Examining the model dependence of the determination of kinetic freeze-out temperature and transverse flow velocity in small collision system

Hai-Ling Lao; Fu-Hu Liu; Bao-Chun Li; Mai-Ying Duan; Roy A. Lacey

doi:10.1007/s41365-018-0504-z

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Examining the model dependence of the determination of kinetic freeze-out temperature and transverse flow velocity in small collision system

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH | Updated：2021-02-07

- Examining the model dependence of the determination of kinetic freeze-out temperature and transverse flow velocity in small collision system
- Nuclear Science and Techniques Vol. 29, Issue 11, Article number: 164(2018)
- Affiliations：
  
  1.Institute of Theoretical Physics & State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China
  2.Departments of Chemistry & Physics, Stony Brook University, Stony Brook, NY 11794, USA
- Author bio：
  
  Corresponding author, fuhuliu@163.com;
  
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- DOI：10.1007/s41365-018-0504-z CLC：
- Published：01 November 2018，
  
  Published Online：04 October 2018，
  
  Received：10 May 2018，
  
  Revised：28 August 2018，
  
  Accepted：29 August 2018
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Hai-Ling Lao, Fu-Hu Liu, Bao-Chun Li, et al. Examining the model dependence of the determination of kinetic freeze-out temperature and transverse flow velocity in small collision system. [J]. Nuclear Science and Techniques 29(11):164(2018)
DOI：

Hai-Ling Lao, Fu-Hu Liu, Bao-Chun Li, et al. Examining the model dependence of the determination of kinetic freeze-out temperature and transverse flow velocity in small collision system. [J]. Nuclear Science and Techniques 29(11):164(2018) DOI： 10.1007/s41365-018-0504-z.

