Non-extensive statistical distributions of charmed meson production in Pb-Pb and pp(p¯) collisions

Non-extensive statistical distributions of charmed meson production in Pb-Pb and pp( <math id="M1"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions 1 2 first-author 1 2 1 2 1 2 corresp ephy@ustc.edu.cn ephy@ustc.edu.cn ephy@ustc.edu.cn State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, China State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, China Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China The mid-rapidity transverse momentum spectra of charmed mesons in Pb-Pb and pp( <math id="M2"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions are analyzed using the Tsallis–Pareto distribution derived from non-extensive statistics. We provide uniform descriptions of both small and large systems over a wide range of collision energies and hadron transverse momenta. By establishing the relationship between the event multiplicity and Tsallis parameters, we observe that there is a significant linear relationship between the thermal temperature and Tsallis q parameter in Pb-Pb collisions at <math id="M3"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> = 2.76 TeV and 5.02 TeV. Further, the slope of the T –( q -1) parameter plot is positively correlated with the hadron mass. In addition, charmed mesons have a higher thermal temperature than light hadrons at the same q -1, indicating that the charm flavor requires a higher temperature to reach the same degree of non-extensivity as light flavors in heavy-ion collisions. The same fit is applied to the transverse momentum spectra of charmed mesons in pp( <math id="M4"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions over a large energy range using the Tsallis–Pareto distribution. It is found that the thermal temperature increases with system energy, whereas the q parameter becomes saturated at the pp( <math id="M5"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) limit, q -1 = 0.142 ± 0.010. In addition, the results of most peripheral Pb-Pb collisions are found to approach the pp( <math id="M6"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) limit, which suggests that more peripheral heavy-ion collisions are less affected by the medium and more similar to pp( <math id="M7"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions. Charmed meson Non-extensive statistics Tsallis–Pareto distribution QGP Heavy-ion collision 1　Introduction 1 According to the Big Bang theory, under the extremely high temperature and high energy density in the early stage of the universe, the quark–gluon plasma (QGP), a new form of matter, was generated by the release of quarks and gluons that had been bound in hadrons by the strong interaction. The masses of the heavy-flavor quarks, m c ˜ 1.3 GeV/ c 2 and m b ˜ 4.8 GeV/ c 2 , are larger than those of light quarks and the quantum chromodynamics (QCD) energy scale (Λ QCD ). Therefore, the generation of heavy-flavor quarks requires sufficiently large energy and momentum transfer by initial hard scattering in heavy-ion collisions and can be calculated using perturbation QCD [ 1 1 , 2 2 ]. Heavy-flavor quarks have a high probability of undergoing relatively complete evolution in QCD materials; thus, they are ideal probes to study the properties of the QGP in heavy-ion collisions. In particular, it is crucial to study the interaction between heavy-flavor quarks and the medium by measuring the nuclear modification factor, flow, and production yield of charmed mesons [ 3 3 - 12 12 3-12 ]. The transverse momentum ( p T ) spectra of identified particles provide valuable data for discoveries based on relativistic heavy-ion collisions [ 13 13 , 14 14 ]. The Boltzmann–Gibbs blast-wave model, which has a compact set of parameters, namely, temperature ( T ) and flow velocity ( β ), has shown that the spectral shape is sensitive to the dynamics of nucleus-nucleus (AA) collisions [ 13 13 , 15 15 ] and can be used to describe the transverse momentum distributions of light-flavor hadrons with different masses. Moreover, the Tsallis blast-wave model, which includes the fluctuation of the initial conditions in the hydrodynamic evolution on an event-by-event basis, has been used to study the π ± , K ± , p( <math