Observation of different scenarios in different temperatures in small and large collision systems

Muhammad Waqas¹¹¹1[email protected], [email protected], Lu-Meng Liu²²²2[email protected], Guang-Xiong Peng^1,3,4³³3Correspondence: [email protected], Muhammad Ajaz⁵⁴⁴4Correspondence: [email protected], [email protected] ¹ School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
² School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
³ Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Beijing 100049, China
⁴ Synergetic Innovation Center for Quantum Effects & Applications, Hunan Normal University, Changsha 410081, China
⁵ Department of Physics, Abdul Wali Khan University Mardan, 23200 Mardan, Pakistan

Abstract

Abstract: We used the modified Hagedron function and analyzed the experimental data measured by the BRAHMS, STAR, PHENIX and ALICE Collaborations in Copper-Copper, Gold-Gold, deuteron-Gold, Lead-Lead, proton-Lead and proton-proton collisions, and extracted the related parameters (kinetic freeze-out temperature, transverse flow velocity, kinetic freeze-out volume, mean transverse momentum and initial temperature) from the transverse momentum spectra of the particles (non-strange and strange particles). We observed that all the above parameters decrease from central to peripheral collisions, except transverse flow velocity which remains unchanged from central to peripheral collisions. The kinetic freeze-out temperature depends on the cross-section interaction of the particle such that larger cross-section of the particle corresponds to smaller $T_{0}$ , and reveals the two kinetic freeze-out scenario, while the initial temperature depends on the mass of the particle and it increase with the particle mass. The transverse flow velocity and mean transverse momentum depends on the mass of the particle and the former decrease while the later increase with the particle mass. In addition, the kinetic kinetic freeze-out volume also decrease with particle mass which reveals the volume differential freeze-out scenario and indicates different freeze-out surfaces for different particles. We also extracted the entropy index-parameter $n$ and the parameter $N_{0}$ , and the former remains almost unchanged while the later decrease from central to peripheral collisions. Furthermore, the kinetic freeze-out temperature, transverse flow velocity, kinetic freeze-out volume, initial temperature, mean transverse momentum and the parameter $N_{0}$ at LHC are larger than that of RHIC, and they show their dependence on the collision cross-section as well as on collision energy at RHIC and LHC.

Keywords: bulk properties of nuclear matter; transverse momentum spectra; strange and non-strange particles; kinetic freeze-out temperature; initial temperature; kinetic freeze-out volume.

PACS numbers: 12.40.Ee, 13.85.Hd, 24.10.Pa

I Introduction

A suitable tool for the production of the hot and dense matter namely Quark-Gluon Plasma (QGP) in the laboratory 1 ; 2 is the high energy collisions. This state of matter is formed in the initial stages of collision, and it survives for a short period of time (7-10 fm/c) and then rapidly transforms into a hadron gas system. Because of the multi-partonic interactions throughout the evolution of the collision, the information about the initial condition of the system gets lost. In order to obtain the final state behavior of such a colliding system, we need the measurement of number and identity of the produced particles along with their energy and momentum spectra. The final state information are very useful to understand the production mechanism of particles and the nature of produced matter in these high energy collisions.

The chemical and kinetic freeze-out are the two symbolic freeze-out conditions where the space-time evolution of the fireball produced in the collision cease. The colliding medium first reaches chemical equilibrium, where the inelastic scattering stops due to expansion of the system which results in the stabilization of the particle chemistry in the fireball. This stage is called chemical freeze-out stage and the temperature at this stage is known as chemical freeze-out temperature $(T_{ch})$ 4 . The relative fractions of the particles are fixed but they still interact with each other, and then stops until the final state interactions between them are no longer effective. This stage is called thermal/kinetic freeze-out stage and the temperature at this stage is known as kinetic freeze-out temperature 5 ; 6 ; 7 ; 8 . In Standard model of heavy ion collisions, the kinetic freeze-out occurs after the chemical freeze-out due to large mean free path of the later which is also claimed in ref. 9 . Generally, the transverse momentum ( $p_{T}$ ) spectra as well as the yields of the produced hadrons constitute some basic measurements for the extraction of parameters of chemical and kinetic freeze-out.

A multiple chemical freeze-out scenario with the early fixing of chemical composition of strange hadron compared to non-strange light hadrons is advocated in ref. 10 . The early chemical freeze-out of strange hadrons is due to their small inelastic cross-section. A natural question arise in mind whether a similar hierarchy structure also occurs at kinetic decoupling, or the mass dependent hierarchy occurs in case of kinetic freeze-out because the variation in medium induced momentum for heavy hadrons would be smaller than the lighter hadrons and therefore, with the decrease of temperature of the fireball, the earlier kinetic decoupling of heavy hadrons is expected.

In the present article, we will analyze the bulk properties in terms of kinetic freeze-out temperature ( $T_{0}$ ), transverse flow velocity ( $\beta_{T}$ ) and kinetic freeze-out volume ( $V$ ). All these mentioned parameters are discussed in detail in various literatures 11 ; 12 ; 13 ; 14 . In the present work, we choose different collision systems such as small and large systems in order to check the dependence of the above parameters on the size of interacting system and different particles are chosen in order to check the differences in different particle emissions.

Before going to the next section, we would like to point out that we also analyse the initial temperature which occurs in the initial stage of collisions and which is also important to study because the phenomenon in the initial stages of collisions and the freeze-out stages is different.

The remainder of the paper consists of formalism and method, results discussions and conclusions.

II Formalism and method

The structure of transverse momentum spectra ( $p_{T}$ ) of the charged particles is very complex, and therefore is distributed into various regions. Especially, when the $p_{T}$ range approaches to 100 GeV/c at LHC collisions 1a . There are different $p_{T}$ regions according to the model analysis 2a . The first $p_{T}$ region include $p_{T}$ $<$ 4-6 GeV/c and second region include 4-6 GeV/c $<$ $p_{T}$ $<$ 17-20 GeV/c, while the third region includes $p_{T}$ $>$ 17-20 GeV/c. It is believed that different $p_{T}$ regions (soft, hard and very hard $p_{T}$ region) signify different interacting mechanisms. There are different explanations due to different models and methods even for the same $p_{T}$ region. In the present work, the maximum $p_{T}$ range is 7-8 GeV/c, therefore we will be limited to the soft and hard process and will skip the very hard process. There is a very soft region also which will be also skipped and it corresponds to $p_{T}$ $<$ 0.5 GeV/c which involves the resonance production that is not described by the model, but one can read ref. 3a ; 4a for more details of very soft and very hard processes. Generally, the soft excitation process is distributed in a narrow $p_{T}$ range of less than 2-3 GeV/c or a little more, and mostly light flavor particles are produced in this region. The soft process have many choices of formalisms such as Blast wave model with Boltzmann Gibb’s statistics 5a ; 6a ; 7a ; 8a , Blast wave model with Tsallis statistics 9a ; 10a ; 11a ; 12a ,Tsallis pareto-type function 12aa , Erlang distribution 13a ; 14a ; 15a , Scwinger mechanism 16a ; 17a ; 18a ; 19a , Hagedorn distribution function 20a , Standard distribution 21a etc. The $p_{T}$ region above 3 GeV/c is contributed for the hard process and is described by Quantum chromodynamics (QCD) calculus 22a ; 23a ; 24a or inverse power law which is also known as Hagedron function

\displaystyle f_{0}(p_{T})=

\displaystyle\frac{1}{N}\frac{\mathrm{d}N}{\mathrm{d}p_{\mathrm{T}}}=Ap_{T}\bigg{(}1+\frac{p_{T}}{p_{0}}\bigg{)}^{-n},

(1)

where $A$ , is the normalization constant, and $p_{0}$ and $n$ are free parameters. This function is revised in different forms in 25a ; 26a ; 27a ; 28a ; 29a ; 30a and each has different significance.

In the present work, we use the Hagedorn function with embeded transverse flow velocity 31a , which can be expressed as

	$\displaystyle f_{0}(p_{T})=$	$\displaystyle\frac{1}{N}\frac{\mathrm{d}N}{\mathrm{d}p_{\mathrm{T}}}=2\pi Cp_{T}\bigg{[}1+<\gamma_{t}>$
	$\displaystyle\frac{(m_{T}-p_{T}<\beta_{T}>)}{nT_{0}}\bigg{]}^{-n}$			(2)

where $C$ stands for the normalization constant that leads the integral in Eq. (2) to be normalized to 1, $m_{T}$ is the transverse mass and is equal to $\sqrt{p_{T}^{2}+m_{0}^{2}}$ , $m_{0}$ is the rest mass of the particle, In Eq. 2, $C$ = $gV/(2\pi)^{2}$ . So Eq. 2 becomes as

	$\displaystyle f_{0}(p_{T})=$	$\displaystyle\frac{1}{N}\frac{\mathrm{d}N}{\mathrm{d}p_{\mathrm{T}}}=\frac{gV}{2\pi}p_{T}\bigg{[}1+<\gamma_{t}>$
	$\displaystyle\frac{(m_{T}-p_{T}<\beta_{T}>)}{nT_{0}}\bigg{]}^{-n}$			(3)

where $g$ is the degeneracy factor which differs for every particle, depends on their spin based on $g_{n}$ =2 $S_{n}$ +1.

