1 Introduction

Temperature is an important concept in high energy nucleus-nucleus collisions. Usually, three types of temperatures which contain the chemical freeze-out temperature, kinetic freeze-out temperature, and effective temperature are used in literature [1-5]. The chemical freeze-out temperature describes the excitation degree of the interacting system at the stage of chemical equilibrium in which the chemical components (relative fractions) of particles are fixed. The kinetic freeze-out temperature describes the excitation degree of the interacting system at the stage of kinetic and thermal equilibrium in which the (transverse) momentum spectra of particles are no longer changed. The effective temperature is not a real temperature. In fact, the effective temperature is related to particle mass and can be extracted from the transverse momentum spectra by using some distribution laws such as the standard (Boltzmann, Fermi-Dirac, and Bose-Einstein), Tsallis, and so forth.

Generally, the chemical freeze-out temperature is usually obtained from the particle ratios [6-8]. It is equal to or larger than the kinetic freeze-out temperature due to the the chemical equilibrium during or earlier than the kinetic equilibrium. The effective temperature is larger than the kinetic freeze-out temperature due to mass and flow effects [9, 10]. Both the chemical freeze-out and effective temperatures in central nucleus-nucleus collisions are larger than those in peripheral collisions due to more violent interactions occurring in central collisions. In fact, central collisions contain more nucleons, and peripheral collisions contains less nucleons. Usually, there are small dissents in the extractions of chemical freeze-out temperature and effective temperature. As for the extraction of kinetic freeze-out temperature, the situations are largely non-uniform.

Currently, four main methods are used in the extraction of kinetic freeze-out temperature, T₀, which are i) the Blast-Wave model with Boltzmann-Gibbs statistics (the BGBW model) [11-13], ii) the Blast-Wave model with Tsallis statistics (the TBW model) [14], iii) the Tsallis distribution with flow effect (the improved Tsallis distribution) [15, 16], and iv) the intercept in T = T₀ + am₀ (the alternative method) [12, 17-20], where m₀ denotes the rest mass and T denotes the effective temperature which can be obtained by different distribution functions. In detail, the alternative method can be divided into a few sub-methods due to different distributions being used. Generally, we are inclined to use the standard and Tsallis distributions in the alternative method due to the standard distribution being closest to the ideal gas model in thermodynamics, and the Tsallis distribution describing a wide spectrum which needs a two- or three-component standard distribution to be fitted [21].

The kinetic freeze-out temperature, T₀, and the mean transverse radial flow velocity, β_T, can be simultaneously extracted by the first three methods. The alternative method needs further treatments in extracting the flow velocity. In our recent works [22-24], the mean transverse flow velocity, β_T, is regarded as the slope in the relation $\langle p_{\text{T}}\rangle ={\langle p_{\text{T}}\rangle }_0+{\beta }_{\text{T}}\overline{m}$, where 〈p_T〉 denotes the mean value of transverse momenta p_T, 〈p_T〉₀ denotes the mean transverse momentum in the case of zero flow velocity, and $\overline{m}$ denotes the mean moving mass. The mean flow velocity, β, is regarded as the slope in the relation $\langle p\rangle ={\langle p\rangle }_0+\beta \overline{m}$, where 〈p〉 denotes the mean value of momenta, p, and 〈p〉₀ denotes the mean momentum in the case of zero flow velocity. Although the mean transverse radial flow and mean transverse flow are not exactly the same, we use the same symbol to denote their velocities and neglect the difference between them. In fact, the mean transverse radial flow contains only the isotropic flow, and the mean transverse flow contains both the isotropic and anisotropic flows. The isotropic flow is mainly caused by isotropic expansion of the interacting system, and the anisotropic flow is mainly caused by anisotropic squeeze between two incoming nuclei.

We are interested in the coincidence and difference among the four methods in the extractions of T₀ and β_T. In this paper, we shall use the four methods to extract T₀ and β_T from the p_T spectra of identified particles produced in central and peripheral gold-gold (Au-Au) collisions at the center-of-mass energy per nucleon pair $\sqrt{s_{\text{NN}}}=200$ GeV (the top RHIC energy) and in central and peripheral lead-lead (Pb-Pb) collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV (one of the LHC energies). The model results on the p_T spectra are compared with the experimental data of the PHENIX [25], STAR [26, 27], and ALICE Collaborations [28, 29], and the model results on T₀ and β_T in different collisions and by different methods are compared each other.

