Effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume in high energy collisions

Effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume in high energy collisions 1 2 3 first-author 1 2 corresp Corresponding author, fuhuliu@163.com; fuhuliu@sxu.edu.cn Corresponding author, fuhuliu@163.com; fuhuliu@sxu.edu.cn fuhuliu@163.com https://orcid.org/0000-0002-2261-6899 1 2 4 Institute of Theoretical Physics and State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China Institute of Theoretical Physics and State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China The transverse momentum spectra of different types of particles produced in central and peripheral gold-gold (Au-Au) and inelastic proton-proton ( pp ) collisions at the Relativistic Heavy Ion Collider (RHIC), as well as in central and peripheral lead-lead (Pb-Pb) and pp collisions at the Large Hadron Collider (LHC) are analyzed by the multi-component standard (Boltzmann-Gibbs, Fermi-Dirac, and Bose-Einstein) distributions. The obtained results from the standard distribution give an approximate agreement with the measured experimental data by the STAR, PHENIX, and ALICE Collaborations. The behavior of the effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume for particles with different masses is obtained, which observes the early kinetic freeze-out of heavier particles as compared to the lighter particles. The parameters of emissions of different particles are observed to be different, which reveals a direct signature of the mass-dependent differential kinetic freeze-out. It is also observed that the peripheral nucleus-nucleus ( AA ) and pp collisions at the same center-of-mass energy per nucleon pair are in good agreement in terms of the extracted parameters. Transverse momentum spectra Effective temperature Kinetic freeze-out temperature Transverse flow velocity Kinetic freeze-out volume 1 Introduction 1 A hot and dense fireball is assumed to form for a brief period of time ( ∼ a few fm/ c ) over an extended region after the initial collisions, which undergoes a collective expansion that leads to the change in the temperature and volume or density of the system. Three types of temperatures namely the initial temperature, chemical freeze-out temperature, and kinetic freeze-out temperature can be found in the literature, which describe the excitation degrees of an interacting system at the stages of initial collisions, chemical freeze-out, and kinetic freeze-out, respectively [ 1 1 - 7 7 1-7 ]. There is another type of temperature, namely the effective temperature, which is not a real temperature and it describes the sum of excitation degrees of the interacting system and the effect of transverse flow at the stage of kinetic freeze-out. In principle, the initial stage of collisions happens earlier than other stages such as the chemical and kinetic freeze-out stages. Naturally, the initial temperature is the highest, and the kinetic freeze-out temperature is the lowest among the three real temperatures, while the chemical freeze-out temperature is in between the initial and kinetic freeze-out temperatures. The collision system does not get rid of the simultaneity for chemical and kinetic freeze-outs, which results in the chemical and kinetic freeze-out temperatures to be the same. The effective temperature is often larger than the kinetic freeze-out temperature but is equal to the kinetic freeze-out temperature in case of zero transverse flow velocity. To understand the given nature of the nuclear force and to break the system into massive fragments [ 8 8 , 9 9 ], it is a good way to make the nucleons interact in nucleus-nucleus ( AA ) collisions at intermediate and high energies. Such a process provokes a liquid-gas type phase transition as a large number of nucleons and other light nuclei are emitted. In AA collisions at higher energies, a phase transition from hadronic matter to quark-gluon plasma (QGP) is expected to occur. The volume occupied by the source of such ejectiles, where the mutual nuclear interactions become negligible (they only feel the Coulombic repulsive force and not the attractive force) is said to be kinetic freeze-out volume and it has been introduced in various statistical and thermodynamic models [ 10 10 , 11 11 ]. Similar to the kinetic freeze-out temperature, the kinetic freeze-out volume also gives the information of the co-existence of phase transition. This is one of the major factors, which are important in the extraction of vital observables such as multiplicity, micro-canonical heat capacity, and its negative branch or shape of caloric curves under the external constraints [ 12 12 - 16 16 12-16 ]. It is conceivable that the temperature (volume) of the interacting system decreases (increases) from the initial state to the final kinetic freeze-out stage. During the evolution process, the transverse flow velocity is present due to the expansion of the interacting system. The study of the dependence of effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume on the collision energy, event centrality, system size, and particle rapidity is very significant. We are very interested in the aforementioned quantities in central and peripheral AA and (inelastic) proton-proton ( pp ) collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) over a wide enough energy range in which QGP is expected to form. Here, we study the dependence of effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume in central and peripheral gold-gold (Au-Au) and lead-lead (Pb-Pb) collisions at the RHIC and LHC energies and compare their peripheral collisions with pp collisions of the same center-of-mass energy per nucleon pair <math id="M1"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> (or the center-of-mass energy <math id="M2"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> for pp collisions). Only 62.4 GeV at the RHIC and 5.02 TeV at the LHC are considered as examples. We present the approach of effective temperature and kinetic freeze-out volume from the transverse momentum spectra of the identified particles produced in the mentioned AA and pp collisions. The kinetic freeze-out temperature and transverse flow velocity are then obtained from particular linear relations. The remainder of this manuscript is structured as follows. The formalism and method are described in Sect. 2. Results and discussion are given in Sect. 3. In Sect. 4, we summarize our main observations and conclusions. 