Table 1. Values of free parameters ( $T$ and $q$ ), normalization constant ( $\sigma_{0}$ ), $\chi^{2}$ , and ndof corresponding to the curves in Figure 1 for $pp$ collisions at 200 GeV, where the spectrum form and particle mass are given together. In the last column, “–” for ndof denotes the case which has the number of data points being less than or equal to 0 and the curve is only to guide the eyes. After rounding, the value of $\chi^{2}$ is taken to be an integer. If the integer is 0, we keep one decimal; If the one decimal is 0.0, we keep two decimals, and so on.

Collisions	Centrality	Particle	$T_{0}$ (GeV)	$\beta_{T}$ (c)	$V(fm^{3})$	$n$	$N_{0}$	$\chi^{2}$ / dof
Fig. 1	0–10%	$\pi^{+}$	$0.057\pm 0.006$	$0.309\pm 0.012$	$2200\pm 150$	$10\pm 1$	$55\pm 3$	0.5/8
Cu-Cu	10–30%	–	$0.052\pm 0.005$	$0.309\pm 0.009$	$2100\pm 130$	$10\pm 1.2$	$33\pm 2$	0.6/8
200 GeV	30–50%	–	$0.049\pm 0.005$	$0.309\pm 0.010$	$2000\pm 130$	$10\pm 1$	$14\pm 1$	1.5/8
	50–70%	–	$0.045\pm 0.004$	$0.309\pm 0.010$	$1874\pm 114$	$10\pm 1.2$	$5.8\pm 0.8$	0.5/8
	0–10%	$\pi^{-}$	$0.057\pm 0.006$	$0.309\pm 0.012$	$2200\pm 150$	$10\pm 1$	$55\pm 3$	0.5/8
	10–30%	–	$0.052\pm 0.005$	$0.309\pm 0.009$	$2100\pm 130$	$10\pm 1.2$	$33\pm 2$	0.6/8
	30–50%	–	$0.049\pm 0.005$	$0.309\pm 0.010$	$2000\pm 130$	$10\pm 1$	$14\pm 1$	1.5/8
	50–70%	–	$0.045\pm 0.004$	$0.309\pm 0.010$	$1874\pm 114$	$10\pm 1.2$	$5.8\pm 0.8$	0.5/8
	0–10%	$K^{+}$	$0.076\pm 0.004$	$0.289\pm 0.009$	$2050\pm 110$	$11\pm 1.4$	$0.036\pm 0.005$	2/7
	10–30%	–	$0.072\pm 0.004$	$0.289\pm 0.008$	$1938\pm 106$	$11\pm 1.3$	$0.009\pm 0.0005$	4/7
	30–50%	–	$0.064\pm 0.006$	$0.289\pm 0.009$	$1841\pm 116$	$11\pm 1.2$	$3.8\times 10^{-3}\pm 5\times 10^{-4}$	3/7
	50–70%	–	$0.060\pm 0.004$	$0.289\pm 0.010$	$1741\pm 111$	$11\pm 1.2$	$3.5\times 10^{-3}\pm 5\times 10^{-4}$	8/7
	0–10%	$K^{-}$	$0.076\pm 0.004$	$0.289\pm 0.009$	$2050\pm 110$	$11\pm 1.4$	$0.036\pm 0.005$	2/7
	10–30%	–	$0.072\pm 0.004$	$0.289\pm 0.008$	$1938\pm 106$	$11\pm 1.3$	$0.009\pm 0.0005$	4/7
	30–50%	–	$0.064\pm 0.006$	$0.289\pm 0.009$	$1841\pm 116$	$11\pm 1.2$	$3.8\times 10^{-3}\pm 5\times 10^{-4}$	3/7
	50–70%	–	$0.060\pm 0.004$	$0.289\pm 0.010$	$1741\pm 111$	$11\pm 1.2$	$3.5\times 10^{-3}\pm 5\times 10^{-4}$	8/7
	0–10%	$p$	$0.058\pm 0.006$	$0.267\pm 0.012$	$1857\pm 123$	$15\pm 1.7$	$8\times 10^{-5}\pm 6\times 10^{-6}$	17/8
	10–30%	–	$0.053\pm 0.005$	$0.267\pm 0.010$	$1730\pm 124$	$15\pm 2$	$5\times 10^{-5}\pm 4\times 10^{-6}$	22/8
	30–50%	–	$0.050\pm 0.004$	$0.267\pm 0.010$	$1600\pm 126$	$15\pm 2$	$2\times 10^{-5}\pm 4\times 10^{-6}$	17/8
	50–70%	–	$0.046\pm 0.005$	$0.267\pm 0.010$	$1500\pm 130$	$15\pm 2.1$	$7\times 10^{-6}\pm 4\times 10^{-7}$	43/8
	0–10%	$\bar{p}$	$0.058\pm 0.006$	$0.267\pm 0.012$	$1857\pm 123$	$15\pm 1.7$	$8\times 10^{-5}\pm 6\times 10^{-6}$	17/8
	10–30%	–	$0.053\pm 0.005$	$0.267\pm 0.010$	$1730\pm 124$	$15\pm 2$	$5\times 10^{-5}\pm 4\times 10^{-6}$	22/8
	30–50%	–	$0.050\pm 0.004$	$0.267\pm 0.010$	$1600\pm 126$	$15\pm 2$	$2\times 10^{-5}\pm 4\times 10^{-6}$	17/8
	50–70%	–	$0.046\pm 0.005$	$0.267\pm 0.010$	$1500\pm 130$	$15\pm 2.1$	$7\times 10^{-6}\pm 4\times 10^{-7}$	43/8
	0–10%	$\Lambda$	$0.076\pm 0.005$	$0.254\pm 0.010$	$1713\pm 108$	$15.4\pm 1.5$	$2.4\pm 0.3$	22/14
	10–20%	–	$0.072\pm 0.004$	$0.254\pm 0.010$	$1610\pm 104$	$15.4\pm 1.8$	$0.165\pm 0.03$	29/14
	20–30%	–	$0.068\pm 0.004$	$0.254\pm 0.011$	$1500\pm 111$	$15.4\pm 2.1$	$0.012\pm 0.004$	20/14
	30–40%	–	$0.064\pm 0.004$	$0.254\pm 0.011$	$1410\pm 120$	$15.4\pm 2.2$	$8\times 10^{-4}\pm 5\times 10^{-5}$	31/14
	40–60%	–	$0.060\pm 0.006$	$0.254\pm 0.010$	$1300\pm 120$	$15.4\pm 2.2$	$4.3\times 10^{-5}\pm 4\times 10^{-6}$	14/14
	0–10%	$\Xi$	$0.076\pm 0.004$	$0.244\pm 0.009$	$1500\pm 115$	$15.5\pm 1.5$	$0.24\pm 0.04$	25/6
	10–20%	–	$0.072\pm 0.004$	$0.244\pm 0.010$	$1420\pm 104$	$15.5\pm 2.2$	$0.016\pm 0.005$	10/6
	20–30%	–	$0.069\pm 0.004$	$0.244\pm 0.012$	$1290\pm 126$	$15.5\pm 2.2$	$0.001\pm 0.0004$	33/6
	30–40%	–	$0.065\pm 0.005$	$0.244\pm 0.011$	$1200\pm 100$	$15.5\pm 2.2$	$8\times 10^{-5}\pm 5\times 10^{-6}$	17/6
	40–60%	–	$0.061\pm 0.004$	$0.244\pm 0.011$	$1120\pm 100$	$15.5\pm 2.2$	$4\times 10^{-6}\pm 6\times 10^{-7}$	15/6
Fig. 2	0–5%	$\pi^{+}$	$0.060\pm 0.005$	$0.325\pm 0.012$	$2357\pm 170$	$10.4\pm 1.5$	$270\pm 32$	0.2/5
Au-Au	5–10%	–	$0.056\pm 0.006$	$0.325\pm 0.010$	$2268\pm 140$	$10.4\pm 1.5$	$210\pm 18$	0.8/5
62.4 GeV	10–20%	–	$0.052\pm 0.004$	$0.325\pm 0.010$	$2150\pm 130$	$10.4\pm 1$	$170\pm 10$	0.5/5
	20–30%	–	$0.048\pm 0.005$	$0.325\pm 0.010$	$2035\pm 120$	$10.4\pm 2$	$146\pm 8$	0.5/5
	30–40%	–	$0.045\pm 0.006$	$0.325\pm 0.011$	$1955\pm 110$	$10.4\pm 1.5$	$106\pm 10$	1.3/5
	40–50%	–	$0.042\pm 0.005$	$0.325\pm 0.011$	$1775\pm 132$	$10.4\pm 1.2$	$62\pm 8$	1.5/5
	50–60%	–	$0.040\pm 0.004$	$0.325\pm 0.009$	$1702\pm 108$	$10.4\pm 1.8$	$38\pm 4$	2.5/5
	60–70%	–	$0.037\pm 0.004$	$0.325\pm 0.011$	$1620\pm 110$	$10.4\pm 1.6$	$21.7\pm 3.5$	5.5/5
	70–80%	–	$0.034\pm 0.004$	$0.325\pm 0.011$	$1530\pm 88$	$10.8\pm 1.6$	$11.6\pm 2$	2.5/5
	0–5%	$K^{+}$	$0.076\pm 0.006$	$0.312\pm 0.010$	$2100\pm 120$	$11\pm 1.5$	$26.4\pm 3.8$	4/5
	5–10%	–	$0.073\pm 0.006$	$0.312\pm 0.010$	$2000\pm 100$	$11\pm 1.3$	$22.4\pm 1.3$	3/5
	10–20%	–	$0.070\pm 0.005$	$0.312\pm 0.011$	$1900\pm 100$	$11\pm 1.4$	$17.7\pm 2$	5/5
	20–30%	–	$0.067\pm 0.005$	$0.312\pm 0.011$	$1813\pm 104$	$11\pm 1.2$	$13\pm 2$	4/5
	30–40%	–	$0.064\pm 0.005$	$0.312\pm 0.010$	$1730\pm 108$	$11\pm 1.4$	$9.2\pm 1.4$	4/5
	40–50%	–	$0.061\pm 0.004$	$0.312\pm 0.010$	$1650\pm 102$	$11\pm 1.3$	$6.4\pm 1.2$	10/5
	50–60%	–	$0.057\pm 0.004$	$0.312\pm 0.010$	$1560\pm 85$	$11\pm 1.2$	$3.8\pm 0.4$	2/5
	60–70%	–	$0.054\pm 0.005$	$0.312\pm 0.009$	$1481\pm 70$	$11\pm 1.1$	$2.2\pm 0.4$	10/5
	70–80%	–	$0.050\pm 0.006$	$0.312\pm 0.009$	$1400\pm 80$	$11\pm 1.2$	$0.9\pm 0.03$	13/5

