Abstract
Transverse momentum spectra of π+, K+, p, , Λ, Ξ or and Ω or or in copper–copper (Cu–Cu), gold–gold (Au–Au) and lead–lead (Pb–Pb) collisions at 200 GeV, 62.4 GeV and 2.76 TeV respectively, are analyzed in different centrality bins by the blast wave model with Tsallis statistics. The model results are approximately in agreement with the experimental data measured by BRAHMS, STAR and ALICE Collaborations in special transverse momentum ranges. Kinetic freeze out temperature, transverse flow velocity and kinetic freezeout volume are extracted from the transverse momentum spectra of the particles. It is observed that and Ω or or have larger kinetic freezeout temperature followed by K+, and Λ than π+ and p due to smaller reaction cross-sections of multi-strange and strange particles than non-strange particles. The present work reveals the scenario of triple kinetic freezeout in collisions at BRAHMS, STAR and ALICE Collaborations, however the transverse flow velocity and kinetic freezeout volume are mass dependent and they decrease with the increasing rest mass of the particle. In addition, the kinetic freezeout temperature, transverse flow velocity and kinetic freezeout volume are decreasing from central to peripheral collisions while the parameter q increase from central to peripheral collisions, indicating the approach of quick equilibrium in the central collisions. Besides, the kinetic freezeout temperature and kinetic freezeout volume are observed to be larger in larger collision system which shows its dependence on the size of the interacting system, while transverse flow velocity increase with increasing energy.
1. Introduction
One of the newest trends in the advancement of relativistic heavy ion collisions is to dig out the new states of strongly interacting matter and to ascertain the quark gluon plasma (QGP) [1–4] anticipated qualitatively by the quantum chromodynamics (QCD) [5–10]. It is considered that the QGP is a state of strongly interacting matter under the extreme conditions of high temperatures and/or high net baryon densities. Nowadays, the QGP can be created in the laboratories during the gold–gold (Au–Au) and copper–copper (Cu–Cu) collisions at relativistic heavy ion collider (RHIC), lead–lead (Pb–Pb) and xenon–xenon (Xe–Xe) collisions at large hadron collider (LHC) and other high energy nucleus–nucleus collisions by increasing the energies and altering the masses of the colliding nuclei. In fact, an extremely high temperature and/or high densities are required for the formation of QGP. In the present work, we will be limited to the concept of temperature. Indeed temperature is a very important concept in thermal and sub-atomic particles [11] due to its wide applications in experimental and theoretical studies. In literature, one can find various kinds of temperatures at various stages of the collision. The degree of excitation of interacting system at the initial stage of collisions is described by the initial temperature. The chemical freezeout temperature describes the excitation degree of chemical freezeout, when the inelastic scattering cease and the particle identities are set until decay [12, 13], followed by the kinetic freezeout temperature which describes the degree of excitation at the kinetic freezeout stage. Since, the present work is mainly focused on the behavior of kinetic freezeout temperature, therefore we will not discuss the other temperatures here, but one can have a look at [14, 15] for their more details. Generally, the freezeout could be very complicated process due to the involvement of duration in time and a hierarchy where various kinds of particles and reactions switch-off at different times. From kinematic point of view, the reactions with lower cross-section is expected to be switched-off at higher densities/temperatures or early in time than the reactions with higher cross-sections. Hence, the chemical freezeout occur earlier in time than the kinetic freezeout, which correspond to the elastic reactions. According to the above statement, it is believed that the charm and strange particles decouple from the system earlier than the lighter hadrons. A series of freezeouts maybe possibly correspond to particular reaction channels [16]. Furthermore, a single [17], double [13, 18] and multiple kinetic freezeout scenarios [19–21] can be found in literature. In addition, the transverse flow velocity (βT) is also an important parameter which reflects the collective expansion of the emission source. Both T0 and βT can be extracted from the transverse momentum (pT) spectra of the particles by using some distribution laws.
Additionally, volume keeps a prominent importance in collision physics. The volume occupied by the sources of ejectiles when the mutual interactions become negligible and they only feel the Coulombic repulsive force is known as kinetic freezeout volume (V). Various freezeout volume correspond to various stages in interaction process but we will only focus on the kinetic freezeout volume in the present work.
The pT spectra of the particles produced in high energy collisions are very important quantities to be measured. The study of pT spectra of particles can give the understanding of the production mechanism and help us to infer the thermodynamic properties of the collision system. Beside, it gives some useful information that contains, but not limited to various types of temperatures and freezeout volumes.
In the present work, we analyzed the pT spectra of π+, K+, p, , Λ, Ξ or and Ω or or by using the blast wave model with Tsallis statistics and extracted the parameters T0, βT and V.
The remainder of the paper consists of method and formalism in section 2, followed by the results and discussion in section 3. In section 4, we summarized our main observations and conclusions.
2. The method and formalism
The structure of pT spectra in high energy collisions are complex processes in which many emission sources are found. A local equilibrium state may possibly be formed by the sources with the same excitation degree, which can be described by the standard distribution. In case of different equilibrium states with different degrees of excitation, we may use different parameters of temperature. In general, the pT spectra in a not too wide pT range can be described by the two or three components standard distribution that reflects the fluctuation of temperature of the interacting system. Meanwhile, a two- or three-components standard distribution can be described by the Tsallis distribution [22–26], blast wave model with Tsallis (TBW) statistics [27–29] or others.