id="M10"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ), ϕ , <math id="M11"><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>Λ</mi><mo stretchy="true">¯</mo></mover><mo stretchy="false">)</mo></mrow></math> , and <math id="M12"><mrow><msup><mi>Ξ</mi><mo>−</mo></msup><mo stretchy="false">(</mo><msup><mrow><mover accent="true"><mi>Ξ</mi><mo stretchy="true">¯</mo></mover></mrow><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></math> spectra in Au-Au collisions at 200 GeV [ 16 16 ]. It was found that the average transverse flow velocity 〈β〉 and temperature at kinetic freeze-out T fro increased with centrality, whereas the non-extensive parameter q showed the opposite behavior. The thermodynamic formulation of non-extensive statistics was recently applied to the data for all light hadrons, and a strong grouping phenomenon in the T –( q -1) parameter space was considered to be a probe for the study of QGP in small and large systems. The non-extensive temperature converges to T ≈ 0.144 ± 0.010 GeV [ 17 17 ], which is consistent with the QCD results [ 18 18 , 19 19 ], indicating that the collisional system may have undergone QGP evolution. However, the non-extensive parameter q ≠ 1, indicating that the conclusion is based entirely on the non-extensive properties. Although the implications and consequences of such an application to particle production are still under investigation, the function is relatively easy to understand [ 20 20 - 22 22 20-22 ]. Heavy flavors are expected to behave differently from light flavors and thus may have different non-extensive thermal parameters. The same concept with a non-extensive statistical distribution function is extended to fit the charmed meson spectra recently measured in the ALICE and STAR experiments. We present the Tsallis–Pareto distribution formula and explain the physical origin of the relationship between the parameters T and q in Sect. 2 of this article. The analysis procedure and fitting results for charmed mesons and the thermal temperature after transverse flow correction are also described in this section. In Sect. 3, the results for the 0–10% (central) and 30–50% (semiperipheral) centrality bins in Pb-Pb collisions at 2.76 TeV and for the 0–10%, 30–50%, and 60–80% (peripheral) centrality bins in Pb-Pb collisions at 5.02 TeV are presented. In the same section, the results obtained in Pb-Pb collisions are compared with the values of T and q measured in proton-(anti)proton ［pp( <math id="M13"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> )］ collisions over a wide energy range. The results and conclusions are summarized in Sect. 4. 2　Tsallis–Pareto distribution and its fit to charmed meson spectra 1 Much work on high-energy particle collisions has focused on the study of the transverse momentum distributions of outgoing particles. In the low- p T regime of the spectra, the conventional exponential distributions can be used to describe the spectral shape, and the formula, assuming vanishing chemical potential at high energies, is given as <math display="block" id="M14"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mi>p</mi><mtext>T</mtext></msub></mrow><mo>)</mo></mrow><mo>∼</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msub><mi>m</mi><mtext>T</mtext></msub></mrow><mi>T</mi></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where m T = <math id="M15"><mrow><msqrt><mrow><msubsup><mi>p</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>+</mo><msup><mi>m</mi><mn>2</mn></msup></mrow></msqrt></mrow></math> is the transverse mass, m is the rest mass of the hadron species, and T is the corresponding temperature. Instead of Eq.