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

2 Formalism and method

The four methods can be found in related references [11-20]. To give a whole representation of this paper, we present directly and concisely the four methods in the following. In the representation, some quantities such as the kinetic freeze-out temperature, the mean transverse (radial) flow velocity, and the effective temperature in different methods are uniformly denoted by T₀, β_T, and T, respectively, though different methods correspond to different values. All of the model descriptions are presented at the mid-rapidity which uses the rapidity y ≈ 0 and results in cosh(y) ≈ 1 which appears in some methods. At the same time, the spin property and chemical potential in the p_T spectra are neglected due to their small influences in high energy collisions. This means that we can give up the Fermi-Dirac and Bose-Einstein distributions and use only the Boltzmann distribution in the case of considering the standard distribution.

According to Refs. [11-13], the BGBW model results in the p_T distribution to be

where C₁ is the normalized constant which results in ${\displaystyle {\int }_0^{\infty }}{f}_1(p_{\text{T}})\text{d}{p}_{\text{T}}=1$, where I₀ and K₁ are the modified Bessel functions of the first and second kinds respectively, $m_{\text{T}}=\sqrt{p_{\text{T}}^2+m_0^2}$ is the transverse mass, ρ= tanh^-1[β(r)] is the boost angle, β(r) = β_S(r/R)ⁿ₀ is a self-similar flow profile, β_S is the flow velocity on the surface of the thermal source, r/R is the relative radial position in the thermal source, and n₀ is a free parameter which is customarily chosen to be 2 [11] due to this quadratic profile resembling the solutions of hydrodynamics closest [30]. Generally, ${\beta }_{\text{T}}=(2/R^2){\displaystyle {\int }_0^R}r\beta (r)\text{d}r=2{\beta }_{\text{S}}/(n_0+2)$. In the case of n₀=2, as used in Ref. [11], we have β_T = 0.5β_S [31].

According to Ref. [14], the TBW model results in the p_T distribution to be

where C₂ is the normalized constant which results in ${\displaystyle {\int }_0^{\infty }}{f}_2(p_{\text{T}})\text{d}{p}_{\text{T}}=1$, q is an entropy index characterizing the degree of non-equilibrium, and ϕ denotes the azimuth. In the case of n₀=1, as used in Ref. [14], we have β_T = 2β_S/(n₀+2)=(2/3)β_S due to the same flow profile as in the BGBW model. We would like to point out that the index -q/(q-1) in Eq. (2) replaced -1/(q-1) in Ref. [14] due to q being very close to 1. In fact, the difference between the results corresponding to -q/(q-1) and -1/(q-1) are small in the Tsallis distribution [32].

According to Refs. [15, 16], the improved Tsallis distribution in terms of p_T is

where C₃ is the normalized constant which results in ${\displaystyle {\int }_0^{\infty }}{f}_3(p_{\text{T}})\text{d}{p}_{\text{T}}=1$, r ≡ γm_T/T₀, s ≡ γβ_Tp_T/T₀, $\gamma =1/\sqrt{1-{\beta }_{\text{T}}^2}$, and I_0-3(s) and K_0-3(r) are the modified Bessel functions of the first and second kinds, respectively.

As for the alternative method [12, 17-20, 22-24], to use the relations T = T₀ + am₀, $\langle p_{\text{T}}\rangle ={\langle p_{\text{T}}\rangle }_0+{\beta }_{\text{T}}\overline{m}$, and $\langle p\rangle ={\langle p\rangle }_0+\beta \overline{m}$, we can choose the standard and Tsallis distributions to fit the p_T spectra of identified particles produced in high energy collisions. Because we give up the Fermi-Dirac and Bose-Einstein distributions, only the Boltzmann distribution is used in the case of considering the standard distribution in the present work. Both the Boltzmann and Tsallis distributions have more than one forms. We choose the form of Boltzmann distribution [33]

and the form of Tsallis distribution [32, 33]

where C_4a and C_4b are the normalized constants which result in ${\displaystyle {\int }_0^{\infty }}{f}_{4a}(p_{\text{T}})\text{d}{p}_{\text{T}}=1$ and ${\displaystyle {\int }_0^{\infty }}{f}_{4b}(p_{\text{T}})\text{d}{p}_{\text{T}}=1$ respectively.

