Energy dependent chemical potentials of light particles and quarks from yield ratios of negative to positive particles in high energy collisions

We collect the yields of negatively and positively charged pions ($\pi^-$ and $\pi^+$), negatively and positively charged kaons ($K^-$ and $K^+$), as well as anti-protons ($\bar p$) and protons ($p$) produced in mid-rapidity interval (in most cases) in central gold-gold (Au-Au), central lead-lead (Pb-Pb), and inelastic or non-single-diffractive proton-proton ($pp$) collisions at different collision energies. The chemical potentials of light particles and quarks are extracted from the yield ratios, $\pi^-/\pi^+$, $K^-/K^+$, and $\bar p/p$, of negative to positive particles over an energy range from a few GeV to more than 10 TeV. At a few GeV ($\sim4$ GeV), the chemical potentials show and the yield ratios do not show different behaviors comparing with those at other energies, though the limiting values of the chemical potentials and the yield ratios at very high energy are zero and one respectively.


Introduction
Chemical potential (µ B ) of baryon in high energy collisions is a very interesting and important topic studied by researchers in the fields of particle and nuclear physics. Combining with temperature (T ch ) at chemical freeze-out, one can study the the quantum chromodynamics (QCD) phase diagram in the plane of T ch − µ B for the phase transition from the hadronic matter to the quark-gluon plasma (QGP) [1][2][3][4]. It is expected that this phase transition possibly happens over a center-ofmass energy ( √ s N N ) range from a few GeV to dozens of GeV. The purpose of the beam energy scan (BES) performed at the Super Proton Synchrotron (SPS) and the Relativistic Heavy Ion Collider (RHIC) is to search for the critical energy at which the phase transition from the hadronic matter to the QGP had happened in all probability [5][6][7][8]. The BES energies at the SPS reach or close to the Alternating Gradient Synchrotron (AGS) energy.
Combining with the AGS, SPS (at its BES), RHIC (at its BES), one can study the QCD phase diagram over an energy range from a few GeV to 200 GeV [1][2][3][4]. In particular, the Large Hadron Collider (LHC) has extended the energy range to a few TeV and even more than 10 TeV [9][10][11][12]. It is convenient for researchers to study the QCD phase transition further. At the same time, the excitation functions of T ch and µ B (the energy dependent T ch and µ B ) can be studied in the mentioned energy range. Generally, the values of T ch and µ B in a given collisions can be obtained from the yield ratios of negative to positive particles, and the yield ratios of different types of particles in a given rapidity interval and transverse momentum range. Although the chemical potentials of other particles such as mesons can also be obtained from the yield ratios of negative to positive particles, few chemical potentials of mesons have been studied in literature.
We are interested in the chemical potentials of different types of particles in high energy collisions, which can be obtained from yield ratios of negative to positive particles in a particular form. We are also interested in the chemical potentials of different flavors of quarks, which can also be obtained from the same yield ratios of negative to positive particles. From the chemical potentials of particles (or quarks), we can evaluate the relative densities of final particles (or produced quarks) at different energies. These relative densities are useful in the understanding of interacting mechanism. Because of the data of yield ratios being very limited, we can only obtain the chemical potentials of light particles and quarks conveniently.

