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

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

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

Muhammad Waqas，

Fu-Hu Liu ，

Li-Li Li，

Haidar Mas’ud Alfanda

Nuclear Science and Techniques

Vol.31, No.11

Article number 109

Published in print 01 Nov 2020

Available online 07 Nov 2020

DOI：10.1007/s41365-020-00821-7

41200

The transverse momentum spectra of different types of particles produced in central and peripheral gold-gold (Au-Au) and inelastic proton-proton (pp) collisions at the Relativistic Heavy Ion Collider (RHIC), as well as in central and peripheral lead-lead (Pb-Pb) and pp collisions at the Large Hadron Collider (LHC) are analyzed by the multi-component standard (Boltzmann-Gibbs, Fermi-Dirac, and Bose-Einstein) distributions. The obtained results from the standard distribution give an approximate agreement with the measured experimental data by the STAR, PHENIX, and ALICE Collaborations. The behavior of the effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume for particles with different masses is obtained, which observes the early kinetic freeze-out of heavier particles as compared to the lighter particles. The parameters of emissions of different particles are observed to be different, which reveals a direct signature of the mass-dependent differential kinetic freeze-out. It is also observed that the peripheral nucleus-nucleus (AA) and pp collisions at the same center-of-mass energy per nucleon pair are in good agreement in terms of the extracted parameters.

Transverse momentum spectraEffective temperatureKinetic freeze-out temperatureTransverse flow velocityKinetic freeze-out volume

1 Introduction

A hot and dense fireball is assumed to form for a brief period of time (∼ a few fm/c) over an extended region after the initial collisions, which undergoes a collective expansion that leads to the change in the temperature and volume or density of the system. Three types of temperatures namely the initial temperature, chemical freeze-out temperature, and kinetic freeze-out temperature can be found in the literature, which describe the excitation degrees of an interacting system at the stages of initial collisions, chemical freeze-out, and kinetic freeze-out, respectively [1-7]. There is another type of temperature, namely the effective temperature, which is not a real temperature and it describes the sum of excitation degrees of the interacting system and the effect of transverse flow at the stage of kinetic freeze-out.

In principle, the initial stage of collisions happens earlier than other stages such as the chemical and kinetic freeze-out stages. Naturally, the initial temperature is the highest, and the kinetic freeze-out temperature is the lowest among the three real temperatures, while the chemical freeze-out temperature is in between the initial and kinetic freeze-out temperatures. The collision system does not get rid of the simultaneity for chemical and kinetic freeze-outs, which results in the chemical and kinetic freeze-out temperatures to be the same. The effective temperature is often larger than the kinetic freeze-out temperature but is equal to the kinetic freeze-out temperature in case of zero transverse flow velocity.

To understand the given nature of the nuclear force and to break the system into massive fragments [8, 9], it is a good way to make the nucleons interact in nucleus-nucleus (AA) collisions at intermediate and high energies. Such a process provokes a liquid-gas type phase transition as a large number of nucleons and other light nuclei are emitted. In AA collisions at higher energies, a phase transition from hadronic matter to quark-gluon plasma (QGP) is expected to occur. The volume occupied by the source of such ejectiles, where the mutual nuclear interactions become negligible (they only feel the Coulombic repulsive force and not the attractive force) is said to be kinetic freeze-out volume and it has been introduced in various statistical and thermodynamic models [10, 11]. Similar to the kinetic freeze-out temperature, the kinetic freeze-out volume also gives the information of the co-existence of phase transition. This is one of the major factors, which are important in the extraction of vital observables such as multiplicity, micro-canonical heat capacity, and its negative branch or shape of caloric curves under the external constraints [12-16].

It is conceivable that the temperature (volume) of the interacting system decreases (increases) from the initial state to the final kinetic freeze-out stage. During the evolution process, the transverse flow velocity is present due to the expansion of the interacting system. The study of the dependence of effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume on the collision energy, event centrality, system size, and particle rapidity is very significant. We are very interested in the aforementioned quantities in central and peripheral AA and (inelastic) proton-proton (pp) collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) over a wide enough energy range in which QGP is expected to form.

Here, we study the dependence of effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume in central and peripheral gold-gold (Au-Au) and lead-lead (Pb-Pb) collisions at the RHIC and LHC energies and compare their peripheral collisions with pp collisions of the same center-of-mass energy per nucleon pair $\sqrt{s_{NN}}$ (or the center-of-mass energy $\sqrt{s}$ for pp collisions). Only 62.4 GeV at the RHIC and 5.02 TeV at the LHC are considered as examples. We present the approach of effective temperature and kinetic freeze-out volume from the transverse momentum spectra of the identified particles produced in the mentioned AA and pp collisions. The kinetic freeze-out temperature and transverse flow velocity are then obtained from particular linear relations.

The remainder of this manuscript is structured as follows. The formalism and method are described in Sect. 2. Results and discussion are given in Sect. 3. In Sect. 4, we summarize our main observations and conclusions.

2 Method and formalism

Generally, two main processes of particle production are under consideration, which includes the soft and hard excitation processes. The soft excitation process corresponds to strong interactions among multiple partons, while the hard excitation process corresponds to a more violent collision between two head-on partons. The soft excitation process has numerous choices of formalisms, including but not limited to the Hagedorn thermal model (Statistical-bootstrap model) [17], the (multi-)standard distribution [18], the Tsallis and related distributions with various formalisms [19], the blast-wave model with Tsallis statistics [20], the blast-wave model with Boltzmann statistics [21-23, 26, 29], and other thermodynamics related models [30-33]. The hard excitation process has very limited choices of formalisms and can be described by the perturbative quantum chromodynamics (pQCD) [34-36].