Table 1. Continue.

Collisions	Centrality	Particle	$T_{0}$ (GeV)	$\beta_{T}$ (c)	$V(fm^{3})$	$n$	$N_{0}$	$\chi^{2}$ / dof
	0–5%	$p$	$0.060\pm 0.005$	$0.280\pm 0.010$	$1930\pm 108$	$8.4\pm 1$	$7\pm 0.8$	9/9
	5–10%	–	$0.056\pm 0.005$	$0.280\pm 0.011$	$1843\pm 110$	$9\pm 0.6$	$6.1\pm 0.6$	2/9
	10–20%	–	$0.053\pm 0.005$	$0.280\pm 0.009$	$1739\pm 120$	$9\pm 0.4$	$5\pm 0.5$	1/9
	20–30%	–	$0.049\pm 0.004$	$0.280\pm 0.010$	$1627\pm 109$	$9\pm 1$	$3.7\pm 0.4$	3/9
	30–40%	–	$0.045\pm 0.006$	$0.280\pm 0.012$	$1500\pm 100$	$11\pm 1$	$2.8\pm 0.5$	2/9
	40–50%	–	$0.042\pm 0.005$	$0.280\pm 0.012$	$1385\pm 100$	$11.5\pm 1.5$	$1.95\pm 0.3$	2/9
	50–60%	–	$0.040\pm 0.005$	$0.280\pm 0.011$	$1300\pm 93$	$13\pm 1.6$	$1.3\pm 0.04$	1/9
	60–70%	–	$0.038\pm 0.004$	$0.280\pm 0.010$	$1212\pm 80$	$15\pm 1.4$	$0.745\pm 0.04$	3/9
	70–80%	–	$0.035\pm 0.005$	$0.280\pm 0.010$	$1100\pm 60$	$16\pm 1.5$	$0.37\pm 0.03$	4/9
	0–5%	$\Lambda$	$0.076\pm 0.006$	$0.260\pm 0.012$	$1800\pm 126$	$15\pm 1.5$	$7.6\pm 0.5$	5/7
	5–10%	–	$0.073\pm 0.006$	$0.260\pm 0.009$	$1700\pm 100$	$15\pm 1.6$	$0.61\pm 0.06$	11/7
	10–20%	–	$0.070\pm 0.006$	$0.260\pm 0.011$	$1610\pm 100$	$15\pm 1.5$	$0.058\pm 0.005$	8/7
	20–30%	–	$0.067\pm 0.005$	$0.260\pm 0.013$	$1523\pm 96$	$15\pm 1.7$	$0.0033\pm 0.0004$	8/7
	30–40%	–	$0.064\pm 0.005$	$0.260\pm 0.010$	$1450\pm 70$	$15\pm 1.4$	$2.5\times 10^{-4}\pm 4\times 10^{-5}$	3/7
	40–60%	–	$0.060\pm 0.004$	$0.260\pm 0.011$	$1329\pm 80$	$15\pm 1.6$	$9.4\times 10^{-6}\pm 5\times 10^{-7}$	16/7
	60–80%	–	$0.055\pm 0.004$	$0.260\pm 0.010$	$1200\pm 85$	$15\pm 1.6$	$2.9\times 10^{-7}\pm 4\times 10^{-8}$	3/7
	0–5%	$\Xi^{-}$	$0.076\pm 0.006$	$0.249\pm 0.012$	$1680\pm 100$	$13\pm 1.2$	$0.33\pm 0.04$	2/5
	5–10%	–	$0.073\pm 0.006$	$0.249\pm 0.009$	$1600\pm 100$	$14\pm 1.2$	$0.033\pm 0.004$	4/5
	10–20%	–	$0.070\pm 0.006$	$0.249\pm 0.011$	$1500\pm 80$	$14\pm 1.3$	$0.0024\pm 0.0005$	4/5
	20–30%	–	$0.067\pm 0.005$	$0.249\pm 0.013$	$1400\pm 70$	$14\pm 1.5$	$1.6\times 10^{-4}\pm 5\times 10^{-5}$	1.5/5
	30–40%	–	$0.064\pm 0.005$	$0.249\pm 0.010$	$1320\pm 70$	$15\pm 1.5$	$4.4\times 10^{-6}\pm 5\times 10^{-7}$	3/5
	40–60%	–	$0.060\pm 0.004$	$0.249\pm 0.011$	$1200\pm 80$	$15\pm 1.5$	$1.4\times 10^{-7}\pm 5\times 10^{-8}$	1.5/5
Fig. 3	0–20%	$\pi^{+}$	$0.058\pm 0.004$	$0.315\pm 0.009$	$2250\pm 100$	$9.8\pm 1.3$	$0.075\pm 0.005$	2/19
d-Au	20–40%	–	$0.055\pm 0.004$	$0.315\pm 0.009$	$2170\pm 104$	$9.8\pm 1.3$	$0.006\pm 0.0004$	2/19
200 GeV	40–60%	–	$0.048\pm 0.005$	$0.315\pm 0.011$	$1954\pm 110$	$9.8\pm 1.2$	$5\times 10^{-5}\pm 5\times 10^{-6}$	0.6/19
	60–80%	–	$0.044\pm 0.004$	$0.315\pm 0.009$	$1843\pm 123$	$9.8\pm 1.4$	$1.8\times 10^{-6}\pm 4\times 10^{-7}$	3/19
	0–20%	$\pi^{-}$	$0.058\pm 0.004$	$0.315\pm 0.009$	$2250\pm 100$	$9.8\pm 1.3$	$0.075\pm 0.005$	2/19
	20–40%	–	$0.055\pm 0.004$	$0.315\pm 0.009$	$2170\pm 104$	$9.8\pm 1.3$	$0.006\pm 0.0004$	2/19
	40–60%	–	$0.048\pm 0.005$	$0.315\pm 0.011$	$1954\pm 110$	$9.8\pm 1.2$	$5\times 10^{-5}\pm 5\times 10^{-6}$	0.6/19
	60–80%	–	$0.044\pm 0.004$	$0.315\pm 0.009$	$1843\pm 123$	$9.8\pm 1.4$	$1.8\times 10^{-6}\pm 4\times 10^{-7}$	3/19
	0–20%	$K^{+}$	$0.078\pm 0.005$	$0.296\pm 0.010$	$2119\pm 110$	$9.4\pm 1.2$	$0.0055\pm 0.0005$	1.4/14
	20–40%	–	$0.074\pm 0.006$	$0.296\pm 0.011$	$2041\pm 121$	$9.4\pm 1.4$	$4.1\times 10^{-4}\pm 5\times 10^{-5}$	1/14
	40–60%	–	$0.067\pm 0.006$	$0.296\pm 0.013$	$1817\pm 116$	$9.4\pm 1.4$	$3.6\times 10^{-6}\pm 5\times 10^{-7}$	1/14
	60–80%	–	$0.062\pm 0.005$	$0.296\pm 0.011$	$1703\pm 120$	$9.8\pm 1.1$	$1.4\times 10^{-7}\pm 6\times 10^{-8}$	3/14
	0–20%	$K^{-}$	$0.078\pm 0.005$	$0.296\pm 0.010$	$2119\pm 110$	$9.4\pm 1.2$	$0.0055\pm 0.0005$	1.4/14
	20–40%	–	$0.074\pm 0.006$	$0.296\pm 0.011$	$2041\pm 121$	$9.4\pm 1.4$	$4.1\times 10^{-4}\pm 5\times 10^{-5}$	1/14
	40–60%	–	$0.067\pm 0.006$	$0.296\pm 0.013$	$1817\pm 116$	$9.4\pm 1.4$	$3.6\times 10^{-6}\pm 5\times 10^{-7}$	1/14
	60–80%	–	$0.062\pm 0.005$	$0.296\pm 0.011$	$1703\pm 120$	$9.8\pm 1.1$	$1.4\times 10^{-7}\pm 6\times 10^{-8}$	3/14
	0–20%	$p$	$0.061\pm 0.006$	$0.283\pm 0.008$	$1920\pm 109$	$11\pm 1.5$	$7.3\times 10^{-4}\pm 6\times 10^{-5}$	4.5/19
	20–40%	–	$0.057\pm 0.004$	$0.283\pm 0.010$	$1834\pm 101$	$11\pm 1.5$	$5.5\times 10^{-5}\pm 5\times 10^{-6}$	2/19
	40–60%	–	$0.048\pm 0.005$	$0.283\pm 0.011$	$1600\pm 106$	$11\pm 1.2$	$5.1\times 10^{-7}\pm 4\times 10^{-8}$	1/19
	60–80%	–	$0.045\pm 0.004$	$0.283\pm 0.010$	$1500\pm 102$	$12\pm 1.4$	$2.4\times 10^{-8}\pm 5\times 10^{-9}$	2/19
	0–20%	$\bar{p}$	$0.061\pm 0.006$	$0.283\pm 0.008$	$1920\pm 109$	$11\pm 1.5$	$7.3\times 10^{-4}\pm 6\times 10^{-5}$	4.5/19
	20–40%	–	$0.057\pm 0.004$	$0.283\pm 0.010$	$1834\pm 101$	$11\pm 1.5$	$5.5\times 10^{-5}\pm 5\times 10^{-6}$	2/19
	40–60%	–	$0.048\pm 0.005$	$0.283\pm 0.011$	$1600\pm 106$	$11\pm 1.2$	$5.1\times 10^{-7}\pm 4\times 10^{-8}$	1/19
	60–80%	–	$0.045\pm 0.004$	$0.283\pm 0.010$	$1500\pm 102$	$12\pm 1.4$	$2.4\times 10^{-8}\pm 5\times 10^{-9}$	2/19