(1), a power-law distribution has been used in high-energy physics to better characterize the spectra with <math id="M16"><mrow><msub><mi>p</mi><mtext>T</mtext></msub><munder accentunder="true"><mo>></mo><mo>˜</mo></munder></mrow></math> 3 GeV/ c . The Tsallis–Pareto distribution derived from non-extensive statistics, which is a generalization of the conventional Boltzmann–Gibbs theory, is thought to describe the full p T spectrum of hadrons. The transverse momentum dependence of identified hadron production from recently published data has been successfully described by non-extensive statistics [ 23 23 ]. In addition, for the theoretical limit <math id="M17"><mrow><mi>q</mi><mo>→</mo><mn>1</mn></mrow></math> , exponential and logarithmic functions can be obtained [ 24 24 - 26 26 24-26 ]. The exploration of this new research direction in statistical mechanics began when Tsallis proposed a non-extensive entropy in 1988 [ 24 24 ]. Non-extensive entropy (the Tsallis entropy) is a generalization of the conventional Boltzmann–Gibbs entropy, and it formed the basis of non-extensive statistics. The Tsallis–Pareto distribution within the non-extensive statistics in [ 17 17 ] is also used in the current analyses: <math display="block" id="M18"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mn>2</mn><mi>π</mi><msub><mi>p</mi><mtext>T</mtext></msub><mtext>d</mtext><msub><mi>p</mi><mtext>T</mtext></msub></mrow></mfrac><mo>=</mo><mi>A</mi><msub><mi>m</mi><mtext>T</mtext></msub><msup><mrow><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow><mrow><msub><mi>T</mi><mi>q</mi></msub></mrow></mfrac><mrow><mo>(</mo><mrow><msub><mi>m</mi><mtext>T</mtext></msub><mo>−</mo><mi>m</mi></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mrow><mo>−</mo><mfrac><mi>q</mi><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where A is a normalizing factor and can reflect the production yield of the hadron p T spectrum. T q is the temperature in the non-extensive statistical theory; the subscript q is omitted for brevity in the following discussion. Note that T q can differ from the temperature T in Eq.(1), but its physical meaning should be the same in the limiting case <math id="M19"><mrow><mi>q</mi><mo>→</mo><mn>1</mn></mrow></math> . The correlations between the parameters T and q have been presented in earlier studies [ 27 27 - 30 30 27-30 ]. Furthermore, the charged particle multiplicity can be derived from the Tsallis–Pareto-distributed transverse momentum, and concrete application to experimental data yields a negative binomial distribution parameter k ˜ <math id="M20"><mi mathvariant="script">O</mi></math> (10) [ 31 31 - 38 38 31-38 ]. In addition, many studies have revealed the importance of measuring the event-by-event multiplicity and its fluctuation. They found that the yield of strange hadrons is positively related to the multiplicity, and a long-range correlation can be observed in small collisional systems as the multiplicity increases [ 39 39 - 43 43 39-43 ]. This study focuses on the relationship between the parameters of heavy-flavor hadrons in the T –( q -1) parameter space using the non-extensive statistics described above. For simplicity, the fluctuations in the number of produced particles can be explained in a one-dimensional relativistic gas model [ 44 44 ], and the Tsallis parameters under consideration are given as <math display="block" id="M21"><mrow><mi>T</mi><mo>=</mo><mfrac><mi>E</mi><mrow><mo>〈</mo><mi>M</mi><mo>〉</mo></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M22"><mrow><mi>q</mi><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mo>〈</mo><mi>M</mi><mo>〉</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>Δ</mi><msup><mi>M</mi><mn>2</mn></msup></mrow><mrow><msup><mrow><mo>〈</mo><mi>M</mi><mo>〉</mo></mrow><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where M is the number of particles at energy E . In the thermodynamic picture, the relationship between T and q can be obtained from Eqs.(3) and (4), assuming that the relative size of the multiplicity fluctuations is constant as in [ 17 17 ]: <math display="block" id="M23"><mrow><mfrac><mrow><mi>Δ</mi><msup><mi>M</mi><mn>2</mn></msup></mrow><mrow><msup><mrow><mo>〈</mo><mi>M</mi><mo>〉</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>≡</mo><msup><mi>ζ</mi><mn>2</mn></msup><mn>.</mn></mrow></math> Thus, we have the following formula in the current analyses: <math display="block" id="M24"><mrow><mi>T</mi><mo>=</mo><mi>E</mi><mrow><mo>(</mo><mrow><msup><mi>ζ</mi><mn>2</mn></msup><mo>−</mo><mrow><mo>(</mo><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> This formula is used to measure the relationship between the Tsallis parameters and event multiplicity in charmed meson production for both small and large systems over a wide range of collision energies and hadron transverse momenta, and the results are compared with the corresponding results for light hadrons. 