The method and formalism
To extract the chemical potentials of light particle and quarks, we need to know the yield ratios of negative to positive particles. Although we can obtain the yield ratios from the normalization constants in the transverse momentum spectra of different particles, the quantity of work is huge if we analyze the spectra over a wide energy range. A direct and convenient method is to collect the values of yield ratios from the experiments performed at the AGS, SPS at its BES, RHIC at its BES, and LHC by productive international collaborations, though some yield ratios are unavailable.
Because of the same formula on the relation between the yield ratio and chemical potential being used in our previous work [35] and the present work, some repetitions are ineluctable to give a whole representation of the present work. According to the statistical arguments based on the chemical and thermal equilibrium, one has the relations between antiproton to proton yield ratios to be [36,37] which is within the thermal and statistical model [36], where is empirically obtained in the framework of a statistical thermal model of non-interacting gas particles with the assumption of standard Maxwell-Boltzmann statistics [1,2,38,39], where √ s N N is in the units of GeV and the "limiting" temperature T lim = 0.164 GeV.
According to Eq. (1), the yield ratios of negative to positive particles for other hadrons with together (anti)protons can be written as where k j denote the yield ratios of negative to positive particles and j = π, K, p, D, and B listed orderly above due to their masses. The symbols µ π , µ K , µ p , µ D , and µ B represent the chemical potentials of π, K, p, D, and B, respectively. To obtain chemical potentials of quarks, the above five hadrons and their antiparticles are enough. Because of the lifetimes of particles contained top quark being very short to measure, we shall not discuss the top quark related particles, top quark itself, and their chemical potentials. 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. The values of µ q are then expected from these relations. According to refs. [40,41], based on the same chemical freeze-out temperature, the yield ratios in terms of quark chemical potentials are According to Eqs. (3) and (4), the chemical potentials of particles and quarks can be obtained, respectively, in terms of yield ratios of negative to positive particles. The chemical potentials of particles are simply given by The chemical potentials of quarks are complicatedly a little. We have Because of the limited data, only the energy dependent chemical potentials of light particles such as π, K, and p, as well as light quarks such as u, d, and s in an energy range covered the AGS, SPS (at its BES), RHIC (at its BES), and LHC are obtained in the present work. That is to say that, in the present work, only the excitation functions of µ π , µ K , µ p , µ u , µ d , and µ s are studied over an energy range from a few GeV to more than 10 TeV. For central Au-Au (Pb-Pb) collisions and INEL or NSD pp colliisons, the energy ranges are not completely corresponding to each other.