2 Method and formalism 1 Generally, two main processes of particle production are under consideration, which includes the soft and hard excitation processes. The soft excitation process corresponds to strong interactions among multiple partons, while the hard excitation process corresponds to a more violent collision between two head-on partons. The soft excitation process has numerous choices of formalisms, including but not limited to the Hagedorn thermal model (Statistical-bootstrap model) [ 17 17 ], the (multi-)standard distribution [ 18 18 ], the Tsallis and related distributions with various formalisms [ 19 19 ], the blast-wave model with Tsallis statistics [ 20 20 ], the blast-wave model with Boltzmann statistics [ 21 21 - 23 23 21-23 , 26 26 , 29 29 ], and other thermodynamics related models [ 30 30 - 33 33 30-33 ]. The hard excitation process has very limited choices of formalisms and can be described by the perturbative quantum chromodynamics (pQCD) [ 34 34 - 36 36 34-36 ]. The experimental data of the transverse momentum ( p T ) spectrum of the particles are fitted using the standard distribution, which is the combination of Boltzmann-Gibbs, Fermi-Dirac, and Bose-Einstein distributions corresponding to the factor S =0, +1 and -1, respectively. The standard distribution at the mid-rapidity can be demonstrated as [ 18 18 ] <math display="block" id="M3"><mtable columnalign="left"><mtr><mtd><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mi>g</mi><msup><mi>V</mi><mo>′</mo></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><msub><mi>p</mi><mtext>T</mtext></msub><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt></mtd></mtr><mtr><mtd><mo>×</mo><msup><mrow><mo stretchy="false">[</mo><mi>exp</mi><mrow><mo stretchy="false">(</mo><mrow><mfrac><msqrt><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></msqrt><mi>T</mi></mfrac></mrow><mo stretchy="false">)</mo></mrow><mo>+</mo><mi>S</mi><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>,</mn></mtd></mtr></mtable></math> where the chemical potential is neglected. Here, N is the experimental number of considered particles, T is the fitted effective temperature, V’ is the fitted kinetic freeze-out volume (i.e. the interaction volume) of the emission source at the kinetic freeze-out stage, g =3 (or 2) is the degeneracy factor for pions and kaons (or protons), and m 0 is the rest mass of the considered particle. As a probability density function, the integral of Eq. (1) is naturally normalized to 1, i.e., we have <math id="M4"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mrow><mtext>T</mtext><mi>max</mi></mrow></msub></mrow></msubsup><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub><mo>=</mo><mn>1</mn></mrow></math> , where p T max denotes the maximum p T . At very high energy, the influence of S =+1 and -1 can be neglected. Only the Boltzmann–Gibbs distribution is sufficient to describe the spectra at the RHIC and LHC. Considering the experimental rapidity range [ y min , y max ] around mid-rapidity, Eq. (1) takes the form <math display="block" id="M5"><mtable columnalign="left"><mtr><mtd><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mi>g</mi><msup><mi>V</mi><mo>′</mo></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><msub><mi>p</mi><mtext>T</mtext></msub><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>y</mi><mrow><mi>min</mi></mrow></msub></mrow><mrow><msub><mi>y</mi><mrow><mi>max</mi></mrow></msub></mrow></msubsup></mrow></mstyle><mo stretchy="false">(</mo><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt><mi>cosh</mi><mi>y</mi><mo>−</mo><mi>μ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><msup><mrow><mo stretchy="false">[</mo><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mrow><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt><mi>cosh</mi><mi>y</mi><mo>−</mo><mi>μ</mi></mrow><mi>T</mi></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>S</mi><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mtext>d</mtext><mi>y</mi><mn>,</mn></mtd></mtr></mtable></math> where the chemical potential μ is particle dependent, which we have studied recently [ 37 37 ]. In high energy collisions, μ j ( j=π , K , and p ) are less than several MeV, which slightly affects V’ compared with that for μ j =0. Then, we may regard μ ≈0 in Eq. (2) at high energies considered in the present study. In Eqs. (1) and (2), only T and V’ are the free parameters. Usually, we have to use the two-component standard distribution because single component standard distribution is not enough for the simultaneous description of very low- (0∼0.2-0.3 GeV/ c ) and low- p T (0.2-0.3∼2-3 GeV/ c or slightly more) regions, which are contributed by the resonance decays and other soft excitation processes, respectively. More than two or multi-component standard distributions can also be used in some cases. We have the simplified multi-component ( l -component) standard distribution to be <math display="block" id="M6"><mtable columnalign="left"><mtr><mtd><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>l</mi></munderover></mstyle><msub><mi>k</mi><mi>i</mi></msub><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mi>i</mi></msub></mrow></mfrac><mfrac><mrow><mi>g</mi><msub><msup><mi>V</mi><mo>′</mo></msup><mi>i</mi></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><msub><mi>p</mi><mtext>T</mtext></msub><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt></mtd></mtr><mtr><mtd><mo>×</mo><msup><mrow><mo stretchy="false">[</mo><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mrow><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt></mrow><mrow><msub><mi>T</mi><mi>i</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>S</mi><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>,</mn></mtd></mtr></mtable></math> where N i and k i denote respectively the particle number and fraction contributed by the i -th component, and T i and <math id="M7"><mrow><msub><msup><mi>V</mi><mo>′</mo></msup><mi>i</mi></msub></mrow></math> denote respectively the effective temperature and kinetic freeze-out volume corresponding to the i -th component. More accurate form of l -component standard distribution can be written as, <math display="block" id="M8"><mtable columnalign="left"><mtr><mtd><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>l</mi></munderover></mstyle><msub><mi>k</mi><mi>i</mi></msub><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mi>i</mi></msub></mrow></mfrac><mfrac><mrow><mi>g</mi><msub><msup><mi>V</mi><mo>′</mo></msup><mi>i</mi></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><msub><mi>p</mi><mtext>T</mtext></msub></mtd></mtr><mtr><mtd><mo>×</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>y</mi><mrow><mi>min</mi></mrow></msub></mrow><mrow><msub><mi>y</mi><mrow><mi>max</mi></mrow></msub></mrow></msubsup></mrow></mstyle><mo stretchy="false">(</mo><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt><mi>cosh</mi><mi>y</mi><mo>−</mo><mi>μ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><msup><mrow><mo stretchy="false">[</mo><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mrow><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt><mi>cosh</mi><mi>y</mi><mo>−</mo><mi>μ</mi></mrow><mrow><msub><mi>T</mi><mi>i</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>S</mi><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mtext>d</mtext><mi>y</mi><mn>.