Examining the model dependence of the determination of kinetic freeze-out temperature and transverse flow velocity in small collision system 1 first-author 1 corresp Corresponding author, fuhuliu@163.com; Corresponding author, fuhuliu@163.com; fuhuliu@163.com 1 1 2 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 Departments of Chemistry & Physics, Stony Brook University, Stony Brook, NY 11794, USA Departments of Chemistry & Physics, Stony Brook University, Stony Brook, NY 11794, USA The transverse momentum distributions of the identified particles produced in small collision systems at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) have been analyzed by four models. The first two models utilize the blast-wave model with different statistics. The last two models employ certain linear correspondences based on different distributions. The four models describe the experimental data measured by the Pioneering High Energy Nuclear Interaction eXperiment (PHENIX), Solenoidal Tracker at RHIC (STAR), and A Large Ion Collider Experiment (ALICE) collaborations equally well. It is found that both the kinetic freeze-out temperature and transverse flow velocity in the central collisions are comparable with those in the peripheral collisions. With the increase of collision energy from that of the RHIC to that of the LHC, the considered quantities typically do not decrease. Comparing with the central collisions, the proton–proton collisions are closer to the peripheral collisions. Kinetic freeze-out temperature Transverse flow velocity Small collision system Central collisions Peripheral collisions 1 Introduction 1 As an important concept in both thermal and subatomic physics, temperature is widely used in experimental measurements and theoretical studies. Contrary to macroscopic thermal physics, temperature in microscopic subatomic physics cannot be measured directly; nevertheless, the temperature measured in thermal physics is manifested by the change of a given quantity of the thermometric material. Instead, we can calculate the temperature by using the methods of particle ratios and transverse momentum ( p T ) spectra. The temperature obtained from particle ratios is typically the chemical freeze-out temperature ( T ch ), which can describe the degree of excitation of the interacting system at the stage of chemical equilibrium. The temperature obtained from the p T spectra with a thermal distribution that does not include the flow effect, is typically an effective temperature ( T eff or T ), which is not a real temperature due to its relation to the particle mass. The temperature obtained from the p T spectra with a thermal distribution that includes the flow effect is typically the kinetic freeze-out temperature ( T kin or T 0 ), which describes the degree of excitation of the interacting system at the stage of kinetic and thermal equilibrium. The chemical freeze-out and kinetic freeze-out are two main stages of the evolution of the interacting system in high energy collisions. At the stage of chemical freeze-out, the chemical components (relative fractions) of the particles are fixed. At the stage of kinetic freeze-out, the p T and momentum ( p ) spectra of the particles are no longer changed. We are interested in the T 0 value, owing to its relation to the p T spectrum of the identified particles, which is one of the quantities measured first in our experiments. At the same time, T 0 is related to the structure of the phase diagram in the T 0 -related spaces, such as T 0 as a function of β T and as a function of <math id="M1"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> , where β T is the mean transverse flow velocity, resulted from the impact and squeeze while <math id="M2"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> denotes the center-of-mass energy per nucleon pair in collisions of nuclei ( <math id="M3"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> in particle collisions such as in proton-proton ( p - p or pp ) collisions). In particular, in the energy ranges available in the beam energy scan (BES) program at the Relativistic Heavy Ion Collider (RHIC) and the BES program at the Super Proton Synchrotron (SPS), the chemical potential ( μ B ) of baryons needs to be considered. Then, the structure of phase diagram in the T 0 versus μ B space can be studied in both the RHIC BES and the SPS BES energy ranges. Generally, μ B can be obtained from the particle ratios and its excitation function has been studied in detail [ 1 1 - 5 5 1-5 ], while T 0 and β T can be obtained from the p T spectra. In Refs. [ 6 6 - 13 13 6-13 ], different methods have been used to obtain T 0 and β T . In our recent studies [ 14 14 - 17 17 14-17 ], we have used a number of models to obtain T 0 and β T in nucleus–nucleus (gold–gold (Au–Au) and lead–lead (Pb–Pb)) collisions at RHIC and Large Hadron Collider (LHC) energies, where the top RHIC energy was <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 while the LHC energy reached a few TeV. Similar results were obtained when a non-zero β T was used in peripheral nucleus-nucleus collisions in the blast-wave model with Boltzmann-Gibbs statistics (BGBW model) [ 6 6 - 8 8 6-8 , 18 18 ] and with Tsallis statistics (TBW model) [ 9 9 , 18 18 , 19 19 ]. Our results show that T 0 ( β T ) in central nucleus-nucleus collisions is comparable to that in peripheral collisions. Similarly, the values of T 0 and β T at the LHC are close to those at the RHIC. It is interesting to compare the results of different models in small collision systems, such as pp and deuteron-gold ( d -Au) collisions at the RHIC, and pp and proton-lead ( p -Pb) collisions at the LHC. In this paper, we use four models to obtain T 0 and β T values from the p T spectra of the identified particles produced in pp and d -Au collisions at the RHIC, and in pp and p -Pb collisions at the LHC. The model results of the p T spectra are compared with each other and with the experimental data of the Pioneering High Energy Nuclear Interaction eXperiment (PHENIX) [ 20 20 ], Solenoidal Tracker at RHIC (STAR) [ 21 21 - 23 23 21-23 ], and A Large Ion Collider Experiment (ALICE) collaborations [ 24 24 , 25 25 ]. Then, similar T 0 and β T values are obtained from the analyses of the experimental data by the four models. The paper is structured as follows: the formalism and method are described in Sect. 2, results and discussion are given in Sect. 3, finally, in Sect. 4, we summarize our main observations and conclusions. 2 Formalism and method 1 In the present work, four models were used for the p T distributions for comparisons in small collision systems; nevertheless, in our recent work [ 14 14 ] they were employed to obtain T 0 and β T values in nucleus–nucleus collisions at RHIC and LHC energies using a different superposition of soft excitation and hard scattering components. In order to provide a comprehensive review of the present work, we discuss the previous studies of the four models as follows. i) BGBW model [ 6 6 - 8 8 6-8 ]: in this model we considered a non-zero β T of the produced particles. According to Refs. [ 6 6 , 7 7 , 8 8 ], the BGBW model gives the p T distribution as <math display="block" id="M5"><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><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><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 N is the number of particles, C 1 is a normalized constant, I 0 and K 1 are modified Bessel functions of the first and second kinds, respectively, <math id="M6"><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, r/R is the relative radial position in the thermal source [ 6 6 ], and n 0 =2 similarly to that in Ref. [ 6 6 ]. The relation between β T and β ( r ) is <math id="M7"><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><mo>=</mo><mn>0.5</mn><msub><mi>β</mi><mtext>S</mtext></msub></mrow></math> . ii) TBW model [ 9 9 ]: in this model we also considered a non-zero β T . According to Ref. [ 9 9 ], the TBW model gives the p T distribution in the form of <math display="block" id="M8"><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><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><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 a normalized constant, q is an entropy index characterizing the degree of non-equilibrium, ϕ denotes the azimuth [ 9 9 ], and n 0 =1 similarly to that in Ref. [ 9 9 ]. In the first two models n 0 is independent: it does not matter if n 0 =1 or n 0 =2 is used. To be compatible with Refs. [ 6 6 ] and [ 9 9 ], we use n 0 =2 in the first model and n 0 =1 in the second model. It should be noted that we use the index - q /( q -1) in Eq. (2) instead of -1/( q -1) in Ref. [ 9 9 ] due to the fact that q is very close to one. This substitution results in a small and negligible difference in the Tsallis distribution [ 19 19 ]. iii) An alternative method, in which the intercept in the T versus m 0 relation is assumed to be T 0 [ 7 7 , 10 10 - 13 13 10-13 ], the slope in the 〈 p T 〉 versus <math id="M9"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> relation is assumed to be β T , and the slope in the 〈 p 〉 versus <math id="M10"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> relation is assumed to be the radial flow velocity β [ 14 14 - 17 17 14-17 ], which does not include the contribution of longitudinal flow. Here, m 0 denotes the rest mass, <math id="M11"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> denotes the mean moving mass (mean energy), 〈...〉 