Kinetic freeze-out temperatures in central and peripheral collisions: Which one is larger?

Kinetic freeze-out temperatures in central and peripheral collisions: Which one is larger? 1 first-author 1 corresp Corresponding author, fuhuliu@163.com; fuhuliu@sxu.edu.cn Corresponding author, fuhuliu@163.com; fuhuliu@sxu.edu.cn fuhuliu@163.com 1 1 Institute of Theoretical Physics & State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China Institute of Theoretical Physics & State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China The kinetic freeze-out temperatures, T 0 , in nucleus-nucleus collisions at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies are extracted by four methods: i) the Blast-Wave model with Boltzmann-Gibbs statistics (the BGBW model), ii) the Blast-Wave model with Tsallis statistics (the TBW model), iii) the Tsallis distribution with flow effect (the improved Tsallis distribution), and iv) the intercept in T = T 0 + am 0 (the alternative method), where m 0 denotes the rest mass and T denotes the effective temperature which can be obtained by different distribution functions. It is found that the relative sizes of T 0 in central and peripheral collisions obtained by the conventional BGBW model which uses a zero or nearly zero transverse flow velocity, β T , are contradictory in tendency with other methods. With a re-examination for β T in the first method, in which β T is taken to be ~(0.40±0.07) c , a recalculation presents a consistent result with others. Finally, our results show that the kinetic freeze-out temperature in central collisions is larger than that in peripheral collisions. Kinetic freeze-out temperature Methods for extraction Central collisions Peripheral collisions 1 Introduction 1 Temperature is an important concept in high energy nucleus-nucleus collisions. Usually, three types of temperatures which contain the chemical freeze-out temperature, kinetic freeze-out temperature, and effective temperature are used in literature [ 1 1 - 5 5 1-5 ]. The chemical freeze-out temperature describes the excitation degree of the interacting system at the stage of chemical equilibrium in which the chemical components (relative fractions) of particles are fixed. The kinetic freeze-out temperature describes the excitation degree of the interacting system at the stage of kinetic and thermal equilibrium in which the (transverse) momentum spectra of particles are no longer changed. The effective temperature is not a real temperature. In fact, the effective temperature is related to particle mass and can be extracted from the transverse momentum spectra by using some distribution laws such as the standard (Boltzmann, Fermi-Dirac, and Bose-Einstein), Tsallis, and so forth. Generally, the chemical freeze-out temperature is usually obtained from the particle ratios [ 6 6 - 8 8 6-8 ]. It is equal to or larger than the kinetic freeze-out temperature due to the the chemical equilibrium during or earlier than the kinetic equilibrium. The effective temperature is larger than the kinetic freeze-out temperature due to mass and flow effects [ 9 9 , 10 10 ]. Both the chemical freeze-out and effective temperatures in central nucleus-nucleus collisions are larger than those in peripheral collisions due to more violent interactions occurring in central collisions. In fact, central collisions contain more nucleons, and peripheral collisions contains less nucleons. Usually, there are small dissents in the extractions of chemical freeze-out temperature and effective temperature. As for the extraction of kinetic freeze-out temperature, the situations are largely non-uniform. Currently, four main methods are used in the extraction of kinetic freeze-out temperature, T 0 , which are i) the Blast-Wave model with Boltzmann-Gibbs statistics (the BGBW model) [ 11 11 - 13 13 11-13 ], ii) the Blast-Wave model with Tsallis statistics (the TBW model) [ 14 14 ], iii) the Tsallis distribution with flow effect (the improved Tsallis distribution) [ 15 15 , 16 16 ], and iv) the intercept in T = T 0 + am 0 (the alternative method) [ 12 12 , 17 17 - 20 20 17-20 ], where m 0 denotes the rest mass and T denotes the effective temperature which can be obtained by different distribution functions. In detail, the alternative method can be divided into a few sub-methods due to different distributions being used. Generally, we are inclined to use the standard and Tsallis distributions in the alternative method due to the standard distribution being closest to the ideal gas model in thermodynamics, and the Tsallis distribution describing a wide spectrum which needs a two- or three-component standard distribution to be fitted [ 21 21 ]. The kinetic freeze-out temperature, T 0 , and the mean transverse radial flow velocity, β T , can be simultaneously extracted by the first three methods. The alternative method needs further treatments in extracting the flow velocity. In our recent works [ 22 22 - 24 24 22-24 ], the mean transverse flow velocity, β T , is regarded as the slope in the relation <math id="M1"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><msub><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo></mrow><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mtext>T</mtext></msub><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , where 〈 p T 〉 denotes the mean value of transverse momenta p T , 〈 p T 〉 0 denotes the mean transverse momentum in the case of zero flow velocity, and <math id="M2"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> denotes the mean moving mass. The mean flow velocity, β , is regarded as the slope in the relation <math id="M3"><mrow><mo>〈</mo><mi>p</mi><mo>〉</mo><mo>=</mo><msub><mrow><mo>〈</mo><mi>p</mi><mo>〉</mo></mrow><mn>0</mn></msub><mo>+</mo><mi>β</mi><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , where 〈 p 〉 denotes the mean value of momenta, p , and 〈 p 〉 0 denotes the mean momentum in the case of zero flow velocity. Although the mean transverse radial flow and mean transverse flow are not exactly the same, we use the same symbol to denote their velocities and neglect the difference between them. In fact, the mean transverse radial flow contains only the isotropic flow, and the mean transverse flow contains both the isotropic and anisotropic flows. The isotropic flow is mainly caused by isotropic expansion of the interacting system, and the anisotropic flow is mainly caused by anisotropic squeeze between two incoming nuclei. We are interested in the coincidence and difference among the four methods in the extractions of T 0 and β T . In this paper, we shall use the four methods to extract T 0 and β T from the p T spectra of identified particles produced in central and peripheral gold-gold (Au-Au) collisions at the center-of-mass energy per nucleon pair <math id="M4"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV (the top RHIC energy) and in central and peripheral lead-lead (Pb-Pb) collisions at <math id="M5"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV (one of the LHC energies). The model results on the p T spectra are compared with the experimental data of the PHENIX [ 25 25 ], STAR [ 26 26 , 27 27 ], and ALICE Collaborations [ 28 28 , 29 29 ], and the model results on T 0 and β T in different collisions and by different methods are compared each other. The rest of this paper is structured as follows. The formalism and method are shortly described in Sect. 2. Results and discussion are given in Sect. 3. Finally, we summarize our main observations and conclusions in Sect. 4. 2 Formalism and method 1 The four methods can be found in related references [ 11 11 - 20 20 11-20 ]. To give a whole representation of this paper, we present directly and concisely the four methods in the following. In the representation, some quantities such as the kinetic freeze-out temperature, the mean transverse (radial) flow velocity, and the effective temperature in different methods are uniformly denoted by T 0 , β T , and T , respectively, though different methods correspond to different values. All of the model descriptions are presented at the mid-rapidity which uses the rapidity y ≈ 0 and results in cosh( y ) ≈ 1 which appears in some methods. At the same time, the spin property and chemical potential in the p T spectra are neglected due to their small influences in high energy collisions. This means that we can give up the Fermi-Dirac and Bose-Einstein distributions and use only the Boltzmann distribution in the case of considering the standard distribution. According to Refs. [ 11 11 - 13 13 11-13 ], the BGBW model results in the p T distribution to be <math display="block" id="M6"><mrow><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><msub><mi>p</mi><mtext>T</mtext></msub><msub><mi>m</mi><mtext>T</mtext></msub><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>R</mi></msubsup></mrow></mstyle><mi>r</mi><mtext>d</mtext><mi>r</mi><mo>×</mo><msub><mi>I</mi><mn>0</mn></msub><mo stretchy="false">[</mo><mfrac><mrow><msub><mi>p</mi><mtext>T</mtext></msub><mi>sinh</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">]</mo><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">[</mo><mfrac><mrow><msub><mi>m</mi><mtext>T</mtext></msub><mi>cosh</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where C 1 is the normalized constant which results in <math id="M7"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>f</mi><mn>1</mn></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 I 0 and K 1 are the modified Bessel functions of the first and second kinds respectively, <math id="M8"><mrow><msub><mi>m</mi><mtext>T</mtext></msub><mo>=</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></mrow></math> is the transverse mass, ρ = tanh -1 [ β ( r )] is the boost angle, β ( r ) = β S ( r/R ) n 0 is a self-similar flow profile, β S is the flow velocity on the surface of the thermal source, r / R is the relative radial position in the thermal source, and n 0 is a free parameter which is customarily chosen to be 2 [ 11 11 ] due to this quadratic profile resembling the solutions of hydrodynamics closest [ 30 30 ]. Generally, <math id="M9"><mrow><msub><mi>β</mi><mtext>T</mtext></msub><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>/</mo><msup><mi>R</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>R</mi></msubsup></mrow></mstyle><mi>r</mi><mi>β</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>r</mi><mo>=</mo><mn>2</mn><msub><mi>β</mi><mtext>S</mtext></msub><mo>/</mo><mo