</mn></mtd></mtr></mtable></math> In Eqs. (3) and (4), only T i , <math id="M9"><mrow><msub><msup><mi>V</mi><mo>′</mo></msup><mi>i</mi></msub></mrow></math> , and k i ( i ≤ l -1) are free parameters. Generally, l =2 or 3 is enough for describing the spectra in a not too wide p T range. Eqs. (1) or (2) and (3) or (4) can be used for the description of p T spectra and for the extraction of effective temperature and kinetic freeze-out volume in very low- and low- p T regions. The high- p T ( > 3-4 GeV/ c ) region contributed by the hard excitation process has to be fitted by the Hagedorn function [ 17 17 ], which is an inverse power law function, given by <math display="block" id="M10"><mrow><msub><mi>f</mi><mi>H</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><mi>A</mi><msub><mi>p</mi><mtext>T</mtext></msub><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><msub><mi>p</mi><mtext>T</mtext></msub></mrow><mrow><msub><mi>p</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup><mn>.</mn></mrow></math> It results from the pQCD [ 34 34 - 36 36 34-36 ], where A is the normalization constant, which depends on the free parameters p 0 and n , and results in <math id="M11"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mrow><mtext>T</mtext><mi>max</mi></mrow></msub></mrow></msubsup><msub><mi>f</mi><mi>H</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>p</mi><mtext>T</mtext></msub><mo>=</mo><mn>1</mn></mrow></math> . While considering the contributions of both the soft and hard excitation processes, we used the superposition in principle <math display="block" id="M12"><mrow><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><msub><mi>f</mi><mi>H</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where, k is the contribution ratio of the soft process and gives a natural result in <math id="M13"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mrow><mtext>T</mtext><mi>max</mi></mrow></msub></mrow></msubsup><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>p</mi><mtext>T</mtext></msub><mo>=</mo><mn>1</mn></mrow></math> . In Eq. (6), the contribution of the soft process is from 0 to ∼2-3 GeV/ c , or even up to ∼3-5 GeV/ c at very high energy, and the hard component contributes to the whole p T range. There is some mixing between the contributions of the two processes in the low- p T region. According to the Hagedorn model [ 17 17 ], the contributions of the two processes can be separated completely. One has another superposition <math display="block" id="M14"><mrow><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>A</mi><mn>1</mn></msub><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>−</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>A</mi><mn>2</mn></msub><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mi>H</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where θ ( x ) is the usual step function and A 1 and A 2 , are the normalization constants, which make A 1 f S ( p 1 )= A 2 f H ( p 1 ). Equation (7) gives the contribution of soft process from 0 to p 1 , while the hard component contributes from p 1 up to the maximum. In the aforementioned two-component functions (Eqs. (6) and (7)), each component ( f S ( p T ) and f H ( p T )) is a traditional distribution. The first component ( f S ( p T )) is one of the Boltzmann-Gibbs, Fermi-Dirac, and Bose-Einstein distributions if we use a given S , such as S =0, +1, or -1. The second component ( f H ( p T )) is the Tsallis-like distribution [ 19 19 ] if we let n =1/( q -1) and p 0 = nT T , where q is the entropy index and T T is the Tsallis temperature. We will use only the first component in Eq. (7) due to the reason that we are not studying a wide p T range in the present work. In the case of neglecting the contribution of the hard component in the low- p T region in Eq. (6), the first component in Eq. (6) gives the same result as that of the first component in Eq. (7). Eq. (4) with l =2, which is the two-component standard distribution, is used in the present work. In addition, considering the treatment of normalization, the real fitted kinetic freeze-out volume should be <math id="M15"><mrow><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><msub><mi>N</mi><mn>1</mn></msub><msub><msup><mi>V</mi><mo>′</mo></msup><mn>1</mn></msub><mo>/</mo><msub><mi>k</mi><mn>1</mn></msub></mrow></math> and <math id="M16"><mrow><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><msub><mi>N</mi><mn>2</mn></msub><msub><msup><mi>V</mi><mo>′</mo></msup><mn>2</mn></msub><mo>/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math> , which will be simply used in the following section. It should be noted that the value of l in the l -component standard distribution has some influences on the free parameters and then on the derived parameters. Generally, l =1 is not enough to fit the particle spectra. For l =2, the influence of the second component is obvious since the contribution of the first component is not sufficient to fit the particle spectra. For l =3, the influence of the third component is rather small because the main contribution is from the first two components, and the contribution of the third component can be neglected. 3 Results and discussion 1 3.1 Comparison with the data 2 Figures 1(a) and 1(b) Figures 1(a) and 1(b) demonstrate the transverse momentum spectra, (1/2 πp T )d 2 N / dp T d y , of the negatively charged particles π - , K - , and <math id="M17"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> produced in (a) central (0–10%), and (b) peripheral (40–80%) Au-Au collisions at <math id="M18"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>62.4</mn></mrow></math> GeV. The circles, triangles, and squares represent the experimental data measured in the mid-rapidity range -0.5 < y < 0 at the RHIC by the STAR Collaboration [ 38 38 ]. The curves represent fitting results by Eq. (4) with l =2. Following each panel, the results of Data/Fit are presented. The values of the related parameters ( T 1 , T 2 , V 1 , V 2 , k 1 , and N 0 ) along with the χ 2 and the number of degrees of freedom (ndof) are given in Table 1 Table 1 . It can be seen that the two-component standard distribution fits approximately the experimental data measured at mid-rapidity in Au-Au collisions at the RHIC. 1 1 Collisions 1 1 Centrality 1 1 Particle 1 1 T 1 (GeV) 1 1 T 2 (GeV) 1 1 V 1 (fm 3 ) 1 1 V 2 (fm 3 ) 1 1 k 1 1 1 N 0 [ σ 0 (mb)] 1 1 χ 2 1 1 ndof 1 1 Figure 1 1 1 0–10% 1 1 π - 1 1 0.141±0.008 1 1 0.285±0.007 