The experimental data of the transverse momentum (p_T) spectrum of the particles are fitted using the standard distribution, which is the combination of Boltzmann-Gibbs, Fermi-Dirac, and Bose-Einstein distributions corresponding to the factor S=0, +1 and -1, respectively. The standard distribution at the mid-rapidity can be demonstrated as [18]

\begin{array}{l} f_{S} (p_{T}) = \frac{1}{N} \frac{d N}{d p_{T}} = \frac{1}{N} \frac{g V^{'}}{{(2 π)}^{2}} p_{T} \sqrt{p_{T}^{2} + m_{0}^{2}} \\ \times {[\exp (\frac{\sqrt{p_{T}^{2} + m_{0}^{2}}}{T}) + S]}^{- 1}, \end{array}

(1)

where the chemical potential is neglected. Here, N is the experimental number of considered particles, T is the fitted effective temperature, V’ is the fitted kinetic freeze-out volume (i.e. the interaction volume) of the emission source at the kinetic freeze-out stage, g=3 (or 2) is the degeneracy factor for pions and kaons (or protons), and m₀ is the rest mass of the considered particle. As a probability density function, the integral of Eq. (1) is naturally normalized to 1, i.e., we have $\int_{0}^{p_{T \max}} f_{S} (p_{T}) d p_{T} = 1$ , where p_{T max} denotes the maximum p_T. At very high energy, the influence of S=+1 and -1 can be neglected. Only the Boltzmann–Gibbs distribution is sufficient to describe the spectra at the RHIC and LHC.

Considering the experimental rapidity range [y_min,y_max] around mid-rapidity, Eq. (1) takes the form

\begin{array}{l} f_{S} (p_{T}) = \frac{1}{N} \frac{g V^{'}}{{(2 π)}^{2}} p_{T} \int_{y_{\min}}^{y_{\max}} (\sqrt{p_{T}^{2} + m_{0}^{2}} \cosh y - μ) \\ \times {[\exp (\frac{\sqrt{p_{T}^{2} + m_{0}^{2}} \cosh y - μ}{T}) + S]}^{- 1} d y, \end{array}

(2)

where the chemical potential μ is particle dependent, which we have studied recently [37]. In high energy collisions, μj (j=π, K, and p) are less than several MeV, which slightly affects V’ compared with that for μj=0. Then, we may regard μ≈0 in Eq. (2) at high energies considered in the present study. In Eqs. (1) and (2), only T and V’ are the free parameters.

Usually, we have to use the two-component standard distribution because single component standard distribution is not enough for the simultaneous description of very low- (0∼0.2-0.3 GeV/c) and low-p_T (0.2-0.3∼2-3 GeV/c or slightly more) regions, which are contributed by the resonance decays and other soft excitation processes, respectively. More than two or multi-component standard distributions can also be used in some cases. We have the simplified multi-component (l-component) standard distribution to be

\begin{array}{l} f_{S} (p_{T}) = \sum_{i = 1}^{l} k_{i} \frac{1}{N_{i}} \frac{g {V^{'}}_{i}}{{(2 π)}^{2}} p_{T} \sqrt{p_{T}^{2} + m_{0}^{2}} \\ \times {[\exp (\frac{\sqrt{p_{T}^{2} + m_{0}^{2}}}{T_{i}}) + S]}^{- 1}, \end{array}

(3)

where Ni and ki denote respectively the particle number and fraction contributed by the i-th component, and Ti and ${V^{'}}_{i}$ denote respectively the effective temperature and kinetic freeze-out volume corresponding to the i-th component.

More accurate form of l-component standard distribution can be written as,

\begin{array}{l} f_{S} (p_{T}) = \sum_{i = 1}^{l} k_{i} \frac{1}{N_{i}} \frac{g {V^{'}}_{i}}{{(2 π)}^{2}} p_{T} \\ \times \int_{y_{\min}}^{y_{\max}} (\sqrt{p_{T}^{2} + m_{0}^{2}} \cosh y - μ) \\ \times {[\exp (\frac{\sqrt{p_{T}^{2} + m_{0}^{2}} \cosh y - μ}{T_{i}}) + S]}^{- 1} d y . \end{array}

(4)

In Eqs. (3) and (4), only Ti, ${V^{'}}_{i}$ , and ki (i≤l-1) are free parameters. Generally, l=2 or 3 is enough for describing the spectra in a not too wide p_T range.

Eqs. (1) or (2) and (3) or (4) can be used for the description of p_T spectra and for the extraction of effective temperature and kinetic freeze-out volume in very low- and low-p_T regions. The high-p_T (>3-4 GeV/c) region contributed by the hard excitation process has to be fitted by the Hagedorn function [17], which is an inverse power law function, given by

f_{H} (p_{T}) = \frac{1}{N} \frac{d N}{d p_{T}} = A p_{T} {(1 + \frac{p_{T}}{p_{0}})}^{- n} .

(5)

It results from the pQCD [34-36], where A is the normalization constant, which depends on the free parameters p₀ and n, and results in $\int_{0}^{p_{T \max}} f_{H} (p_{T}) d p_{T} = 1$ .

While considering the contributions of both the soft and hard excitation processes, we used the superposition in principle

f_{0} (p_{T}) = k f_{S} (p_{T}) + (1 - k) f_{H} (p_{T}),

(6)

where, k is the contribution ratio of the soft process and gives a natural result in $\int_{0}^{p_{T \max}} f_{0} (p_{T}) d p_{T} = 1$ . In Eq. (6), the contribution of the soft process is from 0 to ∼2-3 GeV/c, or even up to ∼3-5 GeV/c at very high energy, and the hard component contributes to the whole p_T range. There is some mixing between the contributions of the two processes in the low-p_T region.