It should be noticed that the above five distributions are only valid for the spectra in a low-p_T range. That is, they describe only the soft excitation process. Even if for the soft process, the Boltzmann distribution is not always enough to fit the p_T spectra in some cases. In fact, two- or three-component Boltzmann distributions can be used if necessary, in which T is the average weight at the effective temperatures obtained from different components. We have

and

where l=2 or 3 denotes the number of components, and ki, C_4ai, and Ti denote the contribution ratio (relative contribution or fraction), normalization constant, and effective temperature related to the i-th component, respectively. As can be seen in the next section, Eqs. (6) and (7) are not needed in the present work because only simple component Boltzmann distribution, i.e. Eq. (4), is used in the analyses. We present here Eqs. (6) and (7) to point out a possible application in future.

For the spectra in a wide p_T range which contains low and high p_T regions, we have to consider the contribution of a hard scattering process. Generally, the contribution of a hard process is parameterized to an inverse power-law

which is resulted from the QCD (quantum chromodynamics) calculation [34-36], where p₀ and n are free parameters, and A is the normalized constant which depends on p₀ and n and results in ${\displaystyle {\int }_0^{\infty }}{f}_{\text{H}}(p_{\text{T}})\text{d}{p}_{\text{T}}=1$.

To describe the spectra in a wide p_T range, we can use a superposition of both contributions of soft and hard processes. The contribution of the soft process is described by one of the BGBW models, the TBW model, the improved Tsallis distribution, the Boltzmann distribution or two- or three-component Boltzmann distributions, and the Tsallis distribution, while the contribution of hard process is described by the inverse power-law. We have the superposition

where k denotes the contribution ratio of the soft process and results naturally in ${\displaystyle {\int }_0^{\infty }}{f}_0(p_{\text{T}})\text{d}{p}_{\text{T}}=1$, and f_S(p_T) denotes one of the five distributions discussed in the four methods.

It should be noted that Eq. (9) and its components f_S(p_T) and f_H(p_T) are probability density functions. The experimental quantity of p_T distribution has mainly three forms, dN/dp_T, d²N/(dydp_T), and (2π p_T)^-1d²N/(dydp^T), where N denotes the number of particles and dy is approximately treated as a constant due to it being usually a given and small value at the mid-rapidity. To connect Eq. (9) with dN/dp_T, we need a normalization constant, N₀. To connect Eq. (9) with d²N/(dydp_T), we need another normalization constant, N₀. To connect Eq. (9) with (2π p_T)^-1d²N/(dydp^T), we have to rewrite Eq. (9) to f₀(p_T)/p_T=[kf_S(p_T)+(1-k)f_H(p_T)]/p_T and compare the right side of the new equation with the data with a new normalization constant, N₀.

3 Results and discussion

Figure 1 presents the transverse momentum spectra, (2π p_T)^-1d²N/(dydp^T), of (a)-(c) positively charged pions (π⁺), positively charged kaons (K⁺), neutral kaons ($K_{\text{S}}^0$ only), and protons (p), as well as (b)-(d) negatively charged pions (π^-), negatively charged kaons (K^-), neutral kaons ($K_{\text{S}}^0$ only), and antiprotons ($\overline{p}$) produced in (a)-(b) central (0–5% and 0–12%) and (c)-(d) peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV, where the spectra for different types of particles and for the same or similar particles in different conditions are multiplied by different amounts shown in the panels for clarity and normalization. The closed symbols represent the experimental data of the PHENIX Collaboration measured in the pseudorapidity range |η|<0.35 [25]. The open symbols represent the STAR data measured in the rapidity range |y|<0.5 [26, 27], where the data for K⁺ and K^- are not available and the data for $K_{\text{S}}^0$ in (a)-(c) and (b)-(d) are the same. The solid, dashed, dotted, dashed-dotted, and dashed-dotted-dotted curves are our results calculated by using the superpositions of i) the BGBW model (Eq. (1)) and inverse power-law (Eq. (8)), ii) the TBW model (Eq. (2)) and inverse power-law, iii) the improved Tsallis distribution (Eq. (3)) and inverse power-law, iv)_a the Boltzmann distribution (Eq. (4)) and inverse power-law, as well as iv)_b the Tsallis distribution (Eq. (5)) and inverse power-law, respectively. These different superpositions are also different methods for fitting the data. The values of free parameters T₀, β_T, k, p₀, and n, normalization constant, N₀, which is used to fit the data by a more accurate method comparing with Ref. [37], and χ² per degree of freedom (χ²/dof) corresponding to the fit of method i) are listed in Table 1; the values of T₀, q, β_T, k, p₀, n, N₀, and χ²/dof corresponding to methods ii) and iii) are listed in Tables 2 and 3 respectively; the values of T, k, p₀, n, N₀, and χ²/dof corresponding to methods iv)_a are listed in Table 4; and the values of T, q, k, p₀, n, N₀, and χ²/dof corresponding to method iv)_b are listed in Table 5. One can see that, in most cases, all of the considered methods describe approximately the p_T spectra of identified particles produced in central and peripheral Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV.