According to references [27–29], the TBW model results in the pT distribution to be
where C stands for the normalization constant that leads the integral in equation (1) to be normalized to 1, g is the degeneracy factor which is different for different particles based on gn = 2Sn +1 (Sn is the spin of the particle), is the transverse mass, m0 is the rest mass of the particle, ϕ represents the azimuthal angle, r is the radial coordinate, V is the freezeout volume, T0 is the kinetic freezeout temperature, R is the maximum r, q is the entropy index and it shows the measure of degree of deviation of the system from an equilibrium state, ρ = tanh−1 [β(r)] is the boost angle, is a self-similar flow profile, βS is the flow velocity on the surface, as mean of , and n0 = 1 or 2 [29]. n0 is regarded a free parameter in some literature [30, 31]. In case we choose n0 as a free parameter, we will have one more free parameter, which is not chosen by us. Furthermore, the index −1/(q − 1) in equation (1) can be replaced by −q/(q − 1) due to the reason that q is being close to 1. This replacement results in a small and negligible divergence in the Tsallis distribution [22, 26].
If the pT spectra is not too wide, it can be described by equation (1) and T0 and βT can be extracted, but in case of wide pT spectra the contribution of hard scattering process will be considered and according to the QCD calculus [32–34] it can be parameterized as an inverse power law.
which is the Hagedorn function, A is the normalization constant, p0 and n are the free parameters. The modified versions of Hagedorn function can be found in literature [35–41].
where as in equations (3)–(5), p0 and n are severally different. In case of wide pT range, the superposition of soft excitation and hard scattering process can be used for its description. If equation (1) describe the soft excitation process, and the hard scattering process is described by one of equations (2)–(5), then the superposition of the two-components can be used to describe the wide pT range.
According to Hagedorn model [42], the usual step function can also be used for the superposition of the two functions, as
where fS and fH are the soft and hard components respectively, k(1 − k) shows the contribution fraction of soft excitation (hard scattering) process which naturally normalizes to 1. A1 and A2 are the normalization constants that synthesize A1 fS(p1) = A1 fH(p1) and θ(x) is the usual step function. The soft component fS(pT) in equations (6) and (7) are the same as f1(pT) in equation (1).
The contribution of soft component in low pT and hard component in high pT in equation (7) are linked with each other at pT = p1.
The soft and hard components in equations (6) and (7) are treated in different ways in the whole pT region. equation (6) refers to the contribution of soft component in the range 0–2 ∼ 3 GeV c−1 or a little more. However in case of the contribution of hard component, though the main contribution in low pT region is soft excitation process, but it covers the whole pT region. In equation (7), in the range from 0 to p1 and from p1 up to maximum are the contributions of soft and hard components respectively and there is no mixed region for the two components.
Equations (6) and (7) are the same if only soft component is included. However if we include both the soft and hard components, small variation in parameters will appear.
3. Results and discussion