stretchy="false">(</mo><msub><mi>n</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></math> . In the case of n 0 =2, as used in Ref. [ 11 11 ], we have β T = 0.5 β S [ 31 31 ]. According to Ref. [ 14 14 ], the TBW model results in the p T distribution to be <math display="block" id="M10"><mrow><msub><mi>f</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><msub><mi>p</mi><mtext>T</mtext></msub><msub><mi>m</mi><mtext>T</mtext></msub><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup></mrow></mstyle><mtext>d</mtext><mi>ϕ</mi><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>R</mi></msubsup></mrow></mstyle><mi>r</mi><mtext>d</mtext><mi>r</mi><msup><mrow><mo>{</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">[</mo><msub><mi>m</mi><mtext>T</mtext></msub><mi>cosh</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>p</mi><mtext>T</mtext></msub><mi>sinh</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>}</mo></mrow><mrow><mo>−</mo><mi>q</mi><mo>/</mo><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mn>,</mn></mrow></math> where C 2 is the normalized constant which results in <math id="M11"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>f</mi><mn>2</mn></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> , q is an entropy index characterizing the degree of non-equilibrium, and ϕ denotes the azimuth. In the case of n 0 =1, as used in Ref. [ 14 14 ], we have β T = 2 β S /( n 0 +2)=(2/3) β S due to the same flow profile as in the BGBW model. We would like to point out that the index - q /( q -1) in Eq. (2) replaced -1/( q -1) in Ref. [ 14 14 ] due to q being very close to 1. In fact, the difference between the results corresponding to - q /( q -1) and -1/( q -1) are small in the Tsallis distribution [ 32 32 ]. According to Refs. [ 15 15 , 16 16 ], the improved Tsallis distribution in terms of p T is <math display="block" id="M12"><mtable columnalign="left"><mtr><mtd><msub><mi>f</mi><mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mn>3</mn></msub><mo>{</mo><mn>2</mn><msub><mi>T</mi><mtext>0</mtext></msub><mo stretchy="false">[</mo><mi>r</mi><msub><mi>I</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><mi>s</mi><msub><mi>I</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mi>K</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>T</mi><mn>0</mn></msub><msup><mi>r</mi><mn>2</mn></msup><msub><mi>I</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>K</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>K</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><mn>4</mn><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>T</mi><mn>0</mn></msub><mi>r</mi><mi>s</mi><msub><mi>I</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>T</mi><mn>0</mn></msub><msup><mi>s</mi><mn>2</mn></msup><msub><mi>K</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>I</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>I</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></mfrac><msub><mi>T</mi><mn>0</mn></msub><msup><mi>r</mi><mn>3</mn></msup><msub><mi>I</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>K</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><mn>3</mn><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><msub><mi>T</mi><mn>0</mn></msub><msup><mi>r</mi><mn>2</mn></msup><mi>s</mi><mo stretchy="false">[</mo><msub><mi>K</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>K</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>I</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mn>3</mn><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><msub><mi>T</mi><mn>0</mn></msub><msup><mi>s</mi><mn>2</mn></msup><mi>r</mi><mo stretchy="false">[</mo><msub><mi>I</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>I</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></mfrac><msub><mi>T</mi><mn>0</mn></msub><msup><mi>s</mi><mn>3</mn></msup><mo stretchy="false">[</mo><msub><mi>I</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><msub><mi>I</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>K</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>}</mo><mn>,</mn></mtd></mtr></mtable></math> where C 3 is the normalized constant which results in <math id="M13"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>f</mi><mn>3</mn></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> , r ≡ γm T / T 0 , s ≡ γβ T p T / T 0 , <math id="M14"><mrow><mi>γ</mi><mo>=</mo><mn>1</mn><mo>/</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msubsup><mi>β</mi><mtext>T</mtext><mn>2</mn></msubsup></mrow></msqrt></mrow></math> , and I 0-3 (s) and K 0-3 ( r ) are the modified Bessel functions of the first and second kinds, respectively. As for the alternative method [ 12 12 , 17 17 - 20 20 17-20 , 22 22 - 24 24 22-24 ], to use the relations T = T 0 + am 0 , <math id="M15"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><msub><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo></mrow><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mtext>T</mtext></msub><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , and <math id="M16"><mrow><mo>〈</mo><mi>p</mi><mo>〉</mo><mo>=</mo><msub><mrow><mo>〈</mo><mi>p</mi><mo>〉</mo></mrow><mn>0</mn></msub><mo>+</mo><mi>β</mi><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , we can choose the standard and Tsallis distributions to fit the p T spectra of identified particles produced in high energy collisions. Because we give up the Fermi-Dirac and Bose-Einstein distributions, only the Boltzmann distribution is used in the case of considering the standard distribution in the present work. Both the Boltzmann and Tsallis distributions have more than one forms. We choose the form of Boltzmann distribution [ 33 33 ] <math display="block" id="M17"><mrow><msub><mi>f</mi><mrow><mn>4</mn><mi>a</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mrow><mn>4</mn><mi>a</mi></mrow></msub><msub><mi>p</mi><mtext>T</mtext></msub><msub><mi>m</mi><mtext>T</mtext></msub><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mfrac><mrow><msub><mi>m</mi><mtext>T</mtext></msub></mrow><mi>T</mi></mfrac><mo stretchy="false">)</mo></mrow></math> and the form of Tsallis distribution [ 32 32 , 33 33 ] <math display="block" id="M18"><mrow><msub><mi>f</mi><mrow><mn>4</mn><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mrow><mn>4</mn><mi>b</mi></mrow></msub><msub><mi>p</mi><mtext>T</mtext></msub><msub><mi>m</mi><mtext>T</mtext></msub><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></mfrac><msub><mi>m</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mi>q</mi><mo>/</mo><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mn>,</mn></mrow></math> where C 4 a and C 4 b are the normalized constants which result in <math id="M19"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>f</mi><mrow><mn>4</mn><mi>a</mi></mrow></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> and <math id="M20"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>f</mi><mrow><mn>4</mn><mi>b</mi></mrow></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> respectively. It should be noticed that the above five distributions are only valid for the spectra in a low- p T range. That is, they describe only the soft excitation process. Even if for the soft process, the Boltzmann distribution is not always enough to fit the p T spectra in some cases. In fact, two- or three-component Boltzmann distributions can be used if necessary, in which T is the average weight at the effective temperatures obtained from different components. We have <math display="block" id="M21"><mrow><msub><mi>f</mi><mrow><mn>4</mn><mi>a</mi></mrow></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><msub><mi>C</mi><mrow><mn>4</mn><mi>a</mi><mi>i</mi></mrow></msub><msub><mi>p</mi><mtext>T</mtext></msub><msub><mi>m</mi><mtext>T</mtext></msub><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mfrac><mrow><msub><mi>m</mi><mtext>T</mtext></msub></mrow><mrow><msub><mi>T</mi><mi>i</mi></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow></math> and <math display="block" id="M22"><mrow><mi>T</mi><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><msub><mi>T</mi><mi>i</mi></msub><mn>,</mn></mrow></math> where l =2 or 3 denotes the number of components, and k i , C 4 ai , and T i denote the contribution ratio (relative contribution or fraction), normalization constant, and effective temperature related to the i -th component, respectively. As can be seen in the next section, Eqs. (6) and (7) are not needed in the present work because only simple component Boltzmann distribution, i.e. Eq. (4), is used in the analyses. We present here Eqs. (6) and (7) to point out a possible application in future. For the spectra in a wide p T range which contains low and high p T regions, we have to consider the contribution of a hard scattering process. Generally, the contribution of a hard process is parameterized to an inverse power-law <math display="block" id="M23"><mrow><msub><mi>f</mi><mtext>H</mtext></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><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></mrow></math> which is resulted from the QCD (quantum chromodynamics) calculation [ 34 34 - 36 36 34-36 ], where p 0 and n are free parameters, and A is the normalized constant which depends on p 0 and n and results in <math id="M24"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>f</mi><mtext>H</mtext></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> . To describe the spectra in a wide p T range, we can use a superposition of both contributions of soft and hard processes. The contribution of the soft process is described by one of the BGBW models, the TBW model, the improved Tsallis distribution, the Boltzmann distribution or two- or three-component Boltzmann distributions, and the Tsallis distribution, while the contribution of hard process is described by the inverse power-law. We have the superposition <math display="block" id="M25"><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><mtext>S</mtext></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><mtext>H</mtext></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where k denotes the contribution ratio of the soft process and results naturally in <math id="M26"><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><msub><mi>f</mi><mn>0</mn></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> , and f S ( p T ) denotes one of the five distributions discussed in the four methods. It should be noted that Eq. (9) and its components f S ( p T ) and f H ( p T ) are probability density functions. The experimental quantity of p T distribution has mainly three forms, d N /d p T , d 2 N /(d y d p T ), and (2π p T ) -1 d 2 N /(d y d p T ), where N denotes the number of particles and d y is approximately treated as a constant due to it being usually a given and small value at the mid-rapidity. To connect Eq. (9) with d N /d p T , we need a normalization constant, N 0 . To connect Eq. (9) with d 2 N /(d y d p T ), we need another normalization constant, N 0 . To connect Eq. (9) with (2π p T ) -1 d 2 N /(d y d p T ), we have to rewrite Eq. (9) to f 0 ( p T )/ p T =[ kf S ( p T )+(1- k )f H ( p T )]/ p T and compare the right side of the new equation with the data with a new normalization constant, N 0 . 