Fig. 1.Full-screen View

(Color online) Transverse momentum spectra of (a)-(c) π⁺, K⁺, $K_{\text{S}}^0$, and p, as well as (b)-(d) π^-, K^-, $K_{\text{S}}^0$, and $\overline{p}$ produced in (a)-(b) central (0–5% and 0–12%) and (c)-(d) peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV, where the spectra for different types of particles and for the same or similar particles in different conditions are multiplied by different amounts shown in the panels for clarity and normalization. The closed symbols represent the experimental data of the PHENIX Collaboration measured in |η| < 0.35 [25]. The open symbols represent the STAR data measured in |y| < 0.5 [26, 27], where the data for K⁺ and K^- are not available and the data for $K_{\text{S}}^0$ in (a)-(c) and (b)-(d) are the same. The solid, dashed, dotted, dashed-dotted, and dashed-dotted-dotted curves are our results calculated by using methods i), ii), iii), iv)_a, and iv)_b, respectively.

Figure 2 is the same as Fig. 1, but it shows the spectra, (1/N_EV) (2πp_T)^-1d²N/(dydp_T), (a)-(c) π⁺(π⁺+π^-), K⁺ (K⁺+K^-), and p ($p+\overline{p}$), as well as (b)-(d) π^- (π⁺+π^-), K^- (K⁺+K^-), and $\overline{p}(p+\overline{p}$) produced in (a)-(b) central (0–5%) and (c)-(d) peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV, where N_EV is on the vertical axis and denotes the number of events, which is usually omitted. The closed (open) symbols represent the experimental data of the ALICE Collaboration measured in |y| < 0.5 [28] (in |η| < 0.8 for the high p_T region and in |y|<0.5 for the low p_T region [29]). The data for π⁺+π^-, K⁺+K^-, and $p+\overline{p}$ in (a)-(c) and (b)-(d) are the same. One can see that, in most cases, all of the considered methods describe approximately the pT spectra of identified particles produced in central and peripheral Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV. Because the values of χ²/dof in most cases are greater than 2 and sometimes as large as 20.5, the fits in Figs. 1 and 2 are only approximate and qualitative. The large values of χ²/dof in the present work are caused by two factors which are the very small errors in the data and large dispersion between the curve and data in some cases. It is hard to reduce the values of χ²/dof in our fits.

Fig. 2.Full-screen View

(Color online) Same as Fig. 1, but showing the spectra of (a)-(c) π⁺ (π⁺+π^-), K⁺ (K⁺+K^-), and p ($p+\overline{p}$), as well as (b)-(d) π^- (π⁺+π^-), K^- (K⁺+K^-), and $\overline{p}$ ($p+\overline{p}$) produced in (a)-(b) central (0–5%) and (c)-(d) peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV, where N_EV on the vertical axis denotes the number of events, which is usually omitted. The closed (open) symbols represent the experimental data of the ALICE Collaboration measured in |y|<0.5 [28] (in |η|<0.8 for high p_T region and in |y|<0.5 for low p_T region [29]). The data for π⁺+π^-, K⁺+K^-, and $p+\overline{p}$ in (a)-(c) and (b)-(d) are the same.

In the above fits, we have an addition term of inverse power-law to account for hard process. This part contributes a small fraction to the p_T spectra, though the contribution coverage is wide. In the fitting procedure, according to the changing tendency of data in a low p_T range from 0 to 2 GeV/c, the part for the soft process can be well constrained first of all, though the contribution of the soft process can even reach 3.5 GeV/c. Then, the part for the hard process can be also constrained conveniently. In addition, in order to get a set of fitted parameters as accurately as possible, we use the least square method in the whole p_T coverage. It seems that different fitted parameters can be obtained in different p_T coverages. We should use a p_T coverage as widely as possible, especially for the extraction of the parameters related to the inverse power-law because a limited p_T coverage can not provide a good constraint of the inverse power-law and thus can easily drive the fitted parameters away from their physical meanings. In fact, for extractions of the effective temperature and transverse flow velocity which are the main topics of the present work, a not too wide p_T coverage, such as 0–2∼3 GeV/c, is enough due to the soft process contributing only in the low p_T region and the changing tendency of data in 0–2 GeV/c that takes part in a main role.

