Chemical potentials of light flavor quarks from yield ratios of negative to positive particles in Au+Au collisions at RHIC

The transverse momentum spectra of $\pi^{-}$, $\pi^{+}$, $K^{-}$, $K^{+}$, $\bar{p}$, and $p$ produced in Au+Au collisions at center-of-mass energy $\sqrt{s_{NN}}=7.7$, 11.5, 19.6, 27, 39, 62.4, 130, and $200$ GeV are analyzed in the framework of a multisource thermal model. The experimental data measured at midrapidity by the STAR Collaboration are fitted by the (two-component) standard distribution. The effective temperature of emission source increases obviously with the increase of the particle mass and the collision energy. At different collision energies, the chemical potentials of up, down, and strange quarks are obtained from the antiparticle to particle yield ratios in given transverse momentum ranges available in experiments. With the increase of logarithmic collision energy, the chemical potentials of light flavor quarks decrease exponentially.


Introduction
The constructions of the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) have been opening a new epoch for the studies of nuclear and quark matters. One of the major goals of the RHC and LHC studies is to obtain information on the quantum chromodynamics (QCD) phase diagram [1]. The phase diagram includes at least a fundamental phase transition between the hadron gas and the quark-gluon plasma (QGP) or quark matter, and is usually plotted as temperature versus baryon chemical potential (T vs µ B ). Nowadays, the detailed characteristics of the phase diagram are not known yet. The experimental and theoretical nuclear physicists have been focusing their attentions on the searching for the critical end point and phase boundary. Lattice QCD calculations indicate a system produced at small µ B or high energies through a crossover at the quark-hadron phase transition [2,3,4]. Based on the lattice QCD [5] and several QCD-based models calculations [6,7,8,9], as well as mathematical extensions of lattice techniques [10,11,12,13], researchers suggest that the transition at larger values of µ B is the first order and the QCD critical end point is existent.
Pinpointing the phase boundary and the critical end point is the central issue to understand the properties of interacting matter under extreme conditions and to map the QCD phase diagram. The matter produced in high-energy heavy-ion collisions can be regarded as an important probe for the phase boundary and the critical end point [6,14]. To this end, the STAR Collaboration at the RHIC has undertaken the first phase of the beam energy scan (BES) program [15,16,17], and starting the second phase from 2018 to 2019 [18]. The program is to vary the collision energy which enable a search for nonmonotonic excitation functions over a broad domain of the phase diagram. Before looking for the evidence of existence of a critical end point and the phase boundary, it is important to know the (T, µ B ) region of phase diagram one can access. The produced particles spectra and yield ratios allow us only to infer the values of temperature and baryon chemical potential [19]. Furthermore, the bulk properties such as rapidity density dN/dy, mean transverse momentum p T , particle ratios, and freeze-out properties may provide insight into the particle production mechanisms at the BES energies. So, it is very important to study these bulk properties systematically, which may reveal the evolution and change about the system created in high-energy heavyion collisions.
As one of the most important measurement quantities, the transverse momentum spectrum includes abundant information which is related to the excitation degree of the collision system and attract many studies. The spectra of identified particles can also provide useful information about temperature, particle ratio, and chemical potential by using thermal and statistical investigations [20]. For any system, one can determine the direction and limitation of mass transfer by comparing the chemical potentials of particles, that is to say the chemical potential is a sign to mark the direction of spontaneous chemical reaction. The chemical potential can also be a criterion for determining whether thermodynamic equilibrium does exist in the interacting region in high-energy collisions. Therefore, the chemical potential is also one of the major problems for investigating the QGP. One can see that the chemical potentials of quarks make an important subject at high energy. We are very interested in measurements or extractions of the chemical potentials of quarks.
In this paper, we extract the chemical potentials of light flavor quarks from the yield ratios of negatively to positively charged particles. By using the standard distribution, the transverse momentum spectra of π − , π + , K − , K + ,p, and p produced in Au+Au collisions at center-of-mass energy (per nucleon pair) √ s N N = 7.7, 11.5, 19.6, 27, 39, 62.4, 130, and 200 GeV measured by the STAR Collaboration in midrapidity interval (|y| < 0.1) [19,21] are described. The collision energy covers main range of the RHIC at its BES.