3 Results and discussion 1 Figure 1 Figure 1 presents the transverse momentum spectra, (2π p T ) -1 d 2 N /(d y d p T ), of (a)-(c) positively charged pions ( π + ), positively charged kaons ( K + ), neutral kaons ( <math id="M27"><mrow><msubsup><mi>K</mi><mtext>S</mtext><mn>0</mn></msubsup></mrow></math> only), and protons ( p ), as well as (b)-(d) negatively charged pions ( π - ), negatively charged kaons ( K - ), neutral kaons ( <math id="M28"><mrow><msubsup><mi>K</mi><mtext>S</mtext><mn>0</mn></msubsup></mrow></math> only), and antiprotons ( <math id="M29"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> ) produced in (a)-(b) central (0–5% and 0–12%) and (c)-(d) peripheral (80–92% and 60–80%) Au-Au collisions at <math id="M30"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV, where the spectra for different types of particles and for the same or similar particles in different conditions are multiplied by different amounts shown in the panels for clarity and normalization. The closed symbols represent the experimental data of the PHENIX Collaboration measured in the pseudorapidity range | η | < 0.35 [ 25 25 ]. The open symbols represent the STAR data measured in the rapidity range | y | < 0.5 [ 26 26 , 27 27 ], where the data for K + and K - are not available and the data for <math id="M31"><mrow><msubsup><mi>K</mi><mtext>S</mtext><mn>0</mn></msubsup></mrow></math> in (a)-(c) and (b)-(d) are the same. The solid, dashed, dotted, dashed-dotted, and dashed-dotted-dotted curves are our results calculated by using the superpositions of i) the BGBW model (Eq. (1)) and inverse power-law (Eq. (8)), ii) the TBW model (Eq. (2)) and inverse power-law, iii) the improved Tsallis distribution (Eq. (3)) and inverse power-law, iv) a the Boltzmann distribution (Eq. (4)) and inverse power-law, as well as iv) b the Tsallis distribution (Eq. (5)) and inverse power-law, respectively. These different superpositions are also different methods for fitting the data. The values of free parameters T 0 , β T , k , p 0 , and n , normalization constant, N 0 , which is used to fit the data by a more accurate method comparing with Ref. [ 37 37 ], and χ 2 per degree of freedom ( χ 2 /dof) corresponding to the fit of method i) are listed in Table 1 Table 1 ; the values of T 0 , q , β T , k , p 0 , n , N 0 , and χ 2 /dof corresponding to methods ii) and iii) are listed in Tables 2 Tables 2 and 3 3 respectively; the values of T , k , p 0 , n , N 0 , and χ 2 /dof corresponding to methods iv) a are listed in Table 4 Table 4 ; and the values of T , q , k , p 0 , n , N 0 , and χ 2 /dof corresponding to method iv) b are listed in Table 5 Table 5 . One can see that, in most cases, all of the considered methods describe approximately the p T spectra of identified particles produced in central and peripheral Au-Au collisions at <math id="M32"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV. 1 1 Fig. 1 1 Cent. 1 1 Main Part. 1 1 T 0 (GeV) 1 1 β T ( c ) 1 1 k 1 1 p 0 (GeV/ c ) 1 1 n 1 1 N 0 1 1 χ 2 /dof 1 1 1(a) 1 1 Central 1 1 π + 1 1 0.113±0.004 1 1 0.413±0.006 1 1 0.988±0.003 1 1 2.075±0.062 1 1 9.015±0.148 1 1 985.309±75.105 1 1 2.708 1 1   1 1   1 1 K + 1 1 0.124±0.005 1 1 0.408±0.006 1 1 0.975±0.004 1 1 1.295±0.058 1 1 7.375±0.128 1 1 65.180±3.873 1 1 7.300 1 1   1 1   1 1 p 1 1 0.127±0.005 1 1 0.392±0.006 1 1 0.989±0.003 1 1 2.485±0.076 1 1 8.775±0.136 1 1 12.611±0.636 1 1 10.729 1 1 1(b) 1 1 Central 1 1 π - 1 1 0.113±0.004 1 1 0.413±0.006 1 1 0.988±0.003 1 1 2.075±0.062 1 1 9.015±0.148 1 1 985.309±75.105 1 1 2.792 1 1   1 1   1 1 K - 1 1 0.124±0.005 1 1 0.408±0.006 1 1 0.975±0.004 1 1 1.295±0.058 1 1 7.375±0.128 1 1 63.109±3.589 1 1 9.267 1 1   1 1   1 1 <math id="M33"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.125±0.005 1 1 0.392±0.006 1 1 0.990±0.003 1 1 2.465±0.076 1 1 8.895±0.136 1 1 10.572±0.599 1 1 20.512 1 1 1(c) 1 1 Peripheral 1 1 π + 1 1 0.160±0.004 1 1 <math id="M34"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.016</mn></mrow></msubsup></mrow></math> 1 1 0.754±0.009 1 1 2.012±0.069 1 1 10.803±0.143 1 1 9.794±0.726 1 1 4.237 1 1   1 1   1 1 K + 1 1 0.238±0.005 1 1 <math id="M35"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.016</mn></mrow></msubsup></mrow></math> 1 1 0.825±0.009 1 1 3.383±0.089 1 1 12.313±0.166 1 1 0.424±0.029 1 1 8.755 1 1   1 1   1 1 p 1 1 0.251±0.005 1 1 <math id="M36"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.016</mn></mrow></msubsup></mrow></math> 1 1 0.851±0.011 1 1 2.006±0.065 1 1 9.466±0.139 1 1 0.167±0.012 1 1 1.434 1 1 1(d) 1 1 Peripheral 1 1 π - 1 1 0.160±0.004 1 1 <math id="M37"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.016</mn></mrow></msubsup></mrow></math> 1 1 0.754±0.009 1 1 2.012±0.069 1 1 10.803±0.143 1 1 9.794±0.726 1 1 3.792 1 1   1 1   1 1 K - 1 1 0.238±0.005 1 1 <math id="M38"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.016</mn></mrow></msubsup></mrow></math> 1 1 0.825±0.009 1 1 3.383±0.089 1 1 12.313±0.166 1 1 0.424±0.029 1 1 7.469 1 1   1 1   1 1 <math id="M39"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.251±0.005 1 1 <math id="M40"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.016</mn></mrow></msubsup></mrow></math> 1 1 0.851±0.011 1 1 2.106±0.066 1 1 9.766±0.141 1 1 0.134±0.007 1 1 0.834 1 1 2(a) 1 1 Central 1 1 π + 1 1 0.128±0.004 1 1 0.434±0.007 1 1 0.992±0.002 1 1 2.775±0.091 1 1 7.435±0.133 1 1 1771.569±112.825 1 1 1.200 1 1   1 1   1 1 K + 1 1 0.187±0.004 1 1 0.390±0.006 1 1 0.993±0.002 1 1 3.575±0.098 1 1 7.135±0.128 1 1 92.874±6.429 1 1 3.647 1 1   1 1   1 1 p 1 1 0.429±0.005 1 1 0.145±0.005 1 1 0.976±0.005 1 1 2.485±0.088 1 1 7.375±0.136 1 1 10.188±0.445 1 1 7.472 1 1 2(b) 1 1 Central 1 1 π - 1 1 0.128±0.004 1 1 0.434±0.007 1 1 0.992±0.002 1 1 2.775±0.091 1 1 7.435±0.133 1 1 1771.569±112.825 1 1 1.221 1 1   1 1   1 1 K - 1 1 0.187±0.004 1 1 0.390±0.006 1 1 0.993±0.002 1 1 3.575±0.098 1 1 7.135±0.128 1 1 92.874±6.429 1 1 3.288 1 1   1 1   1 1 <math id="M41"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.428±0.005 1 1 0.145±0.005 1 1 0.976±0.005 1 1 2.485±0.088 1 1 7.375±0.136 1 1 10.209±0.446 1 1 6.875 1 1 2(c) 1 1 Peripheral 1 1 π + 1 1 0.183±0.004 1 1 <math id="M42"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.017</mn></mrow></msubsup></mrow></math> 1 1 0.909±0.009 1 1 2.793±0.089 1 1 8.985±0.133 1 1 12.144±0.705 1 1 16.627 1 1   1 1   1 1 K + 1 1 0.272±0.004 1 1 <math id="M43"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.017</mn></mrow></msubsup></mrow></math> 1 1 0.835±0.009 1 1 2.375±0.085 1 1 7.885±0.165 1 1 0.707±0.028 1 1 2.808 1 1   1 1   1 1 p 1 1 0.338±0.004 1 1 <math id="M44"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.017</mn></mrow></msubsup></mrow></math> 1 1 0.836±0.009 1 1 1.875±0.078 1 1 7.705±0.138 1 1 0.183±0.011 1 1 2.752 1 1 2(d) 1 1 Peripheral 1 1 π - 1 1 0.183±0.004 1 1 <math id="M45"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.017</mn></mrow></msubsup></mrow></math> 1 1 0.909±0.009 1 1 2.793±0.089 1 1 8.985±0.133 1 1 12.144±0.705 1 1 16.734 1 1   1 1   1 1 K - 1 1 0.272±0.004 1 1 <math id="M46"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.017</mn></mrow></msubsup></mrow></math> 1 1 0.835±0.009 1 1 2.375±0.085 1 1 7.885±0.165 1 1 0.707±0.028 1 1 3.041 1 1   1 1   1 1 <math id="M47"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.342±0.004 1 1 <math id="M48"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.017</mn></mrow></msubsup></mrow></math> 1 1 0.815±0.009 1 1 1.875±0.078 1 1 7.705±0.138 1 1 0.185±0.012 1 1 2.602 1 1 Fig. 1 1 Cent. 1 1 Main Part. 1 1 T 0 (GeV) 1 1 q 1 1 β T ( c ) 1 1 k 1 1 p 0 (GeV/ c ) 1 1 n 1 1 N 0 1 1 χ 2 /dof 1 1 1(a) 1 1 Central 1 1 π + 1 1 0.108±0.004 1 1 1.008±0.005 1 1 0.472±0.010 1 1 0.882±0.008 1 1 1.775±0.069 1 1 9.895±0.143 1 1 486.350±40.221 1 1 4.082 1 1   1 1   1 1 K + 1 1 0.113±0.004 1 1 1.020±0.005 1 1 0.469±0.010 1 1 0.901±0.006 1 1 1.875±0.072 1 1 9.405±0.139 1 1 44.575±2.808 1 1 6.564 1 1   1 1   1 1 p 1 1 0.119±0.004 1 1 1.011±0.004 1 1 0.469±0.008 1 1 0.989±0.003 1 1 2.885±0.082 1 1 9.275±0.136 1 1 7.214±0.368 1 1 1.665 1 1 1(b) 1 1 Central 1 1 π - 1 1 0.108±0.004 1 1 1.008±0.005 1 1 0.472±0.010 1 1 0.882±0.008 1 1 1.775±0.069 1 1 9.895±0.143 1 1 486.350±40.221 1 1 3.856 1 1   1 1   1 1 K - 1 1 0.113±0.004 1 1 1.020±0.005 1 1 0.469±0.010 1 1 0.901±0.006 1 1 1.875±0.072 1 1 9.405±0.139 1 1 42.837±2.808 1 1 5.939 1 1   1 1   1 1 <math id="M49"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.121±0.004 1 1 1.010±0.004 1 1 0.469±0.008 1 1 0.991±0.003 1 1 2.885±0.082 1 1 9.305±0.134 1 1 5.369±0.354 1 1 6.643 1 1 1(c) 1 1 Peripheral 1 1 π + 1 1 0.099±0.004 1 1 1.078±0.005 1 1 <math id="M50"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.036</mn></mrow></msubsup></mrow></math> 1 1 0.862±0.008 1 1 2.198±0.089 1 1 10.982±0.161 1 1 11.341±0.747 1 1 3.221 1 1   1 1   1 1 K + 1 1 0.119±0.004 1 1 1.088±0.004 1 1 <math id="M51"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.036</mn></mrow></msubsup></mrow></math> 1 1 0.985±0.008 1 1 1.983±0.078 1 1 8.253±0.138 1 1 0.589±0.053 1 1 4.002 1 1   1 1   1 1 p 1 1 0.132±0.004 1 1 1.064±0.005 1 1 <math id="M52"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.036</mn></mrow></msubsup></mrow></math> 1 1 0.985±0.004 1 1 2.010±0.088 1 1 7.966±0.129 1 1 0.171±0.014 