Table 1. Continue.

Collisions	Centrality	Particle	$T_{0}$ (GeV)	$\beta_{T}$ (c)	$V(fm^{3})$	$n$	$N_{0}$	$\chi^{2}$ / dof
Fig. 4	0–5%	$\pi^{+}$	$0.108\pm 0.005$	$0.443\pm 0.011$	$3500\pm 140$	$9.8\pm 1$	$2.3\times 10^{5}\pm 5\times 10^{4}$	34/36
Pb-Pb	5–10%	–	$0.104\pm 0.004$	$0.443\pm 0.010$	$3360\pm 143$	$9.8\pm 1$	$1.1\times 10^{5}\pm 4\times 10^{4}$	40/36
2.76 TeV	10–20%		$0.100\pm 0.005$	$0.443\pm 0.010$	$3237\pm 136$	$9.8\pm 1.2$	$4\times 10^{4}\pm 6\times 10^{3}$	21/36
	20–30%	–	$0.096\pm 0.006$	$0.443\pm 0.012$	$3100\pm 124$	$9.8\pm 1.3$	$1.4\times 10^{4}\pm 4\times 10^{3}$	29/36
	30–40%	–	$0.092\pm 0.006$	$0.443\pm 0.010$	$2974\pm 109$	$9.8\pm 1$	$5\times 10^{3}\pm 5\times 10^{2}$	40/36
	40–50%	–	$0.089\pm 0.004$	$0.443\pm 0.012$	$2854\pm 108$	$9.8\pm 1$	$1.6\times 10^{3}\pm 3\times 10^{2}$	35/36
	50–60%	–	$0.086\pm 0.005$	$0.443\pm 0.010$	$2719\pm 120$	$9.8\pm 1.1$	$450\pm 21$	34/36
	60–70%	–	$0.082\pm 0.006$	$0.443\pm 0.012$	$2600\pm 132$	$10.3\pm 1.3$	$150\pm 10$	55/36
	70–80%	–	$0.078\pm 0.004$	$0.443\pm 0.012$	$2500\pm 100$	$10.3\pm 1.2$	$34\pm 4$	71/36
	80–90%	–	$0.074\pm 0.005$	$0.443\pm 0.010$	$2300\pm 161$	$10.3\pm 1.4$	$7.5\pm 0.4$	73/36
	0–5%	$\pi^{-}$	$0.108\pm 0.005$	$0.443\pm 0.011$	$3500\pm 140$	$9.8\pm 1$	$2.3\times 10^{5}\pm 5\times 10^{4}$	34/36
	5–10%	–	$0.104\pm 0.004$	$0.443\pm 0.010$	$3360\pm 143$	$9.8\pm 1$	$1.1\times 10^{5}\pm 4\times 10^{4}$	40/36
	10–20%		$0.100\pm 0.005$	$0.443\pm 0.010$	$3237\pm 136$	$9.8\pm 1.2$	$4\times 10^{4}\pm 6\times 10^{3}$	21/36
	20–30%	–	$0.096\pm 0.006$	$0.443\pm 0.012$	$3100\pm 124$	$9.8\pm 1.3$	$1.4\times 10^{4}\pm 4\times 10^{3}$	29/36
	30–40%	–	$0.092\pm 0.006$	$0.443\pm 0.010$	$2974\pm 109$	$9.8\pm 1$	$5\times 10^{3}\pm 5\times 10^{2}$	40/36
	40–50%	–	$0.089\pm 0.004$	$0.443\pm 0.012$	$2854\pm 108$	$9.8\pm 1$	$1.6\times 10^{3}\pm 3\times 10^{2}$	35/36
	50–60%	–	$0.086\pm 0.005$	$0.443\pm 0.010$	$2719\pm 120$	$9.8\pm 1.1$	$450\pm 21$	34/36
	60–70%	–	$0.082\pm 0.006$	$0.443\pm 0.012$	$2600\pm 132$	$10.3\pm 1.3$	$150\pm 10$	55/36
	70–80%	–	$0.078\pm 0.004$	$0.443\pm 0.012$	$2500\pm 100$	$10.3\pm 1.2$	$34\pm 4$	71/36
	80–90%	–	$0.074\pm 0.005$	$0.443\pm 0.010$	$2300\pm 161$	$10.3\pm 1.4$	$7.5\pm 0.4$	73/36
	0–5%	$K^{+}$	$0.129\pm 0.004$	$0.421\pm 0.009$	$3319\pm 138$	$9.3\pm 1.2$	$1.9\times 10^{4}\pm 4\times 10^{3}$	40/31
	5–10%	–	$0.126\pm 0.004$	$0.421\pm 0.009$	$3319\pm 138$	$9.3\pm 1.2$	$8.6\times 10^{3}\pm 5\times 10^{2}$	24/31
	10–20%		$0.122\pm 0.005$	$0.421\pm 0.011$	$3200\pm 180$	$9.3\pm 1.3$	$3.3\times 10^{3}\pm 5\times 10^{2}$	25/31
	20–30%	–	$0.119\pm 0.006$	$0.421\pm 0.010$	$3000\pm 181$	$9.3\pm 1.3$	$1.2\times 10^{3}\pm 4\times 10^{2}$	17/31
	30–40%	–	$0.115\pm 0.005$	$0.421\pm 0.012$	$2875\pm 130$	$9.3\pm 1.3$	$400\pm 24$	10/31
	40–50%	–	$0.110\pm 0.006$	$0.421\pm 0.011$	$2732\pm 131$	$9.3\pm 1.4$	$130\pm 10$	16/31
	50–60%	–	$0.107\pm 0.005$	$0.421\pm 0.010$	$2600\pm 126$	$9.3\pm 1.4$	$36\pm 5$	20/31
	60–70%	–	$0.102\pm 0.004$	$0.421\pm 0.012$	$2450\pm 142$	$9.5\pm 1.3$	$10\pm 0.4$	12/31
	70–80%	–	$0.098\pm 0.005$	$0.421\pm 0.010$	$2350\pm 126$	$9.8\pm 1.3$	$2.5\pm 0.3$	65/31
	80–90%	–	$0.094\pm 0.005$	$0.421\pm 0.011$	$2220\pm 141$	$10\pm 1.4$	$0.5\pm 0.03$	35/31
	0–5%	$K^{-}$	$0.129\pm 0.004$	$0.421\pm 0.009$	$3319\pm 138$	$9.3\pm 1.2$	$1.9\times 10^{4}\pm 4\times 10^{3}$	40/31
	5–10%	–	$0.126\pm 0.004$	$0.421\pm 0.009$	$3319\pm 138$	$9.3\pm 1.2$	$8.6\times 10^{3}\pm 5\times 10^{2}$	24/31
	10–20%		$0.122\pm 0.005$	$0.421\pm 0.011$	$3200\pm 180$	$9.3\pm 1.3$	$3.3\times 10^{3}\pm 5\times 10^{2}$	25/31
	20–30%	–	$0.119\pm 0.006$	$0.421\pm 0.010$	$3000\pm 181$	$9.3\pm 1.3$	$1.2\times 10^{3}\pm 4\times 10^{2}$	17/31
	30–40%	–	$0.115\pm 0.005$	$0.421\pm 0.012$	$2875\pm 130$	$9.3\pm 1.3$	$400\pm 24$	10/31
	40–50%	–	$0.110\pm 0.006$	$0.421\pm 0.011$	$2732\pm 131$	$9.3\pm 1.4$	$130\pm 10$	16/31
	50–60%	–	$0.107\pm 0.005$	$0.421\pm 0.010$	$2600\pm 126$	$9.3\pm 1.4$	$36\pm 5$	20/31
	60–70%	–	$0.102\pm 0.004$	$0.421\pm 0.012$	$2450\pm 142$	$9.5\pm 1.3$	$10\pm 0.4$	12/31
	70–80%	–	$0.098\pm 0.005$	$0.421\pm 0.010$	$2350\pm 126$	$9.8\pm 1.3$	$2.5\pm 0.3$	65/31
	80–90%	–	$0.094\pm 0.005$	$0.421\pm 0.011$	$2220\pm 141$	$10\pm 1.4$	$0.5\pm 0.03$	35/31
	0–5%	$p$	$0.108\pm 0.005$	$0.403\pm 0.010$	$3100\pm 140$	$9.8\pm 1$	$2.3\times 10^{5}\pm 5\times 10^{4}$	34/32
	5–10%	–	$0.104\pm 0.006$	$0.403\pm 0.011$	$2900\pm 142$	$9\pm 1.2$	$1000\pm 100$	151/32
	10–20%		$0.101\pm 0.004$	$0.403\pm 0.010$	$2768\pm 120$	$9\pm 1.2$	$400\pm 20$	151/32
	20–30%	–	$0.097\pm 0.005$	$0.403\pm 0.010$	$2550\pm 136$	$9\pm 1.2$	$150\pm 10$	148/32
	30–40%	–	$0.094\pm 0.005$	$0.403\pm 0.010$	$2400\pm 130$	$9\pm 1.1$	$47\pm 4$	143/32

Table 1. Continue.