1 1 185±13 1 1 3330±270 1 1 0.88±0.07 1 1 0.080±0.004 1 1 43 1 1 14 1 1 Au-Au 1 1 1 1 K - 1 1 0.199±0.009 1 1 0.316±0.007 1 1 19±1 1 1 2034±162 1 1 0.85±0.11 1 1 0.010±0.003 1 1 174 1 1 13 1 1 62.4 GeV 1 1 1 1 <math id="M21"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.280±0.012 1 1 0.340±0.004 1 1 22±3 1 1 1053±100 1 1 0.92±0.10 1 1 0.020±0.004 1 1 56 1 1 11 1 1 1 1 40–80% 1 1 π - 1 1 0.070±0.006 1 1 0.250±0.006 1 1 32±5 1 1 127±14 1 1 0.70±0.07 1 1 0.025±0.050 1 1 94 1 1 14 1 1 1 1 1 1 K - 1 1 0.239±0.008 1 1 0.260±0.004 1 1 2.3±0.3 1 1 118±18 1 1 0.89±0.09 1 1 0.005±0.001 1 1 40 1 1 13 1 1 1 1 1 1 <math id="M22"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.201±0.007 1 1 0.302±0.005 1 1 3.0±0.2 1 1 69±8 1 1 0.89±0.11 1 1 0.009±0.001 1 1 7 1 1 11 1 1 Figure 2 1 1 0–5% 1 1 π - 1 1 0.267±0.013 1 1 0.624±0.005 1 1 8943±655 1 1 4341±200 1 1 0.93±0.12 1 1 1.770±0.300 1 1 425 1 1 33 1 1 Pb-Pb 1 1 1 1 K - 1 1 0.355±0.014 1 1 0.465±0.006 1 1 1820±250 1 1 5555±300 1 1 0.94±0.12 1 1 0.300±0.040 1 1 776 1 1 32 1 1 5.02 TeV 1 1 1 1 <math id="M23"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.459±0.014 1 1 0.512±0.006 1 1 381±30 1 1 5476±240 1 1 0.94±0.10 1 1 0.325±0.040 1 1 748 1 1 30 1 1 1 1 80–90% 1 1 π - 1 1 0.200±0.009 1 1 0.407±0.004 1 1 154±8 1 1 246±50 1 1 0.70±0.09 1 1 0.060±0.003 1 1 658 1 1 33 1 1 1 1 1 1 K - 1 1 0.198±0.016 1 1 0.420±0.005 1 1 17±2 1 1 264±45 1 1 0.90±0.11 1 1 0.020±0.003 1 1 90 1 1 32 1 1 1 1 1 1 <math id="M24"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.302±0.018 1 1 0.400±0.006 1 1 4.4±0.5 1 1 330±56 1 1 0.92±0.13 1 1 0.008±0.001 1 1 296 1 1 29 1 1 Figure 3(a) 1 1 - 1 1 π - 1 1 0.182±0.006 1 1 0.275±0.005 1 1 65±8 1 1 22±4 1 1 0.68±0.12 1 1 0.350±0.060 1 1 54 1 1 23 1 1 pp 1 1 1 1 K - 1 1 0.160±0.007 1 1 0.255±0.006 1 1 5.0±0.4 1 1 77±10 1 1 0.88±0.15 1 1 0.007±0.001 1 1 4 1 1 13 1 1 62.4 GeV 1 1 1 1 <math id="M25"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.235±0.008 1 1 0.260±0.006 1 1 1.6±0.1 1 1 50±6 1 1 0.95±0.10 1 1 0.008±0.001 1 1 126 1 1 24 1 1 Figure 3(b) 1 1 - 1 1 π - 1 1 0.090±0.008 1 1 0.370±0.005 1 1 16±2 1 1 101±13 1 1 0.64±0.11 1 1 0.016±0.003 1 1 945 1 1 33 1 1 pp 1 1 1 1 K - 1 1 0.850±0.013 1 1 0.370±0.004 1 1 0.80±0.04 1 1 97±12 1 1 0.87±0.11 1 1 0.007±0.001 1 1 666 1 1 31 1 1 5.02 TeV 1 1 1 1 <math id="M26"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.539±0.010 1 1 0.391±0.005 1 1 1.1±0.1 1 1 77±12 1 1 0.90±0.13 1 1 0.003±0.001 1 1 496 1 1 29 To see the contributions of the two components in Eq. (4) with l =2, as examples, Figs. 1(c) and 1(d) Figs. 1(c) and 1(d) show the contributions of the first and second components by the dashed and dotted curves, respectively, and the total contribution is given by the solid curves. Only the results of π - produced in (c) central (0–10%) and (d) peripheral (40–80%) Au–Au collisions at <math id="M27"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>62.4</mn></mrow></math> GeV are presented. The circles represent the same data points as those in Figs. 1(a) and 1(b) Figs. 1(a) and 1(b) . One can see that the first component contributes mainly to the very low- and low- p T region, while the second component contributes to a wider region. There is a large overlap region of the two contributions. The transverse momentum spectra, (1/ N ev )d 2 N /d p T d y , of π - , K - , and <math id="M28"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> produced in (a) central (0–5%), and (b) peripheral (80–90%) Pb-Pb collisions at <math id="M29"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV are shown in Fig. 2 Fig. 2 , where N ev on the vertical axis denotes the number of events. The experimental data of π - , K - , and <math id="M30"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> measured in the mid-rapidity range | y | < 0.5 at the LHC by the ALICE Collaboration [ 39 39 , 40 40 ] are represented by circles, triangles, and squares, respectively. The curves are our results fitted by Eq. (4) with l =2. Following each panel, the results of Data/Fit are presented. The values of the related parameters along with the χ 2 and ndof are given in Table 1 Table 1 . One can see that the two-component standard distribution fits approximately the experimental data measured in the mid-rapidity range in Pb-Pb collisions at the LHC. The fitting in Figs. 1 Figs. 1 and 2 2 for peripheral collisions appear to be worse compared to central collisions. This is caused by a statistical fluctuation and the effect of a cold spectator in peripheral collisions. In the region of the cold spectator, particles are produced by multiple cascade scattering processes which are different from the thermalization processes of particle production in the region of the hot participants. In addition, our fits are done in all ranges of p T < 4.5 GeV/ c . However, as an alternative model, the blast-wave fit takes different cuts of p T for the analysis of different particles (see for instance ref. [ 2 2 ]). These different cuts affect the extraction of parameters, in particular for the analysis of the trends of particles, which is not an ideal treatment. In the next fits, we used all ranges of p T < 4.5 GeV/ c . Figures 3(a) and 3(b) Figures 3(a) and 3(b) show the transverse momentum spectra, E d 3 σ /d p 3 =(1/2 πp T )d 2 σ /d p T d y , of π - , K - , and <math id="M33"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> produced in pp collisions at <math id="M34"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>62.4</mn></mrow></math> GeV and 5.02 TeV, respectively. E and σ on the vertical axis denote the energy and cross-section, respectively. The symbols represent the experimental data measured in the mid-pseudorapidity range | η | < 0.35 by the PHENIX Collaboration [ 41 41 ] and in the mid-rapidity range | y | < 0.5 by the ALICE Collaborations [ 39 39 , 40 40 ]. The curves represent our results, fitted by Eq. (4) with l =2. Following each panel, the results of Data/Fit are presented. The values of the related parameters ( N 0 in Figs. 1 Figs. 1 and 2 2 are replaced by σ 0 in Fig. 3 Fig. 3 ) along with χ 2 and ndof are given in Table 1 Table 1 . One can see that the two-component standard distribution fits approximately the experimental data measured at mid-(pseudo)rapidity in pp collisions at the RHIC and LHC. We would like to point out that the vertical axes of Figs. 1–3 Figs. 1–3 are not the probability density function. We cannot fit them with Eq. (4) with l =2. Hence, we have done a conversion during our fitting. For Fig. 1 Fig. 1 , we have used the relation (1/2 πp T )(d 2 N /d p T d y )=(1/2 πp T ) N 0 f S ( p T )/d y for the conversion, where N 0 is the normalization constant in terms of particle number. For Fig. 2 Fig. 2 , we have used the relation d 2 N /d p T d y = N 0 f S ( p T )/d y