The model and method
To extract the chemical potentials of quarks, we need to know the yield ratios of negatively to positively charged particles. Although we can have the values of yield ratios directly in experiments, whereas they are not complete and comprehensive in some cases. Usually, the transverse momentum spectra of charged particles are given in many experiments. We can get the yield ratios by fitting the available data in a winder transverse momentum range. Then, the values of chemical potentials for the up, down, and strange quarks can be obtained from the yield ratios of π − /π + , K − /K + , and p/p, respectively.
In this paper, the transverse momentum spectra are analyzed in the framework of a multisource thermal model [22], which assumes that various sources are formed in high-energy collisions. These sources are divided into a few groups by different interacting mech-anisms, geometrical relations, or event samples. Each group of sources forms a relatively large emission source which stays in a local thermal equilibrium state at the chemical or kinetic freeze-out. Each emission source is considered to emit particles in its rest frame and treated as a thermodynamic system of relativistic and quantum ideal gas. This means that each emission source can be described by the thermal model or other similar models and distributions. The final-state distribution is attributed to all sources in whole system, which results in a multi-characteristic emission process if we use the standard distribution. This also means that the transverse momentum spectrum can be described by a multicomponent standard distribution in which each component describes a given emission source.
We now structure the multi-component standard distribution. It is assumed that there are l components to be considered. For the i-th component, the standard Boltzmann, Fermi-Dirac and Bose-Einstein distributions [23,24,25,26] can be uniformly shown as where C i is the normalization constant which results in ∞ 0 f i (p T ) = 1; N , m 0 , µ, and T i denote the particle number, the rest mass of the considered particle, the chemical potential of the considered particle, and the effective temperature for the i-th component, respectively; y max is the maximum rapidity and y min is the minimum rapidity; the values of S are 0, +1, and −1, which denote the Boltzmann, Fermi-Dirac, and Bose-Einstein distributions, respectively. Because of the effect of µ on the tendency of curve being small, we neglect its contribution.
In the final state, the transverse momentum spectrum is contributed by the l components, that is where k i (i = 1, 2, · · · , l) is the relative weight contributed by the i-th component. Because of the probability distribution being acquiescently normalized to 1, the coefficient obeys the normalization condition of k i = 1. Considering the relative contribution of each component, we have the mean effective temperature to be T e = k i T i . The T e reflects the mean excitation degree of different sources corresponding to different components and can be used to describe the effective temperature of whole interacting system. It should be noticed that the effective temperature contains the contributions of transverse flow and thermal motion. It is not the "real" temperature of the interacting system. According to the statistical arguments based on the chemical and thermal equilibrium, one has the relations between antiparticle to particle yield ratios to be [27,28] which is within the thermal and statistical model [27].
In a similar way, the yield ratios of antiparticles to particles for other hadrons can be written as where k j (j = π, K, p, D, and B) denote the yield ratios of negatively to positively charged particles obtained from the normalization constants of transverse momentum spectra. The T ch denotes the chemical freeze-out temperature in the reaction system. The µ π , µ K , µ p , µ D , and µ B represent the chemical potentials of π, K, p, D, and B, respectively. In the above discussion, the symbol of a given particle is used for its yield for the purpose of simplicity. Further, we have Let µ q denote the chemical potential of the q-th flavor quark, where q = u, d, s, c, and b represent the up, down, strange-, charm, and bottom quarks, respectively. In principle, we can use the yield ratios of different negatively to positively charged particles to give relations among different µ q . The values of µ q are then expected from these relations. According to refs. [29,30], based on the same chemical freeze-out temperature, the yield ratios in terms of quark chemical potentials are Thus, we have In the framework of a statistical thermal model of non-interacting gas particles with the assumption of standard Maxwell-Boltzmann statistics, there is an empirical expression for the chemical freeze-out temperature [31,32,33,34], where √ s N N is in the units of GeV and the "limiting" temperature T lim = 0.164 GeV.
We would like to point out that the chemical potentials of u, d, and s quarks can be obtained from the yield ratios of negatively to positively charged particles. In some cases, the yield ratios of antiparticles to particles in given p T bins with different bin centers depend on p T , we can obtain the dependences of µ q on p T , that is the p T dependent µ q . Except for the yield ratios of f π − (p T )/f π + (p T ), f K − (p T )/f K + (p T ) and fp(p T /f p (p T ), other combinations can also give µ q if the spectra in the numerator and denominator are in the same experimental conditions.

