Research Article | Open Access
Li-Li Li, Fu-Hu Liu, Muhammad Waqas, Rasha Al-Yusufi, Altaf Mujear, "Excitation Functions of Related Parameters from Transverse Momentum (Mass) Spectra in High-Energy Collisions", Advances in High Energy Physics, vol. 2020, Article ID 5356705, 21 pages, 2020. https://doi.org/10.1155/2020/5356705
Excitation Functions of Related Parameters from Transverse Momentum (Mass) Spectra in High-Energy Collisions
Transverse momentum (mass) spectra of positively and negatively charged pions and of positively and negatively charged kaons, protons, and antiprotons produced at mid-(pseudo)rapidity in various collisions at high energies are analyzed in this work. The experimental data measured in central gold-gold, central lead-lead, and inelastic proton-proton collisions by several international collaborations are studied. The (two-component) standard distribution is used to fit the data and extract the excitation function of effective temperature. Then, the excitation functions of kinetic freeze-out temperature, transverse flow velocity, and initial temperature are obtained. In the considered collisions, the four parameters increase with the increase of collision energy in general, and the kinetic freeze-out temperature appears at the trend of saturation at the top Relativistic Heavy Ion Collider and the Large Hadron Collider.
It is believed that the environment of high temperature and high density is formed in the system evolution process of central nucleus-nucleus (AA) collisions at high energy [1–3], in which quark-gluon plasma (QGP) is possibly created and many particles are produced [4–6]. At present, it is impossible to detect directly the system evolution process of collisions due to very short time interval. Instead, the particle spectra at the stage of kinetic freeze-out can be measured in experiments and the mechanisms of system evolutions and particle productions can be studied indirectly [7–9], though the particle ratios reflect the property at the stage of chemical freeze-out. As for peripheral AA collisions and small collision system, the situation is similar if the multiplicity is high enough due to the small system which also appears collective behavior [10, 11].
Although there are different stages in the system evolution [1–3], the initial state is the most important due to its determining effect to the system evolution. In addition, chemical and kinetic freeze-outs are two important stages in the system evolution. At the stage of chemical freeze-out, the system had a phase transition from QGP to hadronic matter, and the constituents and ratios of various particles do not change anymore. At the stage of kinetic freeze-out, the collisions among various particles are elastic, and the transverse momentum spectra of various particles are fixed [2, 7]. In the small system with low multiplicity, QGP is not expected to create in it due to a very small volume of the violent collision region. From the similar multiplicity at the energy up to 200 GeV, the small system is more similar to peripheral AA collisions, but not to central AA collisions [12, 13]. At the energy down to 10 or several GeV, the situation is different due to the fact that the baryon-dominated effect plays more important role in AA collisions .
The temperatures at the stages of kinetic freeze-out, chemical freeze-out, and initial state are called the kinetic freeze-out temperature ( or ), chemical freeze-out temperature (), and initial temperature (), respectively. Besides, one also has the effective temperature () in which both the contributions of thermal motion and flow effect are included. It is expected that various temperatures can be extracted from particle spectra, which are usually model dependent. Generally, is unavoidably model dependent, and extracted from particle ratios in the statistical thermal model [15–18] is also model dependent. We hope to use a less model-dependent method to extract , , and . The quantities used in the method are expected to relate to experimental data as much as possible, though they can be calculated from models in some cases.
To perform a less model-dependent method, we would like to use the standard distribution or its two-component form to obtain by fitting the experimental transverse momentum () or transverse mass () spectra of various particles. The standard distribution includes the Bose-Einstein, Fermi-Dirac, and Boltzmann distributions, in which the effective temperature parameter is the closest to that in the ideal gas model when comparing with those in other distributions. After the fitting, we hope to extract and from the relation to average () due to the Erlang distribution in the multisource thermal model [19–21] and from the relation to root-mean-square () due to the color string percolation model [22–24]. Obviously, and depend on the data themselves, though they can be calculated from the models.
In this work, the () spectra of positively and negatively charged pions ( and ) and positively and negatively charged kaons ( and ), protons, and antiprotons ( and ) produced at mid-(pseudo)rapidity (mid- or mid-) measured in central gold-gold (Au-Au) collisions at the Alternating Gradient Synchrotron (AGS) by the E866 , E895 [26, 27], and E802 [28, 29] Collaborations and at the Relativistic Heavy Ion Collider (RHIC) by the STAR [30–32] and PHENIX [33, 34] Collaborations, in central lead-lead (Pb-Pb) collisions at the Super Proton Synchrotron (SPS) by the NA49 Collaboration [35–37] and at the Large Hadron Collider (LHC) by the ALICE Collaboration , as well as in inelastic (INEL) proton-proton () collisions at the SPS by the NA61/SHINE Collaboration [39, 40], at the RHIC by the PHENIX Collaboration , and at the LHC by the CMS Collaboration [42, 43], are studied. The (two-component) standard distribution is used to fit the data and to extract , , , and , as well as the excitation functions (energy dependences) of parameters.