for the conversion, where N ev on the vertical axis is neglected because d 2 N /d p T d y is directly regarded as the result per event. For Fig. 3 Fig. 3 , we have used the relation E d 3 σ/d p 3 =(1/2 πp T )(d 2 σ /d p T d y )=(1/2 πp T ) σ 0 f S ( p T )/d y in the conversion, where σ 0 is the normalization constant in terms of the cross-section. From Figs. 1–3 Figs. 1–3 and Table 1 Table 1 , it can be seen that the fitting quality is not great in some cases. It should be pointed out that the model used in these fittings is for soft processes but is used for analyzing p T spectra up to 4.5 GeV/ c . The high values of p T analyzed in this study contain hard processes which could be responsible for the bad fitting as indicated by χ 2 in Table 1 and also in the ratio of data to the fitting of Figs. 1–3 Figs. 1–3 . Then, it may seem necessary to attempt fitting by taking into account the function part corresponding to the hard process. However, the hard process is not necessary for extracting the parameters of the soft process. Although the fittings will be better if we also consider the contribution of the hard process, it is not useful for extracting the parameters considered in the present work. Therefore, we did not consider the contribution of the hard process. 3.2 Discussion on the parameters 2 Considering the contributions of the two components, the effective temperature averaged over the two components is T = k 1 T 1 + k 2 T 2 and the kinetic freeze-out volume by adding the two components is V = V 1 + V 2 . Further, the normalization constants contributed by the first and the second components are k 1 N 0 and k 2 N 0 , respectively. For convenience, we introduced the average p T (〈 p T 〉) and average moving mass ( <math id="M38"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , i.e. average energy in the source rest frame) here. Considering Eq. (4) only, we have <math display="block" id="M39"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mrow><mi>T</mi><mi>max</mi></mrow></msub></mrow></msubsup></mrow></mstyle><msub><mi>p</mi><mtext>T</mtext></msub><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>p</mi><mtext>T</mtext></msub><mn>.</mn></mrow></math> To obtain <math id="M40"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , we can use the Monte Carlo method. Let R 1 and R 2 denote random numbers distributed evenly in [0,1]. A concrete value of p T that satisfies Eq. (4) can be obtained by <math display="block" id="M41"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mtext>T</mtext></msub></mrow></msubsup></mrow></mstyle><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>'</mo><mo stretchy="false">)</mo><mi>d</mi><msub><mi>p</mi><mtext>T</mtext></msub><mo>'</mo><mo><</mo><msub><mi>R</mi><mn>1</mn></msub><mo><</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mtext>T</mtext></msub><mo>+</mo><mi>δ</mi><msub><mi>p</mi><mtext>T</mtext></msub></mrow></msubsup></mrow></mstyle><msub><mi>f</mi><mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>'</mo><mo stretchy="false">)</mo><mi>d</mi><msub><mi>p</mi><mtext>T</mtext></msub><mo>'</mo><mn>,</mn></mrow></math> where δp T denotes a small shift relative to p T . In the source rest frame and under the assumption of isotropic emission, the emission angle θ of the considered particle obeys <math display="block" id="M42"><mrow><msub><mi>f</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>sin</mi><mi>θ</mi><mn>.</mn></mrow></math> which results in <math display="block" id="M43"><mrow><mi>θ</mi><mo>=</mo><mn>2</mn><mi>arcsin</mi><mo stretchy="false">(</mo><msqrt><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow></msqrt><mo stretchy="false">)</mo></mrow></math> in the Monte Carlo method [ 42 42 ]. Then, <math display="block" id="M44"><mrow><mi>m</mi><mo>=</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>/</mo><mi>sin</mi><mi>θ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt><mn>.</mn></mrow></math> After repeating the calculation many times, we can obtain <math id="M45"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> . To study the change in the trends of parameters with the particle mass, Figs. 4(a) and 4(b) Figs. 4(a) and 4(b) show the dependences of T on m 0 for productions of negative charged particles in central and peripheral (a) Au-Au collisions at 62.4 GeV and (b) Pb-Pb collisions at 5.02 TeV, while pp collisions at (a) 62.4 GeV and (b) 5.02 TeV are also studied and compared to peripheral AA collisions of the same energy (per nucleon pair). Correspondingly, Figs. 4(c) and 4(d) Figs. 4(c) and 4(d) show the dependences of 〈 p T 〉 on <math id="M46"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> for the mentioned particles in the considered collisions. The filled, empty, and half-filled symbols represent central AA , peripheral AA , and pp collisions, respectively. The lines represent linear fittings of the relations. The related linear fitting parameters are listed in Table 2 Table 2 , though some of them are not good fitting due to very large χ 2 . The intercept in the linear relation between T and m 0 is regarded as the kinetic freeze-out temperature T 0 , and the slope in the linear relation between 〈 p T 〉 and <math id="M47"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> is regarded as the transverse flow velocity β T . That is, T = am 0 + T 0 [ 26 26 - 28 28 26-28 ] and <math id="M48"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><msub><mi>β</mi><mi>T</mi></msub><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover><mo>+</mo><mi>b</mi></mrow></math> , where a and b are free parameters. 1 1 Figure 1 1 Relation 1 1 <math id="M52"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msqrt></mrow></math> ( <math id="M53"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> ) 1 1 Collisions 1 1 a ( c 2 ), β T ( c ) 1 1 T 0 (GeV), b (GeV/ c ) 1 1 χ 2 1 1 Figure 4(a) 1 1 T-m 0 1 1 62.4 GeV 1 1 Central Au-Au 1 1 0.0679±0.006 1 1 0.2769±0.006 1 1 1 1 1 1 1 1 1 1 1 Peripheral Au-Au 1 1 0.1054±0.005 1 1 0.2022±0.004 1 1 1 1 1 1 1 1 1 1 1 pp 1 1 0.1270±0.005 1 1 0.1543±0.006 1 1 40 1 1 Figure 4(b) 1 1 T-m 0 1 1 5.02 TeV 1 1 Central Pb-Pb 1 1 0.1650±0.004 1 1 0.3593±0.006 1 1 1 1 1 1 1 1 1 1 1 Peripheral Pb-Pb 1 1 0.0994±0.005 1 1 0.3293±0.005 1 1 31 1 1 1 1 1 1 1 1 pp 1 1 0.0829±0.005 1 1 0.3208±0.006 1 1 6 1 1 Figure 4(c) 1 1 <math id="M54"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>−</mo><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> 1 1 62.4 GeV 1 1 Central Au-Au 1 1 0.3857±0.004 1 1 0.1186±0.006 1 1 23 1 1 1 1 1 1 1 1 Peripheral Au-Au 1 1 0.3449±0.006 1 1 0.1381±0.004 1 1 5 1 1 1 1 1 1 1 1 pp 1 1 0.3567±0.006 1 1 0.0983±0.005 1 1 1 1 1 Figure 4(d) 1 1 <math id="M55"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>−</mo><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> 