denotes the theoretical distribution average of the considered quantity, and T is obtained from a Boltzmann distribution [ 18 18 ]. Two steps are required to obtain T 0 and β T . To use the relations T = T 0 + am 0 , <math id="M12"><mrow><mo>〈</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>〉</mo><mo>=</mo><msub><mi>b</mi><mn>1</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="M13"><mrow><mo>〈</mo><mi>p</mi><mo>〉</mo><mo>=</mo><msub><mi>b</mi><mn>2</mn></msub><mo>+</mo><mi>β</mi><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , where a , b 1 , and b 2 are fitted parameters, we choose the form of Boltzmann distribution as [ 18 18 ] <math display="block" id="M14"><mrow><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><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><msub><mi>C</mi><mn>3</mn></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><mn>,</mn></mrow></math> where C 3 is a normalized constant related to the free parameter T and particle mass m 0 via its relation to m T ; nevertheless, the Boltzmann distribution has multiple forms [ 18 18 ]. iv) This model is similar to the third model, but T is obtained from a Tsallis distribution [ 18 18 , 19 19 ]. We choose the Tsallis distribution in the form of [ 18 18 , 19 19 ] <math display="block" id="M15"><mrow><msub><mi>f</mi><mn>4</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><msub><mi>C</mi><mn>4</mn></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 is a normalized constant related to the free parameters T and q , as well as m 0 ; nevertheless, the Tsallis distribution has more than one forms [ 18 18 , 19 19 ]. Similarly to our recent work [ 14 14 ], in both the BGBW and TBW models, a non-zero β T of the produced particles is considered in the peripheral nucleus–nucleus collisions. The peripheral collisions contain a small number of participant nucleons that take part in the violent interactions. This condition is similar to a small collision system, which also contains a small number of participant nucleons. When the cold nuclear effect is neglected, the small collision system is similar to a peripheral collisions. This means that a non-zero β T needs to be considered for the small collision system to maintain consistency; however, the values of β T for a small collision system and peripheral collisions are possibly different. Naturally, it is not unusual if the values of β T in the two types of collisions are nearly the same. From the first model T 0 and β T can be obtained, while from the second model T 0 , β T , and q can be obtained. The first two models are employed to compare their results. Although the forms of the first two models are obviously different, the values of T 0 ( β T ) obtained from them exhibit a little difference only. The last two models are used for comparison as well. The obtained values of the last two models exhibit a little difference as well; however they are still noticeably different. The description of the above models is presented at midrapidity, in which y ≈ 0, where y ≡ 0.5[( E + p z )/( E - p z )], and E and p z denote the energy and longitudinal momentum, respectively. At high p T , y ≈ -lntan( ϑ /2) ≡ η , where ϑ and η denote the emission angle and pseudorapidity of the considered particle, respectively. The effect of the spin and chemical potential on the p T spectra is neglected because they are small at the top RHIC and LHC energies [ 1 1 - 4 4 1-4 ]. Similar to our recent work [ 14 14 ], the kinetic freeze-out temperature, the mean transverse (radial) flow velocity, and the effective temperature in different models are uniformly denoted by T 0 , β T , and T , respectively; however, different values can be obtained by different models. Equations (1)–(4) are the functions describing mainly the contribution of the soft excitation process. These are only valid for the spectra in a narrow p T range, which mainly covers the range of 0 to 2.5–3.5 GeV/ c in most cases or a slightly higher in certain cases. Even for the soft excitation process, the Boltzmann distribution is not sufficient to fit the p T spectra in certain cases. In the case of a two- or three-component Boltzmann distribution, T is the weighted average resulting from different effective temperatures and the corresponding fractions obtained from different components. Generally, in the present work, two main processes in high-energy collisions are considered. Apart from the soft excitation process, the main process is the hard scattering process, which contributes to the spectra in a wide p T range and according to the quantum chromodynamics (QCD) calculations [ 26 26 - 28 28 26-28 ], it can be described by an inverse power-law as <math display="block" id="M16"><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><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><mi>A</mi><msub><mi>p</mi><mtext>T</mtext></msub><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><msub><mi>p</mi><mtext>T</mtext></msub></mrow><mrow><msub><mi>p</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup><mn>,</mn></mrow></math> where p 0 and n are free parameters, and A is a normalized constant related to the free parameters. As a result of the QCD-based calculation, Eq. (5) contributes to the distribution in a range of 0 to high p T . Theoretically, in spite of the overlapping regions in the low p T range between the contributions of Eqs. (1)–(4) and (5), they cannot replace each other. The experimental p T spectra are typically distributed in a wide range. This means that a superposition of both the contributions of soft and hard processes (components) needs to be used to fit the spectra. We use the usual step function for structuring the superposition in order to avoid the entanglement between the contribution ranges of the soft excitation and hard scattering components, such that <math display="block" id="M17"><mtable columnalign="left"><mtr><mtd><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><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><msub><mi>A</mi><mn>1</mn></msub><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>−</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mtext>S</mtext></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mo>+</mo><msub><mi>A</mi><mn>2</mn></msub><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mtext>H</mtext></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where f S ( p T ) denotes one of Eqs. (1)–(4), A 1 and A 2 are constants, ensuring that the contributions of soft and hard components are the same at p T = p 1 , and the step function θ ( x )=1 if x > 0 and θ ( x )=0 if x < 0. The fraction (rate) of the contribution of the soft component is given by <math id="M18"><mrow><mi>k</mi><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msubsup><mrow/><mn>0</mn><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow></msubsup></mrow></mrow></mstyle><msub><mi>A</mi><mn>1</mn></msub><msub><mi>f</mi><mtext>S</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></mrow></math> . Owing to the respective ranges of the different contributions, the selection of parameters in Eqs. (1)–(4) and (5) has no effect on their correlation and dependence on each other. In certain cases, the contribution of the resonance production for pions and the strong stopping effect for the participating nucleons are non-negligible at very low ranges. A very soft component needs to be used for the p T values ranging from 0 to 0.5–1.5 GeV/ c . Let us consider the contribution of the very soft component. Equation (6) can be rewritten as <math display="block" id="M19"><mtable columnalign="left"><mtr><mtd><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><mfrac><mn>1</mn><mi>N</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><msub><mi>A</mi><mrow><mtext>VS</mtext></mrow></msub><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mrow><mtext>VS</mtext></mrow></msub><mo>−</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mrow><mtext>VS</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mo>+</mo><msub><mi>A</mi><mn>1</mn></msub><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>−</mo><msub><mi>p</mi><mrow><mtext>VS</mtext></mrow></msub><mo stretchy="false">)</mo><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>−</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mtext>S</mtext></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mo>+</mo><msub><mi>A</mi><mn>2</mn></msub><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mtext>H</mtext></msub><mo stretchy="false">(</mo><msub><mi>p</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where f VS ( p T ) denotes one of Eqs. (1)–(4) similarly to f S ( p T ), and A VS is a constant ensuring that the contributions of the very soft and soft components are the same at p T = p VS . Let us denote the rates of the very soft and soft components by k VS and k S , respectively. Then, <math id="M20"><mrow><msub><mi>k</mi><mrow><mtext>VS</mtext></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>p</mi><mrow><mtext>VS</mtext></mrow></msub></mrow></msubsup></mrow></mstyle><msub><mi>A</mi><mrow><mtext>VS</mtext></mrow></msub><msub><mi>f</mi><mrow><mtext>VS</mtext></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></mrow></math> and <math id="M21"><mrow><msub><mi>k</mi><mtext>S</mtext></msub><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>p</mi><mrow><mtext>VS</mtext></mrow></msub></mrow><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow></msubsup></mrow></mstyle><msub><mi>A</mi><mn>1</mn></msub><msub><mi>f</mi><mtext>S</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></mrow></math> , where k VS + k S = k (for the definition of k , please refer to the section following Eq. (6)). Although f VS ( p T ) and f S ( p T ) have the same form in Eq. (7), their contribution ranges are different. Similarly, the contribution range of f H ( p T ) is different from those of f VS ( p T ) and f S ( p T ). The three functions have no correlation or dependence in the fitting procedure. We fitted f VS ( p T ) at very soft p T ranging from 0 to 0.5–1.5 GeV/ c , f S ( p T ) at soft p T ranging from 0.5–1.5 GeV/ c to 2.5–3.5 GeV/ c , and f H ( p T ) at hard p T ranging from 2.5–3.5 GeV/ c to the maximum. In the case without f VS ( p T ), Eq. (7) transforms into Eq. (6). Then, we fitted f S ( p T ) in Eq. (6) in the range of 0 to 2.5–3.5 GeV/ c . In the calculation, because of their different fractions, we used the weighted average of parameters in very soft and soft components in Eq. (7) to compare them with the values obtained from Eqs. (6) and (7). 3 Results and discussion 1 In Fig. 1 Fig. 1 , the transverse momentum spectra, 1/(2 πp T )·d 2 N /(d y d p T ), are shown for positively charged pions ( π + ), positively charged kaons ( K + ), and protons ( p ) ( Figs. 3(a) and 3(c) Figs. 3(a) and 3(c) ), as well as for negatively charged pions ( π - ), negatively charged kaons ( K - ), and antiprotons ( <math id="M22"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> )( Figs. 1(b) and 1(d) Figs. 1(b) and 1(d) ) produced in (0–20% ( Figs. 1(a) and 1(b) Figs. 1(a) and 1(b) ) and 60–88% (40–100%) ( Figs. 1(c) and 1(d) Figs. 1(c) and 1(d) ) d –Au collisions at <math id="M23"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV. The closed and open symbols represent the experimental data of the PHENIX and STAR collaborations measured in the pseudorapidity range | ≤ | < 0.35 [ 20 20 ] and the rapidity range | y | < 0.5 [ 21 21 ], respectively. The curves show the results obtained by models i)–iv) and the fitting parameters are given in Tables 1 Tables 1 , 2 2 , 3 3 , 4 4 , respectively, with most of them are fitted by Eq. (6). The numerical values fitted by Eq. (7) are marked by a star at the end of the line, where the results obtained from the very soft and soft components are shown together. It can be seen that the four considered models describe the p T spectra of the identified particles produced in central (0–20%) and peripheral (60–88% and 40–100%) d –Au collisions at <math id="M24"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV similarly well. 1 1 Figure 1 1 Centrality 1 1 Particle 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 (a) 1 1 0–20% 1 1 π + 1 1 0.112±0.006 1 1 0.43±0.01 1 1 0.999±0.001 1 1 5.0±0.3 1 1 15.9±0.4 1 1 4.85±0.51 1 1 37/18 1 1 d –Au 1 1 200 GeV 1 1 K + 1 1 0.128±0.008 1 1 0.42±0.01 1 1 0.994±0.006 1 1 5.8±0.3 1 1 16.5±0.4 1 1 0.64±0.07 1 1 9/15 1 1 PHENIX 1 1 1 1 p 1 1 0.130±0.008 1 1 0.39±0.01 1 1 0.998±0.002 1 1 5.2±0.3 1 1 15.2±0.4 1 1 0.30±0.02 1 1 64/18 1 1 (b) 1 1 0–20% 1 1 π - 1 1 0.121±0.006 1 1 0.43±0.01 1 1 0.999±0.001 1 1 5.4±0.3 1 1 16.7±0.4 1 1 4.30±0.50 1 1 23/18 1 1 1 1 1 1 K - 1 1 0.121±0.008 1 1 0.43±0.01 1 1 0.995±0.004 1 1 6.1±0.3 1 1 17.3±0.4 1 1 0.60±0.06 1 1 7/15 1 1 1 1 1 1 <math id="M25"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.129±0.008 1 1 0.39±0.01 1 1 0.999±0.001 1 1 5.1±0.2 1 1 16.1±0.4 1 1 0.24±0.02 1 1 103/18 1 1 (c) 1 1 60–88% 1 1 π + 1 1 0.104±0.006 1 1 0.43±0.01 1 1 0.998±0.002 1 1 3.5±0.2 1 1 12.9±0.3 1 1 1.29±0.14 1 1 30/18 1 1 1 1 1 1 K + 1 1 0.116±0.008 1 1 0.41±0.01 1 1 0.982±0.008 1 1 6.4±0.2 1 1 17.2±0.4 1 1 0.15±0.02 1 1 12/15 1 1 1 1 1 1 p 1 1 0.119±0.009 1 1 0.37±0.01 1 1 0.996±0.004 1 1 5.5±0.2 1 1 15.6±0.3 1 1 0.07±0.01 1 1 33/18 1 1 (d) 1 1 60–88% 1 1 π - 1 1 0.104±0.006 1 1 0.43±0.01 1 1 0.998±0.002 1 1 3.5±0.2 1 1 12.9±0.3 1 1 1.29±0.08 1 1 36/18 1 1 1 1 1 1 K - 1 1 0.115±0.008 1 1 0.40±0.01 1 1 0.983±0.011 1 1 6.0±0.2 1 1 17.2±0.3 1 1 0.15±0.02 1 1 15/15 1 1 1 1 1 1 <math id="M26"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.119±0.008 1 1 0.37±0.01 1 1 0.997±0.003 1 1 5.5±0.2 1 1 16.6±0.3 1 1 0.05±0.01 1 1 31/18 1 1 (a) 1 1 0–20% 1 1 π + 1 1 0.111±0.006 1 1 0.43±0.01 1 1 0.999±0.001 1 1 4.4±0.2 1 1 15.3±0.3 1 1 9.20±0.99 1 1 21/18 1 1 d –Au 1 1 200 GeV 1 1 p 1 1 0.128±0.008 1 1 0.37±0.01 1 1 0.998±0.002 1 1 5.1±0.2 1 1 15.9±0.3 1 1 0.97±0.10 1 1 18/16 1 1 (b) 1 1 0–20% 1 1 π - 1 1 0.111±0.006 1 1 0.43±0.01 1 1 0.999±0.001 1 1 4.4±0.2 1 1 15.3±0.3 1 1 9.2±1.00 1 1 24/18 1 1 STAR 1 1 1 1 <math id="M27"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.127±0.005 1 1 0.37±0.01 1 1 0.998±0.002 1 1 5.1±0.1 1 1 16.9±0.2 1 1 0.79±0.09 1 1 21/16 1 1 (c) 1 1 40–100% 1 1 π + 1 1 0.103±0.006 1 1 0.42±0.01 1 1 0.999±0.001 1 1 3.7±0.2 1 1 13.4±0.3 1 1 2.78±0.28 1 1 26/18 1 1 1 1 1 1 p 1 1 0.115±0.007 1 1 0.37±0.01 1 1 0.998±0.002 1 1 6.9±0.1 1 1 18.2±0.3 1 1 0.25±0.03 1 1 33/16 1 1 (d) 1 1 40–100% 1 1 π - 1 1 0.103±0.006 1 1 0.42±0.01 1 1 0.999±0.001 1 1 3.7±0.2 1 1 13.4±0.3 1 1 2.78±0.28 1 1 22/18 1 1 1 1 1 1 <math id="M28"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.112±0.006 1 1 0.35±0.01 1 1 0.998±0.002 1 1 6.4±0.1 1 1 18.9±0.3 1 1 0.24±0.02 1 1 39/16 1 1 (a) 1 1 1 1 π + 1 1 0.104±0.006 1 1 0.40±0.01 1 1 0.999±0.001 1 1 2.2±0.1 1 1 11.2±0.3 1 1 0.64±0.07 1 1 22/23 1 1 pp 1 1 200 GeV 1 1 K + 1 1 0.114±0.008 1 1 0.41±0.01 1 1 0.999±0.001 1 1 3.0±0.1 1 1 12.4±0.3 1 1 0.07±0.01 1 1 8/18 1 1 STAR 1 1 1 1 p 1 1 0.116±0.008 1 1 0.34±0.01 1 1 0.999±0.001 1 1 3.1±0.2 1 1 12.6±0.3 1 1 0.05±0.01 1 1 29/22 1 1 (b) 1 1 1 1 π - 1 1 0.104±0.006 1 1 0.40±0.01 1 1 0.999±0.001 1 1 2.2±0.1 1 1 11.3±0.3 1 1 0.64±0.07 1 1 27/23 1 1 1 1 1 1 K - 1 1 0.114±0.008 1 1 0.41±0.01 1 1 0.999±0.001 1 1 3.2±0.1 1 1 13.5±0.3 1 1 0.07±0.01 1 1 4/18 1 1 1 1 1 1 <math id="M29"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.116±0.008 1 1 0.34±0.01 1 1 0.998±0.002 1 1 3.1±0.2 1 1 13.7±0.4 1 1 0.04±0.01 1 1 46/22 1 1 (a) 1 1 0–5% 1 1 π ± 1 1 0.136±0.008 1 1 0.43±0.01 1 1 0.999±0.001 1 1 2.1±0.1 1 1 7.6±0.3 1 1 18.70±1.99 1 1 320/49* 1 1 p –Pb 1 1 5.02 TeV 1 1 K ± 1 1 0.193±0.009 1 1 0.43±0.01 1 1 0.997±0.003 1 1 2.7±0.1 1 1 7.3±0.3 1 1 2.84±0.41 1 1 71/45 1 1 ALICE 1 1 1 1 p + <math id="M30"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.195±0.009 1 1 0.42±0.01 1 1 0.999±0.001 1 1 3.5±0.2 1 1 8.8±0.3 1 1 1.10±0.11 1 1 172/43 1 1 (b) 1 1 80–100% 1 1 π ± 1 1 0.112±0.008 1 1 0.43±0.01 1 1 0.988±0.006 1 1 1.3±0.1 1 1 7.4±0.3 1 1 1.91±0.20 1 1 234/52 1 1 1 1 1 1 K ± 1 1 0.139±0.008 1 1 0.41±0.01 1 1 0.990±0.006 1 1 3.3±0.1 1 1 8.9±0.3 1 1 0.25±0.02 1 1 119/45 1 1 1 1 1 1 p + <math id="M31"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.156±0.009 1 1 0.37±0.01 1 1 0.993±0.006 1 1 3.9±0.1 1 1 10.1±0.3 1 1 0.10±0.01 1 1 225/43 1 1 1 1 1 1 π ± 1 1 0.111±0.008 1 1 0.43±0.01 1 1 0.994±0.005 1 1 1.9±0.1 1 1 8.1±0.3 1 1 3.60±0.35 1 1 382/57 1 1 pp 1 1 2.76 TeV 1 1 K ± 1 1 0.143±0.008 1 1 0.42±0.01 1 1 0.990±0.005 1 1 2.9±0.1 1 1 8.6±0.3 1 1 0.45±0.05 1 1 119/52 1 1 ALICE 1 1 1 1 p + <math id="M32"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.152±0.009 1 1 0.36±0.01 1 1 0.991±0.005 1 1 2.6±0.1 1 1 9.5±0.3 1 1 0.19±0.01 1 1 214/43 1 1 (a) 1 1 0–20% 1 1 π ± 1 1 0.107±0.006 1 1 0.41±0.01 1 1 0.999±0.001 1 1 4.4±0.3 1 1 14.5±0.4 1 1 103.61±11.37 1 1 28/23 1 1 Cu-Cu 1 1 200 GeV 1 1 K ± 1 1 0.122±0.011 1 1 0.41±0.02 1 1 0.997±0.003 1 1 6.1±0.3 1 1 16.3±0.4 1 1 12.52±1.26 1 1 1/10 1 1 1 1 1 1 p + <math id="M33"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.125±0.008 1 1 0.38±0.01 1 1 0.999±0.001 1 1 5.2±0.3 1 1 15.7±0.4 1 1 7.85±0.77 1 1 5/21 1 1 (b) 1 1 40–94% 1 1 π ± 1 1 0.101±0.005 1 1 0.43±0.01 1 1 0.999±0.001 1 1 4.3±0.2 1 1 14.5±0.3 1 1 8.29±0.81 1 1 18/23 1 1 1 1 1 1 K ± 1 1 0.111±0.008 1 1 0.40±0.01 1 1 0.996±0.003 1 1 5.9±0.2 1 1 16.9±0.3 1 1 1.28±0.11 1 1 1/10 1 1 1 1 1 1 p + <math id="M34"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.114±0.009 1 1 0.37±0.01 1 1 0.996±0.003 1 1 6.4±0.2 1 1 19.9±0.2 1 1 0.50±0.05 1 1 15/21 1 1 Figure 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 (a) 1 1 0.108±0.006 1 1 1.025±0.007 1 1 0.46±0.01 1 1 0.991±0.005 1 1 4.8±0.3 1 1 16.2±0.4 1 1 3.86±0.39 1 1 46/17 1 1 d –Au 1 1 0.118±0.008 1 1 1.026±0.008 1 1 0.46±0.01 1 1 0.981±0.006 1 1 5.9±0.3 1 1 16.0±0.4 1 1 0.57±0.06 1 1 24/14 1 1 PHENIX 1 1 0.119±0.008 1 1 1.018±0.007 1 1 0.45±0.01 1 1 0.996±0.004 1 1 5.1±0.2 1 