Figure 1 presents the event centrality dependent double differential mT or pT spectra, 1/Nev[(1/2πmT) d2 N/dydmT] of π+, K+ and p and [(1/2πpT) d2 N/dydpT] of produced in Cu–Cu collisions at the center-of-mass energy per nucleon pair GeV in panels (a)–(d) respectively. The symbols represent the experimental data measured by the BRAHMS and STAR Collaborations [43, 44] and the curves are our fitting results by using the blast-wave model with Tsallis statistics, equation (1). The spectra is distributed in different centrality classes, e:g for π+, K+ and p 0%–10%, 10%–30%, 30%–50% and 50%–70% |y| = 0, while for 0%–10%, 10%–20%, 20%–30%, 30%–40% and 40%–60% |y| < 0.5, where y denotes the rapidity. The corresponding ratio of data/fit is followed in each panel. The related parameters along with χ2 and degree of freedom (dof) are listed in table 1, where the centrality classes and the re-scaled spectra are also presented together. The data is chosen in order to search the differences in different particles emission. One can see that equation (1) fits well the data in Cu–Cu collisions at 200 GeV at the RHIC. It is noticeable that in the present work, we have analyzed the pT distribution in a limited interval and considered the soft excitation process and so parametrization (1) is used. If the pT range becomes wider then hard scattering will be involved and the parametrization of equations (2)–(7) can be used.
Table 1. Values of free parameters T0 and βT, V and q, normalization constant (N0), χ2, and dof corresponding to the curves in figures 1–6.
Collisions | Centrality | Particle | T0 (GeV) | βT (c) | V(fm3) | q | N0 | χ2/dof |
---|---|---|---|---|---|---|---|---|
Figure 1 Cu–Cu 200 GeV | 0%–10% | π+ | 0.104 ± 0.005 | 0.389 ± 0.010 | 4000 ± 270 | 1.05 ± 0.005 | 0.07 ± 0.003 | 1/9 |
10%–30% | — | 0.102 ± 0.005 | 0.380 ± 0.009 | 3800 ± 210 | 1.06 ± 0.004 | 0.04 ± 0.005 | 0.6/9 | |
30%–50% | — | 0.100 ± 0.005 | 0.360 ± 0.010 | 3700 ± 200 | 1.07 ± 0.006 | 0.015 ± 0.004 | 2/9 | |
50%–70% | — | 0.097 ± 0.005 | 0.340 ± 0.010 | 3558 ± 220 | 1.08 ± 0.008 | 0.0065 ± 0.0006 | 1.5/9 | |
0%–10% | K+ | 0.128 ± 0.007 | 0.360 ± 0.013 | 3570 ± 177 | 1.01 ± 0.005 | 0.000 07 ± 0.000 004 | 2/8 | |
10%–30% | — | 0.126 ± 0.005 | 0.353 ± 0.007 | 3490 ± 220 | 1.02 ± 0.006 | 0.000 016 ± 0.000 006 | 2/8 | |
30%–50% | — | 0.124 ± 0.005 | 0.335 ± 0.011 | 3410 ± 180 | 1.03 ± 0.005 | 0.000 015 ± 0.000 006 | 3/8 | |
50%–70% | — | 0.121 ± 0.007 | 0.304 ± 0.010 | 3350 ± 200 | 1.04 ± 0.005 | 0.000 007 ± 0.000 0005 | 11/8 | |
0%–10% | p | 0.105 ± 0.007 | 0.309 ± 0.008 | 3130 ± 165 | 1.03 ± 0.003 | 2.4 × 10−7 ± 4 × 10−8 | 52/9 | |
10%–30% | — | 0.104 ± 0.006 | 0.277 ± 0.009 | 2945 ± 150 | 1.034 ± 0.005 | 1.6 × 10−7 ± 5 × 10−8 | 22/9 | |
30%–50% | — | 0.102 ± 0.007 | 0.247 ± 0.011 | 2870 ± 135 | 1.037 ± 0.006 | 6.6 × 10−8 ± 7 × 10−9 | 16/9 | |
50%–70% | — | 0.100 ± 0.008 | 0.227 ± 0.010 | 2797 ± 180 | 1.04 ± 0.005 | 2.3 × 10−8 ± 5 × 10−9 | 20/9 | |
Figure 2 Cu–Cu 200 GeV | 0%–10% | 0.128 ± 0.007 | 0.361 ± 0.008 | 3550 ± 192 | 1.05 ± 0.004 | 0.02 ± 0.004 | 16/12 | |
10%–20% | — | 0.127 ± 0.005 | 0.342 ± 0.009 | 3465 ± 170 | 1.055 ± 0.005 | 0.0014 ± 0.0005 | 5/12 | |
20%–30% | — | 0.125 ± 0.005 | 0.334 ± 0.009 | 3387 ± 170 | 1.06 ± 0.004 | 9 × 10−5 ± 4 × 10−6 | 4/12 | |
30%–40% | — | 0.123 ± 0.006 | 0.319 ± 0.010 | 3262 ± 164 | 1.064 ± 0.004 | 6 × 10−6 ± 5 × 10−7 | 4/12 | |
40%–60% | — | 0.120 ± 0.007 | 0.306 ± 0.011 | 3100 ± 129 | 1.069 ± 0.005 | 3.6 × 10−7 ± 7 × 10−8 | 8/12 | |
0%–10% | Λ | 0.129 ± 0.009 | 0.303 ± 0.010 | 2990 ± 160 | 1.045 ± 0.005 | 0.006 ± 0.0007 | 7/11 | |
10%–20% | — | 0.127 ± 0.006 | 0.286 ± 0.006 | 2800 ± 155 | 1.048 ± 0.006 | 4.1 × 10−4 ± 2 × 10−5 | 11/11 | |
20%–30% | — | 0.124 ± 0.004 | 0.276 ± 0.011 | 2710 ± 170 | 1.05 ± 0.004 | 3 × 10−5 ± 4 × 10−6 | 2/11 | |