From the above fits one can see that, as a two-component function, Eq. (9) with different soft components can approximately describe the data in a wide p_T coverage. In addition, in our very recent work [37], we used method iii) to describe preliminarily the p_T spectra up to nearly 20 GeV/c. In another work [38], a two-Boltzmann distribution was used to describe the p_T spectra up to nearly 14 GeV/c. Generally, different sets of parameters are needed for different data. In particular, as it is pointed out in Ref. [39], more fitting parameters are needed in order to fit a wider p_T range of particle spectra. In the present work, we fit the particle spectra in a wide p_T range by introducing the inverse power-law to describe the high p_T region. The price to pay is 3 more parameters are added. In the two-component function, the contributions of soft and hard components have little effect in constraining respective free parameters due to different contributive regions, though the contribution fraction of the two components is the main role. This results in the p_T coverage having a small effect on T₀ and β_T. In fact, if we change the boundary of the low p_T region from 2 to 3 or 3.5 GeV/c, the variations of parameters can be neglected due to the tendency of the curve being mainly determined by the data in 0–2 GeV/c. Meanwhile, the data in 2–3.5 GeV/c obey naturally the tendency of the curve due to also the contribution or revision of the hard component. In other words, because of the revision of the hard component, the values of T₀ and β_T are not sensitive to the boundary of low p_T region. Although different p_T coverages obtained in different conditions can drive different fitted curves, these differences appear mainly in the high p_T region and do not largely effect the extraction of T₀ and β_T. In any case, we always use the last square method to extract the fitted parameters. In fact, the method used by us has the minimum randomness in the extractions of the fitted parameters.

It should be noted that although the conventional BGBW and TBW models have only 2–3 parameters to describe the p_T shape and usually fit several spectra simultaneously to reduce the correlation of the parameters, they seems to cover non-simultaneity of the kinetic freeze-outs of different particles. In the present work, although we use 3 more parameters to fit each spectrum individually, we observe an evidence of the mass dependent differential kinetic freeze-out scenario or multiple kinetic freeze-outs scenario [4, 16, 23]. The larger the temperature (mass) is, the earlier the particle produces. The average temperature (flow velocity and entropy index) of the kinetic freeze-outs for different particles is obtained by weighing different T₀ (β_T and q), where the weight factor is the normalization constant of each p_T spectrum. In the case of using the average temperature (flow velocity and entropy index) to fit the pion, kaon, and proton simultaneously to better constrain the parameters, larger values of χ²/dof are obtained.

Based on the descriptions of the p_T spectra, the first three methods can get T₀ and β_T, though the values of parameters are possibly inharmonious due to different methods. In particular, the value of T₀ obtained by method i) in peripheral collisions is larger than that in central collisions, which is different from methods ii) and iii) which obtain an opposite result. According to the conventional treatment in Refs. [11, 14], the values of β_T obtained by methods i) and ii) in peripheral collisions are taken to be nearly zero, which are different from method iii) which obtains a value of about 0.6c in both central and peripheral collisions.

To obtain the values of T₀, β_T, and β by methods iv)_a and iv)_b, we analyze the values of T presented in Tables 4 and 5, and calculate 〈p_T〉, 〈p〉, and $\overline{m}$ based on the values of parameters listed in Tables 4 and 5. In the calculations performed from p_T to 〈p〉 and $\overline{m}$ by the Monte Carlo method, an isotropic assumption in the rest frame of emission source is used [22-24]. In particular, $\overline{m}$ is in fact the mean energy, $\langle \sqrt{p^2+m_0^2}\rangle$.

The relations between T and m₀, 〈p_T〉, and $\overline{m}$, as well as 〈p〉 and $\overline{m}$ are shown in Figs. 3, 4, and 5, respectively, where panels (a) and (b) correspond to methods iv)_a and iv)_b which use the Boltzmann and Tsallis distributions respectively. Different symbols represent central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV and central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV respectively, where the centralities 0–5% and 0–12%, 80–92% and 60–80%, as well as 80–90% and 60–80% can be combined to 0–12%, 60–92%, and 60–90%, respectively. The symbols in Fig. 3 represent values of T listed in Tables 4 and 5 for a different m₀. The symbols in Figs. 4 and 5 represent values of 〈p_T〉 and 〈p〉 for different $\overline{m}$ respectively, which are calculated due to the parameters listed in Tables 4 and 5 and the isotropic assumption in the rest frame of the emission source. The solid and dashed lines in the three figures are the results fitted by the least square method for the positively and negatively charged particles respectively. The values of intercepts, slopes, and χ²/dof are listed in Tables 6 and 7 which correspond to methods iv)_a and iv)_b respectively. One can see that, in most cases, the mentioned relations are described by a linear function. In particular, the intercept in Fig. 3 is regarded as T₀, and the slopes in Figs. 4 and 5 are regarded as β_T and β respectively. The values of T, T₀, β_T, β, and $\overline{m}$ are approximately independent of isospin.