Collisions	Centrality	Particle	$T_{0}$ (GeV)	$\beta_{T}$ (c)	$V(fm^{3})$	$n$	$N_{0}$	$\chi^{2}$ / dof
	40–50%	–	$0.090\pm 0.005$	$0.403\pm 0.011$	$2300\pm 140$	$9.8\pm 1.3$	$17\pm 2$	43/32
	50–60%	–	$0.086\pm 0.004$	$0.403\pm 0.012$	$2200\pm 100$	$10.3\pm 1.1$	$5.5\pm 0.5$	17/32
	60–70%	–	$0.083\pm 0.005$	$0.403\pm 0.012$	$2050\pm 110$	$10.3\pm 1.1$	$1.6\pm 0.3$	35/32
	70–80%	–	$0.080\pm 0.006$	$0.403\pm 0.012$	$1910\pm 120$	$10.7\pm 1.4$	$0.38\pm 0.02$	54/32
	80–90%	–	$0.077\pm 0.004$	$0.403\pm 0.010$	$1800\pm 100$	$11.7\pm 1.2$	$0.08\pm 0.003$	61/32
	0–5%	$\bar{p}$	$0.108\pm 0.005$	$0.403\pm 0.010$	$3100\pm 140$	$9.8\pm 1$	$2.3\times 10^{5}\pm 5\times 10^{4}$	34/32
	5–10%	–	$0.104\pm 0.006$	$0.403\pm 0.011$	$2900\pm 142$	$9\pm 1.2$	$1000\pm 100$	151/32
	10–20%		$0.101\pm 0.004$	$0.403\pm 0.010$	$2768\pm 120$	$9\pm 1.2$	$400\pm 20$	151/32
	20–30%	–	$0.097\pm 0.005$	$0.403\pm 0.010$	$2550\pm 136$	$9\pm 1.2$	$150\pm 10$	148/32
	30–40%	–	$0.094\pm 0.005$	$0.403\pm 0.010$	$2400\pm 130$	$9\pm 1.1$	$47\pm 4$	143/32
	40–50%	–	$0.090\pm 0.005$	$0.403\pm 0.011$	$2300\pm 140$	$9.8\pm 1.3$	$17\pm 2$	43/32
	50–60%	–	$0.086\pm 0.004$	$0.403\pm 0.012$	$2200\pm 100$	$10.3\pm 1.1$	$5.5\pm 0.5$	17/32
	60–70%	–	$0.083\pm 0.005$	$0.403\pm 0.012$	$2050\pm 110$	$10.3\pm 1.1$	$1.6\pm 0.3$	35/32
	70–80%	–	$0.080\pm 0.006$	$0.403\pm 0.012$	$1910\pm 120$	$10.7\pm 1.4$	$0.38\pm 0.02$	54/32
	80–90%	–	$0.077\pm 0.004$	$0.403\pm 0.010$	$1800\pm 100$	$11.7\pm 1.2$	$0.08\pm 0.003$	61/32
	0–10%	$\Lambda$	$0.128\pm 0.005$	$0.381\pm 0.010$	$2900\pm 150$	$7\pm 0.7$	$17\pm 2$	130/14
	10–20%	–	$0.126\pm 0.006$	$0.381\pm 0.010$	$2750\pm 100$	$7\pm 1$	$13\pm 0.4$	74/14
	20–40%	–	$0.120\pm 0.005$	$0.381\pm 0.011$	$2623\pm 110$	$7.3\pm 1.1$	$7.9\pm 0.2$	105/14
	40–60%	–	$0.111\pm 0.006$	$0.381\pm 0.012$	$2500\pm 130$	$9.4\pm 1.2$	$3.5\pm 0.2$	22/14
	60–80%	–	$0.105\pm 0.005$	$0.381\pm 0.012$	$2400\pm 152$	$10.5\pm 1.3$	$1\pm 0.02$	20/14
	0–10%	$\Xi$	$0.128\pm 0.006$	$0.287\pm 0.012$	$2600\pm 180$	$7\pm 0.6$	$5.5\pm 0.3$	39/7
	10–20%	–	$0.125\pm 0.004$	$0.287\pm 0.012$	$2477\pm 150$	$7\pm 0.5$	$4.2\pm 0.4$	25/7
	20–40%	–	$0.120\pm 0.004$	$0.287\pm 0.010$	$2356\pm 144$	$7.6\pm 0.4$	$2.7\pm 0.2$	13/7
	40–60%	–	$0.112\pm 0.005$	$0.287\pm 0.011$	$2207\pm 108$	$8\pm 0.4$	$1\pm 0.2$	32/7
	60–80%	–	$0.105\pm 0.004$	$0.287\pm 0.010$	$2100\pm 120$	$10\pm 0.8$	$0.3\pm 0.03$	57/7
Fig. 5	0–5%	$\pi^{+}+\pi^{-}$	$0.100\pm 0.006$	$0.430\pm 0.010$	$3300\pm 150$	$9\pm 1$	$1700\pm 113$	67/36
p-Pb	5–10%	–	$0.096\pm 0.005$	$0.430\pm 0.011$	$3200\pm 136$	$9\pm 1$	$700\pm 100$	51/36
5.02 TeV	10–20%	–	$0.092\pm 0.004$	$0.430\pm 0.012$	$3100\pm 124$	$9\pm 1.2$	$300\pm 13$	44/36
	20–40%	–	$0.088\pm 0.005$	$0.430\pm 0.011$	$2942\pm 156$	$9\pm 1.1$	$130\pm 16$	49/36
	40–60%	–	$0.083\pm 0.006$	$0.430\pm 0.009$	$2800\pm 140$	$9\pm 1.2$	$37\pm 7$	42/36
	60–80%	–	$0.079\pm 0.005$	$0.430\pm 0.011$	$2647\pm 137$	$9.7\pm 1.3$	$17\pm 3$	82/36
	80–100%	–	$0.075\pm 0.005$	$0.430\pm 0.011$	$2500\pm 141$	$10\pm 1.2$	$3.6\pm 0.5$	67/36
	0–5%	$K^{+}+K^{-}$	$0.120\pm 0.005$	$0.400\pm 0.011$	$3100\pm 145$	$8\pm 0.7$	$128\pm 13$	5/26
	5–10%	–	$0.116\pm 0.006$	$0.400\pm 0.012$	$2974\pm 130$	$8\pm 0.5$	$52\pm 4$	8/26
	10–20%	–	$0.112\pm 0.005$	$0.400\pm 0.012$	$2851\pm 140$	$8.2\pm 0.6$	$23\pm 3$	45/26
	20–40%	–	$0.108\pm 0.005$	$0.400\pm 0.011$	$2714\pm 124$	$8.2\pm 0.5$	$10\pm 0.4$	10/26
	40–60%	–	$0.104\pm 0.005$	$0.400\pm 0.010$	$2600\pm 108$	$8.6\pm 0.6$	$5.2\pm 0.4$	21/26
	60–80%	–	$0.100\pm 0.005$	$0.400\pm 0.010$	$2500\pm 100$	$9\pm 0.7$	$1.2\pm 0.04$	45/26
	80–100%	–	$0.096\pm 0.004$	$0.400\pm 0.011$	$2400\pm 100$	$10.6\pm 1$	$0.3\pm 0.03$	90/26
	0–5%	$p+\bar{p}$	$0.100\pm 0.004$	$0.380\pm 0.012$	$2928\pm 140$	$8.4\pm 0.8$	$20\pm 2$	9/34