30%–40% | — | 0.123 ± 0.006 | 0.267 ± 0.007 | 2587 ± 150 | 1.053 ± 0.004 | 2 × 10−6 ± 4 × 10−7 | 2/11 | |
40%–60% | — | 0.120 ± 0.007 | 0.255 ± 0.006 | 2490 ± 160 | 1.055 ± 0.005 | 1 × 10−7 ± 6 × 10−8 | 3/11 | |
0%–10% | Ξ | 0.143 ± 0.005 | 0.290 ± 0.007 | 2770 ± 220 | 1.037 ± 0.005 | 0.0009 ± 0.000 07 | 2/6 | |
10%–20% | — | 0.142 ± 0.004 | 0.281 ± 0.008 | 2645 ± 200 | 1.043 ± 0.004 | 4.8 × 10−5 ± 5 × 10−6 | 5/6 | |
20%–30% | — | 0.137 ± 0.005 | 0.270 ± 0.009 | 2500 ± 190 | 1.048 ± 0.004 | 3.4 × 10−6 ± 8 × 10−7 | 3/6 | |
30%–40% | — | 0.136 ± 0.006 | 0.258 ± 0.008 | 2400 ± 205 | 1.052 ± 0.004 | 2.2 × 10−7 ± 7 × 10−8 | 3/6 | |
40%–60% | — | 0.134 ± 0.005 | 0.250 ± 0.012 | 2300 ± 210 | 1.053 ± 0.003 | 1.1 × 10−8 ± 7 × 10−9 | 3/6 | |
0%–10% | 0.144 ± 0.005 | 0.278 ± 0.008 | 2400 ± 180 | 1.055 ± 0.004 | 0.0001 ± 0.000 05 | 0.6/1 | ||
10%–20% | — | 0.142 ± 0.006 | 0.270 ± 0.010 | 2313 ± 200 | 1.057 ± 0.005 | 6.7 × 10−6 ± 7 × 10−7 | 0.5/1 | |
20%–30% | — | 0.140 ± 0.005 | 0.260 ± 0.008 | 2200 ± 200 | 1.059 ± 0.004 | 4.3 × 10−7 ± 6 × 10−8 | 1/1 | |
30%–40% | — | 0.138 ± 0.006 | 0.238 ± 0.005 | 2100 ± 200 | 1.060 ± 0.004 | 3 × 10−8 ± 4 × 10−9 | 0.5/1 | |
40%–60% | — | 0.136 ± 0.005 | 0.226 ± 0.010 | 2000 ± 170 | 1.062 ± 0.003 | 1.2 × 10−9 ± 8 × 10−10 | 0.3/1 | |
Figure 3 Au–Au 62.4 GeV | 0%–5% | π+ | 0.112 ± 0.008 | 0.369 ± 0.014 | 5400 ± 300 | 1.01 ± 0.006 | 0.24 ± 0.007 | 7/6 |
5%–10% | — | 0.110 ± 0.007 | 0.359 ± 0.014 | 5200 ± 400 | 1.012 ± 0.008 | 0.19 ± 0.005 | 8/6 | |
10%–20% | — | 0.108 ± 0.008 | 0.344 ± 0.015 | 5125 ± 240 | 1.014 ± 0.009 | 0.15 ± 0.006 | 5/6 | |
20%–30% | — | 0.105 ± 0.005 | 0.331 ± 0.014 | 5000 ± 200 | 1.016 ± 0.008 | 0.11 ± 0.008 | 5/6 | |
30%–40% | — | 0.102 ± 0.010 | 0.320 ± 0.016 | 4850 ± 190 | 1.019 ± 0.008 | 0.077 ± 0.006 | 3/6 | |
40%–50% | — | 0.100 ± 0.009 | 0.306 ± 0.015 | 4700 ± 225 | 1.022 ± 0.01 | 0.053 ± 0.004 | 11/6 | |
50%–60% | — | 0.097 ± 0.007 | 0.290 ± 0.013 | 4600 ± 200 | 1.024 ± 0.01 | 0.034 ± 0.005 | 19/6 | |
60%–70% | — | 0.095 ± 0.008 | 0.270 ± 0.015 | 4430 ± 370 | 1.027 ± 0.007 | 0.019 ± 0.003 | 13/6 | |
70%–80% | — | 0.092 ± 0.008 | 0.252 ± 0.016 | 4260 ± 280 | 1.03 ± 0.014 | 0.010 ± 0.002 | 15/6 | |
0%–5% | K+ | 0.135 ± 0.009 | 0.352 ± 0.014 | 4950 ± 320 | 1.04 ± 0.007 | 0.046 ± 0.004 | 3/6 | |
5%–10% | — | 0.133 ± 0.008 | 0.338 ± 0.012 | 4830 ± 240 | 1.043 ± 0.008 | 0.04 ± 0.005 | 0.4/6 | |
10%–20% | — | 0.132 ± 0.006 | 0.325 ± 0.013 | 4700 ± 220 | 1.045 ± 0.006 | 0.03 ± 0.005 | 2/6 | |
20%–30% | — | 0.130 ± 0.008 | 0.311 ± 0.012 | 4600 ± 200 | 1.048 ± 0.007 | 0.021 ± 0.003 | 1/6 | |
30%–40% | — | 0.127 ± 0.006 | 0.300 ± 0.013 | 4460 ± 200 | 1.05 ± 0.01 | 0.014 ± 0.004 | 2/6 | |
40%–50% | — | 0.125 ± 0.008 | 0.289 ± 0.014 | 4330 ± 200 | 1.052 ± 0.01 | 0.0095 ± 0.0005 | 6/6 | |
50%–60% | — | 0.123 ± 0.007 | 0.275 ± 0.014 | 4200 ± 200 | 1.055 ± 0.01 | 0.0055 ± 0.0007 | 0.4/6 | |
60%–70% | — | 0.120 ± 0.008 | 0.261 ± 0.013 | 4076 ± 250 | 1.058 ± 0.009 | 0.0031 ± 0.0004 | 5/6 | |
70%–80% | — | 0.117 ± 0.007 | 0.237 ± 0.011 | 3900 ± 220 | 1.06 ± 0.01 | 0.0012 ± 0.0003 | 6/6 | |
0%–5% | p | 0.113 ± 0.008 | 0.336 ± 0.012 | 4300 ± 200 | 1.1 ± 0.013 | 0.024 ± 0.005 | 1/10 | |
5%–10% | — | 0.110 ± 0.009 | 0.325 ± 0.013 | 4200 ± 230 | 1.13 ± 0.01 | 0.0215 ± 0.004 | 5/10 | |
10%–20% | — | 0.108 ± 0.007 | 0.318 ± 0.013 | 4040 ± 200 | 1.14 ± 0.008 | 0.018 ± 0.006 | 5/10 | |
20%–30% | — | 0.106 ± 0.008 | 0.307 ± 0.015 | 3900 ± 270 | 1.15 ± 0.01 | 0.0125 ± 0.004 | 11/10 | |
30%–40% | — | 0.104 ± 0.008 | 0.300 ± 0.014 | 3770 ± 185 | 1.16 ± 0.008 | 0.0085 ± 0.0007 | 13/10 | |
40%–50% | — | 0.101 ± 0.009 | 0.288 ± 0.013 | 3600 ± 220 | 1.17 ± 0.008 | 0.005 ± 0.0005 | 19/10 | |