1 1 0.940 1 1 1(d) 1 1 Peripheral 1 1 π - 1 1 0.099±0.004 1 1 1.078±0.005 1 1 <math id="M53"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.036</mn></mrow></msubsup></mrow></math> 1 1 0.862±0.008 1 1 2.198±0.089 1 1 10.982±0.161 1 1 11.341±0.747 1 1 2.924 1 1   1 1   1 1 K - 1 1 0.119±0.004 1 1 1.088±0.004 1 1 <math id="M54"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.036</mn></mrow></msubsup></mrow></math> 1 1 0.985±0.008 1 1 1.983±0.078 1 1 8.253±0.138 1 1 0.589±0.053 1 1 3.652 1 1   1 1   1 1 <math id="M55"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.124±0.004 1 1 1.067±0.005 1 1 <math id="M56"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.036</mn></mrow></msubsup></mrow></math> 1 1 0.983±0.004 1 1 2.018±0.088 1 1 8.166±0.129 1 1 0.144±0.014 1 1 0.552 1 1 2(a) 1 1 Central 1 1 π + 1 1 0.109±0.004 1 1 1.009±0.005 1 1 0.525±0.009 1 1 0.977±0.005 1 1 2.585±0.086 1 1 7.875±0.122 1 1 917.576±91.809 1 1 6.313 1 1   1 1   1 1 K + 1 1 0.145±0.005 1 1 1.004±0.003 1 1 0.500±0.009 1 1 0.984±0.004 1 1 3.255±0.091 1 1 7.508±0.119 1 1 66.904±6.743 1 1 0.580 1 1   1 1   1 1 p 1 1 0.178±0.005 1 1 1.002±0.001 1 1 0.500±0.008 1 1 0.993±0.002 1 1 4.975±0.099 1 1 8.725±0.121 1 1 8.981±0.254 1 1 3.509 1 1 2(b) 1 1 Central 1 1 π - 1 1 0.109±0.004 1 1 1.009±0.005 1 1 0.525±0.009 1 1 0.977±0.005 1 1 2.585±0.086 1 1 7.875±0.122 1 1 917.576±91.809 1 1 6.249 1 1   1 1   1 1 K - 1 1 0.145±0.005 1 1 1.004±0.003 1 1 0.500±0.009 1 1 0.985±0.004 1 1 3.255±0.091 1 1 7.508±0.119 1 1 66.904±6.743 1 1 0.570 1 1   1 1   1 1 <math id="M57"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.178±0.005 1 1 1.002±0.001 1 1 0.500±0.008 1 1 0.993±0.002 1 1 4.975±0.099 1 1 8.725±0.121 1 1 8.981±0.254 1 1 3.253 1 1 2(c) 1 1 Peripheral 1 1 π + 1 1 0.102±0.004 1 1 1.108±0.005 1 1 <math id="M58"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.018</mn></mrow></msubsup></mrow></math> 1 1 0.976±0.005 1 1 3.003±0.089 1 1 8.335±0.118 1 1 15.628±0.563 1 1 10.532 1 1   1 1   1 1 K + 1 1 0.141±0.005 1 1 1.099±0.005 1 1 <math id="M59"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.018</mn></mrow></msubsup></mrow></math> 1 1 0.906±0.005 1 1 1.875±0.071 1 1 7.038±0.109 1 1 0.820±0.063 1 1 1.149 1 1   1 1   1 1 p 1 1 0.172±0.005 1 1 1.076±0.005 1 1 <math id="M60"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.018</mn></mrow></msubsup></mrow></math> 1 1 0.958±0.005 1 1 2.375±0.088 1 1 7.575±0.119 1 1 0.212±0.017 1 1 4.623 1 1 2(d) 1 1 Peripheral 1 1 π - 1 1 0.102±0.004 1 1 1.108±0.005 1 1 <math id="M61"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.018</mn></mrow></msubsup></mrow></math> 1 1 0.976±0.005 1 1 3.003±0.089 1 1 8.335±0.118 1 1 15.628±0.563 1 1 10.481 1 1   1 1   1 1 K - 1 1 0.141±0.005 1 1 1.099±0.005 1 1 <math id="M62"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.018</mn></mrow></msubsup></mrow></math> 1 1 0.906±0.005 1 1 1.875±0.071 1 1 7.038±0.109 1 1 0.820±0.063 1 1 1.279 1 1   1 1   1 1 <math id="M63"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.172±0.005 1 1 1.076±0.005 1 1 <math id="M64"><mrow><msubsup><mrow><mn>0.000</mn></mrow><mrow><mo>−</mo><mn>0</mn></mrow><mrow><mo>+</mo><mn>0.018</mn></mrow></msubsup></mrow></math> 1 1 0.958±0.005 1 1 2.375±0.088 1 1 7.575±0.119 1 1 0.212±0.017 1 1 4.832 1 1 Fig. 1 1 Cent. 1 1 Main Part. 1 1 T 0 (GeV) 1 1 q 1 1 β T ( c ) 1 1 k 1 1 p 0 (GeV/ c ) 1 1 n 1 1 N 0 1 1 χ 2 /dof 1 1 1(a) 1 1 Central 1 1 π + 1 1 0.113±0.006 1 1 1.017±0.007 1 1 0.634±0.009 1 1 0.939±0.008 1 1 2.475±0.088 1 1 11.091±0.165 1 1 746.564±89.743 1 1 2.986 1 1   1 1   1 1 K + 1 1 0.116±0.006 1 1 1.040±0.007 1 1 0.634±0.009 1 1 0.902±0.008 1 1 3.675±0.091 1 1 12.995±0.172 1 1 32.457±5.734 1 1 9.781 1 1   1 1   1 1 p 1 1 0.121±0.006 1 1 1.024±0.007 1 1 0.634±0.009 1 1 0.916±0.008 1 1 2.985±0.090 1 1 11.225±0.162 1 1 5.365±0.677 1 1 1.249 1 1 1(b) 1 1 Central 1 1 π - 1 1 0.113±0.006 1 1 1.017±0.007 1 1 0.634±0.009 1 1 0.939±0.008 1 1 2.475±0.088 1 1 11.091±0.165 1 1 746.564±89.743 1 1 2.700 1 1   1 1   1 1 K - 1 1 0.116±0.006 1 1 1.040±0.007 1 1 0.634±0.009 1 1 0.900±0.008 1 1 3.675±0.091 1 1 12.995±0.172 1 1 31.193±5.698 1 1 8.100 1 1   1 1   1 1 <math id="M65"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.121±0.006 1 1 1.024±0.007 1 1 0.634±0.009 1 1 0.909±0.008 1 1 2.985±0.090 1 1 11.525±0.162 1 1 8.294±1.243 1 1 2.878 1 1 1(c) 1 1 Peripheral 1 1 π + 1 1 0.102±0.006 1 1 1.031±0.007 1 1 0.583±0.009 1 1 0.891±0.008 1 1 2.185±0.086 1 1 10.632±0.148 1 1 10.292±1.860 1 1 3.931 1 1   1 1   1 1 K + 1 1 0.109±0.006 1 1 1.045±0.008 1 1 0.578±0.009 1 1 0.872±0.008 1 1 4.483±0.099 1 1 14.061±0.165 1 1 0.327±0.057 1 1 8.529 1 1   1 1   1 1 p 1 1 0.110±0.006 1 1 1.053±0.008 1 1 0.548±0.008 1 1 0.901±0.008 1 1 3.066±0.095 1 1 11.166±0.126 1 1 0.083±0.005 1 1 2.700 1 1 1(d) 1 1 Peripheral 1 1 π - 1 1 0.102±0.006 1 1 1.031±0.007 1 1 0.583±0.009 1 1 0.891±0.008 1 1 2.185±0.086 1 1 10.532±0.148 1 1 10.771±1.863 1 1 3.751 1 1   1 1   1 1 K - 1 1 0.109±0.006 1 1 1.045±0.008 1 1 0.578±0.009 1 1 0.872±0.008 1 1 4.483±0.099 1 1 14.061±0.165 1 1 0.327±0.057 1 1 7.157 1 1   1 1   1 1 <math id="M66"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.110±0.006 1 1 1.053±0.008 1 1 0.548±0.008 1 1 0.901±0.008 1 1 3.066±0.095 1 1 11.166±0.126 1 1 0.055±0.005 1 1 1.316 1 1 2(a) 1 1 Central 1 1 π + 1 1 0.152±0.004 1 1 1.011±0.004 1 1 0.609±0.010 1 1 0.981±0.007 1 1 2.575±0.094 1 1 7.775±0.145 1 1 1475.441±93.801 1 1 2.682 1 1   1 1   1 1 K + 1 1 0.158±0.004 1 1 1.059±0.008 1 1 0.609±0.010 1 1 0.987±0.006 1 1 3.575±0.102 1 1 7.655±0.144 1 1 58.904±5.207 1 1 1.235 1 1   1 1   1 1 p 1 1 0.194±0.005 1 1 1.069±0.011 1 1 0.609±0.010 1 1 0.987±0.006 1 1 2.885±0.101 1 1 7.375±0.148 1 1 7.792±0.559 1 1 4.833 1 1 2(b) 1 1 Central 1 1 π - 1 1 0.152±0.004 1 1 1.011±0.004 1 1 0.609±0.010 1 1 0.981±0.007 1 1 2.575±0.094 1 1 7.775±0.145 1 1 1475.441±93.801 1 1 2.586 1 1   1 1   1 1 K - 1 1 0.158±0.004 1 1 1.059±0.008 1 1 0.609±0.010 1 1 0.987±0.006 1 1 3.575±0.102 1 1 7.655±0.144 1 1 58.904±5.207 1 1 1.083 1 1   1 1   1 1 <math id="M67"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.194±0.005 1 1 1.069±0.011 1 1 0.609±0.010 1 1 0.987±0.006 1 1 2.885±0.101 1 1 7.375±0.148 1 1 7.792±0.559 1 1 4.482 1 1 2(c) 1 1 Peripheral 1 1 π + 1 1 0.118±0.005 1 1 1.008±0.005 1 1 0.630±0.009 1 1 0.920±0.007 1 1 2.903±0.103 1 1 9.135±0.165 1 1 15.956±0.981 1 1 5.202 1 1   1 1   1 1 K + 1 1 0.143±0.004 1 1 1.011±0.005 1 1 0.602±0.009 1 1 0.901±0.007 1 1 3.003±0.111 1 1 8.335±0.155 1 1 0.530±0.038 1 1 1.880 1 1   1 1   1 1 p 1 1 0.163±0.005 1 1 1.021±0.005 1 1 0.559±0.009 1 1 0.889±0.007 1 1 2.375±0.099 1 1 8.059±0.142 1 1 0.102±0.006 1 1 2.804 1 1 2(d) 1 1 Peripheral 1 1 π - 1 1 0.118±0.005 1 1 1.008±0.005 1 1 0.630±0.009 1 1 0.920±0.007 1 1 2.903±0.103 1 1 9.135±0.165 1 1 15.956±0.981 1 1 5.257 1 1   1 1   1 1 K - 1 1 0.143±0.004 1 1 1.011±0.005 1 1 0.602±0.009 1 1 0.901±0.007 1 1 3.003±0.111 1 1 8.335±0.155 1 1 0.525±0.034 1 1 1.979 1 1   1 1   1 1 <math id="M68"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.163±0.005 1 1 1.021±0.005 1 1 0.559±0.009 1 1 0.889±0.007 1 1 2.375±0.099 1 1 8.059±0.142 1 1 0.101±0.006 1 1 2.942 1 1 Fig. 1 1 Cent. 1 1 Main Part. 1 1 T (GeV) 1 1 k 1 1 p 0 (GeV/ c ) 1 1 n 1 1 N 0 1 1 χ 2 /dof 1 1 1(a) 1 1 Central 1 1 π + 1 1 0.167±0.004 1 1 0.765±0.008 1 1 2.095±0.068 1 1 11.295±0.133 1 1 519.268±39.582 1 1 9.637 1 1   1 1   1 1 K + 1 1 0.235±0.004 1 1 0.752±0.008 1 1 2.915±0.068 1 1 12.335±0.185 1 1 49.650±2.890 1 1 12.847 1 1   1 1   1 1 p 1 1 0.302±0.005 1 1 0.983±0.005 1 1 2.785±0.066 1 1 9.475±0.176 1 1 7.744±0.267 1 1 2.217 1 1 1(b) 1 1 Central 1 1 π - 1 1 0.167±0.004 1 1 0.765±0.008 1 1 2.095±0.068 1 1 11.295±0.133 1 1 519.297±39.582 1 1 9.068 1 1   1 1   1 1 K - 1 1 0.235±0.004 1 1 0.750±0.008 1 1 2.915±0.068 1 1 12.335±0.185 1 1 47.297±2.893 1 1 13.624 1 1   1 1   1 1 <math id="M69"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.296±0.005 1 1 0.981±0.005 1 1 2.715±0.066 1 1 9.675±0.176 1 1 6.516±0.272 1 1 6.399 1 1 1(c) 1 1 Peripheral 1 1 π + 1 1 0.131±0.004 1 1 0.799±0.008 1 1 3.238±0.089 1 1 13.892±0.132 1 1 8.602±0.676 1 1 4.243 1 1   1 1   1 1 K + 1 1 0.185±0.004 1 1 0.702±0.008 1 1 3.483±0.086 1 1 13.083±0.146 1 1 0.556±0.035 1 1 6.799 1 1   1 1   1 1 p 1 1 0.209±0.005 1 1 0.822±0.008 1 1 4.606±0.106 1 1 14.866±0.155 1 1 0.173±0.012 1 1 0.955 1 1 1(d) 1 1 Peripheral 1 1 π - 1 1 0.131±0.004 1 1 0.799±0.008 1 1 3.238±0.089 1 1 13.892±0.132 1 1 8.602±0.676 1 1 4.115 1 1   1 1   1 1 K - 1 1 0.185±0.004 1 1 0.702±0.008 1 1 3.483±0.086 1 1 13.083±0.146 1 1 0.559±0.035 1 1 6.284 1 1   1 1   1 1 <math id="M70"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.209±0.005 1 1 0.822±0.008 1 1 4.606±0.106 1 1 15.279±0.165 1 1 0.139±0.012 1 1 0.627 1 1 2(a) 1 1 Central 1 1 π + 1 1 0.215±0.004 1 1 0.828±0.008 1 1 1.375±0.068 1 1 7.315±0.128 1 1 679.491±44.189 1 1 16.706 1 1   1 1   1 1 K + 1 1 0.299±0.005 1 1 0.972±0.008 1 1 2.945±0.090 1 1 7.685±0.132 1 1 57.722±5.536 1 1 1.889 1 1   1 1   1 1 p 1 1 0.413±0.005 1 1 0.993±0.002 1 1 4.975±0.112 1 1 8.725±0.146 1 1 8.864±0.467 1 1 2.600 1 1 2(b) 1 1 Central 1 1 π - 1 1 0.215±0.004 1 1 0.828±0.008 1 1 1.375±0.068 1 1 7.315±0.128 1 1 679.491±44.189 1 1 16.821 1 1   1 1   1 1 K - 1 1 0.299±0.005 1 1 0.972±0.008 1 1 2.945±0.090 1 1 7.685±0.132 1 1 57.722±5.536 1 1 2.052 1 1   1 1   1 1 <math id="M71"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.413±0.005 1 1 0.993±0.002 1 1 4.975±0.112 1 1 8.725±0.146 1 1 8.864±0.467 1 1 2.433 1 1 2(c) 1 1 Peripheral 1 1 π + 1 1 0.152±0.004 1 1 0.802±0.008 1 1 2.012±0.065 1 1 8.279±0.116 1 1 9.713±0.616 1 1 15.656 1 1   1 1   1 1 K + 1 1 0.219±0.004 1 1 0.803±0.009 1 1 2.035±0.092 1 1 7.595±0.134 1 1 0.822±0.052 1 1 5.123 1 1   1 1   1 1 p 1 1 0.291±0.005 1 1 0.805±0.008 1 1 2.285±0.096 1 1 8.365±0.142 1 1 0.190±0.017 1 1 3.545 1 1 2(d) 1 1 Peripheral 1 1 π - 1 1 0.152±0.004 1 1 0.802±0.008 1 1 2.012±0.065 1 1 8.279±0.116 1 1 9.713±0.616 1 1 15.657 1 1   1 1   1 1 K - 1 1 0.219±0.004 1 1 0.803±0.009 1 1 2.035±0.092 1 1 7.595±0.134 1 1 0.822±0.052 1 1 5.238 1 1   1 1   1 1 <math id="M72"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.296±0.005 1 1 0.805±0.008 1 1 2.285±0.096 1 1 8.365±0.142 1 1 0.188±0.017 1 1 3.391 1 1 Fig. 1 1 Cent. 1 1 Main Part. 1 1 T (GeV) 1 1 q 1 1 k 1 1 p 0 (GeV/ c ) 1 1 n 1 1 N 0 1 1 χ 2 /dof 1 1 1(a) 1 1 Central 1 1 π + 1 1 0.130±0.004 1 1 1.073±0.003 1 