	5–10%	–	$0.096\pm 0.005$	$0.380\pm 0.010$	$2800\pm 120$	$8.4\pm 0.6$	$8.7\pm 0.6$	14/34
	10–20%	–	$0.092\pm 0.005$	$0.380\pm 0.010$	$2700\pm 130$	$8.4\pm 0.6$	$3.5\pm 0.2$	50/34
	20–40%	–	$0.088\pm 0.006$	$0.380\pm 0.010$	$2570\pm 110$	$8.9\pm 0.8$	$1.8\pm 0.03$	86/34
	40–60%	–	$0.084\pm 0.005$	$0.380\pm 0.011$	$2450\pm 109$	$9.5\pm 0.5$	$0.6\pm 0.02$	6/34
	60–80%	–	$0.080\pm 0.005$	$0.380\pm 0.011$	$2323\pm 108$	$10.5\pm 0.6$	$0.24\pm 0.02$	28/34
	80–100%	–	$0.078\pm 0.005$	$0.380\pm 0.010$	$2200\pm 100$	$12\pm 1$	$0.059\pm 0.003$	91/34
	0–5%	$\Lambda+\bar{\Lambda}$	$0.120\pm 0.006$	$0.330\pm 0.009$	$2500\pm 180$	$8.8\pm 0.4$	$15\pm 2$	159/13
	5–10%	–	$0.115\pm 0.006$	$0.330\pm 0.011$	$2350\pm 135$	$8.8\pm 0.3$	$7\pm 0.4$	96/13
	10–20%	–	$0.114\pm 0.004$	$0.330\pm 0.012$	$2220\pm 146$	$8.8\pm 0.4$	$3\pm 0.2$	110/13
	20–40%	–	$0.110\pm 0.005$	$0.330\pm 0.012$	$2220\pm 142$	$9\pm 0.35$	$1.15\pm 0.05$	32/13
	40–60%	–	$0.105\pm 0.004$	$0.330\pm 0.010$	$2100\pm 136$	$9\pm 0.5$	$0.4\pm 0.02$	73/13
	60–80%	–	$0.101\pm 0.005$	$0.330\pm 0.012$	$2000\pm 150$	$9.9\pm 0.7$	$0.14\pm 0.02$	136/13
	80–100%	–	$0.096\pm 0.005$	$0.330\pm 0.010$	$1870\pm 108$	$10.4\pm 0.8$	$0.27\pm 0.05$	376/13

Table 1. Continue.

Collisions	Centrality	Particle	$T_{0}$ (GeV)	$\beta_{T}$ (c)	$V(fm^{3})$	$n$	$N_{0}$	$\chi^{2}$ / dof
p-p 200 GeV	—	$\pi^{+}$	$0.042\pm 0.006$	$0.256\pm 0.008$	$532\pm 30$	$8\pm 0.4$	$63\pm 3$	9/22
	—	$\pi^{-}$	$0.042\pm 0.006$	$0.256\pm 0.008$	$532\pm 30$	$8\pm 0.4$	$63\pm 3$	9/22
	—	$K^{+}$	$0.050\pm 0.004$	$0.230\pm 0.009$	$434\pm 30$	$8\pm 0.4$	$10\pm 0.3$	8/11
	—	$K^{-}$	$0.050\pm 0.004$	$0.230\pm 0.009$	$434\pm 30$	$8\pm 0.4$	$10\pm 0.3$	8/11
	—	$p$	$0.042\pm 0.005$	$0.214\pm 0.010$	$390\pm 31$	$9.8\pm 0.5$	$2\pm 0.2$	9/28
	—	$\bar{p}$	$0.042\pm 0.005$	$0.214\pm 0.010$	$390\pm 23$	$9.8\pm 0.5$	$2\pm 0.2$	9/28
	—	$\phi$	$0.050\pm 0.006$	$0.200\pm 0.011$	$312\pm 21$	$8\pm 0.2$	$0.007\pm 0.0002$	14/8
	—	$\Xi^{-}$	$0.050\pm 0.005$	$0.187\pm 0.010$	$287\pm 21$	$8\pm 0.3$	$0.004\pm 0.0002$	15/6

III Results and discussion

III.1 Comparison with data

Figure 1 shows the $p_{T}$ spectra ( $1/N_{ev}$ [(1/2 $\pi$ $m_{T}$ ) $d^{2}$ $N$ / $dyd$ $m_{T}$ ] or (1/2 $\pi$ $p_{T}$ ) $d^{2}$ $N$ / $dyd$ $p_{T}$ ) of $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ , $\bar{p}$ , $\Lambda$ and $\Xi$ in Copper-Copper (Cu-Cu) collisions at 200 GeV in different centrality bins. The centrality intervals for $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{+}$ , $p$ and $\bar{p}$ are $0-10\%$ , $10-30\%$ , $30-50\%$ and $50-70\%$ . The centrality intervals for $\pi^{+}$ and $\pi^{-}$ $10-30\%$ , $30-50\%$ and $50-70\%$ are scaled by 1/2, 1/4 and 1/6 respectively, while for $K^{+}$ and $K^{-}$ $30-50\%$ and $50-70\%$ are scaled by 0.3. For $p$ and $\bar{p}$ the centrality bins $10-30\%$ , $30-50\%$ and $50-70\%$ are scaled by 0.4, 0.3 and 0.2 respectively. The centrality intervals for $\Lambda$ and $\Xi$ are $0-10\%$ , $10-20\%$ , $20-30\%$ , $30-40\%$ and $40-60\%$ . $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ and $\bar{p}$ are measured at $y=0$ , while for $\Lambda$ and $\Xi$ $|y|<$ 0.5. The symbols are used to represent the experimental data of RHIC measured by the BRAHMS Collaboration, while the curves are our fit results by using the Modified Hageodorn function with embeded transverse flow i:e Eq. (3). The data/fit ratio is given in the figure followed by each panel. It can be seen that Eq. (3) fits the data approximately well. In some cases the fit in the $p_{T}$ range up to 0.5 GeV/c is not good due to the resonance effect. The data for $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ , $\bar{p}$ are taken from ref. 1c , while for $\Lambda$ and $\Xi$ is from ref. 2c .