According to the Hagedorn model [17], the contributions of the two processes can be separated completely. One has another superposition

f_{0} (p_{T}) = A_{1} θ (p_{1} - p_{T}) f_{S} (p_{T}) + A_{2} θ (p_{T} - p_{1}) f_{H} (p_{T}),

(7)

where θ(x) is the usual step function and A₁ and A₂, are the normalization constants, which make A₁fS(p₁)=A₂fH(p₁). Equation (7) gives the contribution of soft process from 0 to p₁, while the hard component contributes from p₁ up to the maximum.

In the aforementioned two-component functions (Eqs. (6) and (7)), each component (fS(p_T) and fH(p_T)) is a traditional distribution. The first component (fS(p_T)) is one of the Boltzmann-Gibbs, Fermi-Dirac, and Bose-Einstein distributions if we use a given S, such as S=0, +1, or -1. The second component (fH(p_T)) is the Tsallis-like distribution [19] if we let n=1/(q-1) and p₀=nT_T, where q is the entropy index and T_T is the Tsallis temperature.

We will use only the first component in Eq. (7) due to the reason that we are not studying a wide p_T range in the present work. In the case of neglecting the contribution of the hard component in the low-p_T region in Eq. (6), the first component in Eq. (6) gives the same result as that of the first component in Eq. (7). Eq. (4) with l=2, which is the two-component standard distribution, is used in the present work. In addition, considering the treatment of normalization, the real fitted kinetic freeze-out volume should be $V_{1} = N_{1} {V^{'}}_{1} / k_{1}$ and $V_{2} = N_{2} {V^{'}}_{2} / (1 - k_{1})$ , which will be simply used in the following section.

It should be noted that the value of l in the l-component standard distribution has some influences on the free parameters and then on the derived parameters. Generally, l=1 is not enough to fit the particle spectra. For l=2, the influence of the second component is obvious since the contribution of the first component is not sufficient to fit the particle spectra. For l=3, the influence of the third component is rather small because the main contribution is from the first two components, and the contribution of the third component can be neglected.

3 Results and discussion

3.1 Comparison with the data

Figures 1(a) and 1(b) demonstrate the transverse momentum spectra, (1/2πp_T)d²N/dp_Tdy, of the negatively charged particles π^-, K^-, and $\bar{p}$ produced in (a) central (0–10%), and (b) peripheral (40–80%) Au-Au collisions at $\sqrt{s_{NN}} = 62.4$ GeV. The circles, triangles, and squares represent the experimental data measured in the mid-rapidity range -0.5<y<0 at the RHIC by the STAR Collaboration [38]. The curves represent fitting results by Eq. (4) with l=2. Following each panel, the results of Data/Fit are presented. The values of the related parameters (T₁, T₂, V₁, V₂, k₁, and N₀) along with the χ² and the number of degrees of freedom (ndof) are given in Table 1. It can be seen that the two-component standard distribution fits approximately the experimental data measured at mid-rapidity in Au-Au collisions at the RHIC.

Fig. 1.

(Color online) (a)(b) Transverse momentum spectra of π^-, K^-, and

\bar{p}

produced in (a) central (0–10%) and (b) peripheral (40–80%) Au-Au collisions at

\sqrt{s_{NN}} = 62.4

GeV. The symbols represent the experimental data measured in the range -0.5<y<0 at the RHIC by the STAR Collaboration [38]. The curves represent fitting results by Eq. (4) with l=2. Following each panel, the results of Data/Fit are presented. (c)(d) As examples, panels (c) and (d) show the contributions of the first and the second components in Eq. (4) with l=2 by the dashed and dotted curves respectively, and the total contribution is given by the solid curves. The circles in panels (c) and (d) represent the same data as those in panels (a) and (b) respectively.

Values of parameters (T₁, T₂, V₁, V₂, k₁, and N₀ (for Figs. 1 and 2) or σ₀ (for Fig. 3)), χ², and the ndof corresponding to the solid curves in Figs. 1–3. From the table, we have k₂=1-k₁, T=k₁T₁+k₂T₂, and V=V₁+V₂. The normalization constants contributed by the first and the second components are k₁N₀ (or k₁σ₀) and k₂N₀, (or k₂σ₀) respectively.