Fig. 3.Full-screen View

(Color online) Relations between T and m₀. Different symbols represent central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV and central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV respectively. The symbols presented in panels (a) and (b) represent the results listed in Tables 4 and 5 and correspond to the fits of Boltzmann and Tsallis distributions respectively, where the closed and open symbols show the results of positively and negatively charged particles respectively. The solid and dashed lines are the results fitted by the least square method for the positively and negatively charged particles respectively, where the intercepts are regarded as T₀.

Fig. 4.Full-screen View

(Color online) Same as Fig. 3, but showing the relations between 〈p_T〉 and $\overline{m}$, and the slopes are regarded as β_T. The symbols presented in panels (a) and (b) represent the results obtained according to the fits of Boltzmann and Tsallis distributions respectively, where the values of parameters are listed in Tables 4 and 5 respectively.

Fig. 5.Full-screen View

(Color online) Same as Fig. 3, but showing the relations between 〈p〉 and $\overline{m}$, and the slopes are regarded as β. The symbols presented in panels (a) and (b) represent the results obtained according to the fits of Boltzmann and Tsallis distributions respectively, where the values of parameters are listed in Tables 4 and 5 respectively.

To compare values of key parameters obtained by different methods for different centralities (both central and peripheral collisions), Figs. 6 and 7 show T₀ and β_T respectively, where panels (a) and (b) correspond to the results for central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV and central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV respectively. The closed and open symbols represent positively and negatively charged particles respectively, which are quoted from Tables 1, 2, 3, 6, and 7 which correspond to methods i), ii), iii), iv)_a, and iv)_b, respectively. In particular, the values of T₀ and β_T in the first three methods are obtained by weighing different particles. One can see that, by using method i), the value of T₀ in central collisions is smaller than that in peripheral collisions, and other methods present a larger T₀ in central collisions. Methods i) and ii) show a nearly zero β_T in peripheral collisions according to Refs. [11, 14], while other methods show a considerable β_T in both central and peripheral collisions.

Fig. 6.Full-screen View

(Color online) Comparisons of T₀ obtained by different methods for different centralities (C), where the values of T₀ in the first three methods are obtained by weighing different particles. Panels (a) and (b) correspond to the results for central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV and central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV respectively.

Fig. 7.Full-screen View

(Color online) Same as Fig. 6, but showing the comparisons of βT obtained by different methods for different centralities.

To explain the inconsistent results in T₀ and β_T for different methods, we re-examine the first two methods. It should be noticed that the same flow profile function, β(r) = β_S(r/R)ⁿ₀, and the same transverse flow velocity, β_T = 2β_S/(n₀+2), are used in the first two methods, though n₀=2 is used in method i) [11] and n₀=1 is used in method ii) [14] with the conventional treatment. As an insensitive quantity, although the radial size R of the thermal source in central collisions can be approximately regarded as the radius of a collision nucleus and in peripheral collisions R is not zero due to a few participant nucleons taking part in the interactions in which we can take approximate R to be 2.5 fm, both methods i) and ii) use a nearly zero β_T in peripheral collisions [11, 14]. If we consider a non-zero β_T in peripheral collisions for methods i) and ii), the situation will be changed.

By using a non-zero β_T in peripheral collisions for methods i) and ii), we re-analyze the data presented in Figs. 1 and 2. At the same time, to see the influences of different n₀ in the self-similar flow profile, we refit the mentioned p_T spectra by the first two methods with n₀=1 and 2 synchronously. The results re-analyzed by us are shown in Figs. 8 and 9 which correspond to 200 GeV Au-Au and 2.76 TeV Pb-Pb collisions respectively. The data points are the same as Figs. 1 and 2 [25-29]. The dotted, solid, dashed, and dotted-dashed curves correspond to the results of method i) with n₀=1 and 2, and of method ii) with n₀=1 and 2, respectively, where the results of method i) with n₀=2 and of method ii) with n₀=1 in central collisions are the same as Figs. 1 and 2. The values of related parameters and χ²/dof are listed in Tables 8 and 9, where the parameters for method i) with n₀=2 and for method ii) with n₀=1 in central collisions repeat those in Tables 1 and 2, which are not listed again. One can see that, after the re-examination, the values of T₀ in central collisions are larger than those in peripheral collisions. The values of β_T in peripheral collisions are no longer zero. These new results are consistent with other methods.