1 0.994±0.003 1 1 1.775±0.069 1 1 8.115±0.148 1 1 508.830±43.650 1 1 1.731 1 1   1 1   1 1 K + 1 1 0.184±0.005 1 1 1.050±0.004 1 1 0.984±0.005 1 1 1.075±0.058 1 1 6.775±0.135 1 1 45.687±2.962 1 1 4.354 1 1   1 1   1 1 p 1 1 0.274±0.004 1 1 1.015±0.003 1 1 0.988±0.003 1 1 2.485±0.088 1 1 8.775±0.152 1 1 8.211±0.194 1 1 3.268 1 1 1(b) 1 1 Central 1 1 π - 1 1 0.130±0.004 1 1 1.073±0.003 1 1 0.994±0.003 1 1 1.775±0.069 1 1 8.115±0.148 1 1 508.830±43.650 1 1 1.648 1 1   1 1   1 1 K - 1 1 0.184±0.005 1 1 1.050±0.004 1 1 0.982±0.005 1 1 1.075±0.058 1 1 6.775±0.135 1 1 42.366±2.868 1 1 2.951 1 1   1 1   1 1 <math id="M73"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.272±0.004 1 1 1.012±0.003 1 1 0.992±0.003 1 1 2.985±0.090 1 1 9.375±0.159 1 1 6.764±0.189 1 1 7.806 1 1 1(c) 1 1 Peripheral 1 1 π + 1 1 0.105±0.004 1 1 1.085±0.005 1 1 0.918±0.005 1 1 1.985±0.075 1 1 10.032±0.155 1 1 8.344±0.606 1 1 1.855 1 1   1 1   1 1 K + 1 1 0.137±0.004 1 1 1.079±0.004 1 1 0.990±0.006 1 1 1.983±0.075 1 1 7.853±0.136 1 1 0.488±0.033 1 1 3.574 1 1   1 1   1 1 p 1 1 0.192±0.005 1 1 1.028±0.006 1 1 0.853±0.008 1 1 2.006±0.056 1 1 9.466±0.155 1 1 0.175±0.012 1 1 1.165 1 1 1(d) 1 1 Peripheral 1 1 π - 1 1 0.105±0.004 1 1 1.085±0.005 1 1 0.918±0.005 1 1 1.985±0.075 1 1 10.032±0.155 1 1 8.344±0.606 1 1 1.635 1 1   1 1   1 1 K - 1 1 0.137±0.004 1 1 1.079±0.004 1 1 0.990±0.006 1 1 1.983±0.075 1 1 7.853±0.136 1 1 0.466±0.030 1 1 2.604 1 1   1 1   1 1 <math id="M74"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.192±0.005 1 1 1.028±0.006 1 1 0.853±0.008 1 1 2.106±0.059 1 1 9.766±0.158 1 1 0.140±0.012 1 1 0.715 1 1 2(a) 1 1 Central 1 1 π + 1 1 0.170±0.005 1 1 1.066±0.005 1 1 0.992±0.007 1 1 2.775±0.062 1 1 7.275±0.185 1 1 711.631±55.063 1 1 6.847 1 1   1 1   1 1 K + 1 1 0.264±0.006 1 1 1.030±0.005 1 1 0.993±0.002 1 1 3.575±0.108 1 1 7.135±0.203 1 1 62.036±5.422 1 1 0.548 1 1   1 1   1 1 p 1 1 0.409±0.006 1 1 1.002±0.001 1 1 0.993±0.002 1 1 4.975±0.112 1 1 8.725±0.206 1 1 8.968±0.417 1 1 2.813 1 1 2(b) 1 1 Central 1 1 π - 1 1 0.170±0.005 1 1 1.066±0.005 1 1 0.992±0.007 1 1 2.775±0.062 1 1 7.275±0.185 1 1 711.631±55.063 1 1 6.813 1 1   1 1   1 1 K - 1 1 0.264±0.006 1 1 1.030±0.005 1 1 0.993±0.002 1 1 3.575±0.108 1 1 7.135±0.203 1 1 62.036±5.422 1 1 0.654 1 1   1 1   1 1 <math id="M75"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.409±0.006 1 1 1.002±0.001 1 1 0.993±0.002 1 1 4.975±0.112 1 1 8.725±0.206 1 1 8.968±0.417 1 1 2.651 1 1 2(c) 1 1 Peripheral 1 1 π + 1 1 0.117±0.004 1 1 1.099±0.005 1 1 0.972±0.005 1 1 3.003±0.098 1 1 8.335±0.196 1 1 10.635±0.595 1 1 7.995 1 1   1 1   1 1 K + 1 1 0.173±0.005 1 1 1.069±0.005 1 1 0.905±0.006 1 1 2.375±0.071 1 1 7.575±0.192 1 1 0.725±0.043 1 1 1.674 1 1   1 1   1 1 p 1 1 0.263±0.005 1 1 1.035±0.005 1 1 0.911±0.006 1 1 1.875±0.065 1 1 7.265±0.146 1 1 0.139±0.009 1 1 2.285 1 1 2(d) 1 1 Peripheral 1 1 π - 1 1 0.117±0.004 1 1 1.099±0.005 1 1 0.972±0.005 1 1 3.003±0.098 1 1 8.335±0.196 1 1 10.635±0.595 1 1 7.904 1 1   1 1   1 1 K - 1 1 0.173±0.005 1 1 1.069±0.005 1 1 0.905±0.006 1 1 2.375±0.071 1 1 7.575±0.192 1 1 0.725±0.043 1 1 1.875 1 1   1 1   1 1 <math id="M76"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.263±0.005 1 1 1.035±0.005 1 1 0.911±0.006 1 1 1.875±0.065 1 1 7.265±0.146 1 1 0.144±0.009 1 1 2.255 Figure 2 Figure 2 is the same as Fig. 1 Fig. 1 , but it shows the spectra, (1/ N EV ) (2πp T ) -1 d 2 N /(d y d p T ), (a)-(c) π + (π + +π - ), K + (K + +K - ), and p ( <math id="M82"><mrow><mi>p</mi><mo>+</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow></math> ), as well as (b)-(d) π - (π + +π - ), K - (K + +K - ), and <math id="M83"><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow></math> ) produced in (a)-(b) central (0–5%) and (c)-(d) peripheral (80–90% and 60–80%) Pb-Pb collisions at <math id="M84"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV, where N EV is on the vertical axis and denotes the number of events, which is usually omitted. The closed (open) symbols represent the experimental data of the ALICE Collaboration measured in | y | < 0.5 [ 28 28 ] (in | η | < 0.8 for the high p T region and in | y | < 0.5 for the low p T region [ 29 29 ]). The data for π + + π - , K + + K - , and <math id="M85"><mrow><mi>p</mi><mo>+</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow></math> in (a)-(c) and (b)-(d) are the same. One can see that, in most cases, all of the considered methods describe approximately the p T spectra of identified particles produced in central and peripheral Pb-Pb collisions at <math id="M86"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV. Because the values of χ 2 /dof in most cases are greater than 2 and sometimes as large as 20.5, the fits in Figs. 1 Figs. 1 and 2 2 are only approximate and qualitative. The large values of χ 2 /dof in the present work are caused by two factors which are the very small errors in the data and large dispersion between the curve and data in some cases. It is hard to reduce the values of χ 2 /dof in our fits. In the above fits, we have an addition term of inverse power-law to account for hard process. This part contributes a small fraction to the p T spectra, though the contribution coverage is wide. In the fitting procedure, according to the changing tendency of data in a low p T range from 0 to 2 GeV/ c , the part for the soft process can be well constrained first of all, though the contribution of the soft process can even reach 3.5 GeV/ c . Then, the part for the hard process can be also constrained conveniently. In addition, in order to get a set of fitted parameters as accurately as possible, we use the least square method in the whole p T coverage. It seems that different fitted parameters can be obtained in different p T coverages. We should use a p T coverage as widely as possible, especially for the extraction of the parameters related to the inverse power-law because a limited p T coverage can not provide a good constraint of the inverse power-law and thus can easily drive the fitted parameters away from their physical meanings. In fact, for extractions of the effective temperature and transverse flow velocity which are the main topics of the present work, a not too wide p T coverage, such as 0–2 ∼ 3 GeV/ c , is enough due to the soft process contributing only in the low p T region and the changing tendency of data in 0–2 GeV/ c that takes part in a main role. From the above fits one can see that, as a two-component function, Eq. (9) with different soft components can approximately describe the data in a wide p T coverage. In addition, in our very recent work [ 37 37 ], we used method iii) to describe preliminarily the p T spectra up to nearly 20 GeV/ c . In another work [ 38 38 ], a two-Boltzmann distribution was used to describe the p T spectra up to nearly 14 GeV/ c . Generally, different sets of parameters are needed for different data. In particular, as it is pointed out in Ref. [ 39 39 ], more fitting parameters are needed in order to fit a wider p T range of particle spectra. In the present work, we fit the particle spectra in a wide p T range by introducing the inverse power-law to describe the high p T region. The price to pay is 3 more parameters are added. In the two-component function, the contributions of soft and hard components have little effect in constraining respective free parameters due to different contributive regions, though the contribution fraction of the two components is the main role. This results in the p T coverage having a small effect on T 0 and β T . In fact, if we change the boundary of the low p T region from 2 to 3 or 3.5 GeV/ c , the variations of parameters can be neglected due to the tendency of the curve being mainly determined by the data in 0–2 GeV/ c . Meanwhile, the data in 2–3.5 GeV/ c obey naturally the tendency of the curve due to also the contribution or revision of the hard component. In other words, because of the revision of the hard component, the values of T 0 and β T are not sensitive to the boundary of low p T region. Although different p T coverages obtained in different conditions can drive different fitted curves, these differences appear mainly in the high p T region and do not largely effect the extraction of T 0 and β T . In any case, we always use the last square method to extract the fitted parameters. In fact, the method used by us has the minimum randomness in the extractions of the fitted parameters. It should be noted that although the conventional BGBW and TBW models have only 2–3 parameters to describe the p T shape and usually fit several spectra simultaneously to reduce the correlation of the parameters, they seems to cover non-simultaneity of the kinetic freeze-outs of different particles. In the present work, although we use 3 more parameters to fit each spectrum individually, we observe an evidence of the mass dependent differential kinetic freeze-out scenario or multiple kinetic freeze-outs scenario [ 4 4 , 16 16 , 23 23 ]. The larger the temperature (mass) is, the earlier the particle produces. The average temperature (flow velocity and entropy index) of the kinetic freeze-outs for different particles is obtained by weighing different T 0 ( β T and q ), where the weight factor is the normalization constant of each p T spectrum. In the case of using the average temperature (flow velocity and entropy index) to fit the pion, kaon, and proton simultaneously to better constrain the parameters, larger values of χ 2 /dof are obtained. Based on the descriptions of the p T spectra, the first three methods can get T 0 and β T , though the values of parameters are possibly inharmonious due to different methods. In particular, the value of T 0 obtained by method i) in peripheral collisions is larger than that in central collisions, which is different from methods ii) and iii) which obtain an opposite result. According to the conventional treatment in Refs. [ 11 11 , 14 14 ], the values of β T obtained by methods i) and ii) in peripheral collisions are taken to be nearly zero, which are different from method iii) which obtains a value of about 0.6 c in both central and peripheral collisions. To obtain the values of T 0 , β T , and β by methods iv) a and iv) b , we analyze the values of T presented in Tables 4 Tables 4 and 5 5 , and calculate 〈 p T 〉, 〈 p 〉, and <math id="M92"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> based