1 15.9±0.4 1 1 0.25±0.02 1 1 19/17 1 1 (b) 1 1 0.108±0.006 1 1 1.025±0.007 1 1 0.46±0.01 1 1 0.992±0.005 1 1 4.8±0.3 1 1 16.4±0.4 1 1 3.86±0.39 1 1 56/17 1 1 1 1 0.118±0.008 1 1 1.026±0.008 1 1 0.46±0.01 1 1 0.983±0.009 1 1 5.9±0.3 1 1 17.0±0.4 1 1 0.57±0.06 1 1 34/14 1 1 1 1 0.118±0.008 1 1 1.018±0.007 1 1 0.45±0.01 1 1 0.996±0.004 1 1 5.1±0.2 1 1 16.3±0.4 1 1 0.20±0.02 1 1 36/17 1 1 (c) 1 1 0.088±0.006 1 1 1.045±0.008 1 1 0.46±0.01 1 1 0.994±0.004 1 1 3.5±0.2 1 1 13.7±0.3 1 1 1.02±0.10 1 1 34/17 1 1 1 1 0.090±0.008 1 1 1.029±0.008 1 1 0.46±0.01 1 1 0.955±0.011 1 1 6.4±0.3 1 1 17.5±0.4 1 1 0.13±0.01 1 1 9/14 1 1 1 1 0.098±0.008 1 1 1.012±0.007 1 1 0.44±0.01 1 1 0.990±0.006 1 1 5.5±0.2 1 1 15.9±0.2 1 1 0.06±0.01 1 1 37/17 1 1 (d) 1 1 0.088±0.006 1 1 1.045±0.008 1 1 0.46±0.01 1 1 0.994±0.006 1 1 3.5±0.2 1 1 13.7±0.3 1 1 1.02±0.10 1 1 46/17 1 1 1 1 0.090±0.008 1 1 1.029±0.008 1 1 0.46±0.01 1 1 0.957±0.011 1 1 6.7±0.3 1 1 18.6±0.4 1 1 0.13±0.01 1 1 11/14 1 1 1 1 0.097±0.008 1 1 1.012±0.007 1 1 0.43±0.01 1 1 0.992±0.006 1 1 5.5±0.2 1 1 17.1±0.3 1 1 0.05±0.01 1 1 48/17 1 1 (a) 1 1 0.106±0.006 1 1 1.020±0.008 1 1 0.46±0.01 1 1 0.995±0.004 1 1 4.4±0.2 1 1 15.6±0.4 1 1 7.07±0.76 1 1 38/17 1 1 d –Au 1 1 0.115±0.008 1 1 1.010±0.007 1 1 0.38±0.01 1 1 0.998±0.002 1 1 4.4±0.2 1 1 15.6±0.4 1 1 0.96±0.10 1 1 35/11* 1 1 (b) 1 1 0.106±0.006 1 1 1.020±0.008 1 1 0.46±0.01 1 1 0.995±0.004 1 1 4.4±0.2 1 1 15.6±0.4 1 1 7.07±0.76 1 1 39/17 1 1 STAR 1 1 0.116±0.008 1 1 1.008±0.005 1 1 0.44±0.01 1 1 0.997±0.003 1 1 5.1±0.2 1 1 17.7±0.4 1 1 0.73±0.07 1 1 44/15 1 1 (c) 1 1 0.085±0.006 1 1 1.038±0.008 1 1 0.46±0.01 1 1 0.996±0.004 1 1 3.7±0.2 1 1 13.9±0.3 1 1 2.50±0.25 1 1 33/17 1 1 1 1 0.090±0.008 1 1 1.008±0.007 1 1 0.35±0.01 1 1 0.998±0.002 1 1 6.9±0.2 1 1 19.6±0.4 1 1 0.31±0.02 1 1 29/11* 1 1 (d) 1 1 0.085±0.006 1 1 1.038±0.008 1 1 0.46±0.01 1 1 0.996±0.004 1 1 3.7±0.2 1 1 13.9±0.3 1 1 2.54±0.25 1 1 48/17 1 1 1 1 0.094±0.008 1 1 1.016±0.007 1 1 0.44±0.01 1 1 0.996±0.004 1 1 5.9±0.2 1 1 19.4±0.3 1 1 0.19±0.02 1 1 53/15 1 1 (a) 1 1 0.089±0.006 1 1 1.023±0.008 1 1 0.44±0.01 1 1 0.997±0.003 1 1 2.2±0.1 1 1 11.2±0.4 1 1 0.62±0.06 1 1 41/22 1 1 pp 1 1 0.098±0.008 1 1 1.029±0.009 1 1 0.43±0.01 1 1 0.996±0.004 1 1 3.0±0.2 1 1 12.8±0.4 1 1 0.07±0.01 1 1 29/17 1 1 STAR 1 1 0.104±0.009 1 1 1.006±0.001 1 1 0.39±0.01 1 1 0.996±0.004 1 1 3.1±0.2 1 1 13.5±0.4 1 1 0.05±0.01 1 1 55/21 1 1 (b) 1 1 0.089±0.006 1 1 1.023±0.008 1 1 0.44±0.01 1 1 0.997±0.003 1 1 2.2±0.1 1 1 11.5±0.4 1 1 0.62±0.06 1 1 52/22 1 1 1 1 0.098±0.008 1 1 1.029±0.009 1 1 0.43±0.01 1 1 0.996±0.004 1 1 3.0±0.2 1 1 13.8±0.4 1 1 0.07±0.01 1 1 26/17 1 1 1 1 0.104±0.009 1 1 1.006±0.001 1 1 0.39±0.01 1 1 0.996±0.004 1 1 3.1±0.2 1 1 13.9±0.4 1 1 0.04±0.01 1 1 84/21 1 1 (a) 1 1 0.107±0.007 1 1 1.001±0.001 1 1 0.48±0.01 1 1 0.999±0.001 1 1 2.2±0.1 1 1 7.7±0.3 1 1 20.98±2.00 1 1 323/47* 1 1 p –Pb 1 1 0.188±0.009 1 1 1.012±0.006 1 1 0.48±0.01 1 1 0.995±0.004 1 1 2.7±0.2 1 1 7.8±0.3 1 1 2.78±0.29 1 1 436/44 1 1 ALICE 1 1 0.198±0.009 1 1 1.013±0.008 1 1 0.47±0.01 1 1 0.999±0.001 1 1 3.5±0.2 1 1 9.1±0.3 1 1 1.10±0.10 1 1 223/42 1 1 (b) 1 1 0.089±0.006 1 1 1.001±0.001 1 1 0.45±0.01 1 1 0.999±0.001 1 1 1.4±0.1 1 1 7.3±0.3 1 1 2.21±0.20 1 1 606/43* 1 1 1 1 0.113±0.008 1 1 1.023±0.006 1 1 0.45±0.01 1 1 0.976±0.010 1 1 3.3±0.2 1 1 9.1±0.3 1 1 0.23±0.02 1 1 325/44 1 1 1 1 0.115±0.009 1 1 1.002±0.001 1 1 0.45±0.01 1 1 0.982±0.010 1 1 3.9±0.2 1 1 10.6±0.3 1 1 0.09±0.01 1 1 493/42 1 1 1 1 0.089±0.006 1 1 1.001±0.001 1 1 0.45±0.01 1 1 0.999±0.001 1 1 1.7±0.1 1 1 7.8±0.3 1 1 4.00±0.31 1 1 485/48* 1 1 pp 1 1 0.113±0.008 1 1 1.013±0.006 1 1 0.48±0.01 1 1 0.975±0.010 1 1 2.9±0.1 1 1 9.0±0.3 1 1 0.46±0.05 1 1 376/51 1 1 ALICE 1 1 0.116±0.008 1 1 1.004±0.001 1 1 0.44±0.01 1 1 0.975±0.010 1 1 2.5±0.2 1 1 9.9±0.3 1 1 0.20±0.02 1 1 494/42 1 1 (a) 1 1 0.101±0.007 1 1 1.027±0.009 1 1 0.47±0.02 1 1 0.999±0.001 1 1 4.4±0.2 1 1 14.8±0.4 1 1 66.17±7.10 1 1 27/22 1 1 Cu–Cu 1 1 0.110±0.008 1 1 1.026±0.008 1 1 0.46±0.02 1 1 0.996±0.004 1 1 6.2±0.3 1 1 16.5±0.4 1 1 12.23±1.20 1 1 3/9 1 1 1 1 0.114±0.008 1 1 1.020±0.007 1 1 0.45±0.01 1 1 0.999±0.001 1 1 5.2±0.3 1 1 16.3±0.4 1 1 6.02±0.60 1 1 4/20 1 1 (b) 1 1 0.085±0.007 1 1 1.052±0.008 1 1 0.47±0.02 1 1 0.999±0.001 1 1 4.3±0.2 1 1 14.7±0.3 1 1 6.58±0.68 1 1 19/22 1 1 1 1 0.090±0.008 1 1 1.029±0.008 1 1 0.47±0.02 1 1 0.996±0.004 1 1 6.0±0.3 1 1 16.7±0.3 1 1 1.08±0.01 1 1 3/9 1 1 1 1 0.095±0.008 1 1 1.012±0.008 1 1 0.46±0.01 1 1 0.992±0.004 1 1 6.6±0.2 1 1 21.3±0.4 1 1 0.38±0.04 1 1 12/20 1 1 Figure 1 1 Centrality 1 1 Particle 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 (a) 1 1 0–20% 1 1 π + 1 1 0.179±0.006 1 1 0.992±0.005 1 1 4.9±0.2 1 1 16.8±0.3 1 1 3.70±0.35 1 1 28/17* 1 1 d –Au 1 1 200 GeV 1 1 K + 1 1 0.243±0.009 1 1 0.976±0.011 1 1 5.9±0.2 1 1 16.9±0.3 1 1 0.60±0.05 1 1 39/16 1 1 PHENIX 1 1 1 1 p 1 1 0.293±0.009 1 1 0.991±0.006 1 1 5.1±0.2 1 1 15.8±0.3 1 1 0.25±0.02 1 1 24/19 1 1 (b) 1 1 0–20% 1 1 π - 1 1 0.179±0.006 1 1 0.993±0.006 1 1 4.8±0.2 1 1 16.8±0.3 1 1 3.70±0.35 1 1 29/17* 1 1 1 1 1 1 K - 1 1 0.240±0.009 1 1 0.974±0.011 1 1 5.6±0.2 1 1 16.9±0.3 1 1 0.58±0.05 1 1 37/16 1 1 1 1 1 1 <math id="M35"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.290±0.009 1 1 0.993±0.005 1 1 5.0±0.2 1 1 16.5±0.3 1 1 0.20±0.02 1 1 30/19 1 1 (c) 1 1 60–88% 1 1 π + 1 1 0.148±0.006 1 1 0.995±0.005 1 1 3.5±0.1 1 1 13.5±0.2 1 1 1.14±0.01 1 1 61/17* 1 1 1 1 1 1 K + 1 1 0.200±0.009 1 1 0.950±0.011 1 1 6.4±0.3 1 1 17.7±0.3 1 1 0.15±0.01 1 1 18/16 1 1 1 1 1 1 p 1 1 0.247±0.009 1 1 0.993±0.005 1 1 5.3±0.2 1 1 15.5±0.3 1 1 0.07±0.01 1 1 42/19 1 1 (d) 1 1 60–88% 1 1 π - 1 1 0.148±0.006 1 1 0.995±0.004 1 1 3.5±0.1 1 1 13.5±0.2 1 1 1.14±0.01 1 1 70/17* 1 1 1 1 1 1 K - 1 1 0.200±0.009 1 1 0.954±0.012 1 1 6.2±0.3 1 1 17.9±0.3 1 1 0.14±0.01 1 1 17/16 1 1 1 1 1 1 <math id="M36"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.247±0.009 1 1 0.993±0.005 1 1 5.0±0.2 1 1 16.5±0.3 1 1 0.05±0.01 1 1 28/19 1 1 (a) 1 1 0–20% 1 1 π + 1 1 0.172±0.007 1 1 0.999±0.001 1 1 4.1±0.1 1 1 15.0±0.3 1 1 7.70±0.70 1 1 42/17* 1 1 d –Au 1 1 200 GeV 1 1 p 1 1 0.208±0.009 1 1 0.999±0.001 1 1 5.8±0.2 1 1 16.5±0.3 1 1 1.07±0.10 1 1 28/15* 1 1 (b) 1 1 0–20% 1 1 π - 1 1 0.172±0.007 1 1 0.999±0.001 1 1 4.1±0.1 1 1 15.0±0.3 1 1 7.70±0.70 1 1 36/17* 1 1 STAR 1 1 1 1 <math id="M37"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.253±0.008 1 1 0.997±0.003 1 1 5.2±0.2 1 1 17.2±0.3 1 1 0.73±0.06 1 1 33/17 1 1 (c) 1 1 40–100% 1 1 π + 1 1 0.143±0.007 1 1 0.998±0.002 1 1 3.1±0.1 1 1 12.4±0.2 1 1 2.63±0.24 1 1 59/17* 1 1 1 1 1 1 p 1 1 0.219±0.009 1 1 0.991±0.005 1 1 5.8±0.2 1 1 17.8±0.3 1 1 0.27±0.02 1 1 37/17 1 1 (d) 1 1 40–100% 1 1 π - 1 1 0.143±0.007 1 1 0.998±0.002 1 1 3.1±0.1 1 1 12.4±0.2 1 1 2.63±0.24 1 1 49/17* 1 1 1 1 1 1 <math id="M38"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.217±0.009 1 1 0.992±0.005 1 1 5.4±0.1 1 1 18.5±0.3 1 1 0.22±0.02 1 1 30/17 1 1 (a) 1 1 1 1 π + 1 1 0.144±0.007 1 1 0.999±0.001 1 1 2.0±0.1 1 1 10.8±0.3 1 1 0.55±0.33 1 1 35/22* 1 1 pp 1 1 200 GeV 1 1 K + 1 1 0.203±0.009 1 1 0.989±0.007 1 1 3.3±0.1 1 1 13.3±0.3 1 1 0.07±0.01 1 1 31/19 1 1 STAR 1 1 1 1 p 1 1 0.234±0.009 1 1 0.997±0.003 1 1 3.2±0.2 1 1 13.6±0.3 1 1 0.05±0.01 1 1 121/23 1 1 (b) 1 1 1 1 π - 1 1 0.144±0.007 1 1 0.999±0.001 1 1 2.0±0.1 1 1 10.8±0.3 1 1 0.55±0.33 1 1 47/22* 1 1 1 1 1 1 K - 1 1 0.203±0.009 1 1 0.991±0.005 1 1 3.1±0.1 1 1 13.7±0.3 1 1 0.07±0.01 1 1 21/19 1 1 1 1 1 1 <math id="M39"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.230±0.009 1 1 0.996±0.004 1 1 3.1±0.2 1 1 14.3±0.3 1 1 0.04±0.01 1 1 91/23 1 1 (a) 1 1 0–5% 1 1 π ± 1 1 0.163±0.008 1 1 0.999±0.001 1 1 2.0±0.1 1 1 7.7±0.3 1 1 22.14±2.10 1 1 852/49* 1 1 p –Pb 1 1 5.02 TeV 1 1 K ± 1 1 0.297±0.008 1 1 0.992±0.005 1 1 3.4±0.1 1 1 8.6±0.3 1 1 2.87±0.29 1 1 110/44* 1 1 ALICE 1 1 1 1 p + <math id="M40"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.381±0.009 1 1 0.997±0.003 1 1 3.2±0.1 1 1 9.4±0.3 1 1 1.17±0.01 1 1 138/42* 1 1 (b) 1 1 80–100% 1 1 π ± 1 1 0.123±0.009 1 1 0.999±0.001 1 1 1.4±0.1 1 1 7.2±0.3 1 1 2.23±0.19 1 1 935/49* 1 1 1 1 1 1 K ± 1 1 0.212±0.010 1 1 0.995±0.005 1 1 3.8±0.1 1 1 9.0±0.3 1 1 0.28±0.02 1 1 403/44* 1 1 1 1 1 1 p + <math id="M41"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.235±0.010 1 1 0.997±0.003 1 1 3.3±0.1 1 1 9.8±0.3 1 1 0.11±0.01 1 1 128/42* 1 1 1 1 1 1 π ± 1 1 0.123±0.008 1 1 0.997±0.003 1 1 1.7±0.1 1 1 7.9±0.3 1 1 4.14±0.31 1 1 688/54* 1 1 pp 1 1 2.76 TeV 1 1 K ± 1 1 0.205±0.009 1 1 0.988±0.010 1 1 2.8±0.1 1 1 8.7±0.3 1 1 0.49±0.06 1 1 178/51* 1 1 ALICE 1 1 1 1 p + <math id="M42"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.241±0.009 1 1 0.995±0.005 1 1 2.6±0.1 1 1 9.5±0.3 1 1 0.20±0.02 1 1 104/42* 1 1 (a) 1 1 0–20% 1 1 π ± 1 1 0.179±0.008 1 1 0.999±0.001 1 1 4.2±0.1 1 1 14.5±0.2 1 1 72.07±7.00 1 1 35/22* 1 1 Cu–Cu 1 1 200 GeV 1 1 K ± 1 1 0.231±0.010 1 1 0.991±0.006 1 1 5.9±0.2 1 1 16.9±0.3 1 1 12.37±1.21 1 1 5/11 1 1 1 1 1 1 p + <math id="M43"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.296±0.009 1 1 0.999±0.001 1 1 5.4±0.2 1 1 17.0±0.4 1 1 5.81±0.67 1 1 6/22 1 1 (b) 1 1 40–94% 1 1 π ± 1 1 0.139±0.006 1 1 0.999±0.001 1 1 3.8±0.1 1 1 13.9±0.2 1 1 10.95±0.89 1 1 23/22* 1 1 1 1 1 1 K ± 1 1 0.179±0.009 