Fig. 8.Full-screen View

(Color online) Reanalyzing the transverse momentum spectra [25-27] collected in Fig. 1 by the first two methods. The dotted, solid, dashed, and dotted-dashed curves are our results calculated by using method i) with n₀=1 and 2, as well as method ii) with n₀=1 and 2, respectively. The results for central collisions obtained by method i) with n₀=2 and by method ii) with n₀=1 are the same as Fig. 1.

Fig. 9.Full-screen View

(Color online) Same as Fig. 8, but reanalyzing the transverse momentum spectra [28, 29] collected in Fig. 2 by the first two methods. The results for central collisions obtained by method i) with n₀=2 and by method ii) with n₀=1 are the same as Fig. 2.

To give new comparisons for T₀ and β_T, the new results obtained by the first two methods are shown in Figs. 10 and 11 respectively, where the results corresponding to method i) for central collisions with n₀=2 and to method ii) for central collisions with n₀=1 are the same as those in Figs. 6 and 7. Combing Figs. 6, 7, 10, and 11, one can see that the four methods show approximately the consistent results. These comparisons enlighten us to use the first two methods in peripheral collisions by a non-zero β_T. After the re-examination for β_T in peripheral collisions, we obtain a relatively larger T₀ in central collisions for the four methods. In particular, the parameter T₀ at the LHC is slightly larger than or nearly equal to that at the RHIC, not only for central collisions but also for peripheral collisions. Except for method iii), the methods show a slightly larger or nearly invariant β_T in central collisions when compared with peripheral collisions, and when LHC comparing data from LHC with the RHIC, while method iii) shows nearly the same β_T in different centralities and at different energies.

Fig. 10.Full-screen View

(Color online) Comparisons of T₀ obtained by the first two methods with n₀=1 and 2 for different centralities. Panels (a) and (b) correspond to the results for central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV and central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV respectively. The values of T₀ are obtained by weighing different particles and the results for central collisions obtained by method i) with n₀=2 and by method ii) with n₀=1 are the same as Fig. 6

Fig. 11.Full-screen View

(Color online) Same as Fig. 10, but showing the comparisons of β_T obtained by the first two methods for different centralities. The results for central collisions obtained by method i) with n₀=2 and by method ii) with n₀=1 are the same as Fig. 7

We would like to point out that, in the re-examination for β_T in methods i) and ii), we have assumed both β_T in central and peripheral collisions to be non-zero. In most cases [11, 14], both the conventional BGBW and TBW models used non-zero β_T in central collisions and zero (or almost zero) β_T in peripheral collisions. In the case of using a non-zero or zero (or almost zero) β_T in peripheral collisions, we can obtain a relatively smaller or larger T₀ compared with central collisions. Indeed, the selection of β_T in peripheral collisions is an important issue in both the BGBW and TBW models. In fact, β_T is a sensitive quantity which can affect T₀. The larger β_T that is selected, the smaller T₀ that is needed. The main correlation is between β_T and T₀, and the effect of n₀ is very small. In Figs. 1 and 2, we have used a zero β_T for peripheral collisions and obtained a harmonious result on the relative size of T₀ with Ref. [28] in which β_T (0.35c) for peripheral collisions is nearly a half of that (0.65c) for central collisions, and n₀ is also different from ours. While in Figs. 8 and 9, we have used a non-zero and slightly smaller β_T for peripheral collisions and obtained a different result from Ref. [28].

In order to make the conclusion more convincing, we can only fit the low p_T region of the particle spectra using the four methods with the same p_T cut to decrease the number of free fitting parameters. When the p_T cut increases from 2 to 3.5 GeV/c, T₀ (or T) increases or both T₀ (or T) and β_T increase slightly. The relative size of T₀ (β_T) obtained above for central and peripheral collisions is unchanged. In particular, β_T is also a sensitive quantity. For peripheral collisions, a zero or non-zero β_T in the first two methods can give different results. In our opinion, in central and peripheral collisions, it depends on β_T if we want to determine which T₀ is larger. We are inclined to use a non-zero β_T for peripheral collisions due to the small system which is similar to peripheral collisions in number of participant nucleons also showing collective expansion [40].