2.1　 Application of Tsallis–Pareto distribution to charmed meson spectra 2 In this study, we analyze the transverse momentum dependence of charmed meson production in pp( <math id="M25"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) and AA collisions measured by the ALICE, CDF, and STAR collaborations [ 45 45 - 52 52 45-52 ]. We find that the charmed meson data from small and large collision systems with broad selection criteria can be investigated simultaneously. D 0 and D * spectra were measured by the STAR Collaboration in pp collisions at 200 GeV for 0.4 < p T < 8.8 GeV/ c and at 500 GeV for 1.5 < p T < 17.7 GeV/ c . D 0 spectra were measured by the CDF Collaboration in p <math id="M26"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> collisions at 1.96 TeV for 5.8 < p T < 16.0 GeV/ c and by the ALICE Collaboration in pp collisions at 7 TeV for 1.5 < p T < 13.8 GeV/ c . Figure 1 Figure 1 shows these p T spectra of D mesons, together with our fit results at four energies (200 GeV, 500 GeV, 1.96 TeV, and 7 TeV) in pp( <math id="M27"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions. The fit parameters and χ 2 / ndf are listed in Table 1 Table 1 . The solid curves from the Tsallis–Pareto distributions describe the data well. Error bars denote the quadratic sums of the statistical and systematic uncertainties. Data are scaled by factors of 10 n for better visibility. 1 1 <math id="M30"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> (GeV) 1 1 Centrality 1 1 Charmed meson 1 1 A 1 1 T 1 1 q 1 1 χ 2 / ndf 1 1 Pb-Pb, 2760 1 1 0–10% 1 1 D 0 1 1 1.279± 0.516 1 1 0.239±0.030 1 1 1.166± 0.014 1 1 5.15/6 1 1 1 1 1 1 D + 1 1 1.052± 0.350 1 1 0.201± 0.024 1 1 1.180± 0.013 1 1 9.13/7 1 1 1 1 1 1 D *+ 1 1 0.515± 0.212 1 1 0.240±0.031 1 1 1.179± 0.012 1 1 2.9/7 1 1 1 1 30–50% 1 1 D 0 1 1 0.178± 0.076 1 1 0.278±0.041 1 1 1.169±0.020 1 1 0.88/5 1 1 1 1 1 1 D + 1 1 0.052± 0.025 1 1 0.322± 0.055 1 1 1.151±0.025 1 1 0.51/5 1 1 1 1 1 1 D *+ 1 1 0.084± 0.045 1 1 0.250±0.049 1 1 1.205± 0.025 1 1 0.34/5 1 1 Pb-Pb, 5020 1 1 0–10% 1 1 D 0 1 1 1.957± 0.551 1 1 0.240± 0.020 1 1 1.187±0.008 1 1 14.95/9 1 1 1 1 1 1 D + 1 1 0.768±0.277 1 1 0.245± 0.025 1 1 1.190± 0.008 1 1 5.16/9 1 1 1 1 1 1 D *+ 1 1 0.675± 0.209 1 1 0.258± 0.026 1 1 1.184±0.010 1 1 6.12/9 1 1 1 1 30–50% 1 1 D 0 1 1 0.145± 0.040 1 1 0.328± 0.026 1 1 1.175± 0.008 1 1 2.12/9 1 1 1 1 1 1 D + 1 1 0.088± 0.024 1 1 0.311± 0.024 1 1 1.179± 0.007 1 1 5.35/9 1 1 1 1 1 1 D *+ 1 1 0.057± 0.020 1 1 0.331±0.034 1 1 1.185± 0.008 1 1 3.76/9 1 1 1 1 60–80% 1 1 D 0 1 1 0.010± 0.003 1 1 0.427± 0.042 1 1 1.151± 0.012 1 1 2.52/8 1 1 1 1 1 1 D + 1 1 0.005± 0.001 1 1 0.402± 0.043 1 1 1.170± 0.013 1 1 2.21/8 1 1 1 1 1 1 D *+ 1 1 0.004± 0.002 1 1 0.430± 0.069 1 1 1.156± 0.017 1 1 4.48/8 1 1 pp, 200 1 1 1 1 D 0 + D * 1 1 3.8e-4 ± 5.3e-5 1 1 0.322± 0.022 1 1 1.081± 0.011 1 1 4.39/7 1 1 pp, 500 1 1 1 1 D 0 + D * 1 1 6.8e-4 ± 1.7e-4 1 1 0.310± 0.020 1 1 1.132± 0.006 1 1 4.13/10 1 1 p( <math id="M31"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ), 1960 1 1 1 1 D 0 1 1 1.8e-3±1.7e-3 1 1 0.386± 0.058 1 1 1.143± 0.011 1 1 0.87/3 1 1 pp, 7000 1 1 1 1 D 0 1 1 1.6e-3 ± 7.6e-4 1 1 0.494± 0.062 1 1 1.139± 0.023 1 1 1.75/6 In Fig. 2 Fig. 2 , the fits are applied to the transverse momentum distributions of charmed mesons for different centrality bins in AA collisions [ 49 49 - 52 52 49-52 ]. The fits of the transverse momentum distributions of prompt D 0 , D + , and D *+ mesons in Pb-Pb collisions at 2.76 TeV are shown in Fig. 2(a) Fig. 2(a) . Solid circles, diamonds, and triangles represent D 0 , D + , and D *+ , respectively. The 0–10% (solid) and 30–50% (open) centrality bins are shown in Fig. 2(a) Fig. 2(a) [ 49 49 ], where the D 0 and D *+ production yields at 0–10% and 30–50% are scaled by factors of 10 and 0.05, respectively. For the 0–10% centrality bin, fits are performed in the range 0 < p T < 20.0 GeV/ c for D 0 and in the range 0 < p T < 30.0 GeV/ c for D + and D *+ . For the 30–50% centrality bin, the same fitting procedures were performed in the range 0 < p T < 14.0 GeV/ c for D 0 , D + , and D *+ . The vertical bars represent the sums of the statistical and systematic uncertainties. Table 1 Table 1 shows that the fit parameters of the charmed meson