmann distribution.The fitted parameter values and the corresponding  2 /dof are given in Figure 1: The transverse momentum distributions of  + / − ,  + / − , and / produced in Au-Au collisions at √  = 130 GeV with different centrality classes.The symbols represent the experimental data of the PHENIX collaboration [38,39], and the curves are our results calculated by the two-component revised Boltzmann distributions.4. The symbols represent the experimental data of the PHENIX collaboration [40] and the curves are our results calculated by the two-component revised Boltzmann distribution.We see again that the model describes well the experimental data.Correspondingly, the values of parameters and  2 /dof are given in Table 3. Once more, for emissions of charged hadrons, the temperature parameter increases with increase of particle mass.In particular, for emissions of /, the temperature parameter increases with increase of impact centrality.
Based on the above successful descriptions on the transverse momentum distributions of particles and antiparticles, The curves are our results calculated by (4).The mean values (including their standard deviations) of   ,   , and   are given in Table 4.We see that   ,   , and   do not obviously depend on transverse momentum range and impact centrality.The relations between the chemical potentials of quarks and the transverse momentum in d-Au and Au-Au collisions at √  = 200 GeV with different centralities are presented in Figure 7.The curves are our results calculated by (4).The mean values of   ,   , and   are given in Table 4.In some cases the general trend of curves is that   (  ) increases with increase of transverse momentum and   decreases with increase of transverse momentum.However, it is hard to say that there is a difference in depending on the transverse momentum, because these differences are in fact statistical fluctuations in the calculation.One can see that   ,   , and   do depend nonobviously on impact centrality.
Figure 8 shows the correlations between  (  ,   , and   ) and   in Au-Au collisions at √  = 200 GeV with different centrality classes.The curves are our results obtained by (4).The mean values of different  are shown in Table 4. Once more, all  do depend nonobviously on transverse momentum range and impact centrality.The estimated mean values of   ,   , and   from the fittings are 16.6 ± 0.3, 15.7 ± 0.3, and 9.6 ± 0.3 MeV, respectively.We see that the difference between   and   is indeed small.

Discussions and Conclusions
In the above calculations, we have used the twocomponent revised Boltzmann distribution in which  2   exp(−√ 2  +  2 0 /) is used to replace   exp(−√ 2  +  2 0 / ) in the original Boltzmann distribution.This revision can give a better description on the tail part of the transverse momentum distribution, while the original one underestimates particle production at high transverse momentum.To reproduce a high probability for particles with high transverse momentum, another revision which uses   √ 2  +  2 0 exp(−√ 2  +  2 0 /) instead of   exp(−√ 2  +  2 0 /) was proposed some years ago [42].In the two revisions, the normalizations have to be reconsidered, respectively.Spectral distributions of final-state particles are very important observations in high energy collisions.Researching the transverse momentum distributions of final-state particles provides a method for us to understand the evolution of interacting systems and the generation mechanism of final-state particles.In previous literature, a number of models and formulas have been used to describe different particles produced in different collisions at different energies.The present work is an approach to give good descriptions of the transverse momentum distributions with a few free parameters, and chemical potentials of quarks are obtained from the antiparticle and particle yield ratios in a wide transverse momentum range.
The present work is somewhat similar to the previous search [36,43,44] but with some different models, parameters, and collisions, where the transverse momentum spectrums of final-state products produced in nucleusnucleus and proton-proton collisions at high energy have distributions result in different temperatures, we need to correct the temperatures to a standard one when we compare these results.
The present work provides a new variety and professional analysis for collisions that may give a more complete picture of the RHIC and helpful in understanding quark chemical potentials behavior, as well as the relationship between the chemical potentials of quarks and the transverse momentum with a few free parameters and different centralities.at different transverse momentums, it does not mean that there is dependence of quark chemical potential on transverse momentum.
In conclusion, the transverse momentum distributions of charged particles produced in Au-Au and d-Au collisions at RHIC energies have been studied by the multisource thermal model.It is shown that the two-component revised Boltzmann distribution is successful in the descriptions of experimental data.For emissions of charged particles, the temperature parameter increases with increases of particle  mass and incident energy.In most cases, the temperature parameter does not obviously depend on impact centrality.
In the study of phase transition or thermodynamical and chemical equilibrium, it is quite important to achieve a thorough understanding of the quark chemical potentials.In the present work, we have used a practical method for calculating the quark chemical potentials.The antiparticle to particle yield ratios obtained by the model give the quark chemical potentials in a wide transverse momentum range.We would like to point out that the considered three types of quark chemical potentials do not show an obvious change with the increase of transverse momentum and impact centrality in nucleus-nucleus collisions at RHIC energies, and in most cases the mean values of   and   are small and the mean values of   are smaller.
The difference between   and   is small due to the small mass difference between up and down quarks.The small values of   ,   , and   at RHIC energies indicate that the chemical potentials of other quarks are impossibly large.This renders that the interactions among quarks in the interacting system are not too strong, and the interacting system may stay at the state of QGP.The small chemical potentials of quarks result in small chemical potentials of hadrons.According to the general relation between temperature and hadron chemical potential, we have a high temperature of interacting system, which is expected by the formation of QGP.