Compared with peripheral collisions, the larger T₀ in central collisions renders more deposition of collision energy and higher excitation of the interacting system due to more participating nucleons taking part in the violent collisions. Compared with the top RHIC energy, the larger T₀ at the LHC energy also renders more deposition of collision energy and higher excitation of interacting system due to higher $\sqrt{s_{\text{NN}}}$ at the LHC. At the same time, from the top RHIC to the LHC energies, a nearly invariant T₀ reflects the limiting deposition of collision energy. Compared with peripheral collisions, the slightly larger or nearly the same β_T in central collisions renders similar expansion in both the centralities. At the same time, at the top RHIC and LHC energies, the two systems also show similar expansion due to similar β_T.

It should be noted that, although Eq. (2) [14] does not implement the azimuthal integral over the freeze-out surface which gives rise to the modified Bessel functions in Eq. (1), it does not affect the extractions of kinetic freeze-out parameters due to the application of numerical integral. Although Eq. (3) [15, 16] assumes a single, infinitesimally thin shell of fixed flow velocity and also does not perform the integral over the freeze-out surface, it can extract the mean trend of kinetic freeze-out parameters. As for the alternative method [12, 17-20, 22-24], it assumes non-relativistic flow velocities in the expressions used to extract the freeze-out parameters, which is the case that β_T is indeed not too large at the top RHIC and LHC energies.

4 Conclusion

We summarize here our main observations and conclusions.

(a) The p_T spectra of π^±, K^±, $K_{\text{S}}^0$, p, and $\overline{p}$ produced in central (0–5% and 0–12%) and peripheral (80–92% and 60–80%) Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV and in central (0–5%) and peripheral (80–90% and 60–80%) Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76$ TeV have been analyzed by a few different superpositions in which the distributions related to the extractions of T₀ and β_T are used for the soft component and the inverse power law is used for the hard component. We have used five distributions for the soft component, i) the Blast-Wave model with Boltzmann-Gibbs statistics, ii) the Blast-Wave model with Tsallis statistics, iii) the Tsallis distribution with flow effect, iv)_a the Boltzmann distribution, and iv)_b the Tsallis distribution. The first three distributions are in fact three methods for the extractions of T₀ and β_T. The last two distributions are used in the fourth method, i.e. the alternative method.

(b) The experimental data measured by the PHENIX, STAR, and ALICE Collaborations are fitted by the model results. Our calculations show that the parameter T₀ obtained by method i) with the conventional treatment for central collisions is smaller than that for peripheral collisions, which is inconsistent with the results obtained by other model methods. In the conventional treatment, the parameter β_T in peripheral collisions is taken to be nearly zero, which results in a larger T₀ than normal. By using the conventional treatment, both methods i) and ii) show a nearly zero β_T value in the peripheral collisions according to Refs. [11, 14], while other methods show a considerable β_T in both central and peripheral collisions.

(c) In central and peripheral collisions, we have to select a suitable β_T so that we can determine which T₀ is larger. We are inclined to use a non-zero β_T for peripheral collisions due to the small system also showing collective expansion. We have given a re-examination for β_T in peripheral collisions in methods i) and ii) in which β_T is taken to be (0.40±0.07)c. By using a non-zero β_T, the first two methods show approximately consistent results with other methods, not only for T₀ but also for β_T, though method iii) gives a larger β_T. We have uniformly obtained a larger T₀ in central collisions by the four methods. In particular, the parameter T₀ at the LHC is larger than or equal to that at the RHIC. Except for method iii), the methods show a slightly larger or nearly invariant β_T in central collisions compared to peripheral collisions, and at the LHC compared with the RHIC.

(d) The new results obtained by the widely used Blast-Wave model with Boltzmann-Gibbs or Tsallis statistics are in agreement with those obtained by the newly used alternative method which uses the Boltzmann or Tsallis distribution. This consistency confirms the validity of the alternative method. The result that the central collisions have a larger T₀ renders more deposition of collision energy and higher excitation of the interacting system due to more participating nucleons taking part in the violent collisions. From the RHIC to LHC, the slightly increased or nearly invariant T₀ renders the limiting or maximum deposition of collisions energy. From central to peripheral collisions and from the RHIC to LHC, the slightly increased or nearly invariant β_T renders the limiting or maximum blast of the interacting system.