50%–60% | — | 0.099 ± 0.007 | 0.276 ± 0.011 | 3500 ± 180 | 1.18 ± 0.005 | 0.003 ± 0.0002 | 13/10 | |
60%–70% | — | 0.097 ± 0.008 | 0.260 ± 0.012 | 3400 ± 180 | 1.19 ± 0.005 | 0.0015 ± 0.0004 | 12/10 | |
70%–80% | — | 0.095 ± 0.009 | 0.246 ± 0.014 | 3330 ± 200 | 1.2 ± 0.005 | 6.5 × 10−4 ± 7 × 10−5 | 4/10 | |
Figure 4 Au–Au 62.4 GeV | 0%–5% | 0.135 ± 0.007 | 0.347 ± 0.007 | 4900 ± 300 | 1.034 ± 0.005 | 0.023 ± 0.005 | 1/10 | |
5%–10% | — | 0.132 ± 0.005 | 0.338 ± 0.007 | 4800 ± 240 | 1.038 ± 0.004 | 0.002 ± 0.0004 | 2/10 | |
10%–20% | — | 0.130 ± 0.007 | 0.322 ± 0.008 | 4670 ± 360 | 1.045 ± 0.004 | 0.000 16 ± 0.000 05 | 1/10 | |
20%–30% | — | 0.125 ± 0.006 | 0.309 ± 0.009 | 4500 ± 245 | 1.047 ± 0.005 | 1.1 × 10−5 ± 4 × 10−6 | 2/10 | |
30%–40% | — | 0.122 ± 0.006 | 0.296 ± 0.007 | 4380 ± 300 | 1.053 ± 0.005 | 1 × 10−6 ± 3 × 10−7 | 1/10 | |
40%–60% | — | 0.120 ± 0.006 | 0.282 ± 0.009 | 4200 ± 215 | 1.056 ± 0.006 | 4.5 × 10−8 ± 4 × 10−9 | 2/10 | |
60%–80% | — | 0.118 ± 0.007 | 0.274 ± 0.010 | 4100 ± 180 | 1.058 ± 0.005 | 1.6 × 10−9 ± 6 × 10−10 | 3/10 | |
0%–5% | Λ | 0.136 ± 0.006 | 0.224 ± 0.010 | 3790 ± 260 | 1.036 ± 0.004 | 0.02 ± 0.004 | 4/8 | |
5%–10% | — | 0.135 ± 0.007 | 0.215 ± 0.012 | 3600 ± 300 | 1.037 ± 0.004 | 1.6 × 10−4 ± 4 × 10−5 | 9/8 | |
10%–20% | — | 0.133 ± 0.005 | 0.205 ± 0.011 | 3470 ± 185 | 1.039 ± 0.005 | 1.4 × 10−4 ± 5 × 10−5 | 7/8 | |
20%–30% | — | 0.132 ± 0.006 | 0.195 ± 0.010 | 3356 ± 200 | 1.040 ± 0.005 | 8 × 10−6 ± 5 × 10−7 | 11/8 | |
30%–40% | — | 0.130 ± 0.005 | 0.185 ± 0.011 | 3210 ± 170 | 1.042 ± 0.006 | 7 × 10−7 ± 4 × 10−8 | 3/8 | |
40%–60% | — | 0.128 ± 0.007 | 0.172 ± 0.013 | 3100 ± 190 | 1.044 ± 0.005 | 2.7 × 10−8 ± 6 × 10−9 | 19/8 | |
60%–80% | — | 0.125 ± 0.006 | 0.165 ± 0.010 | 3000 ± 170 | 1.050 ± 0.004 | 7 × 10−10 ± 8 × 10−11 | 5/8 | |
0%–5% | 0.151 ± 0.006 | 0.215 ± 0.007 | 3000 ± 260 | 1.015 ± 0.004 | 0.0014 ± 0.0003 | 1/6 | ||
5%–10% | — | 0.149 ± 0.007 | 0.206 ± 0.009 | 2800 ± 200 | 1.02 ± 0.004 | 0.000 12 ± 0.000 05 | 3/7 | |
10%–20% | — | 0.147 ± 0.005 | 0.197 ± 0.010 | 2680 ± 156 | 1.024 ± 0.004 | 8.5 × 10−7 ± 4 × 10−8 | 1/7 | |
20%–40% | — | 0.145 ± 0.006 | 0.184 ± 0.009 | 2600 ± 140 | 1.025 ± 0.005 | 5.5 × 10−7 ± 3 × 10−8 | 2/7 | |
40%–60% | — | 0.143 ± 0.005 | 0.180 ± 0.008 | 2500 ± 144 | 1.028 ± 0.005 | 1.6 × 10−8 ± 6 × 10−9 | 2/6 | |
60%–80% | — | 0.141 ± 0.006 | 0.170 ± 0.010 | 2400 ± 150 | 1.029 ± 0.004 | 4.8 × 10−10 ± 7 × 10−11 | 4/5 | |
0%–20% | 0.153 ± 0.006 | 0.170 ± 0.008 | 2400 ± 200 | 1.025 ± 0.004 | 0.0001 ± 0.000 03 | 1/1 | ||
20%–40% | — | 0.150 ± 0.007 | 0.160 ± 0.012 | 2260 ± 170 | 1.035 ± 0.005 | 3 × 10−7 ± 7 × 10−8 | 0.4/1 | |
40%–60% | — | 0.148 ± 0.005 | 0.150 ± 0.011 | 2160 ± 138 | 1.055 ± 0.008 | 4 × 10−9 ± 6 × 10−10 | 1/0 | |
Figure 5 Pb–Pb 2.76 TeV | 0%–5% | π+ | 0.126 ± 0.005 | 0.440 ± 0.013 | 8660 ± 400 | 1.02 ± 0.006 | 290 ± 45 | 215/37 |
5%–10% | — | 0.124 ± 0.005 | 0.430 ± 0.011 | 8530 ± 370 | 1.029 ± 0.006 | 120 ± 27 | 191/37 | |
10%–20% | — | 0.122 ± 0.005 | 0.416 ± 0.012 | 8400 ± 290 | 1.034 ± 0.005 | 50 ± 7 | 211/37 | |
20%–30% | — | 0.120 ± 0.005 | 0.407 ± 0.009 | 8260 ± 300 | 1.042 ± 0.005 | 17 ± 4 | 203/37 | |
30%–40% | — | 0.117 ± 0.006 | 0.399 ± 0.012 | 8000 ± 360 | 1.046 ± 0.004 | 6 ± 1 | 274/37 | |
40%–50% | — | 0.115 ± 0.007 | 0.389 ± 0.010 | 7800 ± 330 | 1.054 ± 0.006 | 1.8 ± 0.3 | 186/37 | |
50%–60% | — | 0.113 ± 0.006 | 0.378 ± 0.011 | 7620 ± 300 | 1.059 ± 0.005 | 0.5 ± 0.03 | 181/37 | |
60%–70% | — | 0.110 ± 0.005 | 0.364 ± 0.010 | 7500 ± 290 | 1.062 ± 0.004 | 0.13 ± 0.04 | 339/37 | |
70%–80% | — | 0.108 ± 0.005 | 0.350 ± 0.012 | 7280 ± 280 | 1.069 ± 0.005 | 0.03 ± 0.002 | 255/37 | |