1 1 5.02 TeV 1 1 Central Pb-Pb 1 1 0.4260±0.006 1 1 0.1178±0.005 1 1 37 1 1 1 1 1 1 1 1 Peripheral Pb-Pb 1 1 0.4371±0.005 1 1 0.0465±0.004 1 1 2 1 1 1 1 1 1 1 1 pp 1 1 0.4048±0.006 1 1 0.1331±0.005 1 1 1 Note that the relation T = am 0 + T 0 [ 26 26 - 28 28 26-28 ] is used because the intercept should be the kinetic freeze-out temperature T 0 which corresponds to the emission of massless particles for which, there is no influence of the flow effect. The relation <math id="M56"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><msub><mi>β</mi><mtext>T</mtext></msub><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover><mo>+</mo><mi>b</mi></mrow></math> was used in our previous works [ 22 22 - 25 25 22-25 ] for the same dimensions of 〈 p T 〉 and <math id="M57"><mrow><msub><mi>β</mi><mtext>T</mtext></msub><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> . The interpretation of slope a in T = am 0 + T 0 and the intercept b in <math id="M58"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><msub><mi>β</mi><mtext>T</mtext></msub><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover><mo>+</mo><mi>b</mi></mrow></math> is not clear to us. Possibly, am 0 reflects the effective temperature contributed by The flow effect and b reflects the average transverse momentum contributed by the thermal motion. From Fig. 4 Fig. 4 and Table 2 Table 2 , one can see that T ( T 0 or β T ) is larger in the central AA collisions as compared to peripheral AA collisions, and peripheral AA collisions are comparable with the pp collisions at the same <math id="M59"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> ( <math id="M60"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> ). The mass-dependent or differential kinetic freeze-out scenario for T is observed, as T increased with the increase in m 0 . The present work confirms various mass dependent or differential kinetic freeze-out scenarios [ 2 2 , 3 3 , 20 20 , 43 43 , 44 44 ]. Because T 0 ( β T ) is obtained from the linear relation between T and m 0 (〈 p T 〉 and <math id="M61"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> ), it seems that there is no conclusion for the mass dependence. However, if we first fit π - and K - , and then include <math id="M62"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> , we can see that T 0 ( β T ) increases (decreases slightly) with increasing the mass. Thus, we observe the mass dependence or differential kinetic freeze-out. It should be noted that although Fig. 4 Fig. 4 also shows the enhancement of T when m 0 increases, this has been observed in many experiments and was reported for the first time by NA44 Collaboration [ 45 45 ] as evidence of the flow. This result was from a fit of p T to a thermal model for π - , K - , and <math id="M63"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> . This indicates that the use of the two-component source model is unnecessary to observe the enhancement of T when m 0 increases. Although one can arrive at the same conclusion using a single-component source model, the two-component source model can describe well the p T spectra. In addition, including the hard component, the model can describe better the p T spectra. The mass dependence of T ( T 0 ) and β T exists because it reflects the mass dependence of 〈 p T 〉. We do not think that the mass dependence of T ( T 0 ) and β T is a model dependence, though the values of T ( T 0 ) and β T themselves are model dependent. In our fittings, we have used the same p T range for π - , K - , and <math id="M64"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> , while in the blast-wave fitting, different p T ranges were used for the three types of particles [ 2 2 ]. The treatment by the latter increases the flexibility in the selection of parameters. Figure 5(a) Figure 5(a) shows the dependences of kinetic freeze-out volume V on rest mass m 0 for production of negatively charged particles in central and peripheral Au-Au collisions at <math id="M65"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>62.4</mn></mrow></math> GeV as well as in pp collisions at <math id="M66"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>62.4</mn></mrow></math> GeV, while Fig. 5(b) Fig. 5(b) shows the dependences of V on m 0 for negatively charged particles produced in central and peripheral Pb-Pb collisions at <math id="M67"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV as well as in pp collisions at <math id="M68"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV. The filled, empty, and half filled symbols represent the central AA , peripheral AA , and pp collisions, respectively, and they represent the results weighted by different contribution fractions (volumes) in two components listed in Table 1 Table 1 . It can be seen from Fig. 5 Fig. 5 that V in central AA collisions for all the particles are larger than those in peripheral AA collisions, which shows more participant nucleons and larger expansion in central AA collisions as compared to that in the peripheral AA collisions. Meanwhile, V in pp collisions is less than that in peripheral AA collisions of the same <math id="M69"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> ( <math id="M70"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> ), which is caused by fewer participant nucleons (less multiplicity) in pp collisions. It is also observed that V decreases with an increase of m 0 . This leads to a volume dependent or differential freeze-out scenario and indicates different freeze-out surfaces for different particles, depending on their masses that show early freeze-out of heavier particles as compared to the lighter particles [ 10 10 , 11 11 ]. Figure 6 Figure 6 shows the dependences of T on V for the production of negatively charged particles in (a) central and peripheral Au-Au collisions as well as in pp collisions at 62.4 GeV, and in (b) central and peripheral Pb-Pb collisions as well as in pp collisions at 5.02 TeV. The filled, empty, and half-filled symbols represent central AA , peripheral AA , and pp collisions, respectively. One can see that T decreases with the increase in V in the central and peripheral AA and pp collisions. This result is natural due to the fact that a large V corresponds to a long kinetic freeze-out time and then a cool system and a low T . As we have not done any systematic analysis of the mass dependence of T 0 ( β T ) in the present work, we shall not study the relation between T 0 ( β T ) and V , though we can still predict the trend. As a supplement, our recent work [ 46 46 ] reported the mass dependence (slight dependence) of T 0 ( β T ) using the same method as used in the present work, but using the Tsallis distribution as the "thermometer". We understand