1 1 0.994±0.005 1 1 4.9±0.1 1 1 16.2±0.3 1 1 1.16±0.13 1 1 1/9* 1 1 1 1 1 1 p + <math id="M44"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.245±0.008 1 1 0.999±0.001 1 1 4.8±0.2 1 1 17.7±0.3 1 1 0.43±0.06 1 1 3/20* 1 1 Figure 1 1 Centrality 1 1 Particle 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 (a) 1 1 0–20% 1 1 π + 1 1 0.134±0.008 1 1 1.082±0.009 1 1 0.994±0.005 1 1 4.8±0.2 1 1 16.3±0.4 1 1 3.93±0.36 1 1 32/18 1 1 d –Au 1 1 200 GeV 1 1 K + 1 1 0.189±0.009 1 1 1.052±0.010 1 1 0.980±0.010 1 1 6.1±0.2 1 1 16.9±0.4 1 1 0.57±0.06 1 1 13/15 1 1 PHENIX 1 1 1 1 p 1 1 0.272±0.009 1 1 1.015±0.007 1 1 0.999±0.001 1 1 5.5±0.2 1 1 15.3±0.4 1 1 0.28±0.02 1 1 49/18 1 1 (b) 1 1 0–20% 1 1 π - 1 1 0.134±0.008 1 1 1.082±0.009 1 1 0.993±0.005 1 1 4.8±0.2 1 1 16.8±0.4 1 1 3.68±0.36 1 1 33/18 1 1 1 1 1 1 K - 1 1 0.189±0.009 1 1 1.052±0.010 1 1 0.982±0.011 1 1 6.0±0.2 1 1 17.1±0.4 1 1 0.56±0.06 1 1 18/15 1 1 1 1 1 1 <math id="M45"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.273±0.009 1 1 1.015±0.007 1 1 0.999±0.001 1 1 5.4±0.2 1 1 16.0±0.4 1 1 0.18±0.02 1 1 44/18 1 1 (c) 1 1 60–88% 1 1 π + 1 1 0.108±0.008 1 1 1.099±0.009 1 1 0.999±0.001 1 1 3.6±0.1 1 1 13.1±0.3 1 1 1.20±0.12 1 1 37/18 1 1 1 1 1 1 K + 1 1 0.141±0.009 1 1 1.083±0.011 1 1 0.984±0.012 1 1 6.6±0.2 1 1 17.5±0.4 1 1 0.15±0.02 1 1 16/15 1 1 1 1 1 1 p 1 1 0.194±0.010 1 1 1.035±0.010 1 1 0.996±0.004 1 1 5.8±0.2 1 1 15.9±0.4 1 1 0.07±0.01 1 1 22/18 1 1 (d) 1 1 60–88% 1 1 π - 1 1 0.108±0.008 1 1 1.099±0.009 1 1 0.999±0.001 1 1 3.5±0.1 1 1 12.9±0.3 1 1 1.17±0.12 1 1 36/18 1 1 1 1 1 1 K - 1 1 0.141±0.009 1 1 1.083±0.011 1 1 0.979±0.012 1 1 6.5±0.2 1 1 17.8±0.4 1 1 0.13±0.02 1 1 11/15 1 1 1 1 1 1 <math id="M46"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.194±0.010 1 1 1.035±0.010 1 1 0.998±0.002 1 1 5.8±0.2 1 1 16.6±0.4 1 1 0.05±0.01 1 1 30/18 1 1 (a) 1 1 0–20% 1 1 π + 1 1 0.129±0.008 1 1 1.076±0.009 1 1 0.997±0.003 1 1 4.4±0.1 1 1 15.4±0.4 1 1 8.03±0.80 1 1 26/18 1 1 d –Au 1 1 200 GeV 1 1 p 1 1 0.221±0.009 1 1 1.005±0.005 1 1 0.999±0.001 1 1 5.9±0.2 1 1 16.8±0.3 1 1 1.01±0.09 1 1 23/13* 1 1 (b) 1 1 0–20% 1 1 π - 1 1 0.129±0.008 1 1 1.076±0.009 1 1 0.997±0.003 1 1 4.4±0.1 1 1 15.4±0.4 1 1 8.03±0.80 1 1 27/18 1 1 STAR 1 1 1 1 <math id="M47"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.260±0.009 1 1 1.009±0.005 1 1 0.999±0.001 1 1 5.7±0.2 1 1 17.3±0.3 1 1 0.68±0.07 1 1 46/16 1 1 (c) 1 1 40–100% 1 1 π + 1 1 0.104±0.008 1 1 1.089±0.009 1 1 0.998±0.002 1 1 3.4±0.1 1 1 13.1±0.3 1 1 2.62±0.25 1 1 32/18 1 1 1 1 1 1 p 1 1 0.173±0.009 1 1 1.011±0.005 1 1 0.999±0.001 1 1 6.3±0.2 1 1 17.0±0.3 1 1 0.31±0.03 1 1 33/13* 1 1 (d) 1 1 40–100% 1 1 π - 1 1 0.104±0.008 1 1 1.089±0.009 1 1 0.998±0.002 1 1 3.4±0.1 1 1 13.1±0.3 1 1 2.62±0.25 1 1 26/18 1 1 1 1 1 1 <math id="M48"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.189±0.009 1 1 1.036±0.005 1 1 0.999±0.001 1 1 5.4±0.2 1 1 17.7±0.3 1 1 0.19±0.02 1 1 35/16 1 1 (a) 1 1 1 1 π + 1 1 0.120±0.008 1 1 1.051±0.009 1 1 0.997±0.003 1 1 2.1±0.1 1 1 10.9±0.3 1 1 0.60±0.05 1 1 44/23 1 1 pp 1 1 200 GeV 1 1 K + 1 1 0.153±0.009 1 1 1.057±0.011 1 1 0.997±0.003 1 1 3.5±0.1 1 1 13.2±0.3 1 1 0.07±0.01 1 1 14/18 1 1 STAR 1 1 1 1 p 1 1 0.190±0.009 1 1 1.019±0.009 1 1 0.997±0.003 1 1 3.3±0.1 1 1 13.3±0.4 1 1 0.05±0.01 1 1 32/22 1 1 (b) 1 1 1 1 π - 1 1 0.120±0.008 1 1 1.056±0.009 1 1 0.997±0.003 1 1 2.1±0.1 1 1 11.0±0.3 1 1 0.56±0.05 1 1 45/23 1 1 1 1 1 1 K - 1 1 0.153±0.009 1 1 1.057±0.011 1 1 0.998±0.002 1 1 3.5±0.1 1 1 13.9±0.3 1 1 0.07±0.01 1 1 7/18 1 1 1 1 1 1 <math id="M49"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.190±0.009 1 1 1.019±0.009 1 1 0.997±0.003 1 1 3.3±0.1 1 1 13.9±0.4 1 1 0.04±0.01 1 1 42/22 1 1 (a) 1 1 0–5% 1 1 π ± 1 1 0.156±0.008 1 1 1.031±0.012 1 1 0.999±0.001 1 1 2.2±0.1 1 1 7.7±0.3 1 1 21.20±1.91 1 1 934/45* 1 1 p –Pb 1 1 5.02 TeV 1 1 K ± 1 1 0.262±0.008 1 1 1.059±0.011 1 1 0.995±0.005 1 1 3.0±0.1 1 1 7.8±0.3 1 1 2.78±0.28 1 1 261/45 1 1 ALICE 1 1 1 1 p + <math id="M50"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.351±0.009 1 1 1.035±0.009 1 1 0.999±0.001 1 1 3.4±0.1 1 1 9.1±0.3 1 1 1.09±0.01 1 1 97/43 1 1 (b) 1 1 80–100% 1 1 π ± 1 1 0.111±0.008 1 1 1.042±0.009 1 1 0.998±0.002 1 1 1.4±0.1 1 1 7.3±0.3 1 1 2.15±0.20 1 1 389/49* 1 1 1 1 1 1 K ± 1 1 0.171±0.008 1 1 1.068±0.012 1 1 0.986±0.010 1 1 3.8±0.1 1 1 9.3±0.3 1 1 0.23±0.02 1 1 282/45 1 1 1 1 1 1 p + <math id="M51"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.192±0.009 1 1 1.056±0.011 1 1 0.993±0.005 1 1 3.4±0.1 1 1 9.8±0.3 1 1 0.10±0.01 1 1 230/43 1 1 1 1 1 1 π ± 1 1 0.112±0.008 1 1 1.042±0.004 1 1 0.997±0.003 1 1 1.7±0.1 1 1 7.9±0.3 1 1 3.90±0.36 1 1 461/54* 1 1 pp 1 1 2.76 TeV 1 1 K ± 1 1 0.175±0.009 1 1 1.071±0.011 1 1 0.985±0.010 1 1 2.8±0.1 1 1 8.7±0.3 1 1 0.44±0.06 1 1 253/52 1 1 ALICE 1 1 1 1 p + <math id="M52"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> 1 1 0.223±0.009 1 1 1.029±0.008 1 1 0.988±0.010 1 1 2.6±0.1 1 1 9.5±0.3 1 1 0.19±0.02 1 1 373/43 1 1 (a) 1 1 0–20% 1 1 π + 1 1 0.131±0.007 1 1 1.070±0.006 1 1 0.999±0.001 1 1 4.4±0.2 1 1 14.6±0.3 1 1 73.32±8.01 1 1 23/23 1 1 Cu–Cu 1 1 200 GeV 1 1 K + 1 1 0.173±0.011 1 1 1.055±0.010 1 1 0.997±0.003 1 1 6.4±0.2 1 1 16.9±0.4 1 1 12.42±1.31 1 1 1/10 1 1 1 1 1 1 p 1 1 0.250±0.009 1 1 1.018±0.006 1 1 0.995±0.005 1 1 5.4±0.2 1 1 16.4±0.3 1 1 6.30±0.75 1 1 6/21 1 1 (b) 1 1 40–94% 1 1 π + 1 1 0.105±0.006 1 1 1.096±0.006 1 1 0.999±0.001 1 1 4.4±0.3 1 1 14.1±0.4 1 1 7.90±0.83 1 1 23/23 1 1 1 1 1 1 K + 1 1 0.139±0.009 1 1 1.076±0.009 1 1 0.998±0.002 1 1 5.2±0.3 1 1 15.9±0.4 1 1 1.08±0.12 1 1 1/10 1 1 1 1 1 1 p 1 1 0.197±0.009 1 1 1.042±0.006 1 1 0.995±0.005 1 1 5.1±0.2 1 1 17.9±0.3 1 1 0.43±0.04 1 1 9/21 Similarly to Fig. 1 Fig. 1 , Figs. 2(a) and 2(b) Figs. 2(a) and 2(b) show the spectra of π + , K + , and p , as well as π - , K - , and <math id="M57"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> , respectively, produced in pp collisions at <math id="M58"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV. The closed and open symbols represent the experimental data of the STAR collaboration measured in the range of -0.5 < y < 0 and at | y | < 0.5, respectively [ 22 22 , 23 23 ]. The fitting parameters are given in Tables 1 Tables 1 , 2 2 , 3 3 , 4 4 . It can be seen that the four considered models describe the p T spectra of the identified particles produced in pp collisions at <math id="M59"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV similarly well. Figure 3 Figure 3 is similar to Fig. 1 Fig. 1 , and it shows the spectra of π + + π - , K + + K - , and <math id="M62"><mrow><mi>p</mi><mo>+</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow></math> produced in 0–5% ( Fig. 3(a) Fig. 3(a) ) and 80–100% ( Fig. 3(b) Fig. 3(b) ) p –Pb collisions at <math id="M63"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV. The symbols represent the experimental data of the ALICE collaboration measured in the range of -0.5 < y < 0 [ 24 24 ]. It can be seen in most cases that the four considered models describe the p T spectra of the identified particles produced in p –Pb collisions at <math id="M64"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV similarly well. Similarly to Fig. 1 Fig. 1 , Fig. 4 Fig. 4 shows the spectra, (1/ N EV ) ·1/(2 πp T )·d 2 N /(d y d p T , of π + + π - , K + + K - , and <math id="M65"><mrow><mi>p</mi><mo>+</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow></math> produced in pp collisions at <math id="M66"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV, where N EV denotes the number of events and it is typically omitted. The symbols represent the experimental data of the ALICE collaboration measured in | y | < 0.5 for low- p T particles and in | η | < 0.8 for high- p T particles [ 25 25 ]. The four considered models describe the p T spectra of the identified particles produced in pp collisions at <math id="M67"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>2.76</mn></mrow></math> TeV similarly well in most of the cases. It should be noted that although we have used several free parameters in each fit, these parameters are restricted and irrelevant. A small number of them (1–3 parameters) are sensitive to the very soft component, which describes the very low p T range from 0 to 0.5–1.5 GeV/ c in certain cases. The same number of parameters (1–3) are sensitive to the soft component describing the low p T range from 0.5–1.5 GeV/ c to 2.5–3.5 GeV/ c in certain cases or typically in the range from 0 to 2.5–3.5 GeV/ c . While the final two parameters ( p 0 and n ) are sensitive to the hard component describing the wide p T range from 2.5–3.5 GeV/ c to the maximum. In certain cases, the data in the very low p T range is not available. In these cases, the number of free parameters are reduced by 1–3, and the low p T range from 0 to 2.5–3.5 GeV/ c can be used. The last two models use the relations between T and m 0 , 〈 p T 〉 and <math id="M70"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , as well as 〈 p 〉 and <math id="M71"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> . Due to the mass dependences of the relations, these are not suitable to fit all particles simultaneously in the low p T range. In