Collisions	Centrality	Particle	T₁ (GeV)	T₂ (GeV)	V₁ (fm³)	V₂ (fm³)	k₁	N₀ [σ₀ (mb)]	χ²	ndof
Figure 1	0–10%	π^-	0.141±0.008	0.285±0.007	185±13	3330±270	0.88±0.07	0.080±0.004	43	14
Au-Au		K^-	0.199±0.009	0.316±0.007	19±1	2034±162	0.85±0.11	0.010±0.003	174	13
62.4 GeV		$\bar{p}$	0.280±0.012	0.340±0.004	22±3	1053±100	0.92±0.10	0.020±0.004	56	11
	40–80%	π^-	0.070±0.006	0.250±0.006	32±5	127±14	0.70±0.07	0.025±0.050	94	14
		K^-	0.239±0.008	0.260±0.004	2.3±0.3	118±18	0.89±0.09	0.005±0.001	40	13
		$\bar{p}$	0.201±0.007	0.302±0.005	3.0±0.2	69±8	0.89±0.11	0.009±0.001	7	11
Figure 2	0–5%	π^-	0.267±0.013	0.624±0.005	8943±655	4341±200	0.93±0.12	1.770±0.300	425	33
Pb-Pb		K^-	0.355±0.014	0.465±0.006	1820±250	5555±300	0.94±0.12	0.300±0.040	776	32
5.02 TeV		$\bar{p}$	0.459±0.014	0.512±0.006	381±30	5476±240	0.94±0.10	0.325±0.040	748	30
	80–90%	π^-	0.200±0.009	0.407±0.004	154±8	246±50	0.70±0.09	0.060±0.003	658	33
		K^-	0.198±0.016	0.420±0.005	17±2	264±45	0.90±0.11	0.020±0.003	90	32
		$\bar{p}$	0.302±0.018	0.400±0.006	4.4±0.5	330±56	0.92±0.13	0.008±0.001	296	29
Figure 3(a)	-	π^-	0.182±0.006	0.275±0.005	65±8	22±4	0.68±0.12	0.350±0.060	54	23
pp		K^-	0.160±0.007	0.255±0.006	5.0±0.4	77±10	0.88±0.15	0.007±0.001	4	13
62.4 GeV		$\bar{p}$	0.235±0.008	0.260±0.006	1.6±0.1	50±6	0.95±0.10	0.008±0.001	126	24
Figure 3(b)	-	π^-	0.090±0.008	0.370±0.005	16±2	101±13	0.64±0.11	0.016±0.003	945	33
pp		K^-	0.850±0.013	0.370±0.004	0.80±0.04	97±12	0.87±0.11	0.007±0.001	666	31
5.02 TeV		$\bar{p}$	0.539±0.010	0.391±0.005	1.1±0.1	77±12	0.90±0.13	0.003±0.001	496	29

To see the contributions of the two components in Eq. (4) with l=2, as examples, Figs. 1(c) and 1(d) show the contributions of the first and second components by the dashed and dotted curves, respectively, and the total contribution is given by the solid curves. Only the results of π^- produced in (c) central (0–10%) and (d) peripheral (40–80%) Au–Au collisions at $\sqrt{s_{NN}} = 62.4$ GeV are presented. The circles represent the same data points as those in Figs. 1(a) and 1(b). One can see that the first component contributes mainly to the very low- and low-p_T region, while the second component contributes to a wider region. There is a large overlap region of the two contributions.

The transverse momentum spectra, (1/N_ev)d²N/dp_Tdy, of π^-, K^-, and $\bar{p}$ produced in (a) central (0–5%), and (b) peripheral (80–90%) Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV are shown in Fig. 2, where Nev on the vertical axis denotes the number of events. The experimental data of π^-, K^-, and $\bar{p}$ measured in the mid-rapidity range |y|<0.5 at the LHC by the ALICE Collaboration [39, 40] are represented by circles, triangles, and squares, respectively. The curves are our results fitted by Eq. (4) with l=2. Following each panel, the results of Data/Fit are presented. The values of the related parameters along with the χ² and ndof are given in Table 1. One can see that the two-component standard distribution fits approximately the experimental data measured in the mid-rapidity range in Pb-Pb collisions at the LHC.

Fig. 2.

(Color online) Transverse momentum spectra of π^-, K^-, and

produced in (a) central (0–5%) and (b) peripheral (80–90%) Pb-Pb collisions at

\sqrt{s_{NN}} = 5.02

TeV. The symbols represent the experimental data measured at |y|<0.5 at the LHC by the ALICE Collaboration [39, 40]. The curves represent fitting by Eq. (4) with l=2. Following each panel, the results of Data/Fit are presented.

The fitting in Figs. 1 and 2 for peripheral collisions appear to be worse compared to central collisions. This is caused by a statistical fluctuation and the effect of a cold spectator in peripheral collisions. In the region of the cold spectator, particles are produced by multiple cascade scattering processes which are different from the thermalization processes of particle production in the region of the hot participants. In addition, our fits are done in all ranges of p_T<4.5 GeV/c. However, as an alternative model, the blast-wave fit takes different cuts of p_T for the analysis of different particles (see for instance ref. [2]). These different cuts affect the extraction of parameters, in particular for the analysis of the trends of particles, which is not an ideal treatment.

In the next fits, we used all ranges of p_T<4.5 GeV/c. Figures 3(a) and 3(b) show the transverse momentum spectra, Ed³σ/dp³=(1/2πp_T)d²σ/dp_Tdy, of π^-, K^-, and $\bar{p}$ produced in pp collisions at $\sqrt{s} = 62.4$ GeV and 5.02 TeV, respectively. E and σ on the vertical axis denote the energy and cross-section, respectively. The symbols represent the experimental data measured in the mid-pseudorapidity range |η|<0.35 by the PHENIX Collaboration [41] and in the mid-rapidity range |y|<0.5 by the ALICE Collaborations [39, 40]. The curves represent our results, fitted by Eq. (4) with l=2. Following each panel, the results of Data/Fit are presented. The values of the related parameters (N₀ in Figs. 1 and 2 are replaced by σ₀ in Fig. 3) along with χ² and ndof are given in Table 1. One can see that the two-component standard distribution fits approximately the experimental data measured at mid-(pseudo)rapidity in pp collisions at the RHIC and LHC.

Fig. 3.

(Color online) Transverse momentum spectra of π^-, K^-, and

produced in pp collisions at (a)

\sqrt{s} = 62.4

GeV and (b)

\sqrt{s} = 5.02

TeV. The symbols represent the experimental data measured at |η|<0.35 by the PHENIX Collaboration [41] and at |y|<0.5 by the ALICE Collaborations [39, 40]. The curves represent our results, fitted by Eq. (4) with l=2. Following each panel, the results of Data/Fit are presented.