that with increasing m 0 (decreasing V ), T 0 would increase naturally, and β T would decrease slightly. From Figs. 4–6 Figs. 4–6 , one can see that T , T 0 , β T , and V obtained from collisions at the LHC are larger than those obtained from the collisions at the RHIC. This is expected due to more violent collisions happening at higher energy. However, from the RHIC to LHC, the increase in the collision energy is considerably large, and the increases in T , T 0 , β T , and V are relatively small. This reflects the penetrability of the projectiles in the transparent target. In addition, pions correspond to a larger V than protons in some cases. This is caused by the fact that pions have larger β T and thus, reach larger distance than protons due to the smaller m 0 in the case of the former at similar momenta for pions and protons at the kinetic freeze-out. This hypothesis is true because V is a reflection of multiplicity, and experimental results indicate an enhancement in the hadron source with the multiplicity. The result that pions correspond to a much larger V than protons indicates that the protons cease to interact while pions are still interacting. One may think that pions and protons stop interacting in different V , where large V corresponds to long interaction time. As protons have larger m 0 than pions, protons are left behind as the system evolved from the origin of collisions to the radial direction, which is the behavior of hydrodynamics [ 47 47 ]. This results in the volume dependent freeze-out scenario that shows the early freeze-out of heavier particles as compared to the lighter particles [ 10 10 , 11 11 ]. Thus, pions correspond to larger interacting volumes than protons, at the kinetic freeze-out stage. To further study the dependences of T and V on centrality and collisions energy, Table 3 Table 3 compiles the values of average T (〈 T 〉) and average V (〈 V 〉) for different types of collisions at the RHIC and LHC. These averages are obtained by different particle weights due to different contribution fractions ( V ) of π - , K - , and <math id="M71"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> . One can see that 〈 T 〉 and 〈 V 〉 at the LHC are larger than those at the RHIC. Generally, the value of T lies between T ch and T 0 . In particular, T ch in central AA collisions is approximately 160 MeV, and T 0 in central AA collisions is less than 130 MeV [ 30 30 , 31 31 , 48 48 , 49 49 ]. However, the values of 〈 T 〉 in Table 3 Table 3 are larger because of Eq. (4) was used. Eq. (4) contains the contributions of both thermal motion and flow effect, which can be regarded as a different "thermometer" from literature [ 30 30 , 31 31 , 48 48 - 50 50 48-50 ] and results in different T that is beyond the general range of [ T ch , T 0 ]. 1 1 Figure 1 1 <math id="M73"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> ( <math id="M74"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> ) 1 1 Collisions 1 1 ⟨ T ⟩ (GeV) 1 1 V (fm 3 ) 1 1 Figure 1(a) 1 1 62.4 GeV 1 1 Central Au-Au 1 1 0.303±0.007 1 1 2610±218 1 1 Figure 1(b) 1 1 1 1 Peripheral Au-Au 1 1 0.247±0.007 1 1 130±17 1 1 Figure 3(a) 1 1 1 1 pp 1 1 0.214±0.006 1 1 77±10 1 1 Figure 2(a) 1 1 5.02 TeV 1 1 Central Pb-Pb 1 1 0.478±0.009 1 1 10002±658 1 1 Figure 2(b) 1 1 1 1 Peripheral Pb-Pb 1 1 0.374±0.008 1 1 344±54 1 1 Figure 3(b) 1 1 1 1 pp 1 1 0.360±0.006 1 1 100±13 Even for T 0 (the intercept in Table 2 Table 2 for Figs. 4(a) and 4(b) Figs. 4(a) and 4(b) ) obtained from T = am 0 + T 0 , one can see the larger values. This is caused by the use of a different "thermometers". If other fitting functions are used [ 29 29 - 33 33 29-33 ], the obtained T 0 will be larger or smaller depending on the fitting function. For β T (the slope in Table 2 Table 2 for Figs. 4(c) and 4(d) Figs. 4(c) and 4(d) ) obtained from <math id="M75"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><msub><mi>β</mi><mtext>T</mtext></msub><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover><mo>+</mo><mi>b</mi></mrow></math> , one can see different values in the case of other methods ("thermometers") [ 20 20 - 22 22 20-22 , 26 26 ]. Anyhow, the relative sizes of T 0 ( β T ) obtained from the present work for different events centralities, system sizes, and collision energies are useful and significant. Generally, T 0 ≤ T ch . However, because of different "thermometers", we cannot simply compare the two temperatures. Although the absolute values of T ( T 0 ) and β T obtained in the present work are possibly inconsistent with other results, the relative values are worth considering. Similar is true for V . The present work shows that V in central and peripheral Pb-Pb and pp collisions at 5.02 TeV is also larger than that in central and peripheral Au-Au and pp collisions at 62.4 GeV. This shows a strong dependence of the parameters on the collision energy. Furthermore, V in central and peripheral Pb-Pb collisions is larger than that in central and peripheral Au-Au collisions also shows parameter dependence on the size of the system, though this dependence can be neglected due to a small difference in the size. The dependence of collision energy and system size is not discussed here in detail because of the unavailability of a wide range of analyses but it can be focused in future work. 3.3 Further discussion 2 Before the summary and conclusions, we would like to point out that the method that the related parameters can be extracted from the p T spectra of the identified particles seems approximately effective in high energy collisions. At high energy (dozens of GeV and above), the particle-dependent chemical potential μ is less than several MeV, which affects the parameters less. Eqs. (1)–(4) can be used in the present work. We believe that our result on the source volume for pp collisions being larger than that (∼34 fm 3 ) by the femtoscopy with two-pion Bose-Einstein correlations [ 51 51 ] is caused by the use of different methods. At intermediate and low energies, the method used here seems unsuitable due to the fact that the particle dependent μ at kinetic freeze-out is large and unavailable. In general, the particles of different species develop μ differently from chemical freeze-out to kinetic freeze-out. This seems to result in more difficulty in applying Eqs. (1)–(4) at intermediate and low energies. μ has less influence on the extraction of source volume due to its less influence on the data normalization or multiplicity. As we know, the source volume is proportional to the data normalization or multiplicity. Although we can obtain the normalization or multiplicity from a model, the obtained value is almost independent of the model. In other words, the normalization or multiplicity reflects the data, but not the model itself. Different