Results and discussion
The energy dependent double-differential p T spectra of π − , π + , K − , K + ,p, and p produced in central Au+Au collisions at √ s N N = 7.7, 11.5, 19.6, 27, 39, 62.4, 130, and 200 GeV as a function of p T at the midrapidity |y| < 0.1 are presented in Fig. 1, where the centrality interval at 130 GeV is of 0-6% and at other energies is of 0-5%. The different symbols represent the data measured by the STAR Collaboration [19,21], and the curves are the results calculated here by the one-or two-component standard distribution. In the calculation, the values of the free parameters (T 1 , k 1 , and T 2 ), the normalization constant (N 0 ), and χ 2 obtained by fitting the data are listed in Table 1 with value of degrees of freedom (dof), where N 0 is concretely explained in the caption. One can see that the calculated results are in good agreement with the data. From the parameter values, one can see that the effective temperature increases with the increase of the particle mass and the collision energy for emissions of the six types of particles.
Based on the above successful descriptions of the transverse momentum spectra of antiparticles and particles, we can use Eq. (7) and the curves in Fig. 1 to study the p T -dependent chemical potentials of light flavor quarks. The solid, dashed and dotted curves in Fig. 2 are our results of p T -dependent µ u , µ d , and µ s , where µ is used in the vertical axis to replace µ q which are marked in the panel for different styles of curves, and k j in Eq. (7) are obtained by the yield ratios of antiparticles to particles in given p T bins with different bin centers. From Figs. 2(a) to 2(h), the collision energies are 7.7, 11.5, 19.6, 27, 39, 62.4, 130, and 200 GeV, respectively. One can see that, at low energy, µ u and µ d are similar, and they depart from µ s . At high energy, all µ q are close to zero. With the increase of p T , µ u has small fluctuations, µ d and µ s have large fluctuations, and the differences among different µ q increase obviously.
To smooth the fluctuations in p T -dependent µ q at different energies, we use the normalization constants, N 0 , for the calculations of yield ratios of different particles in Fig. 1 (see Table 1) and obtain the dependence of µ q on √ s N N . This treatment is in fact to perform integral for yield over a wide p T range at different energies. Figure 3 shows the correlations between µ q and √ s N N , where µ is used in the vertical axis to replace µ q which are marked in the panel for different styles of symbols. One can see that, with the increase of logarithmic collision energy, an exponential decrease of the chemical potentials of light flavor quarks is observed. Corresponding to the solid, dashed and dotted curves which fit to the dependences of µ u , µ d , and µ s on √ s N N , respectively, we have with the χ 2 /dof = 5.988/6, 5.976/6, and 5.988/6, respectively.
The similarity in up and down quark masses renders the similarity in their chemical potentials. The difference between the chemical potentials of up (or down) and strange quarks is caused by the difference between their masses. At the lowest BES energy the chemical potentials of light flavor quarks are around 100 MeV, while at the top RHIC energy these quantities are around a few MeV. The decrease is obvious, which indicates the change of mean free path of produced quarks in the middle state. If the produced quarks at the lowest BES energy have a small mean free path which looks as if a liquid-like middle state is formed, the produced quarks at the top RHIC energy should have a large mean free path which looks as if a gas-like middle state is formed. The main difference at different energies is different mean free paths of the produced quarks. To search for the critical energy at which the change from a liquid-like middle state to a gas-like middle state had happen is beyond the focus of the present work.
From Eq. (9) we can obtain a linear relation between ln µ q and ln √ s N N , where the intercept a and slope −|b| can be obtained from the parameters in Eq. (9). In particular, −|b| is close to −1. The large negative slope renders an obvious anticorrelation between ln µ q and ln √ s N N . That is to say that ln µ q decreases obviously with the increase of ln √ s N N . It is expected that ln µ q will be smaller at higher energy or larger at lower energy. In particular, at the LHC, ln µ q will be negative due to µ q will be less than 1 MeV.  [19,21] and the curves represent the results calculated by us using the one-or two-component standard distribution. The values of parameters can be found in Table 1.      Table 1). The solid, dashed and dotted curves are fitted results corresponding to µ u , µ d and µ s , respectively.
We would like to point out that, although we have used the one-or two-component standard distribution in the descriptions of transverse momentum spectra and the effective temperatures have been used, the transverse momentum dependent chemical potentials of light flavor quarks obtained by us are independent of models and parameters due to the yield ratios of antiparticles to particles being absolutely main factor. In fact, we could use the yield ratios of data themselves to obtain the chemical potentials. The reason why we use the function form instead of data themselves is to extend the transverse momentum spectrum to a wide range where the data is not available. In our opinion, using a wide spectrum in theory could obtain a more accurate result.

Conclusions
In summary, we find a good description of the transverse momentum spectra of charged particles produced in central Au+Au collisions at the RHIC at its BES energies, though the spectra range is not too wide. It is shown that the one-or two-component standard distribution is successful in the descriptions of the data measured in midrapidity interval by the STAR Collaboration, though other distributions are also acceptable.
The effective temperature parameter increases with the increase of the particle mass and the collision energy.
At different energies within the RHIC at its BES, the transverse momentum dependent chemical potentials of light flavor quarks are obtained due to the yield ratios of negatively to positively charged particles in given transverse momentum bins with different bin centers, though the fluctuations are large. At low energy, the chemical potentials of up and down quarks are similar, and they depart from that of strange-quark. At high energy, the three chemical potentials are close to zero. With the increase of the transverse momentum, the differences among the three chemical potentials increase obviously.
From the lowest BES energy to the top RHIC energy, with the increase of logarithmic collision energy, an exponential decrease of the chemical potentials of light flavor quarks is observed. The similarity in up and down quark masses renders the similarity in their chemical potentials. The difference between the chemical potentials of up (or down) and strange quarks is caused by the difference between their masses. The chemical potentials of light flavor quarks change from ∼ 100 MeV to a few MeV. This obvious decrease indicates that the mean free path of interacting particles changes from a small value to a large one.

Data Availability
All data are quoted from the mentioned references. All MATLAB or FORTRAN codes used in the calculations are available to supply if necessary.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.