We would like to point out that the vertical axes of Figs. 1–3 are not the probability density function. We cannot fit them with Eq. (4) with l=2. Hence, we have done a conversion during our fitting. For Fig. 1, we have used the relation (1/2πp_T)(d²N/dp_Tdy)=(1/2πp_T)N₀fS(p_T)/dy for the conversion, where N₀ is the normalization constant in terms of particle number. For Fig. 2, we have used the relation d²N/dp_Tdy=N₀fS(p_T)/dy for the conversion, where Nev on the vertical axis is neglected because d²N/dp_Tdy is directly regarded as the result per event. For Fig. 3, we have used the relation Ed³σ/dp³=(1/2πp_T)(d²σ/dp_Tdy)=(1/2πp_T)σ₀fS(p_T)/dy in the conversion, where σ₀ is the normalization constant in terms of the cross-section.

From Figs. 1–3 and Table 1, it can be seen that the fitting quality is not great in some cases. It should be pointed out that the model used in these fittings is for soft processes but is used for analyzing p_T spectra up to 4.5 GeV/c. The high values of p_T analyzed in this study contain hard processes which could be responsible for the bad fitting as indicated by χ² in Table 1 and also in the ratio of data to the fitting of Figs. 1–3. Then, it may seem necessary to attempt fitting by taking into account the function part corresponding to the hard process. However, the hard process is not necessary for extracting the parameters of the soft process. Although the fittings will be better if we also consider the contribution of the hard process, it is not useful for extracting the parameters considered in the present work. Therefore, we did not consider the contribution of the hard process.

3.2 Discussion on the parameters

Considering the contributions of the two components, the effective temperature averaged over the two components is T=k₁T₁+k₂T₂ and the kinetic freeze-out volume by adding the two components is V=V₁+V₂. Further, the normalization constants contributed by the first and the second components are k₁N₀ and k₂N₀, respectively.

For convenience, we introduced the average p_T (〈p_T〉) and average moving mass ( $\bar{m}$ , i.e. average energy in the source rest frame) here. Considering Eq. (4) only, we have

〈 p_{T} 〉 = \int_{0}^{p_{T \max}} p_{T} f_{S} (p_{T}) d p_{T} .

(8)

To obtain $\bar{m}$ , we can use the Monte Carlo method. Let R₁ and R₂ denote random numbers distributed evenly in [0,1]. A concrete value of p_T that satisfies Eq. (4) can be obtained by

\int_{0}^{p_{T}} f_{S} (p_{T}') d p_{T}' < R_{1} < \int_{0}^{p_{T} + δ p_{T}} f_{S} (p_{T}') d p_{T}',

(9)

where δp_T denotes a small shift relative to p_T. In the source rest frame and under the assumption of isotropic emission, the emission angle θ of the considered particle obeys

f_{θ} (θ) = \frac{1}{2} \sin θ .

(10)

which results in

θ = 2 \arcsin (\sqrt{R_{2}})

(11)

in the Monte Carlo method [42]. Then,

m = \sqrt{{(p_{T} / \sin θ)}^{2} + m_{0}^{2}} .

(12)

After repeating the calculation many times, we can obtain $\bar{m}$ .

To study the change in the trends of parameters with the particle mass, Figs. 4(a) and 4(b) show the dependences of T on m₀ for productions of negative charged particles in central and peripheral (a) Au-Au collisions at 62.4 GeV and (b) Pb-Pb collisions at 5.02 TeV, while pp collisions at (a) 62.4 GeV and (b) 5.02 TeV are also studied and compared to peripheral AA collisions of the same energy (per nucleon pair). Correspondingly, Figs. 4(c) and 4(d) show the dependences of 〈p_T〉 on $\bar{m}$ for the mentioned particles in the considered collisions. The filled, empty, and half-filled symbols represent central AA, peripheral AA, and pp collisions, respectively. The lines represent linear fittings of the relations. The related linear fitting parameters are listed in Table 2, though some of them are not good fitting due to very large χ². The intercept in the linear relation between T and m₀ is regarded as the kinetic freeze-out temperature T₀, and the slope in the linear relation between 〈p_T〉 and $\bar{m}$ is regarded as the transverse flow velocity β_T. That is, T=am₀+T₀ [26-28] and $〈 p_{T} 〉 = β_{T} \bar{m} + b$ , where a and b are free parameters.

Fig. 4.

(Color online) Dependences of (a)(b) T on m₀ and (c)(d) ⟨p_T⟩ on

\bar{m}

for negatively charged particles produced in (a)(c) central and peripheral Au-Au collisions as well as pp collisions at 62.4 GeV, and in (b)(d) central and peripheral Pb-Pb collisions as well as pp collisions at 5.02 TeV. The filled, empty, and half-filled symbols represent the parameter values from central AA, peripheral AA, and pp collisions, respectively. The lines are linear fits for the parameter values

Values of slopes, intercepts, and χ² in the linear relations T=am₀+T₀ and

〈 p_{T} 〉 = β_{T} \bar{m} + b

, where T, m₀ (

\bar{m}

), and 〈p_T〉 are in the units of GeV, GeV/c² and GeV/c respectively.