80%–90% | — | 0.106 ± 0.006 | 0.340 ± 0.011 | 7140 ± 210 | 1.073 ± 0.006 | 0.006 ± 0.0005 | 255/37 | |
0%–5% | K+ | 0.142 ± 0.007 | 0.422 ± 0.009 | 8100 ± 300 | 1.045 ± 0.004 | 40 ± 7 | 21/32 | |
5%–10% | — | 0.140 ± 0.006 | 0.410 ± 0.013 | 7900 ± 320 | 1.047 ± 0.005 | 20 ± 4 | 32/32 | |
10%–20% | — | 0.136 ± 0.005 | 0.396 ± 0.011 | 7760 ± 270 | 1.058 ± 0.008 | 7 ± 0.2 | 29/32 | |
20%–30% | — | 0.133 ± 0.005 | 0.382 ± 0.012 | 7600 ± 220 | 1.062 ± 0.005 | 2.5 ± 0.5 | 40/32 | |
30%–40% | — | 0.130 ± 0.007 | 0.375 ± 0.010 | 7500 ± 200 | 1.07 ± 0.006 | 0.8 ± 0.03 | 23/32 | |
40%–50% | — | 0.128 ± 0.005 | 0.361 ± 0.013 | 7350 ± 230 | 1.076 ± 0.006 | 0.25 ± 0.04 | 10/32 | |
50%–60% | — | 0.125 ± 0.006 | 0.350 ± 0.011 | 7220 ± 210 | 1.08 ± 0.005 | 0.07 ± 0.004 | 13/32 | |
60%–70% | — | 0.124 ± 0.006 | 0.340 ± 0.010 | 7100 ± 180 | 1.085 ± 0.006 | 0.018 ± 0.003 | 104/32 | |
70%–80% | — | 0.122 ± 0.005 | 0.327 ± 0.012 | 6940 ± 220 | 1.089 ± 0.005 | 0.004 ± 0.0005 | 15/32 | |
80%–90% | — | 0.120 ± 0.007 | 0.317 ± 0.010 | 6810 ± 265 | 1.09 ± 0.006 | 0.000 82 ± 0.000 04 | 11/32 | |
0%–5% | p | 0.127 ± 0.007 | 0.410 ± 0.009 | 7467 ± 260 | 1.065 ± 0.005 | 9.82 ± 0.6 | 316/33 | |
5%–10% | — | 0.125 ± 0.006 | 0.400 ± 0.011 | 7300 ± 300 | 1.07 ± 0.005 | 3.79 ± 0.4 | 335/33 | |
10%–20% | — | 0.123 ± 0.005 | 0.388 ± 0.012 | 7170 ± 190 | 1.078 ± 0.007 | 1.49 ± 0.2 | 273/33 | |
20%–30% | — | 0.121 ± 0.006 | 0.373 ± 0.013 | 7000 ± 215 | 1.082 ± 0.004 | 0.49 ± 0.05 | 223/33 | |
30%–40% | — | 0.119 ± 0.005 | 0.361 ± 0.009 | 6800 ± 200 | 1.084 ± 0.004 | 0.17 ± 0.03 | 168/33 | |
40%–50% | — | 0.116 ± 0.007 | 0.348 ± 0.011 | 6692 ± 192 | 1.087 ± 0.006 | 0.052 ± 0.004 | 116/33 | |
50%–60% | — | 0.113 ± 0.006 | 0.338 ± 0.010 | 6520 ± 175 | 1.088 ± 0.004 | 0.0145 ± 0.004 | 35/33 | |
60%–70% | — | 0.110 ± 0.006 | 0.325 ± 0.012 | 6400 ± 200 | 1.09 ± 0.004 | 0.004 ± 0.0004 | 19/33 | |
70%–80% | — | 0.108 ± 0.007 | 0.311 ± 0.011 | 6300 ± 160 | 1.092 ± 0.005 | 0.000 84 ± 0.000 04 | 24/33 | |
80%–90% | — | 0.105 ± 0.005 | 0.291 ± 0.013 | 6250 ± 250 | 1.094 ± 0.004 | 0.000 17 ± 0.000 06 | 113/33 | |
Figure 6 Pb–Pb 2.76 TeV | 0%–10% | 0.142 ± 0.007 | 0.420 ± 0.015 | 8020 ± 380 | 1.045 ± 0.004 | 0.47 ± 0.07 | 33/17 | |
10%–20% | — | 0.140 ± 0.006 | 0.410 ± 0.012 | 7900 ± 295 | 1.05 ± 0.005 | 0.35 ± 0.06 | 16/17 | |
20%–40% | — | 0.138 ± 0.006 | 0.400 ± 0.010 | 7770 ± 257 | 1.057 ± 0.005 | 0.20 ± 0.04 | 13/17 | |
40%–60% | — | 0.136 ± 0.008 | 0.387 ± 0.013 | 7640 ± 245 | 1.062 ± 0.006 | 0.072 ± 0.003 | 16/17 | |
60%–80% | — | 0.133 ± 0.006 | 0.374 ± 0.010 | 7510 ± 215 | 1.069 ± 0.004 | 0.019 ± 0.003 | 120/17 | |
0%–10% | Λ | 0.143 ± 0.006 | 0.400 ± 0.011 | 7010 ± 335 | 1.033 ± 0.004 | 0.055 ± 0.005 | 39/15 | |
10%–20% | — | 0.141 ± 0.005 | 0.389 ± 0.010 | 6900 ± 270 | 1.035 ± 0.005 | 0.042 ± 0.0055 | 15/15 | |
20%–40% | — | 0.138 ± 0.005 | 0.375 ± 0.009 | 6770 ± 256 | 1.038 ± 0.004 | 0.026 ± 0.004 | 10/15 | |
40%–60% | — | 0.136 ± 0.007 | 0.355 ± 0.012 | 6660 ± 240 | 1.039 ± 0.005 | 0.01 ± 0.002 | 52/15 | |
60%–80% | — | 0.133 ± 0.006 | 0.302 ± 0.015 | 6560 ± 195 | 1.041 ± 0.004 | 0.002 78 ± 0.000 06 | 68/15 | |
0%–10% | Ξ | 0.157 ± 0.008 | 0.385 ± 0.015 | 6560 ± 260 | 1.03 ± 0.004 | 0.0155 ± 0.003 | 8/8 | |
10%–20% | — | 0.155 ± 0.006 | 0.375 ± 0.013 | 6460 ± 240 | 1.032 ± 0.005 | 0.0118 ± 0.004 | 17/8 | |
20%–40% | — | 0.153 ± 0.007 | 0.342 ± 0.010 | 6320 ± 320 | 1.034 ± 0.006 | 0.0075 ± 0.000 04 | 18/8 | |
40%–60% | — | 0.150 ± 0.005 | 0.302 ± 0.012 | 6200 ± 210 | 1.036 ± 0.006 | 0.003 ± 0.0002 | 17/8 | |
60%–80% | — | 0.146 ± 0.008 | 0.287 ± 0.009 | 6154 ± 194 | 1.04 ± 0.005 | 0.000 65 ± 0.000 05 | 39/8 | |