methods do not affect the source volume considerably due to the normalization or multiplicity being one of the main factors, if not the only one. In the case of using a significant μ , neglecting the radial flow, and using T , there is no considerable influence on the normalization or multiplicity, then on the source volume. In addition, although we use the method of linear relation to obtain T 0 and β T in the present work, we used the blast-wave model [ 20 20 , 21 21 , 26 26 , 29 29 ] to obtain the two parameters in our previous works [ 22 22 , 43 43 , 44 44 ]. Besides, we could add indirectly the flow velocity in the treatment of standard distribution [ 52 52 ]. Because of different "thermometers" (fit functions) being used in different methods, the "measured" temperatures have different values, though the same trend can be observed in the same or similar collisions. The results obtained from different "thermometers" can be checked with each other. In particular, we obtained a higher temperature, though it is also the kinetic freeze-out temperature and describes the excitation degree of emission source at the kinetic freeze-out stage. We cannot compare the T 0 obtained in the present work with T ch used in literature directly due to different "thermometers". We found that the present work gives the same trend for main parameters when we compare them with our previous works [ 22 22 , 43 43 , 44 44 ], which used the blast-wave model [ 20 20 , 21 21 , 26 26 , 29 29 ]. It may be possible that the relative size of the main parameters in central and peripheral collisions as well as in AA and pp collisions will be the same if we use the standard distribution and the blast-wave model. It should be pointed out that although we have studied some parameters at the stage of kinetic freeze-out, the parameters at the stage of chemical freeze-out are lacking in this study. In fact, the parameters at the stage of chemical freeze-out are more important [ 53 53 - 58 58 53-58 ] to map the phase diagram in which μ is an essential factor. Both the T ch and μ are the most important parameters at the chemical freeze-out stage. In the extensive statistics and/or axiomatic/generic non-extensive statistics [ 53 53 - 55 55 53-55 ], one may discuss the chemical and/or kinetic freeze-out parameters systematically. Reference [ 56 56 ] has tried to advocate a new parametrization procedure rather than the standard χ 2 procedure with yields. The authors constructed the mean value of conserved charges and have utilized their ratios to extract T ch and μ . Reference [ 57 57 ] evaluated systematic error arising due to the chosen set of particle ratios and constraints. A centrality dependent study for the chemical freeze-out parameters [ 58 58 ] could be obtained. Meanwhile, with the help of the single-freeze-out model in the chemical equilibrium framework [ 59 59 , 60 60 ], reference [ 61 61 ] studied the centrality dependence of freeze-out temperature fluctuations in high energy AA collisions. We are very interested to do a uniform study on the chemical and kinetic freeze-out parameters in the future. Meanwhile, the distribution characteristics of various particles produced in high energy collisions are very abundant [ 62 62 - 65 65 62-65 ], and the methods of modeling analysis are multiple. We hope to study the spectra of multiplicities, transverse energies, and transverse momenta of various particles produced in different collisions by a uniform method, in which the probability density function contributed by each participant parton is considered carefully. 4 Summary and conclusions 1 We summarize here our main observations and conclusions (a) Main parameters extracted from the transverse momentum spectra of identified particles produced in central and peripheral Au-Au collisions at 62.4 GeV and Pb-Pb collisions at 5.02 TeV were studied. Furthermore, the same analysis was done for pp collisions at both RHIC and LHC energies. The two-component standard distribution was used, which included both the very soft and soft excitation processes. The effective temperature, kinetic freeze-out temperature, transverse flow velocity, and kinetic freeze-out volume were found to be larger in central collisions as compared to that in the peripheral collisions, which shows higher excitation and larger expansion in central collisions. (b) Effective temperatures in central and peripheral Au-Au (Pb-Pb) collisions at the RHIC (LHC) increased with increasing the particle mass, which showed a mass-dependent differential kinetic freeze-out scenario at RHIC and LHC energies. The kinetic freeze-out temperature is also expected to increase with increasing the particle’s mass. The kinetic freeze-out volume decreased with the increase of particle mass that showed different values for different particles and indicated a volume dependent differential kinetic freeze-out scenario. The transverse flow velocity is expected to decrease slightly with the increase of particle mass. (c) Effective (kinetic freeze-out) temperatures in peripheral Au-Au and pp collisions at 62.4 GeV as well as in peripheral Pb-Pb and pp collisions at 5.02 TeV were respectively similar and had a similar trend, which showed similar thermodynamic nature of the parameters in peripheral AA and pp collisions at the same center-of-mass energy (per nucleon pair). Effective (kinetic) freeze-out) temperatures in both central and peripheral AA and pp collisions decreased with an increase in the kinetic freeze-out volume. The transverse flow velocity is expected to increase slightly with the increase of the kinetic freeze-out volume in the considered energy range. (d) Effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume in central and peripheral AA and pp collisions at the LHC were larger than those at the RHIC, which showed their dependence on collision energy. 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Tech . 29 , 136 ( 2018 ). https://doi.org/10.1007/s41365-018-0469-y https://doi.org/10.1007/s41365-018-0469-y Multiplicity dependence of charged particle, φ meson, and multi-strange particle productions in p+p collisions at s=200 GeV from PYTHIA simulation 10.1007/s41365-020-00821-7 This work was supported by the National Natural Science Foundation. of China (Nos. 11575103 and 11947418), the Chinese Government Scholarship (China Scholarship Council), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) (No. 201802017), the Shanxi Provincial Natural Science Foundation (No. 201901D111043), and the Fund for Shanxi “1331 Project” Key Subjects Construction. 2020-08-02 2020-08-25 2020-09-17 2020-11-07 2020-11-01 2021-04-10 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-020-00821-7 http://dx.doi.org/10.1007/s41365-020-00821-7 . 2020 Nuclear Science and Techniques 11 31