Figure	Relation	$\sqrt{s_{N N}}$ ( $\sqrt{s}$ )	Collisions	a (c²), β_T (c)	T₀ (GeV), b (GeV/c)	χ²
Figure 4(a)	T-m₀	62.4 GeV	Central Au-Au	0.0679±0.006	0.2769±0.006	1
			Peripheral Au-Au	0.1054±0.005	0.2022±0.004	1
			pp	0.1270±0.005	0.1543±0.006	40
Figure 4(b)	T-m₀	5.02 TeV	Central Pb-Pb	0.1650±0.004	0.3593±0.006	1
			Peripheral Pb-Pb	0.0994±0.005	0.3293±0.005	31
			pp	0.0829±0.005	0.3208±0.006	6
Figure 4(c)	$〈 p_{T} 〉 - \bar{m}$	62.4 GeV	Central Au-Au	0.3857±0.004	0.1186±0.006	23
			Peripheral Au-Au	0.3449±0.006	0.1381±0.004	5
			pp	0.3567±0.006	0.0983±0.005	1
Figure 4(d)	$〈 p_{T} 〉 - \bar{m}$	5.02 TeV	Central Pb-Pb	0.4260±0.006	0.1178±0.005	37
			Peripheral Pb-Pb	0.4371±0.005	0.0465±0.004	2
			pp	0.4048±0.006	0.1331±0.005	1

Note that the relation T=am₀+T₀ [26-28] is used because the intercept should be the kinetic freeze-out temperature T₀ which corresponds to the emission of massless particles for which, there is no influence of the flow effect. The relation $〈 p_{T} 〉 = β_{T} \bar{m} + b$ was used in our previous works [22-25] for the same dimensions of 〈p_T〉 and $β_{T} \bar{m}$ . The interpretation of slope a in T=am₀+T₀ and the intercept b in $〈 p_{T} 〉 = β_{T} \bar{m} + b$ is not clear to us. Possibly, am₀ reflects the effective temperature contributed by The flow effect and b reflects the average transverse momentum contributed by the thermal motion.

From Fig. 4 and Table 2, one can see that T (T₀ or β_T) is larger in the central AA collisions as compared to peripheral AA collisions, and peripheral AA collisions are comparable with the pp collisions at the same $\sqrt{s_{NN}}$ ( $\sqrt{s}$ ). The mass-dependent or differential kinetic freeze-out scenario for T is observed, as T increased with the increase in m₀. The present work confirms various mass dependent or differential kinetic freeze-out scenarios [2, 3, 20, 43, 44]. Because T₀ (β_T) is obtained from the linear relation between T and m₀ (〈p_T〉 and $\bar{m}$ ), it seems that there is no conclusion for the mass dependence. However, if we first fit π^- and K^-, and then include $\bar{p}$ , we can see that T₀ (β_T) increases (decreases slightly) with increasing the mass. Thus, we observe the mass dependence or differential kinetic freeze-out.

It should be noted that although Fig. 4 also shows the enhancement of T when m₀ increases, this has been observed in many experiments and was reported for the first time by NA44 Collaboration [45] as evidence of the flow. This result was from a fit of p_T to a thermal model for π^-, K^-, and $\bar{p}$ . This indicates that the use of the two-component source model is unnecessary to observe the enhancement of T when m₀ increases. Although one can arrive at the same conclusion using a single-component source model, the two-component source model can describe well the p_T spectra. In addition, including the hard component, the model can describe better the p_T spectra.

The mass dependence of T (T₀) and β_T exists because it reflects the mass dependence of 〈p_T〉. We do not think that the mass dependence of T (T₀) and β_T is a model dependence, though the values of T (T₀) and β_T themselves are model dependent. In our fittings, we have used the same p_T range for π^-, K^-, and $\bar{p}$ , while in the blast-wave fitting, different p_T ranges were used for the three types of particles [2]. The treatment by the latter increases the flexibility in the selection of parameters.

Figure 5(a) shows the dependences of kinetic freeze-out volume V on rest mass m₀ for production of negatively charged particles in central and peripheral Au-Au collisions at $\sqrt{s_{NN}} = 62.4$ GeV as well as in pp collisions at $\sqrt{s} = 62.4$ GeV, while Fig. 5(b) shows the dependences of V on m₀ for negatively charged particles produced in central and peripheral Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV as well as in pp collisions at $\sqrt{s} = 5.02$ TeV. The filled, empty, and half filled symbols represent the central AA, peripheral AA, and pp collisions, respectively, and they represent the results weighted by different contribution fractions (volumes) in two components listed in Table 1.

Fig. 5.

(Color online) Dependences of V on m₀ for negatively charged particles produced in (a) central and peripheral Au-Au collisions as well as pp collisions at 62.4 GeV, and in (b) central and peripheral Pb-Pb collisions as well as pp collisions at 5.02 TeV. The filled, empty, and half-filled symbols represent the parameter values from central AA, peripheral AA, and pp collisions, respectively.

It can be seen from Fig. 5 that V in central AA collisions for all the particles are larger than those in peripheral AA collisions, which shows more participant nucleons and larger expansion in central AA collisions as compared to that in the peripheral AA collisions. Meanwhile, V in pp collisions is less than that in peripheral AA collisions of the same $\sqrt{s_{NN}}$ ( $\sqrt{s}$ ), which is caused by fewer participant nucleons (less multiplicity) in pp collisions. It is also observed that V decreases with an increase of m₀. This leads to a volume dependent or differential freeze-out scenario and indicates different freeze-out surfaces for different particles, depending on their masses that show early freeze-out of heavier particles as compared to the lighter particles [10, 11].

Figure 6 shows the dependences of T on V for the production of negatively charged particles in (a) central and peripheral Au-Au collisions as well as in pp collisions at 62.4 GeV, and in (b) central and peripheral Pb-Pb collisions as well as in pp collisions at 5.02 TeV. The filled, empty, and half-filled symbols represent central AA, peripheral AA, and pp collisions, respectively. One can see that T decreases with the increase in V in the central and peripheral AA and pp collisions. This result is natural due to the fact that a large V corresponds to a long kinetic freeze-out time and then a cool system and a low T.