0%–10% | Ω | 0.158 ± 0.007 | 0.350 ± 0.013 | 6120 ± 175 | 1.04 ± 0.004 | 0.0012 ± 0.0003 | 0.3/3 | |
10%–20% | — | 0.155 ± 0.005 | 0.330 ± 0.010 | 6000 ± 220 | 1.054 ± 0.005 | 0.0008 ± 0.000 04 | 1/3 | |
20%–40% | — | 0.153 ± 0.006 | 0.280 ± 0.009 | 5800 ± 200 | 1.06 ± 0.006 | 0.0005 ± 0.000 03 | 3/3 | |
40%–60% | — | 0.149 ± 0.006 | 0.220 ± 0.014 | 5700 ± 240 | 1.062 ± 0.008 | 0.0002 ± 0.000 03 | 7/3 | |
60%–80% | — | 0.147 ± 0.005 | 0.190 ± 0.011 | 5500 ± 214 | 1.065 ± 0.007 | 3.5 × 10−5 ± 5 × 10−6 | 3/2 |
The normalization constant N0 is used for comparison of the fit function fS(pT) or fS(mT) and the experimental spectra, and the normalization constant C is used to let the integral of equation (1) be unity. The two normalization constants are not the same, though C can be absorbed in N0. We have used both C and N0 to present a clear description.
Figure 2 is similar to figure 1, but it shows the pT spectra of Λ, Ξ and in panels (a)–(c) with |y| < 0.5 in Cu–Cu collisions at 200 GeV. The spectra of Λ, Ξ and is distributed in different centrality classes of 0%–10%, 10%–20%, 20%–30%, 30%–40% and 40%–60%. The symbols represent the experimental data of STAR [43] collaboration while the curves are the results of our fitting by using equation (1). Each panel is followed by result of its data/fit. One can see that the model results describe approximately the experimental data in special pT ranges at RHIC.
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Standard image High-resolution imageFigures 3 and 4 are similar to figure 1, but it shows the pT spectra of π+, K+, p and in figures 3(a)–(d) and Λ, and in figures 4(a)–(c) in Au–Au collisions at 62.4 GeV. The spectra is distributed in different centrality classes, e.g for π+, K+ and p 0%–5%, 5%–10%, 10%–20%, 20%–30%, 30%–40%, 40%–50%, 50%–60%, 60%–70% and 70%–80% , while for , Λ, and 0%–5%, 5%–10%, 10%–20%, 20%–30%, 30%–40%, 40%–60% and 60%–80%, and for and is 0%–20%, 20%–40% and 40%–60%. The symbols represent the experimental data of STAR Collaboration [31, 45], while the curves are the results of our fitting by using equation (1). Each panel is followed by result of its data/fit. One can see that the model results describe approximately the experimental data in special pT ranges at RHIC. The special pT range refers to the contribution of soft excitation process. Furthermore the data for , Λ, Ξ and in Cu–Cu collisions and in Au–Au collision is cut at higher at high pT.
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Standard image High-resolution imageFigures 5 and 6 are the same as figure 1, but it shows the pT spectra of π+, K+, p and in panels of figures 5(a)–(d) and Λ, Ξ and Ω in figures 6(a)–(c) in Pb–Pb collisions at 2.76 TeV in |y| < 0.5. The spectra is distributed in different centrality classes, e.g. for π+, K+ and p 0%–5%, 5%–10%, 10%–20%, 20%–30%, 30%–40%, 40%–50%, 50%–60%, 60%–70%, 70%–80% and 80%–90% , while for , Λ, Ξ and Ω 0%–10%, 10%–20%, 20%–40%, 40%–60% and 60%–80%. The symbols represent the experimental data of ALICE Collaboration [46, 47] (the data for π+, K+ and p is taken from reference [46] and data for , Λ, Ξ and Ω is taken from [47]), while the curves are the results of our fitting by using equation (1). Each panel is followed by result of its data/fit. One can see that the model results describe the experimental data in special pT ranges at LHC. We comment here by the way that the description of the pion spectra at LHC is poor due to the non-inclusion of the resonance generation in low pT region. In figures 1–6, pT spectra of some particles are re-scaled in some centrality bins which are showed in the corresponding figures.