The remainder of this paper is structured as follows. The formalism and method are shortly described in Section 2. Results and discussion are given in Section 3. In Section 4, we summarize our main observations and conclusions.
2. Formalism and Method
In high-energy collisions, the soft excitation and hard scattering processes are two main processes of particle productions. Most light flavor particles are produced in the soft excitation process and distributed in a narrow range which is less than GeV/ or a little more. Some light flavor particles are produced in the hard scattering process and distributed in a wide range. In collisions at not too high energies, the contribution of the hard scattering process can be neglected and the main contributor that produced particles is the soft excitation process. In collisions at a high energy, the contribution of the hard scattering process cannot be neglected, though the main contributor that produced particles is also the soft excitation process. It is expected that the contribution fraction of the hard scattering process increases with the increase of collision energy.
The contributions of soft excitation and hard scattering processes can be described by similar or different probability density functions. Generally, the hard scattering process does not contribute mainly to the temperature and flow velocity due to its small fraction in a narrow range. We can neglect the contribution of the hard scattering process if we study the spectra in a not too wide range. On the contribution of the soft excitation process, we have more than one functions to describe the spectra. These functions include, but are not limited to, the standard distribution , the Tsallis statistics [44–47], the Erlang distribution [19–21], the Schwinger mechanism [48–51], the blast-wave model with Boltzmann statistics [52, 53], the blast-wave model with Tsallis statistics [54–56], the Hagedorn thermal distribution , and their superposition with two- or three-component. These functions also describe partly the spectra of the hard scattering process in most cases.
In our opinion, in the case of fitting the data with acceptable representations, various distributions show similar behaviors which result in similar () with different parameters. To be the closest to the temperature concept in the ideal gas model, we choose the standard distribution in which the chemical potential and spin property are included. That is, one has the probability density function in terms of to be  where is the rest mass, denotes the particle number, () is the minimum (maximum) value in the rapidity interval, (+1) is for bosons (fermions), and is the normalization constant. Similarly, the probability density function in terms of is In some cases, the independent variable in Equation (3) is replaced by which starts at 0. Both and show the same distribution shape. As probability density functions, the integrals of Equations (1) and (3) are naturally normalized to 1, respectively.
The chemical potential in Equations (1) and (3) is particle dependent. For the particle type (, , and in this work), its chemical potential is expressed by [34, 58, 59] where denotes the ratio of negative to positive particle numbers, is empirically the chemical freeze-out temperature in the statistical thermal model [15–18], GeV is the limiting or saturation temperature , and is the center-of-mass energy per nucleon pair in the units of GeV.
Generally, one needs one or two standard distributions to fit the () spectra in a narrow range. In particular, if the resonance decays contribute a large fraction, a two-component distribution is indeed needed. Or, if the hard scattering process contributes a sizable fraction in the considered () range, a two-component distribution is also needed. In the case of using the two-component standard distribution in which the contributions from resonance decay are naturally included in the first component which covers the spectra in the low- region ( GeV/), one has the probability density functions of and to be respectively, where denotes the contribution fraction of the first component, and  and  are given in Equations (1) and (3), respectively. The integrals of Equations (6) and (7) are also normalized to 1, respectively. Correspondingly, is averaged by weighting the two fractions. The temperature defined in Equation (8) reflects the common effective temperature of the two components in the case that the two components are assumed to stay in equilibrium.
According to the Hagedorn model , one may also use the usual step function to superpose the two standard distributions, where if and if . Thus, we have new probability density functions of and to be respectively, where and are constants which result in the two components to be equal to each other at and . The integrals of Equations (9) and (10) should be normalized to 1, respectively, due to the fact that they are probability density functions. The contribution fractions of the first component in Equations (9) and (10) are respectively, where and denote the maximum and , respectively. Equation (8) is also suitable for the superposition in terms of the Hagedorn model .
The two superpositions show respective advantages and disadvantages. The first superposition can fit the data by a smooth curve. However, there are correlations in determining and . The second superposition can determine and without correlations. However, the curves are possibly not smooth at or . In the case of obtaining and , it does not matter which superposition is used, though the two are slightly different. In this work, we use the first superposition to obtain smooth curves. One has due to
Based on the spectrum, we may use the same parameters to obtain and from the related formula of distribution.