Fig. 6.

(Color online) Same as Fig. 5, but showing the dependences of T on V

As we have not done any systematic analysis of the mass dependence of T₀ (β_T) in the present work, we shall not study the relation between T₀ (β_T) and V, though we can still predict the trend. As a supplement, our recent work [46] reported the mass dependence (slight dependence) of T₀ (β_T) using the same method as used in the present work, but using the Tsallis distribution as the "thermometer". We understand that with increasing m₀ (decreasing V), T₀ would increase naturally, and β_T would decrease slightly.

From Figs. 4–6, one can see that T, T₀, β_T, and V obtained from collisions at the LHC are larger than those obtained from the collisions at the RHIC. This is expected due to more violent collisions happening at higher energy. However, from the RHIC to LHC, the increase in the collision energy is considerably large, and the increases in T, T₀, β_T, and V are relatively small. This reflects the penetrability of the projectiles in the transparent target. In addition, pions correspond to a larger V than protons in some cases. This is caused by the fact that pions have larger β_T and thus, reach larger distance than protons due to the smaller m₀ in the case of the former at similar momenta for pions and protons at the kinetic freeze-out. This hypothesis is true because V is a reflection of multiplicity, and experimental results indicate an enhancement in the hadron source with the multiplicity.

The result that pions correspond to a much larger V than protons indicates that the protons cease to interact while pions are still interacting. One may think that pions and protons stop interacting in different V, where large V corresponds to long interaction time. As protons have larger m₀ than pions, protons are left behind as the system evolved from the origin of collisions to the radial direction, which is the behavior of hydrodynamics [47]. This results in the volume dependent freeze-out scenario that shows the early freeze-out of heavier particles as compared to the lighter particles [10, 11]. Thus, pions correspond to larger interacting volumes than protons, at the kinetic freeze-out stage.

To further study the dependences of T and V on centrality and collisions energy, Table 3 compiles the values of average T (〈T〉) and average V (〈V〉) for different types of collisions at the RHIC and LHC. These averages are obtained by different particle weights due to different contribution fractions (V) of π^-, K^-, and $\bar{p}$ . One can see that 〈T〉 and 〈V〉 at the LHC are larger than those at the RHIC. Generally, the value of T lies between T_ch and T₀. In particular, T_ch in central AA collisions is approximately 160 MeV, and T₀ in central AA collisions is less than 130 MeV [30, 31, 48, 49]. However, the values of 〈T〉 in Table 3 are larger because of Eq. (4) was used. Eq. (4) contains the contributions of both thermal motion and flow effect, which can be regarded as a different "thermometer" from literature [30, 31, 48-50] and results in different T that is beyond the general range of [T_ch,T₀].

Values of 〈T〉 and 〈V〉 for different types of collisions at the RHIC and LHC. The average values are obtained by different weights due to different contribution fractions (V) of π^-, K^-, and

\bar{p}

Figure	$\sqrt{s_{NN}}$ ( $\sqrt{s}$ )	Collisions	⟨T⟩ (GeV)	V (fm³)
Figure 1(a)	62.4 GeV	Central Au-Au	0.303±0.007	2610±218
Figure 1(b)		Peripheral Au-Au	0.247±0.007	130±17
Figure 3(a)		pp	0.214±0.006	77±10
Figure 2(a)	5.02 TeV	Central Pb-Pb	0.478±0.009	10002±658
Figure 2(b)		Peripheral Pb-Pb	0.374±0.008	344±54
Figure 3(b)		pp	0.360±0.006	100±13

Even for T₀ (the intercept in Table 2 for Figs. 4(a) and 4(b)) obtained from T=am₀+T₀, one can see the larger values. This is caused by the use of a different "thermometers". If other fitting functions are used [29-33], the obtained T₀ will be larger or smaller depending on the fitting function. For β_T (the slope in Table 2 for Figs. 4(c) and 4(d)) obtained from $〈 p_{T} 〉 = β_{T} \bar{m} + b$ , one can see different values in the case of other methods ("thermometers") [20-22, 26]. Anyhow, the relative sizes of T₀ (β_T) obtained from the present work for different events centralities, system sizes, and collision energies are useful and significant. Generally, T₀≤T_ch. However, because of different "thermometers", we cannot simply compare the two temperatures.

Although the absolute values of T (T₀) and β_T obtained in the present work are possibly inconsistent with other results, the relative values are worth considering. Similar is true for V. The present work shows that V in central and peripheral Pb-Pb and pp collisions at 5.02 TeV is also larger than that in central and peripheral Au-Au and pp collisions at 62.4 GeV. This shows a strong dependence of the parameters on the collision energy. Furthermore, V in central and peripheral Pb-Pb collisions is larger than that in central and peripheral Au-Au collisions also shows parameter dependence on the size of the system, though this dependence can be neglected due to a small difference in the size. The dependence of collision energy and system size is not discussed here in detail because of the unavailability of a wide range of analyses but it can be focused in future work.

3.3 Further discussion

Before the summary and conclusions, we would like to point out that the method that the related parameters can be extracted from the p_T spectra of the identified particles seems approximately effective in high energy collisions. At high energy (dozens of GeV and above), the particle-dependent chemical potential μ is less than several MeV, which affects the parameters less. Eqs. (1)–(4) can be used in the present work. We believe that our result on the source volume for pp collisions being larger than that (∼34 fm³) by the femtoscopy with two-pion Bose-Einstein correlations [51] is caused by the use of different methods.