), 3 (
b), and 3(c)), as well as  − ,  − ,and  (Figures3(d), 3(e), and 3(f)), the temperature parameter increases with increase of particle mass.For emissions of charged particles, the values of temperature parameters do not change obviously with centrality class, and for each particle in different centrality classes, the temperature parameter closes to a certain value.The transverse momentum distributions of identified charged particles produced in Au-Au collisions at √  = 200 GeV with different centrality classes and magnifications are shown in Figure

Figure 2 :
Figure 2: The same as Figure 1, but showing the results in Au-Au collisions at √  = 200 GeV.The symbols represent the experimental data of the PHENIX collaboration [40].

Figure 5 :
Figure 5: Correlations between chemical potentials of quarks and the transverse momentum in Au-Au collisions at √  = 130 GeV with different centrality classes.The curves are our results calculated by (4).

Figure 6 :
Figure 6: The same as Figure 5, but showing the results in Au-Au collisions at √  = 200 GeV.

Figure 7 :
Figure 7: Correlations between  (  ,   , and   ) and   in d-Au and Au-Au collisions at √  = 200 GeV with different centrality classes.The curves are our calculated results.

Figure 8 :
Figure 8: The same as Figure 7, but showing only the results in Au-Au collisions at √  = 200 GeV for more centrality classes.

Table 1 :
Parameter values corresponding to the curves in Figures1 and 2for Au-Au collisions at √  = 130 and 200 GeV, respectively.The values for positively/negatively charged particles are given in terms of "the first value/the second value" in the case of the two values being different.The relative errors estimated for  1,2 and  1 are less than 6% and 0.5%, respectively.
a The last point is not included in the calculation of  2 /dof due to the low statistics.

Table 2 :
Parameter values corresponding to the curves in Figure3for Au-Au and d-Au collisions at √  = 200 GeV.The values for positively/negatively charged particles are given in terms of "the first value/the second value" in the case of the two values being different.The relative errors estimated for  1,2 and  1 are less than 6% and 0.5%, respectively.

Table 3 :
The same as that for Table2, but showing the results corresponding to the curves in Figure4for Au-Au collisions at √  = 200 GeV.
a The last point is not included in the calculation of  2 /dof due to the low statistics.bThelast two points are not included in the calculation of  2 /dof due to the low statistics.

Table 2 .
In the calculation of  2 /dof, we take the errors to be a half of the symbol size due to that for some data the errors are not available.One can see that our calculated results are in good agreement with the experimental data.From Table2we see clearly that for emissions of  + ,  + , and  (Figures3(a

Table 4 :
Mean values (  ,   , and   ) of chemical potentials obtained from the curves in Figures5-8, where the errors are the standard deviations.
(4)can use(4)to study the chemical potentials of quarks.Figures5 and 6present the dependencies of the chemical potentials of (a) up, (b) down, and (c) strange quarks on the transverse momentum in Au-Au collisions at √  = 130 and 200 GeV with different centrality classes, respectively.