Fig. 2 is similar to fig.1, but it shows the $p_{T}$ spectra of $\pi^{+}$ , $K^{+}$ , $p$ , $\Lambda$ and $\Xi^{-}$ in Au-Au collisions at 62.4 GeV. The centrality is distributed in $0-5\%$ , $0-10\%$ , $10-20\%$ , $20-30\%$ , $30-40\%$ , $40-50\%$ , $50-60\%$ , $60-70\%$ and $70-80\%$ intervals for $\pi^{+}$ , $K^{+}$ and $p$ and the centrality bins $0-10\%$ , $10-20\%$ , $20-30\%$ , $30-40\%$ , $40-50\%$ , $50-60\%$ , $60-70\%$ and $70-80\%$ are scaled by 1/4, 1/15, 1/50, 1/80, 1/500, 1/1300, 1/2700 and 1/5400 respectively. $\Lambda$ is distributed into $0-5\%$ , $0-10\%$ , $10-20\%$ , $20-30\%$ , $30-40\%$ , $40-60\%$ and $60-80\%$ centrality intervals, while the centrality distribution for $\Xi^{-}$ is $0-5\%$ , $0-10\%$ , $10-20\%$ , $20-30\%$ , $30-40\%$ and $40-60\%$ . One can see that the measurement of the STAR Collaboration is well fitted by the modified Hagedorn function. The data/fit ratio is followed in each panel in the figure. The data for $\pi^{+}$ , $K^{+}$ and $p$ are taken from ref. 3c , while for $\Lambda$ and $\Xi$ is taken from ref. 4c .

The $p_{T}$ spectra of the given particles are demonstrated in fig.3 and 4 in Deuteron-Gold (d-Au) collisions at 200 GeV and in Lead-Lead (Pb-Pb) collisions at 2.76 TeV respectively. In d-Au collisions $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ and $\bar{p}$ , and in Pb-Pb collisions $\pi^{+}$ , $K^{+}$ , $p$ , $\Lambda$ and $\Xi$ are analyzed. The centrality in d-Au is distributed in various centrality classes such that $0-20\%$ , $20-40\%$ , $40-60\%$ and $60-88\%$ . However the centrality for $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ and $\bar{p}$ in Pb-Pb collisions is distributed in $0-5\%$ , $0-10\%$ , $10-20\%$ , $20-30\%$ , $30-40\%$ , $40-50\%$ , $50-60\%$ , $60-70\%$ , $70-80\%$ and $80-90\%$ , and except $0-5\%$ , the other centrality bins are scaled by 1/2, 1/5, 1/10, 1/19, 1/36, 1/50, 1/70, 1/78 and 1/88 respectively. The centrality bins for $\Lambda$ and $\Xi$ are $0-10\%$ , $10-20\%$ , $20-40\%$ , $40-60\%$ and $60-80\%$ and except $0-10\%$ , the remaining centrality class are scaled by 1/2, 1/4, 1/6 and 1/8 respectively. The well fitting results of the model to the experimental data of ALICE collaboration can be seen, and the data/fit ratio are followed in each panel. The data in fig. 3 are taken from ref. 5c , while in fig.4 for $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ and $\bar{p}$ is taken from ref. 6c , and $\Lambda$ and $\Xi^{-}$ are taken from ref. 7c .

Fig.5 shows the $p_{T}$ of the given particles in proton-Lead (p-Pb) and proton-proton ( $pp$ ) collisions. Panel (a)-(d) shows the $p_{T}$ spectra of the particles in p-Pb collisions in different centrality intervals at 5.02 TeV, and panel (e) represent the $p_{T}$ spectra of the particles in $pp$ collisions at 200 GeV. Various centrality bins for $\pi^{+}+\pi^{-}$ , $K^{+}+K^{-}$ , $p+\bar{p}$ and $\Lambda+\bar{\Lambda}$ are $0-5\%$ , $0-10\%$ , $10-20\%$ , $20-40\%$ , $40-60\%$ , $60-80\%$ , and $80-100\%$ , and $0-10\%$ , $10-20\%$ , $20-40\%$ , $40-60\%$ , $60-80\%$ , and $80-100\%$ in panel (a)-(c) are scaled by 1/2, 1/4, 1/7, 1/10, 1/13 and 1/16 respectively. Panel (e) represents the $p_{T}$ spectra of $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ , $\bar{p}$ , $\phi$ and $\Xi^{-}$ , and the spectra of $p$ and $\bar{p}$ , $\phi$ and $\Xi^{-}$ are scaled by 2, 50 and 10 respectively. One can see that the modified Hagedorn function fits the experimental data well. The results of the data/fit ratio are given in figure followed by each panel. The data for $\pi^{+}+\pi^{-}$ , $K^{+}+K^{-}$ , $p+\bar{p}$ and $\Lambda+\bar{\Lambda}$ in panels (a-e) is taken from ref. 8c , while the data for $\pi^{+}$ , $\pi^{-}$ , $K^{+}$ , $K^{-}$ , $p$ , $\bar{p}$ , and $\phi$ and $\Xi^{-}$ are taken from ref. 9c ; 10c ; 11c respectively.

III.2 Tendencies of parameters

In order to study the tendency of parameters, the dependences of $T_{0}$ on centrality in (a)-(e) and $T_{0}$ on $m_{0}$ in (f) with different symbols are displayed in Figure 6. The values of the two parameters are cited from Table 1. Each panel represent the results from different collisions. The symbols in the figure are used in order to represent different particles. The parameters trend from left to right shows the behavior of $T_{0}$ from central to peripheral collisions from panel (a)-(e) while panel (f) shows its behavior with increasing mass of the particle. In panels (a)-(e) one can see that $T_{0}$ for all the particles is larger in central collisions and it decrease as the centrality decrease due the reason that more energy is deposited to the system in central collisions due to the involvement of large number of participants which decrease towards periphery. In addition, from panel (a)-(f), $T_{0}$ is larger for the strange particles and the separate decoupling of the strange and non-strange particles is observed. The larger $T_{0}$ for the strange particles is due to their smaller cross-section interaction, and according to kinematics, the reactions with lower cross-section is expected to be switched-off at higher temperatures/densities or early in time than the reactions with higher cross-sections. We also believe that the charm particles may decouple earlier than the strange particles and it is possible that a series of freeze-outs correspond to particular reaction channels 12c , but it is a regret that we don’t have the data for charm particles. Furthermore, one can see that $T_{0}$ is larger at LHC than that of RHIC, and even at RHIC $pp$ is the smallest collisions system and the observed $T_{0}$ in it is very smaller than the rest which is a clear evidence of $T_{0}$ on the size of the interacting system. From the above observation, one thing is clear that there is a strong dependence of $T_{0}$ on the collision energy because Pb-Pb and p-Pb collisions are the largest systems and they have larger $T_{0}$ , and $pp$ is the smallest collision system among them and has smaller vale for $T_{0}$ . However Cu-Cu, Au-Au and d-Au at RHIC, and p-Pb and Pb-Pb at LHC are different systems and have almost the same values for $T_{0}$ respectively, and this is due to the effect of collision energy. So we can conclude that $T_{0}$ is dependent on the size of the collision system as well as the collision energy.