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Standard image High-resolution imageTo study the changing tendencies of parameters, figure 7 shows the dependence of T0 on the rest mass of the particle as well as on centrality. Different symbols represent different particles and the trend of symbols from up to downwards show the trend from central toward peripheral collisions in panels (a)–(c). One can see that T0 of the emission source extracted in different centrality intervals is the same for π+ and p, similarly it is the same for K+, and Λ, and Ξ or and or or Ω. In the analyzed particles, Ξ or and or or Ω are the heaviest particles and they freezeout earlier than the rest of others. π+ is the lightest particle and it freezeout after Ξ or , or or Ω, K+, and Λ. p is heavier than π, K+ and but it freezeout after K+ and (same result of larger T0 for K+ and than p can be found in [48], although the main idea is different from our present work). Eventually, the freezeout of p occurs at the same time with π. At present, there is no clear dependence of T0 on mass of the particle. Therefore we believe that the recent result of T0 of the particles may be dependent on the production cross-section of the interacting particle, such that, smaller the production cross section of the interacting particle, larger will be its source T0 and the particle will freezeout earlier. π and p are non-strange particles and have larger production cross-section, while K+, and Λ are strange and Ξ or and or or Ω are the multi-strange particles and the later has the smallest production cross-section and they freezeout earlier than the strange particles. The above statement reveals the scenario of triple kinetic freezeout due to the separate decoupling of the non-strange, strange and multi-strange particles and this scenario is observed by us (single, double and multi-kinetic freezeout scenarios are already studied in different literatures). Furthermore, T0 depends on centrality and it shows a decreasing trend as we go from central to peripheral collisions. In central collision, large number of particles get involved in interactions due to large participant collision cross-section of the interacting system and deposits more energy in the system which results in larger kinetic freezeout temperature due to higher degree of excitation of the system [14, 15, 49]. However the cross section of the interacting system decrease with decreasing centrality that leads the decrease of kinetic freezeout temperature. The kinetic freezeout temperature is observed to be larger in Pb–Pb than in Au–Au collisions and in the later case it is larger than Cu–Cu collisions.
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Standard image High-resolution imageFigure 8 is similar to figure 7, but it represents the dependence of βT on the rest mass of the particle and on centrality. One can see that greater the mass of the particle is, smaller the value for βT. It is also observed that βT is decreasing with decreasing the event centrality due to decreasing the participant collision cross-section of the interacting system. In larger cross-section interacting system, large amount of energy is stored due to the participation of large number of nucleons in the interaction and thus the system undergoes a rapid expansion but this expansion becomes more and more steady from center to periphery. βT is found to be slightly larger in Pb–Pb than in Cu–Cu and in the later it is larger than in Au–Au which shows its dependence on the center of mass energy of the system.
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Standard image High-resolution imageFigure 9 is also similar to figure 7, but it exhibits the dependence of kinetic freezeout volume on rest mass of and centrality. The volume differential scenario is observed as the the kinetic freezeout volume decreases for heavier particles, which shows the early freezeout of the heavy particles than the lighter ones and this maybe the tip-off of different freezeout surfaces for different particles. The kinetic freezeout volume is observed to be larger in Pb–Pb than in Au–Au collisions and in the later it is larger than Cu–Cu collisions. The kinetic freezeout volume has a decreasing trend from central to peripheral collisions due to decreasing the number of participant nucleons from central to periphery. Because the mass of K+ and is almost identical and the relative parameter values give very close values to each other, so the K+ data points are hidden beneath .
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Standard image High-resolution imageFigure 10 shows the dependence of q on centrality. The more central the collision is, less will be q, which indicates to a quick approach of equilibrium in the central collisions. The range of increase of 'q' from central collisions to periphery is 0.001 to 0.03. Similarly, figure 11 shows the dependence of N0 on centrality. N0 has a physical significance and it reflects the multiplicity. N0 decreases with the event centrality which obviously shows lower multiplicity from central to periphery.
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Standard image High-resolution image4. Conclusions
The main observations and conclusions are summarized here.
- (a)The transverse momentum spectra of different particle species are analyzed by the blast wave model with Tsallis statistics and the bulk properties in terms of the kinetic freezeout temperature, transverse flow velocity and kinetic freezeout volume are extracted.
- (b)The kinetic freezeout temperature (T0) is observed to be dependent on the cross-section of the interacting particle and interacting system as well. Larger the production cross-section of the interacting particle, smaller will be T0. However the case is reverse for participant collision cross-section of the interacting system where larger cross-section leads to larger T0.
- (c)A triple kinetic freezeout scenario is observed due to the separate decoupling of non-strange, strange and strange particles.
- (d)The transverse flow velocity (βT) and kinetic freezeout volume (V) are observed to be mass dependant. Larger the mass of the particle, smaller the βT and V are additionally both of them decrease from central to peripheral collisions.
- (e)V is larger in Pb–Pb collisions than in Au–Au collisions and in Au–Au collisions it is larger than Cu–Cu collisions which shows its dependence on collision energy βT exhibits the dependence on the center of mass of the interacting system as it is larger in Pb–Pb than in Cu–Cu collisions and in the later case it is larger than in Au–Au collisions. The volume for Pb–Pb collisions is about twice the one for Cu–Cu. This is natural due to the large size for Pb.
- (f)The entropy index q is observed to increases from central to peripheral collisions while the normalization constant N0 decreases from central to peripheral collisions.
Acknowledgments
The authors would like to thank support from the National Natural Science Foundation of China (Grant Nos. 11875052, 11575190, and 11135011).
Compliance with ethical standards
The authors declare that they are in compliance with ethical standards regarding the content of this paper.
Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).