on the values of parameters listed in Tables 4 Tables 4 and 5 5 . In the calculations performed from p T to 〈 p 〉 and <math id="M93"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> by the Monte Carlo method, an isotropic assumption in the rest frame of emission source is used [ 22 22 - 24 24 22-24 ]. In particular, <math id="M94"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> is in fact the mean energy, <math id="M95"><mrow><mo>〈</mo><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt><mo>〉</mo></mrow></math> . The relations between T and m 0 , 〈 p T 〉, and <math id="M96"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , as well as 〈 p 〉 and <math id="M97"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> are shown in Figs. 3 Figs. 3 , 4 4 , and 5 5 , respectively, where panels (a) and (b) correspond to methods iv) a and iv) b which use the Boltzmann and Tsallis distributions respectively. Different symbols represent central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at <math id="M98"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV and central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at <math id="M99"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV respectively, where the centralities 0–5% and 0–12%, 80–92% and 60–80%, as well as 80–90% and 60–80% can be combined to 0–12%, 60–92%, and 60–90%, respectively. The symbols in Fig. 3 Fig. 3 represent values of T listed in Tables 4 Tables 4 and 5 5 for a different m 0 . The symbols in Figs. 4 Figs. 4 and 5 5 represent values of 〈 p T 〉 and 〈 p 〉 for different <math id="M100"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> respectively, which are calculated due to the parameters listed in Tables 4 Tables 4 and 5 5 and the isotropic assumption in the rest frame of the emission source. The solid and dashed lines in the three figures are the results fitted by the least square method for the positively and negatively charged particles respectively. The values of intercepts, slopes, and χ 2 /dof are listed in Tables 6 Tables 6 and 7 7 which correspond to methods iv) a and iv) b respectively. One can see that, in most cases, the mentioned relations are described by a linear function. In particular, the intercept in Fig. 3 Fig. 3 is regarded as T 0 , and the slopes in Figs. 4 Figs. 4 and 5 5 are regarded as β T and β respectively. The values of T , T 0 , β T , β , and <math id="M101"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> are approximately independent of isospin. 1 1 Figure 1 1 Relation 1 1 Type and main particles 1 1 Centrality 1 1 Intercept 1 1 Slope 1 1 χ 2 /dof 1 1 3(a) 1 1 T - m 0 1 1 Au-Au positive 1 1 Central 1 1 0.147±0.007 1 1 0.168±0.012 1 1 2.625 1 1   1 1   1 1 negative 1 1 Central 1 1 0.149±0.010 1 1 0.160±0.016 1 1 4.618 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.125±0.017 1 1 0.096±0.028 1 1 14.910 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.125±0.017 1 1 0.096±0.028 1 1 14.910 1 1   1 1   1 1 Pb-Pb positive 1 1 Central 1 1 0.179±0.003 1 1 0.248±0.005 1 1 0.424 1 1   1 1   1 1 negative 1 1 Central 1 1 0.179±0.003 1 1 0.248±0.005 1 1 0.424 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.130±0.005 1 1 0.174±0.008 1 1 1.142 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.128±0.003 1 1 0.180±0.005 1 1 0.394 1 1 4(a) 1 1 <math id="M102"><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 Au-Au positive 1 1 Central 1 1 0.147±0.018 1 1 0.436±0.013 1 1 0.864 1 1   1 1   1 1 negative 1 1 Central 1 1 0.152±0.023 1 1 0.430±0.017 1 1 1.312 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.163±0.041 1 1 0.362±0.036 1 1 4.734 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.163±0.041 1 1 0.362±0.036 1 1 4.734 1 1   1 1   1 1 Pb-Pb positive 1 1 Central 1 1 0.133±0.004 1 1 0.492±0.002 1 1 0.024 1 1   1 1   1 1 negative 1 1 Central 1 1 0.133±0.004 1 1 0.492±0.002 1 1 0.024 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.130±0.013 1 1 0.438±0.010 1 1 0.499 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.125±0.010 1 1 0.443±0.007 1 1 0.285 1 1 5(a) 1 1 <math id="M103"><mrow><mo>〈</mo><mi>p</mi><mo>〉</mo><mo>−</mo><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> 1 1 Au-Au positive 1 1 Central 1 1 0.230±0.028 1 1 0.683±0.021 1 1 0.865 1 1   1 1   1 1 negative 1 1 Central 1 1 0.239±0.035 1 1 0.673±0.026 1 1 1.313 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.255±0.064 1 1 0.568±0.056 1 1 4.746 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.255±0.064 1 1 0.568±0.056 1 1 4.746 1 1   1 1   1 1 Pb-Pb positive 1 1 Central 1 1 0.209±0.006 1 1 0.771±0.003 1 1 0.024 1 1   1 1   1 1 negative 1 1 Central 1 1 0.209±0.006 1 1 0.771±0.003 1 1 0.024 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.203±0.020 1 1 0.686±0.015 1 1 0.496 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.196±0.015 1 1 0.694±0.011 1 1 0.283 1 1 Figure 1 1 Relation 1 1 Type and main particles 1 1 Centrality 1 1 Intercept 1 1 Slope 1 1 χ 2 /dof 1 1 3(b) 1 1 T - m 0 1 1 Au-Au positive 1 1 Central 1 1 0.101±0.009 1 1 0.181±0.014 1 1 3.059 1 1   1 1   1 1 negative 1 1 Central 1 1 0.102±0.008 1 1 0.179±0.013 1 1 2.533 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.087±0.006 1 1 0.110±0.009 1 1 1.708 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.087±0.006 1 1 0.110±0.009 1 1 1.708 1 1   1 1   1 1 Pb-Pb positive 1 1 Central 1 1 0.124±0.011 1 1 0.300±0.017 1 1 2.877 1 1   1 1   1 1 negative 1 1 Central 1 1 0.124±0.011 1 1 0.300±0.017 1 1 2.877 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.088±0.008 1 1 0.184±0.013 1 1 2.258 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.088±0.008 1 1 0.184±0.013 1 1 2.258 1 1 4(b) 1 1 <math id="M104"><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 Au-Au positive 1 1 Central 1 1 0.154±0.013 1 1 0.427±0.010 1 1 0.270 1 1   1 1   1 1 negative 1 1 Central 1 1 0.160±0.018 1 1 0.420±0.013 1 1 0.495 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.174±0.049 1 1 0.373±0.040 1 1 4.116 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.174±0.049 1 1 0.373±0.040 1 1 4.116 1 1   1 1   1 1 Pb-Pb positive 1 1 Central 1 1 0.131±0.001 1 1 0.493±0.001 1 1 0.001 1 1   1 1   1 1 negative 1 1 Central 1 1 0.131±0.001 1 1 0.493±0.001 1 1 0.001 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.140±0.011 1 1 0.445±0.008 1 1 0.148 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.140±0.011 1 1 0.445±0.008 1 1 0.148 1 1 5(b) 1 1 <math id="M105"><mrow><mo>〈</mo><mi>p</mi><mo>〉</mo><mo>−</mo><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> 1 1 Au-Au positive 1 1 Central 1 1 0.240±0.021 1 1 0.670±0.015 1 1 0.269 1 1   1 1   1 1 negative 1 1 Central 1 1 0.251±0.028 1 1 0.659±0.021 1 1 0.494 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.272±0.077 1 1 0.584±0.063 1 1 4.111 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.272±0.077 1 1 0.584±0.063 1 1 4.111 1 1   1 1   1 1 Pb-Pb positive 1 1 Central 1 1 0.205±0.002 1 1 0.772±0.001 1 1 0.001 1 1   1 1   1 1 negative 1 1 Central 1 1 0.205±0.002 1 1 0.772±0.001 1 1 0.001 1 1   1 1   1 1 positive 1 1 Peripheral 1 1 0.220±0.017 1 1 0.697±0.012 1 1 0.148 1 1   1 1   1 1 negative 1 1 Peripheral 1 1 0.220±0.017 1 1 0.697±0.012 1 1 0.148 To compare values of key parameters obtained by different methods for different centralities (both central and peripheral collisions), Figs. 6 Figs. 6 and 7 7 show T 0 and β T respectively, where panels (a) and (b) correspond to the results for central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at <math id="M110"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV and central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at <math id="M111"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV respectively. The closed and open symbols represent positively and negatively charged particles respectively, which are quoted from Tables 1 Tables 1 , 2 2 , 3 3 , 6 6 , and 7 7 which correspond to methods i), ii), iii), iv) a , and iv) b , respectively. In particular, the values of T 0 and β T in the first three methods are obtained by weighing different particles. One can see that, by using method i), the value of T 0 in central collisions is smaller than that in peripheral collisions, and other methods present a larger T 0 in central collisions. Methods i) and ii) show a nearly zero β T in peripheral collisions according to Refs. [ 11 11 , 14 14 ], while other methods show a considerable β T in both central and peripheral collisions. To explain the inconsistent results in T 0 and β T for different methods, we re-examine the first two methods. It should be noticed that the same flow profile function, β ( r ) = β S ( r/R ) n 0 , and the same transverse flow velocity, β T = 2 β S /( n 0 +2), are used in the first two methods, though n 0 =2 is used in method i) [ 11 11 ] and n 0 =1 is used in method ii) [ 14 14 ] with the conventional treatment. As an insensitive quantity, although the radial size R of the thermal source in central collisions can be approximately regarded as the radius of a collision nucleus and in peripheral collisions R is not zero due to a few participant nucleons taking part in the interactions in which we can take approximate R to be 2.5 fm, both methods i) and ii) use a nearly zero β T in peripheral collisions [ 11 11 , 14 14 ]. If we consider a non-zero β T in peripheral collisions for methods i) and ii), the situation will be changed. By using a non-zero β T in peripheral collisions for methods i) and ii), we re-analyze the data presented in Figs. 1 Figs. 1 and 2 2 . At the same time, to see the influences of different n 0 in the self-similar flow profile, we refit the mentioned p T spectra by the first two methods with n 0 =1 and 2 synchronously. The results re-analyzed by us are shown in Figs. 8 Figs. 8 and 9 9 which correspond to 200 GeV Au-Au and 2.76 TeV Pb-Pb collisions respectively. The data points are the same as Figs. 1 Figs. 1 and 2 2 [ 25 25 - 29 29 25-29 ]. The dotted, solid, dashed, and dotted-dashed curves correspond to the results of method i) with n 0 =1 and 2, and of method ii) with n 0 =1 and 2, respectively, where the results of method i) with n 0 =2 and of method ii) with n 0 =1 in central collisions are the same as Figs. 1 Figs. 1 and 2 2 . The values of related parameters and χ 2 /dof are listed in Tables 8 Tables 8 and 9 9 , where the parameters for method i) with n 0 =2 and for method ii) with n 0 =1 in central collisions repeat those in Tables 1 Tables 1 and 2 2 , which are not listed again. One can see that, after the re-examination, the values of T 0 in central collisions are larger than those in peripheral collisions. The values of β T in peripheral collisions are no longer zero. These new results are consistent with other methods. 