spectra in the 0–10% centrality bin have smaller uncertainties with more data points than those in the 30–50% centrality bin at 2.76 TeV. Figure 2(b), (c), and (d) Figure 2(b), (c), and (d) show the transverse momentum distributions d N /d p T of D 0 mesons (solid circles), D + mesons (diamonds), and D *+ mesons (triangles) in the 0–10%, 30–50%, and 60–80% centrality bins, respectively, in Pb-Pb collisions at 5.02 TeV [ 50 50 ]. The vertical bars represent the quadratic sums of the statistical and systematic uncertainties, and symbols are placed at the center of the bin. The solid curves representing the Tsallis–Pareto distributions describe the data well. For visibility, the D 0 and D *+ distributions in the three centrality bins are scaled by factors of 10 and 1/10, respectively. To more physically constrain the D + and D *+ yields at p T = 1.5 GeV/ c , we applied a D + / D 0 and D *+ / D 0 ratio of approximately 0.5 from PYTHIA and performed the fit. The ratio obtained from PYTHIA is consistent with the experimental data [ 50 50 ]. The fitted T , q , and A parameters and χ 2 / ndf values are listed in Table 1 Table 1 . The T and q parameters after transverse flow correction are shown in Fig. 5 Fig. 5 . In addition, we also applied the same Tsallis–Pareto fits to the transverse momentum spectra of π ± , K ± , and p( <math id="M32"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) in 0–5%, 5–10%, 10–20%, 20–30%, 30–40%, 40–50%, 50–60%, 60–70%, 70–80%, and 80–90% centrality bins in Pb-Pb collisions at 2.76 and 5.02 TeV [ 51 51 , 52 52 ]. The fit parameters after transverse flow correction are also shown in Fig. 5 Fig. 5 . we studied ϕ , Λ 0 , and Ξ for different centralities at 2.76 TeV using the same method. The point-by-point statistical and systematic uncertainties were added as a quadratic sum when we performed these fits. 2.2　 Thermal temperature with flow correction 2 The phenomenological model can describe almost all hadronic spectra by beginning with thermalization and collective flow as basic assumptions [ 15 15 ]. The mass dependence of the effective temperature T has been described by introducing a Gaussian parameterization [ 53 53 - 56 56 53-56 ] and can be interpreted as the presence of a radial flow. The velocity of the radial flow, which is generated by violent nucleon-nucleon collisions in two colliding nuclei and developed in both the QGP phase and hadronic rescattering, increases the transverse momentum of particles in proportion to their mass [ 15 15 , 57 57 , 58 58 ]. Many models have been used to investigate the radial flow [ 59 59 ]; a radial flow model [ 15 15 , 58 58 ] we can use in this analysis is written as <math display="block" id="M33"><mrow><mi>T</mi><mo>=</mo><msub><mi>T</mi><mrow><mtext>fro</mtext></mrow></msub><mo>+</mo><mi>m</mi><msup><mrow><mo>〈</mo><msub><mi>u</mi><mtext>t</mtext></msub><mo>〉</mo></mrow><mn>2</mn></msup><mn>,</mn></mrow></math> where T fro is the hadron kinetic freeze-out temperature, and 〈 u t 〉 is a measure of the strength of the (average radial) transverse flow. The relationship between the average transverse velocity 〈 β t 〉 and 〈 u t 〉 is given as <math display="block" id="M34"><mrow><mo>〈</mo><msub><mi>β</mi><mtext>t</mtext></msub><mo>〉</mo><mo>=</mo><mfrac><mrow><mo>〈</mo><msub><mi>u</mi><mtext>t</mtext></msub><mo>〉</mo></mrow><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo>〈</mo><msub><mi>u</mi><mtext>t</mtext></msub><mo>〉</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mn>.</mn></mrow></math> Note that although the T value according to the non-extensive statistical theory can differ from the usual temperature in Eq.(1), the flow correction of the spectral temperatures is independent of the statistical model. In addition, the following function is used to study the collectivity of charmed mesons produced in heavy-ion collisions [ 5 5 ]: <math display="block" id="M35"><mrow><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>N</mi></mrow><mrow><mn>2</mn><mi>π</mi><msub><mi>m</mi><mtext>T</mtext></msub><mtext>d</mtext><msub><mi>m</mi><mtext>T</mtext></msub><mtext>d</mtext><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mtext>d</mtext><mi>N</mi><mo>/</mo><mtext>d</mtext><mi>y</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>T</mi><mo stretchy="false">(</mo><msub><mi>m</mi><mn>0</mn></msub><mo>+</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mo stretchy="false">(</mo><msub><mi>m</mi><mtext>T</mtext></msub><mo>−</mo><mi>m</mi><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi></mrow></msup><mn>,</mn></mrow></math> where m 0 is the rest mass of the hadron species. This method is used to analyze the charmed meson spectra and to understand the collective velocity of the radial flow from the data in Pb-Pb collisions at 2.76 TeV. The function in Eq.