It should be noted that, since we aim to extract the parameters in a less model-dependent way, we shall obtain and from the combination of data points and fit function in this paper. In fact, we may divide () spectrum into two or three regions according to the measured and unmeasured () ranges. To obtain and , we may use the data points in the measured () range and only use the fit function to extrapolate to the unmeasured () range.
In each nucleon-nucleon collision in AA and pp collisions, the projectile and target participant sources contribute equally to . In the framework of the multisource thermal model [19–21], each projectile and target source contribute a fraction of 1/2 to , i.e., which is contributed together by the thermal motion and flow effect. Let () denote the contribution fraction of thermal motion (flow effect); we define empirically where is the mean Lorentz factor of the considered particles and is a parameterized representation in this paper due to our comparison with the results [12, 13] from the blast-wave model [52–56]. In Equation (18), is in the units of GeV as that in Equation (5).
In a recent work , it is shown that the effective temperature is proportional to and the kinetic freeze-out temperature is proportional to the effective temperature, though the effective temperature used in Ref.  is different from this paper. This confirms the relation of (Equation (16)) used in this paper. Considering each projectile and target source contributing [19–21], we have concretely . The remainder in is naturally contributed by the transverse flow. This confirms Equations (16) and (17) to be justified, though is an empirical representation.
To continue this work, we need some assumptions and a coordinate system. In the source rest frame, the particles are assumed to emit isotropically. Meanwhile, the interactions among various sources are neglected, which affects slightly the () spectra, through which affects largely anisotropic flows . A right-handed coordinate system – is established in the source rest frame, where the axis is along the beam direction, the plane is the transverse plane, and the plane is the reaction plane.
We can obtain by a Monte Carlo (MC) method. Let denote a random number distributed evenly in ; each concrete satisfies where denotes a small shift relative to . Each concrete emission angle satisfies due to the fact that obeys the probability density function in in the case of isotropic assumption in the source rest frame. The solution of the equation is Equation (20). We give up to use rapidity due to the fact that it is unnecessary here. Each concrete momentum , energy , and Lorentz factor can be obtained by respectively. After multiple repeating calculations due to the MC method, we have where denotes the mean for a given type of particle.
In addition, each concrete azimuthal angle satisfies due to the fact that obeys the probability density function in in the case of isotropic assumption in the source rest frame. The solution of the equation is Equation (25). Each concrete momentum components , , and can be obtained by respectively. By using the components and , , and , we can obtain other quantities such as (pseudo)rapidity and event structure  which are beyond the focus of this work and will not be studied anymore.
According to the color string percolation model [22–24], one has Meanwhile, we have the relation between the three components , , and of the momentum to be in which the root-mean-square components , , and are used. Naturally, can be given by one of the root-mean-square components.
We would like to point out that the above isotropic assumption is only performed in the source rest frame. It is expected that many sources are formed in high-energy collisions according to the multisource model [19–21]. These sources distribute at different rapidities in the rapidity space, which appear at the effect of longitudinal flow. The two-component and spectra render that these sources stay in two different excitation states or have two different decay mechanisms. The interactions among these sources also affect anisotropic flows in transverse plane .
3. Results and Discussion
Figures 1(a)–1(q) show the () spectra, , of , , , , , and produced at mid- or mid- in central Au-Au collisions at different , where the particle types, or intervals, centrality classes, and collision energies are marked in the panels. The closed and open symbols represent, respectively, the experimental data of positively and negatively charged particles measured by the E866 , E895 [26, 27], E802 [28, 29], STAR [30–32], and PHENIX [33, 34] Collaborations marked in the panels, where in Figures 1(a)–1(d), the data for and are taken from the E866 Collaboration  and the data for are taken from the E895 Collaboration [26, 27]. The solid and dashed curves are our results fitted by Equation (6) or (7) for positively and negatively charged particles, respectively. The values of free parameters (, if available, and ), derived parameter (), normalization constant (), , and degree-of-freedom (dof) are listed in Table 1. The dot-dashed curves are our results fitted by using the single component function with the weighted average parameter which will be discussed later. The dotted curves and asterisks in Figures 1(a)–1(f) represent the MC results for with high ( particles) and low ( particles) statistics, respectively, which will be discussed at the end of this section. In the fitting process for the solid and dashed curves, the least squares method is used to determine the best parameter values. The experimental global uncertainties used in the calculation of are taken to be the root sum square of statistical uncertainties and point-by-point systematic uncertainties. The best parameters are determined due to the limitation of the minimum . The global uncertainties of parameters are obtained using the method of statistical simulation . We note that per dof (/dof) in a few cases is larger than 10, which renders that the fit is not too good. One can see that the (two-component) standard distribution fits approximately the () spectra of , , , and measured at mid- or mid- in central Au-Au collisions over an energy range from 2.7 to 200 GeV in most cases.