1 1 Fig. 1 1 Cent. 1 1 Main Part. 1 1 T 0 (GeV) 1 1 β T ( c ) 1 1 k 1 1 p 0 (GeV/ c ) 1 1 n 1 1 N 0 1 1 χ 2 /dof 1 1 8(a) 1 1 Central 1 1 π + 1 1 0.138±0.005 1 1 0.452±0.008 1 1 0.964±0.006 1 1 2.375±0.069 1 1 10.365±0.188 1 1 633.869±62.976 1 1 3.369 1 1 Au-Au 1 1   1 1 K + 1 1 0.169±0.005 1 1 0.412±0.008 1 1 0.901±0.006 1 1 1.998±0.058 1 1 9.675±0.185 1 1 54.966±3.838 1 1 5.502 1 1 n 0 =1 1 1   1 1 p 1 1 0.198±0.005 1 1 0.398±0.008 1 1 0.995±0.002 1 1 2.485±0.072 1 1 8.075±0.171 1 1 8.457±0.646 1 1 5.274 1 1 8(b) 1 1 Central 1 1 π - 1 1 0.138±0.005 1 1 0.452±0.008 1 1 0.964±0.006 1 1 2.375±0.069 1 1 10.365±0.188 1 1 633.869±62.976 1 1 3.277 1 1   1 1   1 1 K - 1 1 0.169±0.005 1 1 0.412±0.008 1 1 0.901±0.006 1 1 2.098±0.060 1 1 9.835±0.188 1 1 54.759±3.823 1 1 6.405 1 1   1 1   1 1 <math id="M114"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.198±0.005 1 1 0.397±0.008 1 1 0.994±0.002 1 1 2.185±0.070 1 1 7.975±0.168 1 1 7.096±0.649 1 1 12.058 1 1 8(c) 1 1 Peripheral 1 1 π + 1 1 0.115±0.005 1 1 0.415±0.008 1 1 0.901±0.008 1 1 2.512±0.079 1 1 11.123±0.173 1 1 11.713±0.591 1 1 4.455 1 1   1 1   1 1 K + 1 1 0.145±0.005 1 1 0.415±0.008 1 1 0.888±0.008 1 1 3.923±0.082 1 1 12.923±0.178 1 1 0.482±0.077 1 1 6.711 1 1   1 1   1 1 p 1 1 0.157±0.006 1 1 0.353±0.008 1 1 0.947±0.008 1 1 3.316±0.069 1 1 11.016±0.169 1 1 0.142±0.015 1 1 1.444 1 1 8(d) 1 1 Peripheral 1 1 π - 1 1 0.115±0.005 1 1 0.415±0.008 1 1 0.901±0.008 1 1 2.512±0.079 1 1 11.123±0.173 1 1 11.713±0.591 1 1 3.800 1 1   1 1   1 1 K - 1 1 0.145±0.005 1 1 0.415±0.008 1 1 0.888±0.008 1 1 3.923±0.082 1 1 12.923±0.178 1 1 0.482±0.077 1 1 5.907 1 1   1 1   1 1 <math id="M115"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.157±0.006 1 1 0.353±0.008 1 1 0.945±0.008 1 1 3.316±0.069 1 1 11.528±0.169 1 1 0.112±0.011 1 1 0.904 1 1 8(c) 1 1 Peripheral 1 1 π + 1 1 0.103±0.005 1 1 0.395±0.008 1 1 0.896±0.008 1 1 2.012±0.063 1 1 10.203±0.185 1 1 14.240±1.308 1 1 2.956 1 1 Au-Au 1 1   1 1 K + 1 1 0.117±0.006 1 1 0.383±0.008 1 1 0.901±0.008 1 1 3.983±0.071 1 1 12.993±0.195 1 1 0.636±0.033 1 1 4.221 1 1 n 0 =2 1 1   1 1 p 1 1 0.118±0.006 1 1 0.355±0.008 1 1 0.905±0.008 1 1 3.268±0.066 1 1 11.506±0.186 1 1 0.170±0.012 1 1 1.093 1 1 8(d) 1 1 Peripheral 1 1 π - 1 1 0.103±0.005 1 1 0.395±0.008 1 1 0.896±0.008 1 1 2.012±0.063 1 1 10.203±0.185 1 1 14.240±1.308 1 1 2.652 1 1   1 1   1 1 K - 1 1 0.117±0.006 1 1 0.383±0.008 1 1 0.901±0.008 1 1 3.983±0.071 1 1 12.993±0.195 1 1 0.636±0.033 1 1 3.879 1 1   1 1   1 1 <math id="M116"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.118±0.006 1 1 0.355±0.008 1 1 0.905±0.008 1 1 3.268±0.066 1 1 11.926±0.186 1 1 0.128±0.012 1 1 0.589 1 1 9(a) 1 1 Central 1 1 π + 1 1 0.149±0.005 1 1 0.473±0.008 1 1 0.922±0.008 1 1 1.535±0.056 1 1 7.276±0.104 1 1 1465.409±127.197 1 1 3.815 1 1 Pb-Pb 1 1   1 1 K + 1 1 0.235±0.005 1 1 0.399±0.008 1 1 0.938±0.008 1 1 1.295±0.055 1 1 6.114±0.101 1 1 77.086±7.666 1 1 1.463 1 1 n 0 =1 1 1   1 1 p 1 1 0.338±0.005 1 1 0.332±0.006 1 1 0.991±0.002 1 1 2.285±0.082 1 1 6.485±0.108 1 1 10.152±0.330 1 1 11.411 1 1 9(b) 1 1 Central 1 1 π - 1 1 0.149±0.005 1 1 0.473±0.008 1 1 0.922±0.008 1 1 1.535±0.056 1 1 7.276±0.104 1 1 1465.409±127.197 1 1 3.751 1 1   1 1   1 1 K - 1 1 0.235±0.005 1 1 0.399±0.008 1 1 0.938±0.008 1 1 1.295±0.055 1 1 6.114±0.101 1 1 77.157±7.674 1 1 1.229 1 1   1 1   1 1 <math id="M117"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.338±0.005 1 1 0.332±0.006 1 1 0.991±0.002 1 1 2.285±0.082 1 1 6.485±0.108 1 1 10.152±0.330 1 1 10.234 1 1 9(c) 1 1 Peripheral 1 1 π + 1 1 0.127±0.005 1 1 0.473±0.008 1 1 0.934±0.008 1 1 2.793±0.078 1 1 8.765±0.138 1 1 14.233±0.756 1 1 8.290 1 1   1 1   1 1 K + 1 1 0.169±0.004 1 1 0.453±0.008 1 1 0.902±0.008 1 1 2.665±0.074 1 1 7.995±0.129 1 1 0.723±0.050 1 1 2.448 1 1   1 1   1 1 p 1 1 0.180±0.005 1 1 0.436±0.008 1 1 0.918±0.008 1 1 2.995±0.092 1 1 8.599±0.132 1 1 0.167±0.014 1 1 3.944 1 1 9(d) 1 1 Peripheral 1 1 π - 1 1 0.127±0.005 1 1 0.473±0.008 1 1 0.934±0.008 1 1 2.793±0.078 1 1 8.765±0.138 1 1 14.233±0.756 1 1 8.285 1 1   1 1   1 1 K - 1 1 0.169±0.004 1 1 0.453±0.008 1 1 0.902±0.008 1 1 2.665±0.074 1 1 7.995±0.129 1 1 0.723±0.050 1 1 2.686 1 1   1 1   1 1 <math id="M118"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.180±0.005 1 1 0.436±0.008 1 1 0.918±0.008 1 1 2.995±0.092 1 1 8.599±0.132 1 1 0.167±0.014 1 1 4.196 1 1 9(c) 1 1 Peripheral 1 1 π + 1 1 0.116±0.004 1 1 0.410±0.008 1 1 0.941±0.007 1 1 2.393±0.058 1 1 8.185±0.153 1 1 17.976±0.731 1 1 4.533 1 1 Pb-Pb 1 1   1 1 K + 1 1 0.184±0.005 1 1 0.367±0.008 1 1 0.908±0.007 1 1 2.375±0.056 1 1 7.585±0.145 1 1 0.702±0.044 1 1 1.120 1 1 n 0 =2 1 1   1 1 p 1 1 0.204±0.005 1 1 0.343±0.008 1 1 0.919±0.007 1 1 2.178±0.055 1 1 7.515±0.145 1 1 0.172±0.015 1 1 1.791 1 1 9(d) 1 1 Peripheral 1 1 π - 1 1 0.116±0.004 1 1 0.410±0.008 1 1 0.941±0.007 1 1 2.393±0.058 1 1 8.185±0.153 1 1 17.976±0.731 1 1 4.601 1 1   1 1   1 1 K - 1 1 0.184±0.005 1 1 0.367±0.008 1 1 0.908±0.007 1 1 2.375±0.056 1 1 7.585±0.145 1 1 0.702±0.044 1 1 1.232 1 1   1 1   1 1 <math id="M119"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.204±0.005 1 1 0.343±0.008 1 1 0.919±0.007 1 1 2.178±0.055 1 1 7.515±0.145 1 1 0.172±0.015 1 1 1.963 1 1 Fig. 1 1 Cent. 1 1 Main Part. 1 1 T 0 (GeV) 1 1 q 1 1 β T ( c ) 1 1 k 1 1 p 0 (GeV/ c ) 1 1 n 1 1 N 0 1 1 χ 2 /dof 1 1 8(c) 1 1 Peripheral 1 1 π + 1 1 0.079±0.004 1 1 1.069±0.006 1 1 0.405±0.009 1 1 0.924±0.006 1 1 2.192±0.083 1 1 10.379±0.189 1 1 9.197±0.912 1 1 1.715 1 1 Au-Au 1 1   1 1 K + 1 1 0.089±0.005 1 1 1.063±0.005 1 1 0.389±0.009 1 1 0.921±0.006 1 1 3.602±0.096 1 1 12.282±0.165 1 1 0.491±0.052 1 1 4.499 1 1 n 0 =1 1 1   1 1 p 1 1 0.095±0.005 1 1 1.028±0.005 1 1 0.389±0.009 1 1 0.902±0.007 1 1 3.810±0.102 1 1 12.568±0.171 1 1 0.134±0.010 1 1 1.457 1 1 8(d) 1 1 Peripheral 1 1 π - 1 1 0.079±0.004 1 1 1.069±0.006 1 1 0.405±0.009 1 1 0.924±0.006 1 1 2.192±0.083 1 1 10.379±0.189 1 1 9.197±0.912 1 1 1.445 1 1   1 1   1 1 K - 1 1 0.089±0.005 1 1 1.061±0.005 1 1 0.389±0.009 1 1 0.921±0.006 1 1 3.602±0.096 1 1 12.282±0.165 1 1 0.486±0.052 1 1 3.127 1 1   1 1   1 1 <math id="M120"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.095±0.005 1 1 1.028±0.005 1 1 0.389±0.009 1 1 0.908±0.007 1 1 3.810±0.102 1 1 12.868±0.171 1 1 0.100±0.010 1 1 0.670 1 1 8(a) 1 1 Central 1 1 π + 1 1 0.091±0.003 1 1 1.010±0.005 1 1 0.401±0.008 1 1 0.985±0.003 1 1 3.591±0.091 1 1 12.035±0.173 1 1 683.617±48.090 1 1 3.630 1 1 Au-Au 1 1   1 1 K + 1 1 0.103±0.005 1 1 1.008±0.004 1 1 0.395±0.007 1 1 0.961±0.004 1 1 2.675±0.103 1 1 10.327±0.089 1 1 51.119±5.034 1 1 5.703 1 1 n 0 =2 1 1   1 1 p 1 1 0.118±0.005 1 1 1.009±0.004 1 1 0.374±0.005 1 1 0.997±0.002 1 1 3.385±0.168 1 1 8.895±0.108 1 1 9.706±0.421 1 1 6.866 1 1 8(b) 1 1 Central 1 1 π - 1 1 0.091±0.003 1 1 1.010±0.005 1 1 0.401±0.008 1 1 0.985±0.003 1 1 3.591±0.091 1 1 12.035±0.173 1 1 683.617±48.090 1 1 3.362 1 1   1 1   1 1 K - 1 1 0.103±0.005 1 1 1.008±0.004 1 1 0.395±0.007 1 1 0.961±0.004 1 1 2.675±0.103 1 1 10.327±0.159 1 1 49.059±5.034 1 1 6.731 1 1   1 1   1 1 <math id="M121"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.118±0.005 1 1 1.009±0.004 1 1 0.374±0.005 1 1 0.997±0.002 1 1 3.385±0.168 1 1 9.095±0.112 1 1 7.862±0.422 1 1 15.669 1 1 8(c) 1 1 Peripheral 1 1 π + 1 1 0.073±0.004 1 1 1.025±0.004 1 1 0.398±0.008 1 1 0.943±0.004 1 1 2.653±0.091 1 1 11.093±0.169 1 1 10.627±0.888 1 1 3.602 1 1   1 1   1 1 K + 1 1 0.082±0.005 1 1 1.033±0.005 1 1 0.380±0.008 1 1 0.891±0.005 1 1 3.683±0.092 1 1 12.553±0.170 1 1 0.470±0.005 1 1 4.498 1 1   1 1   1 1 p 1 1 0.085±0.005 1 1 1.009±0.005 1 1 0.359±0.008 1 1 0.910±0.005 1 1 3.950±0.093 1 1 12.756±0.181 1 1 0.150±0.013 1 1 1.306 1 1 8(d) 1 1 Peripheral 1 1 π - 1 1 0.073±0.004 1 1 1.025±0.004 1 1 0.398±0.008 1 1 0.943±0.004 1 1 2.653±0.091 1 1 11.093±0.169 1 1 10.627±0.888 1 1 3.239 1 1   1 1   1 1 K - 1 1 0.082±0.005 1 1 1.033±0.005 1 1 0.380±0.008 1 1 0.891±0.005 1 1 3.683±0.092 1 1 12.553±0.170 1 1 0.470±0.052 1 1 3.570 1 1   1 1   1 1 <math id="M122"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.085±0.005 1 1 1.009±0.005 1 1 0.359±0.008 1 1 0.910±0.005 1 1 3.950±0.093 1 1 13.018±0.181 1 1 0.117±0.011 1 1 0.647 1 1 9(c) 1 1 Peripheral 1 1 π + 1 1 0.089±0.004 1 1 1.041±0.005 1 1 0.446±0.010 1 1 0.929±0.006 1 1 2.403±0.075 1 1 8.398±0.169 1 1 14.318±0.567 1 1 12.971 1 1 Pb-Pb 1 1   1 1 K + 1 1 0.099±0.005 1 1 1.065±0.005 1 1 0.446±0.010 1 1 0.926±0.006 1 1 2.375±0.071 1 1 7.468±0.153 1 1 0.650±0.062 1 1 1.544 1 1 n 0 =1 1 1   1 1 p 1 1 0.110±0.005 1 1 1.030±0.005 1 1 0.446±0.010 1 1 0.894±0.007 1 1 2.415±0.077 1 1 8.005±0.161 1 1 0.157±0.014 1 1 2.881 1 1 9(d) 1 1 Peripheral 1 1 π - 1 1 0.089±0.004 1 1 1.041±0.005 1 1 0.446±0.010 1 1 0.929±0.006 1 1 2.403±0.075 1 1 8.398±0.169 1 1 14.318±0.567 1 1 12.947 1 1   1 1   1 1 K - 1 1 0.099±0.005 1 1 1.065±0.005 1 1 0.446±0.010 1 1 0.926±0.006 1 1 2.375±0.071 1 1 7.468±0.153 1 1 0.650±0.062 1 1 1.724 1 1   1 1   1 1 <math id="M123"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.110±0.005 1 1 1.030±0.005 1 1 0.446±0.010 1 1 0.894±0.007 1 1 2.415±0.077 1 1 8.005±0.161 1 1 0.157±0.014 1 1 3.065 1 1 9(a) 1 1 Central 1 1 π + 1 1 0.099±0.005 1 1 1.006±0.004 1 1 0.435±0.006 1 1 0.989±0.003 1 1 2.775±0.085 1 1 7.515±0.158 1 1 1099.140±107.121 1 1 2.897 1 1 Pb-Pb 1 1   1 1 K + 1 1 0.113±0.005 1 1 1.002±0.001 1 1 0.435±0.006 1 1 0.984±0.003 1 1 3.575±0.101 1 1 7.735±0.115 1 1 73.563±7.358 1 1 3.623 1 1 n 0 =2 1 1   1 1 p 1 1 0.155±0.005 1 1 1.002±0.001 1 1 0.419±0.004 1 1 0.996±0.002 1 1 4.975±0.109 1 1 8.225±0.128 1 1 10.566±0.284 1 1 15.778 1 1 9(b) 1 1 Central 1 1 π - 1 1 0.099±0.005 1 1 1.006±0.004 1 1 0.435±0.006 1 1 0.989±0.003 1 1 2.775±0.085 1 1 7.515±0.158 1 1 1099.140±107.121 1 1 2.955 1 1   1 1   1 1 K - 1 1 0.113±0.005 1 1 1.002±0.001 1 1 0.435±0.006 1 1 0.984±0.003 1 1 3.575±0.101 1 1 7.735±0.115 1 1 73.563±7.358 1 1 3.282 1 1   1 1   1 1 <math id="M124"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.155±0.005 1 1 1.002±0.001 1 1 0.419±0.004 1 1 