Results and discussion
The yield ratios, π − /π + , K − /K + , andp/p, of negative to positive particles produced in mid-rapidity interval (in most cases) in central Au-Au, central Pb-Pb, and INEL or NSD pp collisions at the AGS, SPS, RHIC, and LHC are shown in Figs. 1(a), 1(b), and 1(c), respectively. The circles, squares, triangles, and starts denote the data measured in Au-Au collisions in midrapidity interval from |y| < 0.05 to |y| < 0.4 and centrality 0-5% by the E895, E866, and E917 Collaborations [13][14][15] at the AGS, in mid-rapidity interval |y| < 0.4 and centrality 0-10% by the E802 and E866 Collaboration [16,17] at the AGS, in mid-pseudorapidity interval |η| < 0.35 and centrality 0-5% by the PHENIX Collaboration [18][19][20], and in mid-rapidity interval from |y| < 0.1 to |y| < 0.5 and centrality from 0-5% to 0-10% by the STAR Collaborations [3,[21][22][23] at the RHIC, respectively. The circles, squares, and triangles with aclinal crosses denote the data measured in Pb-Pb collisions in mid-rapidity interval from 0 < y < 0.2 or |y| < 0.1 to |y| < 0.6 and centrality from 0-5% to 0-7.2% by the NA49 Collaboration [24][25][26][27] at the SPS, in mid-rapidity interval from |y| < 0.5 to |y| < 0.85 and centrality 0-3.7% by the NA44 Collaboration [28] at the SPS, and in mid-rapidity interval |y| < 0.5 and centrality 0-5% by the ALICE Collaboration [29] at the LHC, respectively. The circles, squares, triangles, and stars with diagonal crosses denote the data measured in the forward rapidity region (in the center-of-mass system) in INEL pp collisions by the NA61/SHINE Collaboration [30] at the SPS, in mid-rapidity interval |y| < 0.1 in NSD pp collisions by the STAR Collaboration [3,31] at the RHIC, in mid-rapidity interval |y| < 0.5 in INEL pp collisions by the ALICE Collaboration [32] at the LHC, and in mid-rapidity interval |y| < 1 in INEL pp collisions by the CMS Collaboration [33,34] at the LHC, respectively. The solid and dashed curves in Fig. 1(a) are the results fitted by us for the √ s N N dependent π − /π + in central Au-Au (Pb-Pb) and INEL or NSD pp collisions respectively. The solid curves in Figs. 1(b) and 1(c) are the results fitted by us for the √ s N N dependent K − /K + andp/p respectively, for the combining central Au-Au (Pb-Pb) and INEL or NSD pp collisions. One can see that, with the increase of √ s N N , π − /π + decreases obviously in central Au-Au (Pb-Pb) collisions and it increases obviously in INEL or NSD pp collisions, and K − /K + andp/p increase obviously in both central Au-Au (Pb-Pb) and INEL or NSD pp collisions. The limiting values of the three yield ratios is one at very high energy. The solid and dashed curves in Fig. 1(a) can be empirically described by and In particular, the NA61/SHINE Collaboration does not perform the results in mid-rapidity interval, but in the forward rapidity region (in the center-of-mass system). The solid and dashed curves in Fig. 1(a) are the results fitted by us for the √ s N N dependent π − /π + in central Au-Au (Pb-Pb) and INEL or NSD pp collisions respectively. The solid curves in Figs. 1(b) and 1(c) are the results fitted by us for the √ s N N dependent K − /K + andp/p respectively, for the combining central Au-Au (Pb-Pb) and INEL or NSD pp collisions. 1(b) and 1(c) can be empirically described by and respectively, with χ 2 /dof to be 2.735 and 7.715 respectively. According to these functions, by using Eqs. (5) and (6), the chemical potentials of light particles and quarks can be obtained. Figures 2(a), 2(b), and 2(c) present respectively the chemical potentials, µ π , µ K , and µ p , of π, K, and p produced in mid-rapidity interval (in most cases) in central Au-Au (Pb-Pb) and INEL or NSD pp collisions at high energies. The symbols denote the derivative data obtained from Fig. 1  s N N over a range from more than a few GeV to more than 10 TeV, µ π increases obviously in central Au-Au (Pb-Pb) collisions and it decreases obviously in INEL or NSD pp collisions, and µ K and µ p decrease obviously in both central Au-Au (Pb-Pb) and INEL or NSD pp collisions. The limiting values of the three types of chemical potentials are zero at very high energy. Figure 3 is the same as Fig. 2, but Figs. 3(a), 3(b), and 3(c) present respectively the chemical potentials, µ u , µ d , and µ s , of u, d, and s quarks, which are derived from mid-rapidity interval (in most cases) in central Au-Au (Pb-Pb) and INEL or NSD pp collisions at high energies. The symbols denote the derivative data obtained from Fig. 1 according to Eq. (6), where different symbols correspond to different collaborations marked in the panels which are the same as Figs. 1 and 2. For the purpose of comparison, the normal, medium, and small symbols with diagonal crosses denote the derivative data in INEL or NSD pp collisions obtained by T ch , 0.9T ch , and 0.8T ch in Eq. (6), respectively. The curves are the derivative results obtained from the curves in Fig. 1 according to Eq. (6). The solid and dashed curves in Fig.  3(a) are the derivative results for central Au-Au (Pb-Pb) and INEL or NSD pp collisions respectively. The solid curves in Figs. 3(b) and 3(c) are the derivative results for the combining central Au-Au (Pb-Pb) and INEL or NSD pp collisions. One can see that, with the increase of √ s N N over a range from more than a few GeV to more than 10 TeV, µ u , µ d , and µ s decrease obviously in both central Au-Au (Pb-Pb) and INEL or NSD pp collisions.
The limiting values of the three chemical potentials are zero at very high energy.
In Fig. 2, at a few GeV (∼ 4 GeV), some curves show different behaviors comparing with those at other energies, which are not observed from the curves of the yield ratios in Fig. 1. At the same time, the curves in Fig. 3 also show different behaviors at a few GeV. Indeed, this energy is a special energy. In our opinion, these special behaviors are appeared due to this energy being the initial energy of limiting fragmentation of collision nuclei. This energy is also the energy at which the phase transition from a liquid-like state to a gaslike state in hot nuclei is expected to happen initially, where the liquid-like state is a state in which the meanfree-path of interacting nucleons is relatively short, and the gas-like state is a state in which the mean-free-path of interacting nucleons is relatively long. In addition, the density of baryon number in nucleus-nucleus collisions at this energy has a large value. Because of these particular factors, the collisions at this energy present different mechanisms from other energies.
From more than a few GeV to dozens of GeV, the cases of k π > 1 and µ π < 0 in central Au-Au (Pb-Pb) collisions are different from other particles and INEL or NSD pp collisions. These render the resonant production of pions in central Au-Au (Pb-Pb) collisions, which does not exist in the production of other particles or in INEL or NSD pp collisions. At the RHIC and LHC, the behaviors of k π and µ π in central Au-Au (Pb-Pb) collisions are close to other particles or INEL or NSD pp collisions due to the increase of hard scattering component of pions, which are similar to other particles. From more than a few GeV to more than 10 TeV, the yield ratios approach to one and the chemical potentials approach to zero. These render that the mean-free-path of produced particles (quarks) becomes large and the viscous effect becomes weakly at the LHC. The interacting system changes completely from the liquid-like state to the gas-like state at the early and medium stage   Fig. 1 according to Eq. (6). In particular, the NA61/SHINE Collaboration does not perform the results in mid-rapidity interval, but in the forward rapidity region (in the center-of-mass system). The normal, medium, and small symbols with diagonal crosses denote the derivative data in INEL or NSD pp collisions obtained by T ch , 0.9T ch , and 0.8T ch in Eq. (6), respectively. The curves surrounded the symbols are the derivative results obtained from the curves in Fig. 1 according to Eq. (6).
in collisions at very high energy.