We would like to point out that we have observed two kinetic freeze-out scenarios which is agreement with 13c , but in contrast with our recent work 13ca . In fact, there are different freeze-out scenarios in literature which include one, two, three or multiple kinetic freeze-out scenarios. No doubt, the process of high energy collisions is complex and it is very interesting to see whether one, two, or multiple kinetic freeze-out scenarios exist in it. Due to the fact that different scenarios are reported in different literature, it is an open question up to now.

Fig. 7 shows similar to the fig. 6, but it shows the dependence of $\beta_{T}$ on centrality from panel (a)-(e), while panel (f) shows the dependence of $\beta_{T}$ on $m_{0}$ . One can see that $\beta_{T}$ remains invariant as we go from central to peripheral collision due to the reason that collective behavior from central to peripheral collisions does not change. In addition, $\beta_{T}$ from panel (a)-(f)is larger for the lighter particles and it decrease for the heavier particles and it is a natural hydrodynamical behavior because heavier particles are left behind in the system. We also noticed that like $T_{0}$ , $\beta_{T}$ in Cu-Cu, Au-Au and d-Au at RHIC, and Pb-Pb and p-Pb at LHC are close to each other due to the effect of their dependence on both the collision cross-section and collision energy.

To check the dependence of $V$ on centrality (mass), fig. 8 is represented. It is similar to that of fig. 6 and fig. 7. One can see that $V$ decreases from central to peripheral collisions in panels (a)-(e) due to fact that the participant nucleons decreases as we go from central to peripheral collisions depending on the interaction volume. The system which contains more participants reaches to equilibrium quickly because there are large number of secondary collisions by the re-scattering of partons in central collisions, and the system goes away from equilibrium states as it goes from central to peripheral collision due to reason that the number of participants in the system decreases. Furthermore, we observed that $V$ from panel (a)-(f) is dependent of $m_{0}$ . Heavier the particle is, smaller is the value of $V$ , which reveals a volume differential freeze-out scenario, and may indicate that there are different freeze-out surfaces for different particles. Like $T_{0}$ and $\beta_{T}$ , $V$ at LHC is larger than at RHIC and in $pp$ collisions $V$ is the smallest among all these systems, nucleus-nucleus collision systems at RHIC and LHC have nearly same $V$ respectively and this is due to their energy difference.

Fig. 9 represents the behavior of the parameter $n$ . $n$ =1/(q-1), and q is the entropy index, which is referred to equilibrium degree. In general, $q$ =1 corresponds to equilibrium state, where $q$ $>$ 1 corresponds to non-equilibrium state. Smaller $q$ corresponds to large $n$ . One can see that the parameter $n$ is larger in almost all the cases which means that the system is in equilibrium state, and also it remains constant in most cases from central to peripheral collision except for proton that increases from central to peripheral collisions which is not understood by us.

The dependence of the parameter $N_{0}$ on centrality ( $m_{0}$ ). $N_{0}$ is a normalization constant but it has its significance too. It reflects the multiplicity. One can see that the parameter $N_{0}$ decreases from central to peripheral collisions which indicates that the multiplicity decreases as the system goes from central to peripheral collisions. It can also be seen that $N_{0}$ is dependent on the size of the interacting system. As we can see that in Au-Au collisions it is larger than Cu-Cu and d-Au collisions, and in Pb-Pb collisions it has the highest value followed by p-Pb collisions. In $pp$ collisions $N_{0}$ is the smallest due to being it is the smallest system among them. This shows that the multiplicity depends on the size of the interacting system and also have the collision energy dependence as the collision energy of the AA collisions at RHIC and LHC are different.

The mean transverse momentum ( $<p_{T}>$ ) dependence on centrality ( $m_{0}$ ) is demonstrated in fig. 11. One can see that $<p_{T}>$ decrease as the system goes from central to peripheral collisions and this is due to the fact that the system gains large momentum (energy) in central collisions where further multiple scattering happens, which decrease when the system goes towards periphery. Like $T_{0}$ and $\beta_{T}$ , $<p_{T}>$ also depends on the size of the interacting system and also there is an effect on the behavior of collision energy. We also analyze root-mean-square $p_{T}$ ( $\sqrt{<p^{2}_{T}>}$ ) over $\sqrt{2}$ ( $\sqrt{<p^{2}_{T}>}$ / $\sqrt{2}$ ) in fig. 12. and showed its behavior with changing the centrality ( $m_{0}$ ). ( $\sqrt{<p^{2}_{T}>}$ / $\sqrt{2}$ ) represents the initial temperature ( $T_{i}$ ) of the interacting system according to the string percolation model 14c ; 15c ; 16c . It can be obviously seen that the initial temperature is larger in central collisions, however it decrease from central to peripheral collision systems, and similar to $T_{0}$ it depends on the size of the interacting system and is also effected by the collision energy. Moreover, we have observed that the initial temperature depends on mass of the particle. The more massive the particle is, the larger is the initial temperature.

We would like to point out that the initial temperature depends on mass of the particle, and the more heavier the particle is, the larger the initial temperature, however the kinetic freeze-out temperature depends on the cross-section interaction of the particles. The larger the cross-section of the particle is, the smaller the value of kinetic freeze-out temperature. The difference in the two temperatures is due to the reason that they occur at two different process in the system evolution. Furthermore, it is also possible that the centrality dependence of the two types of temperatures is also different if one gets the increasing trend from central to peripheral for $T_{0}$ which is observed in our recent work 17c ; 18c . In fact the trend dependence of $T_{0}$ is also an open question in high energy collisions because different literature give different trend.

Before going to the summary and conclusions, we would like to point out that the initial temperature is observed to be larger than the kinetic freeze-out temperature. The former is followed by the chemical freeze-out temperature which can be expresses as

\displaystyle T_{ch}=\frac{T_{\lim}}{1+\exp[2.60-\ln(\sqrt{s_{NN}})/0.45]}

(4)

where $T_{lim}$ =0.1584 GeV. The chemical freeze-out temperature is followed by the effective temperature and then the kinetic freeze-out temperature and this order is in agreement with the order of time evolution of the interacting system.

IV Summary and Conclusions

We summarize here our main observations and conclusions.

(a) The transverse momentum spectra of strange and non-strange hadrons produced in Cu-Cu, Au-Au, d-Au, Pb-Pb, p-Pb and $pp$ collisions have been studied by the modified Hagedorn model. The results are well in agreement with the experimental data BRAHM, STAR, PHENIX and ALICE Collaborations at RHIC and LHC.

(b) $T_{0}$ and $T_{i}$ are larger in central collisions and they decrease from central to peripheral collisions due to the decrease in the participant nucleons towards periphery which results in lower degree of excitation of the system in the peripheral collisions. $<p_{T}>$ also decrease towards periphery due to the reason that the energy (momentum) transfer becomes lower in the system from central to peripheral collisions. In addition $V$ is larger in central collisions and it decrease towards periphery due to the reason of decreasing of partons re-scattering towards periphery.

(c) $\beta_{T}$ depends on the mass of the particle. Heavier the particle is, smaller is the value of $\beta_{T}$ is. $\beta_{T}$ remains unchanged from central to peripheral collisions because the collective flow from central to peripheral collisions does not change.

(d) The parameter $N_{0}$ represents the multiplicity and it decrease from central to peripheral collisions.

(e) $T_{0}$ depends on the cross-section interaction of the particle and therefore the strange and non-strange particles have separate freeze-out and it reveals the scenario of two kinetic freeze-out temperature, while the initial temperature depends on the mass of the particle and this is due to the reason that both of them occurs at different stages in the evolution system.

(f) $V$ depends on the mass of the particle. The heavier particle has smaller $V$ which reveals the volume differential freeze-out scenario and indicates a separate freeze-out surface for each particle.

(g) $T_{0}$ , $T_{i}$ , $\beta_{T}$ , $V$ , $<p_{T}>$ and $N_{0}$ are dependent on the size of the interacting system because all these parameters have larger values at LHC collision systems than at RHIC collision systems and in $pp$ collisions it has the lowest values. The mentioned parameters in Cu-Cu, d-Au, and Au-Au are different collision systems with different energies (200 GeV, 62.4 GeV and 200 GeV for Cu-Cu, Au-Au and d-Au respectively), but the values of the parameters obsered in these systems are nearly equal, due to their dependence on both the collision energy and collision cross-section. Similar behavior is observed in p-Pb and Pb-Pb collisions.

Acknowledgments

This research was funded by the National Natural Science Foundation of China grant number 11875052, 11575190, and 11135011.

Author Contributions All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.]

Compliance with Ethical Standards

Ethical Approval The authors declare that they are in compliance with ethical standards regarding the content of this paper.

Disclosure The funding agencies have no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript, or in the decision to publish the results.

Conflict of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.

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