principle, simultaneous fits of all particles can be performed by using the first two models. In the case of simultaneous fits, a larger χ 2 can be obtained due to the same set of parameters. Although we fit different particle spectra by different sets of parameters, the mean value of a given parameter can be obtained by weighting different yields of the considered particles. Thus, the weighted mean parameter can be regarded as a parameter suitable for the simultaneous fit of all particles. Therefore, both the simultaneous and non-simultaneous fits can be used in the analysis of the particle spectra. Based on the descriptions of the p T spectra, the first two models can conveniently provide T 0 and β T ; however, the values of parameters are possibly not the same according to different models. To obtain the values of T 0 , β T , and β by models iii) and iv), we analyze the values of T presented in Tables 3 Tables 3 and 4 4 , and calculate 〈 p T 〉, 〈 p 〉, and <math id="M72"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> based on these values. That is, we derived 〈 p T 〉, 〈 p 〉, and <math id="M73"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> by using a more complex fitting of Boltzmann and Tsallis distributions in the p T range from 0 to p 1 ; however, as the data was unavailable in certain regions the simple counting of published spectra could not be used. Based on an isotropic assumption in the rest frame of the emission source and using a Monte Carlo method, we can perform the calculations according to p T to obtain 〈 p 〉 and <math id="M74"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> [ 15 15 - 17 17 15-17 ]. It can seen that there are other constraints in the statistical fits due to the excluding contribution of the hard component and the selecting reference frame of the emission source. The relations between T and m 0 , 〈 p T 〉 and <math id="M75"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> , as well as 〈 p 〉 and <math id="M76"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> are shown in Figs. 5 Figs. 5 , 6 6 , and 7 7 , respectively, where panels (a) and (b) correspond to models iii) and iv) using Boltzmann and Tsallis distributions, respectively. The symbols in Fig. 5 Fig. 5 represent values of T listed in Tables 3 Tables 3 and 4 4 for different m 0 values. The symbols in Figs. 6 Figs. 6 and 7 7 represent values of 〈 p T 〉 and 〈 p 〉 for different <math id="M77"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> values, respectively, which are calculated from the parameters listed in Tables 3 Tables 3 and 4 4 and by an isotropic assumption in the rest frame of the emission source. The error bars in the three figures represent overall errors. Although the method of least squares was used to provide an appropriate connection, the lines in the three figures connect the points for a better visibility in each event sample. The intercept in Fig. 5 Fig. 5 provides T 0 , and β T and β can be obtained from the slopes in Figs. 6 Figs. 6 and 7 7 , respectively. The values of T , T 0 , β T , β , and <math id="M78"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> can be considered independent of isospin. To compare values of key parameters obtained by different models for different event samples, in the following we discuss the qualitative dependences of T 0 and β T on the centrality. From Tables 1 Tables 1 and 2 2 , we can obtain T 0 and β T in the first two models by weighting the yields of different particles. From the intercept in Fig. 5 Fig. 5 , T 0 can be obtained in the last two models, while from the slope in Fig. 6 Fig. 6 (or 7 7 ), we can obtain β T (or β ) in the last two models. Generally, the four models present similar results, and in certain cases these results are in agreement with each other within errors. In central d -Au and p -Pb collisions, T 0 is relatively greater than that in peripheral collisions. Ranging from the RHIC to LHC energies, T 0 shows a slight increase or it is nearly invariant. Ranging from the peripheral to central collisions and from the RHIC to LHC energies, both β T values show a slight increase or they are nearly invariant. These conditions are in agreement with our recent work, which studied Au-Au collisions at the RHIC and Pb-Pb collisions at the LHC [ 14 14 ] by a slightly different superposition. In particular, the absolute values of T 0 and β T do not show obvious change in the range from d -Au ( p -Pb) to Au-Au (Pb-Pb) collisions, except for a systematical increase (≤5%) due to different superpositions. In pp collisions, the dependences of T 0 and β T on <math id="M83"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> are similar to those in peripheral nuclear ( d -Au, Au-Au, p -Pb, and Pb-Pb) collisions. Apparently, in the above discussions T 0 and β T are related to the physical properties of an expanding thermal system, which, in the present work, is a high-energy collision system with a given impact parameter. If a mini-bias data sample is considered, T 0 and β T are the averages over various impact parameters. In particular, T 0 and β T in central (peripheral) collisions are the averages over a given centrality range. For pp collisions without choosing a centrality, T 0 and β T are the averages over a given data sample and they are related to the physical properties of the sample. In terms of excitation degree, characterized by T 0 , nuclear collisions, such as d –Au and Au–Au collisions at the RHIC and p –Pb and Pb–Pb collisions at the LHC, show similar excitation degree at the kinetic freeze-out; however, the excitation degree in central collisions is slightly higher than that in peripheral collisions. The excitation degree depends on the heaviest nucleus, but independent of the total nucleus, minimum nucleus, numbers of participating nucleons, and binary collisions in nuclear collisions at a given energy. To confirm the above statement of the heaviest nucleus, instead of using the total nucleus to determine T 0 , in the following we analyze copper–copper (Cu-Cu) collisions. Figure 8 Figure 8 shows the spectra of π + + π - , K + + K - , and <math id="M84"><mrow><mi>p</mi><mo>+</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow></math> produced in 0–20% ( Fig. 8(a) Fig. 8(a) ) and 40–94% (60–92%, 60–94%, and 40–60%) ( Fig. 8(b) Fig. 8(b) ) Cu-Cu collisions at <math id="M85"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV. The closed and open symbols represent the experimental data of the PHENIX and STAR collaborations measured in | ≤ | < 0.35 and | y | < 0.5, respectively [ 29 29 , 30 30 ], where the data of the 0–20% collisions are obtained by combining different centralities (0–5%, 5–10%, and 10–20%) to match with those in Fig. 1 Fig. 1 , and the data measured by different collaborations are connected by scaling different amounts. The fit parameters are given in Tables 1 Tables 1 , 2 2 , 3 3 , 4 4 , where the values of N 0 are obtained from the scaled spectra, instead of the original spectra. It can be seen that the four considered models approximately describe the p T spectra of the identified particles produced in the central (0–20%) and peripheral (40–94%) Cu–Cu collisions at <math id="M86"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV. Figures 9(a) and 9(b) Figures 9(a) and 9(b) show the relationships of T and m 0 , as well as 〈 p T 〉 and <math id="M89"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> (〈 p 〉 and <math id="M90"><mrow><mover accent="true"><mi>m</mi><mo stretchy="true">¯</mo></mover></mrow></math> ), according to the parameter values of Cu–Cu collisions at <math id="M91"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></mrow></math> GeV. It can be seen that the mentioned relationships show nearly linear tendencies in most cases. In particular, the intercept in Fig. 9(a) Fig. 9(a) represents T 0 , and the slopes related to 〈 p T 〉 and 〈 p 〉 in Fig. 9(b) Fig. 9(b) represent β T and β , respectively. For a qualitative comparison of the results obtained in different types of collisions, Tables 1 Tables 1 and 2 2 , as well as Figs. 5 Figs. 5 , 6 6 , and 9 9 are examined by comparing with the values of T 0 and β T . It can be seen that the T 0 values in central Cu-Cu collisions are slightly smaller than those in central d -Au (or p -Pb) collisions because the size of Cu is smaller than that of Au (or Pb). This is a direct and strong evidence for the statement that the heaviest nucleus needs to be considered instead of the total nucleus to determine T 0 . In addition, the T 0 in peripheral Cu-Cu collisions are nearly equal to those in peripheral d -Au ( p -Pb) collisions and in pp collisions. Apparently, the dependence of β T on the size of heaviest nucleus is undefined; however, β T in central collisions is comparable with that in peripheral collisions. The good agreement of the results obtained in the small system and nucleus–nucleus collisions reveal certain universalities in the hadroproduction process, as it is demonstrated in Refs. [ 31 31 - 35 35 31-35 ]. The universality in the hadroproduction process appears in different quantities observed [ 36 36 ] in different types of collisions (including proton-proton, proton-nucleus, and nucleus-nucleus collisions) and/or at different energies (available in the range from SPS BES to LHC) [ 31 31 - 35 35 31-35 ]. These quantities include, but not limited to mean multiplicity, rapidity or pseudorapidity density, multiplicity or transverse momentum distribution, and event patterns in different spaces under certain conditions. The present work confirms that the universality in the hadroproduction process possibly exists in thermal parameters at kinetic freeze-out in different types of collisions ranging from RHIC to LHC energies [ 14 14 ]. Although the blast-wave model and the related distributions have no contributions from resonance decays and strong stopping effects, a two-component form can be used to describe the spectra in very low and low p T ranges. In addition, in d -Au and p -Pb collisions, the cold nuclear modification effects on the p T spectra are not considered by us; however, a few of them widening the p T spectra of the identified particles due to the multiple cascade collisions in the cold spectator region. If the contribution of the effects of cold nuclear modification on the p T spectra is excluded, smaller T 0 and β T can be obtained. The comparison with pp collisions reveals that the contribution of cold nuclear modification effects on T 0 and β T is not obvious because peripheral nuclear collisions and pp collisions have similar values. Furthermore, the contribution of cold nuclear modification effects on T 0 and β T in Au-Au (Pb-Pb) collisions at the RHIC (LHC) is not obvious as well [ 14 14 ]. The Tsallis function is connected to the thermal model via its fits to the two- or three-component Boltzmann distribution [ 37 37 ]. Index q represents the degree of non-equilibrium among two or three states described by Boltzmann distributions, and the Tsallis temperature describes the fluctuations of Boltzmann temperatures. These explanations on the level of drawing curves of p T reveal that the interacting systems at the RHIC and LHC stays in a transitional region from the extensive system to the non-extensive system. There is no obvious boundary to distinguish the extensive system and the non-extensive system for a given interacting system in the considered energy range. Nevertheless, at the RHIC and its beam energy scan energies or similar energies, the generic axiomatic non-extensive statistics is used to obtain the chemical freeze-out temperature and the baryon chemical potential [ 38 38 - 40 40 38-40 ]. This indicates that the Boltzmann–Gibbs and Tsallis statistics are not always necessary or applicable, which suggests that the interacting systems at the considered energies are complex, and more studies are needed in the future. In central collisions at RHIC and LHC energies, the kinetic freeze-out temperature obtained from the four models is T 0 ≈120 MeV. It is lower than the chemical freeze-out temperature T ch ≈160 MeV [ 1 1 - 4 4 1-4 ]. This confirms that the kinetic freeze-out occurs later than the chemical freeze-out at the considered energies. As an approximate treatment, we consider an ideal fluid, in which the time-evolution of the temperature follows T f = T i ( τ i / τ f ) 1/3 , where T i and τ i are the initial temperature and proper time, respectively [ 41 41 , 42 42 ], and T f and τ f denote the final temperature and time, respectively. When considering T i =300 MeV and τ i =1 fm [ 42 42 ], the chemical freeze-out occurs at τ ch ≈ 6.6 fm and the kinetic freeze-out occurs at τ 0 ≈ 15.6 fm. When considering peripheral collisions, the kinetic freeze-out occurs at T 0 ≈105 MeV and τ 0 ≈23.3 fm. For instance, if a non-ideal fluid is considered, and the viscosity to entropy density ratio η/s is considered as 0.2, the time delay for the two freeze-outs is small, compared with the ideal fluid. Let us summarize the main contributions of the present work as follows. Before reconsidering the first two models, applying a nearly zero β T in them, the four models do not exhibit similar results. After reconsidering the first two models, applying a non-zero β T in them, the four models exhibit similar results. By comparing the central nuclear collisions, the proton–proton collisions are found to be closer to the peripheral nuclear collisions, especially in terms of T 0 and β T . The T 0 ( β T ) value in the central collisions is comparable with that in the peripheral collisions, and T 0 ( β T ) value in collisions at the LHC is comparable with that at the RHIC. At any rate, T 0 ( β T ) value in the central collisions is not smaller than that in the peripheral collisions, and T 0 ( β T ) value at the LHC is not smaller than that at the RHIC. Before the final conclusions, it should be emphasized that the comparisons of different models and the obtained T 0 and β T values in small collision system presented in this study are significant and useful owing to the collective expansion in a small system [ 43 43 ]. This also indicates that a large β T (~ 0.4 c ) is applied in peripheral nuclear collisions and pp collisions. As we know, certain models [ 6 6 - 8 8 6-8 , 44 44 - 52 52 44-52 ] are used to obtain T 0 and β T , and it is difficult to obtain the similar results compared to others [ 53 53 - 59 59 53-59 ] from these models with the increase of quantities. Although the present work provides similar results to [ 53 53 - 59 59 53-59 ] by the four models, the first and third models are preferred as they use a Boltzmann distribution, which is closer to the well-known ideal gas model. In addition, the hard component has no contribution to T 0 and β T due to its non-thermal production. Instead, the very soft and soft components, which contribute fitly in the very low and low p T regions, are used to obtain T 0 and β T . Thus, the third and fourth fits are suitable, because they can be applied for massive particles and in very low and low p T ranges. In addition, complex physics processes, high-energy collisions contain abundant information. This information includes, but is not limited to, electromagnetic field effects [ 60 60 ], strong magnetic field effects [ 61 61 ], and particular effects of strangeness [ 62 62 ]. The determination of T 0 and β T can be affected by these effects; hence, the search for the QCD critical point [ 63 63 ]. As a study at the exploratory stage of development, the present work still has needs to be improved with the highest possible accuracy. Further studies needs to be focused on the accurate determinations of T 0 and β T . In addition, the accurate determination of other types of temperatures, such as the effective temperature, chemical freeze-out temperature, and initial temperature, and comparisons of their dependences on the centrality and collision energy is also in the focus of our research. 4 Conclusion 1 As a conclusion, the transverse momentum distributions of π + , π - , K + , K - , p , and <math id="M95"><mover accent="true"><mi>p</mi><mo>¯</mo></mover></math> produced in pp and d -Au collisions at the RHIC, as well as in pp and p -Pb collisions at the LHC, have been analyzed by four models. The first two models utilize the blast-wave model with Boltzmann-Gibbs statistics and with Tsallis statistics, respectively. The last two models employ certain linear correspondences, in which the Boltzmann and Tsallis distributions are used to obtain the effective temperatures. These models and distributions describe only the contribution of the soft excitation process. For the hard scattering process, the inverse power law is uniformly used. The experimental data measured by the PHENIX, STAR, and ALICE collaborations are fitted by the model results. We used a non-zero β T in the first two methods. The four models present similar results. Both T 0 and β T in central collisions are comparable with those in peripheral collisions. With the increase of collision energy ranging from that of the RHIC to that of the LHC, the considered quantities typically do not decrease. Comparing with the central nuclear collisions, the pp collisions are closer to the peripheral nuclear collisions. 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Xu , Search for the QCD critical point with fluctuations of conserved quantities in relativistic heavy-ion collisions at RHIC: An overview . Nucl. Sci. Tech . 28 , 112 ( 2017 ). doi: 10.1007/s41365-017-0257-0 http://doi.org/10.1007/s41365-017-0257-0 Search for the QCD critical point with fluctuations of conserved quantities in relativistic heavy-ion collisions at RHIC: An overview We highly appreciate the communications with Dr. Muhammad Waqas. 10.1007/s41365-018-0504-z This work was supported by the National Natural Science Foundation of China (Nos. 11575103 and 11747319), the Shanxi Provincial Natural Science Foundation (No. 201701D121005), the Fund for Shanxi "1331 Project" Key Subjects Construction, and the US DOE (DE-FG02-87ER40331.A008). 2018-05-10 2018-08-28 2018-08-29 2018-10-04 2018-11-01 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/s41365-018-0504-z http://dx.doi.org/10.1007/s41365-018-0504-z . 2018 Nuclear Science and Techniques 11 29

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