Conclusions
In summary, we have collected the yield ratios, π − /π + , K − /K + , andp/p, of negative to positive particles produced in mid-rapidity interval (in most cases) in central Au-Au (Pb-Pb) and INEL or NSD pp collisions over an energy range from a few GeV to more than 10 TeV. It is shown that, with the increase of √ s N N , π − /π + decreases obviously in central Au-Au (Pb-Pb) collisions and it increases obviously in INEL or NSD pp collisions, and K − /K + andp/p increase obviously in both central Au-Au (Pb-Pb) and INEL or NSD pp collisions. The limiting values of the three yield ratios is one at very high energy.
The chemical potentials of light particles and quarks are extracted from the yield ratios. With the increase of √ s N N over a range from more than a few GeV to more than 10 TeV, µ π increases obviously in central Au-Au (Pb-Pb) collisions and it decreases obviously in INEL or NSD pp collisions, and µ K and µ p decrease obviously in both central Au-Au (Pb-Pb) and INEL or NSD pp collisions. Meanwhile, µ u , µ d , and µ s decrease obviously in both central Au-Au (Pb-Pb) and INEL or NSD pp collisions. The limiting values of the chemical potentials of the three types of light particles and the three flavors of light quarks are zero at very high energy.
At a few GeV (∼ 4 GeV), some curves of the chemical potentials show different behaviors which are not observed from the curves of the yield ratios. These special behaviors are appeared due to this energy being the initial energy of limiting fragmentation of collision nuclei. This energy is also the energy at which the phase transition from the liquid-like state to the gas-like state in hot nuclei is expected to happen. At the same time, the density of baryon number in nucleus-nucleus collisions at this energy has a large value. These particular factors render different collision mechanisms at this energy. The interacting system changes completely from the liquid-like state to the gas-like state at the early and medium stage in collisions at very high energy.

Data Availability
All data are quoted from the mentioned references. As a phenomenological work, this paper does not report new data.

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