0.996±0.002 1 1 4.975±0.109 1 1 8.225±0.128 1 1 9.983±0.278 1 1 14.519 1 1 9(c) 1 1 Peripheral 1 1 π + 1 1 0.079±0.004 1 1 1.045±0.005 1 1 0.405±0.008 1 1 0.976±0.004 1 1 3.003±0.095 1 1 8.335±0.129 1 1 14.692±0.760 1 1 7.361 1 1   1 1   1 1 K + 1 1 0.086±0.005 1 1 1.053±0.005 1 1 0.399±0.008 1 1 0.928±0.004 1 1 2.375±0.089 1 1 7.475±0.121 1 1 0.760±0.084 1 1 0.975 1 1   1 1   1 1 p 1 1 0.102±0.005 1 1 1.025±0.005 1 1 0.385±0.007 1 1 0.940±0.006 1 1 2.675±0.092 1 1 7.965±0.126 1 1 0.177±0.014 1 1 2.380 1 1 9(d) 1 1 Peripheral 1 1 π - 1 1 0.079±0.004 1 1 1.045±0.005 1 1 0.405±0.008 1 1 0.976±0.004 1 1 3.003±0.095 1 1 8.335±0.129 1 1 14.692±0.760 1 1 7.488 1 1   1 1   1 1 K - 1 1 0.086±0.005 1 1 1.053±0.005 1 1 0.399±0.008 1 1 0.928±0.004 1 1 2.375±0.089 1 1 7.475±0.121 1 1 0.760±0.084 1 1 1.069 1 1   1 1   1 1 <math id="M125"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.102±0.005 1 1 1.025±0.005 1 1 0.385±0.007 1 1 0.940±0.006 1 1 2.675±0.092 1 1 7.965±0.126 1 1 0.171±0.014 1 1 2.410 To give new comparisons for T 0 and β T , the new results obtained by the first two methods are shown in Figs. 10 Figs. 10 and 11 11 respectively, where the results corresponding to method i) for central collisions with n 0 =2 and to method ii) for central collisions with n 0 =1 are the same as those in Figs. 6 Figs. 6 and 7 7 . Combing Figs. 6 Figs. 6 , 7 7 , 10 10 , and 11 11 , one can see that the four methods show approximately the consistent results. These comparisons enlighten us to use the first two methods in peripheral collisions by a non-zero β T . After the re-examination for β T in peripheral collisions, we obtain a relatively larger T 0 in central collisions for the four methods. In particular, the parameter T 0 at the LHC is slightly larger than or nearly equal to that at the RHIC, not only for central collisions but also for peripheral collisions. Except for method iii), the methods show a slightly larger or nearly invariant β T in central collisions when compared with peripheral collisions, and when LHC comparing data from LHC with the RHIC, while method iii) shows nearly the same β T in different centralities and at different energies. We would like to point out that, in the re-examination for β T in methods i) and ii), we have assumed both β T in central and peripheral collisions to be non-zero. In most cases [ 11 11 , 14 14 ], both the conventional BGBW and TBW models used non-zero β T in central collisions and zero (or almost zero) β T in peripheral collisions. In the case of using a non-zero or zero (or almost zero) β T in peripheral collisions, we can obtain a relatively smaller or larger T 0 compared with central collisions. Indeed, the selection of β T in peripheral collisions is an important issue in both the BGBW and TBW models. In fact, β T is a sensitive quantity which can affect T 0 . The larger β T that is selected, the smaller T 0 that is needed. The main correlation is between β T and T 0 , and the effect of n 0 is very small. In Figs. 1 Figs. 1 and 2 2 , we have used a zero β T for peripheral collisions and obtained a harmonious result on the relative size of T 0 with Ref. [ 28 28 ] in which β T (0.35 c ) for peripheral collisions is nearly a half of that (0.65 c ) for central collisions, and n 0 is also different from ours. While in Figs. 8 Figs. 8 and 9 9 , we have used a non-zero and slightly smaller β T for peripheral collisions and obtained a different result from Ref. [ 28 28 ]. In order to make the conclusion more convincing, we can only fit the low p T region of the particle spectra using the four methods with the same p T cut to decrease the number of free fitting parameters. When the p T cut increases from 2 to 3.5 GeV/ c , T 0 (or T ) increases or both T 0 (or T ) and β T increase slightly. The relative size of T 0 ( β T ) obtained above for central and peripheral collisions is unchanged. In particular, β T is also a sensitive quantity. For peripheral collisions, a zero or non-zero β T in the first two methods can give different results. In our opinion, in central and peripheral collisions, it depends on β T if we want to determine which T 0 is larger. We are inclined to use a non-zero β T for peripheral collisions due to the small system which is similar to peripheral collisions in number of participant nucleons also showing collective expansion [ 40 40 ]. Compared with peripheral collisions, the larger T 0 in central collisions renders more deposition of collision energy and higher excitation of the interacting system due to more participating nucleons taking part in the violent collisions. Compared with the top RHIC energy, the larger T 0 at the LHC energy also renders more deposition of collision energy and higher excitation of interacting system due to higher <math id="M128"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> at the LHC. At the same time, from the top RHIC to the LHC energies, a nearly invariant T 0 reflects the limiting deposition of collision energy. Compared with peripheral collisions, the slightly larger or nearly the same β T in central collisions renders similar expansion in both the centralities. At the same time, at the top RHIC and LHC energies, the two systems also show similar expansion due to similar β T . It should be noted that, although Eq. (2) [ 14 14 ] does not implement the azimuthal integral over the freeze-out surface which gives rise to the modified Bessel functions in Eq. (1), it does not affect the extractions of kinetic freeze-out parameters due to the application of numerical integral. Although Eq. (3) [ 15 15 , 16 16 ] assumes a single, infinitesimally thin shell of fixed flow velocity and also does not perform the integral over the freeze-out surface, it can extract the mean trend of kinetic freeze-out parameters. As for the alternative method [ 12 12 , 17 17 - 20 20 17-20 , 22 22 - 24 24 22-24 ], it assumes non-relativistic flow velocities in the expressions used to extract the freeze-out parameters, which is the case that β T is indeed not too large at the top RHIC and LHC energies. 4 Conclusion 1 We summarize here our main observations and conclusions. (a) The p T spectra of π ± , K ± , <math id="M129"><mrow><msubsup><mi>K</mi><mtext>S</mtext><mn>0</mn></msubsup></mrow></math> , p , and <math id="M130"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> produced in central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at <math id="M131"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV and in central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at <math id="M132"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV have been analyzed by a few different superpositions in which the distributions related to the extractions of T 0 and β T are used for the soft component and the inverse power law is used for the hard component. We have used five distributions for the soft component, i) the Blast-Wave model with Boltzmann-Gibbs statistics, ii) the Blast-Wave model with Tsallis statistics, iii) the Tsallis distribution with flow effect, iv) a the Boltzmann distribution, and iv) b the Tsallis distribution. The first three distributions are in fact three methods for the extractions of T 0 and β T . The last two distributions are used in the fourth method, i.e. the alternative method. (b) The experimental data measured by the PHENIX, STAR, and ALICE Collaborations are fitted by the model results. Our calculations show that the parameter T 0 obtained by method i) with the conventional treatment for central collisions is smaller than that for peripheral collisions, which is inconsistent with the results obtained by other model methods. In the conventional treatment, the parameter β T in peripheral collisions is taken to be nearly zero, which results in a larger T 0 than normal. By using the conventional treatment, both methods i) and ii) show a nearly zero β T value in the peripheral collisions according to Refs. [ 11 11 , 14 14 ], while other methods show a considerable β T in both central and peripheral collisions. (c) In central and peripheral collisions, we have to select a suitable β T so that we can determine which T 0 is larger. We are inclined to use a non-zero β T for peripheral collisions due to the small system also showing collective expansion. We have given a re-examination for β T in peripheral collisions in methods i) and ii) in which β T is taken to be (0.40±0.07) c . By using a non-zero β T , the first two methods show approximately consistent results with other methods, not only for T 0 but also for β T , though method iii) gives a larger β T . We have uniformly obtained a larger T 0 in central collisions by the four methods. In particular, the parameter T 0 at the LHC is larger than or equal to that at the RHIC. Except for method iii), the methods show a slightly larger or nearly invariant β T in central collisions compared to peripheral collisions, and at the LHC compared with the RHIC. (d) The new results obtained by the widely used Blast-Wave model with Boltzmann-Gibbs or Tsallis statistics are in agreement with those obtained by the newly used alternative method which uses the Boltzmann or Tsallis distribution. This consistency confirms the validity of the alternative method. The result that the central collisions have a larger T 0 renders more deposition of collision energy and higher excitation of the interacting system due to more participating nucleons taking part in the violent collisions. From the RHIC to LHC, the slightly increased or nearly invariant T 0 renders the limiting or maximum deposition of collisions energy. 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Tech . 28 , 99 ( 2017 ). doi: 10.1007/s41365-017-0245-4 http://doi.org/10.1007/s41365-017-0245-4 Collective flow and hydrodynamics in large and small systems at the LHC 10.1007/s41365-018-0425-x This work was supported by the National Natural Science Foundation of China (Nos. 11575103 and 11747319), the Shanxi Provincial Natural Science Foundation (No. 201701D121005), and the Fund for Shanxi "1331 Project" Key Subjects Construction. 2018-01-15 2018-02-10 2018-02-18 2018-04-26 2018-06-01 2021-04-11 © 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. 2018 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/s41365-018-0425-x http://dx.doi.org/10.1007/s41365-018-0425-x . 2018 Nuclear Science and Techniques 06 29