At intermediate and low energies, the method used here seems unsuitable due to the fact that the particle dependent μ at kinetic freeze-out is large and unavailable. In general, the particles of different species develop μ differently from chemical freeze-out to kinetic freeze-out. This seems to result in more difficulty in applying Eqs. (1)–(4) at intermediate and low energies. μ has less influence on the extraction of source volume due to its less influence on the data normalization or multiplicity.

As we know, the source volume is proportional to the data normalization or multiplicity. Although we can obtain the normalization or multiplicity from a model, the obtained value is almost independent of the model. In other words, the normalization or multiplicity reflects the data, but not the model itself. Different methods do not affect the source volume considerably due to the normalization or multiplicity being one of the main factors, if not the only one. In the case of using a significant μ, neglecting the radial flow, and using T, there is no considerable influence on the normalization or multiplicity, then on the source volume.

In addition, although we use the method of linear relation to obtain T₀ and β_T in the present work, we used the blast-wave model [20, 21, 26, 29] to obtain the two parameters in our previous works [22, 43, 44]. Besides, we could add indirectly the flow velocity in the treatment of standard distribution [52]. Because of different "thermometers" (fit functions) being used in different methods, the "measured" temperatures have different values, though the same trend can be observed in the same or similar collisions. The results obtained from different "thermometers" can be checked with each other.

In particular, we obtained a higher temperature, though it is also the kinetic freeze-out temperature and describes the excitation degree of emission source at the kinetic freeze-out stage. We cannot compare the T₀ obtained in the present work with T_ch used in literature directly due to different "thermometers". We found that the present work gives the same trend for main parameters when we compare them with our previous works [22, 43, 44], which used the blast-wave model [20, 21, 26, 29]. It may be possible that the relative size of the main parameters in central and peripheral collisions as well as in AA and pp collisions will be the same if we use the standard distribution and the blast-wave model.

It should be pointed out that although we have studied some parameters at the stage of kinetic freeze-out, the parameters at the stage of chemical freeze-out are lacking in this study. In fact, the parameters at the stage of chemical freeze-out are more important [53-58] to map the phase diagram in which μ is an essential factor. Both the T_ch and μ are the most important parameters at the chemical freeze-out stage. In the extensive statistics and/or axiomatic/generic non-extensive statistics [53-55], one may discuss the chemical and/or kinetic freeze-out parameters systematically.

Reference [56] has tried to advocate a new parametrization procedure rather than the standard χ² procedure with yields. The authors constructed the mean value of conserved charges and have utilized their ratios to extract T_ch and μ. Reference [57] evaluated systematic error arising due to the chosen set of particle ratios and constraints. A centrality dependent study for the chemical freeze-out parameters [58] could be obtained. Meanwhile, with the help of the single-freeze-out model in the chemical equilibrium framework [59, 60], reference [61] studied the centrality dependence of freeze-out temperature fluctuations in high energy AA collisions.

We are very interested to do a uniform study on the chemical and kinetic freeze-out parameters in the future. Meanwhile, the distribution characteristics of various particles produced in high energy collisions are very abundant [62-65], and the methods of modeling analysis are multiple. We hope to study the spectra of multiplicities, transverse energies, and transverse momenta of various particles produced in different collisions by a uniform method, in which the probability density function contributed by each participant parton is considered carefully.

4 Summary and conclusions

We summarize here our main observations and conclusions

(a) Main parameters extracted from the transverse momentum spectra of identified particles produced in central and peripheral Au-Au collisions at 62.4 GeV and Pb-Pb collisions at 5.02 TeV were studied. Furthermore, the same analysis was done for pp collisions at both RHIC and LHC energies. The two-component standard distribution was used, which included both the very soft and soft excitation processes. The effective temperature, kinetic freeze-out temperature, transverse flow velocity, and kinetic freeze-out volume were found to be larger in central collisions as compared to that in the peripheral collisions, which shows higher excitation and larger expansion in central collisions.

(b) Effective temperatures in central and peripheral Au-Au (Pb-Pb) collisions at the RHIC (LHC) increased with increasing the particle mass, which showed a mass-dependent differential kinetic freeze-out scenario at RHIC and LHC energies. The kinetic freeze-out temperature is also expected to increase with increasing the particle’s mass. The kinetic freeze-out volume decreased with the increase of particle mass that showed different values for different particles and indicated a volume dependent differential kinetic freeze-out scenario. The transverse flow velocity is expected to decrease slightly with the increase of particle mass.

(c) Effective (kinetic freeze-out) temperatures in peripheral Au-Au and pp collisions at 62.4 GeV as well as in peripheral Pb-Pb and pp collisions at 5.02 TeV were respectively similar and had a similar trend, which showed similar thermodynamic nature of the parameters in peripheral AA and pp collisions at the same center-of-mass energy (per nucleon pair). Effective (kinetic) freeze-out) temperatures in both central and peripheral AA and pp collisions decreased with an increase in the kinetic freeze-out volume. The transverse flow velocity is expected to increase slightly with the increase of the kinetic freeze-out volume in the considered energy range.

(d) Effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume in central and peripheral AA and pp collisions at the LHC were larger than those at the RHIC, which showed their dependence on collision energy. Also, central (peripheral) Pb-Pb collisions rendered slightly larger effective (kinetic freeze-out) temperature, transverse flow velocity, and kinetic freeze-out volume than central (peripheral) Au-Au collisions. This showed the dependence of the parameters on the size of the system, which could be neglected for Pb–Pb and Au–Au collisions due to their small difference in the size.

References

[1]

N. Xu,

(for the STAR Collaboration), An overview of STAR experimental results

. Nucl. Phys. A 931, 1 (2014). https://doi.org/10.1016/j.nuclphysa.2014.10.022