(9) is applied to the π , K , p ( <math id="M36"><mrow><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> ), ϕ , Λ 0 , Ξ , Ω , and D 0 spectra for 0–10% and 30–50% centrality at 2.76 TeV. For Λ 0 , Ξ , and Ω , the semiperipheral results are obtained at 20–40% centrality. The fitted parameters are shown in Fig. 3 Fig. 3 . The solid and open circles represent the results for 0–10% and 30–50% centrality, respectively. The slope parameters obtained for D 0 mesons and strange hadrons at 0–10% and 30–50% centrality are 0.113± 0.037 and 0.107± 0.036, respectively. In addition, the linear fits of the ϕ , Λ 0 , Ξ , Ω , and D 0 data points in Fig. 3 Fig. 3 show a smaller slope than those of the π , K , and p( <math id="M37"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) data points, indicating that the former may freeze out earlier and gain less transverse collectivity during system evolution. All fits were performed up to m T - m 0 values of < 1 GeV/ c 2 for π , K , and p ( <math id="M38"><mrow><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> ), < 2 GeV/ c 2 for ϕ , Λ 0 , Ξ , and Ω , and < 3 GeV/ c 2 for D 0 . The fitted Tsallis–Pareto T parameters in Sect. 2.1 are plotted as a function of the corresponding masses of π ± , K ± , and p ( <math id="M39"><mrow><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> ) in the 0–5%, 5–10%, 10–20%, 20–30%, 30–40%, 40–50%, 50–60%, 60–70%, 70–80%, and 80–90% centrality bins in Pb-Pb collisions at 2.76 TeV and 5.02 TeV. The 〈 u t 〉 values from Eq.(7) were extracted, and the 〈 β t 〉 distributions as a function of 〈d N ch /d η 〉 in Pb-Pb collisions at 2.76 and 5.02 TeV are indicated by open squares and open diamonds, respectively, in Fig. 4 Fig. 4 . The D 0 meson is added to the above process at 0–10% and 30–50% centrality in Pb-Pb collisions at 2.76 TeV and is extrapolated to a lower centrality with the same centrality dependence as light-flavor hadrons. We plot the relationship between 〈 β t 〉 and 〈d N ch /d η 〉 as solid circles in Fig. 4 Fig. 4 . We finally obtained the thermal temperatures after flow correction of the spectral temperatures for charmed mesons, as shown in Sect. 3. Figure 4 Figure 4 reveals that the multiplicity dependence of the average radial flow velocity is linear as a function of log 〈d N ch /d η 〉. Furthermore, the average radial flow velocity increases with increasing multiplicity, which is consistent with an earlier blast-wave analysis [ 17 17 ]. In addition, the values can be extracted for π ± , K ± , and p( <math id="M40"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) using the non-extensive statistical theory: <math display="block" id="M41"><mrow><mo>〈</mo><msub><mi>β</mi><mtext>t</mtext></msub><mo>〉</mo><mo>=</mo><mrow><mo>(</mo><mrow><mn>0.286</mn><mo>±</mo><mn>0.013</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>0.065</mn><mo>±</mo><mn>0.006</mn></mrow><mo>)</mo></mrow><mi>log</mi><mo>〈</mo><mtext>d</mtext><msub><mi>N</mi><mrow><mtext>ch</mtext></mrow></msub><mo>/</mo><mtext>d</mtext><mi>η</mi><mo>〉</mo><mn>.</mn></mrow></math> The linear dependence for charmed mesons is <math display="block" id="M42"><mrow><mo>〈</mo><msub><mi>β</mi><mtext>t</mtext></msub><mo>〉</mo><mo>=</mo><mrow><mo>(</mo><mrow><mn>0.129</mn><mo>±</mo><mn>0.037</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>0.065</mn><mo>±</mo><mn>0.000</mn></mrow><mo>)</mo></mrow><mi>log</mi><mo>〈</mo><mtext>d</mtext><msub><mi>N</mi><mrow><mtext>ch</mtext></mrow></msub><mo>/</mo><mtext>d</mtext><mi>η</mi><mo>〉</mo><mn>.</mn></mrow></math> 3　Results and discussion 1 The measured transverse momentum distributions of charmed mesons in pp( <math id="M43"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions from the ALICE, CDF, and STAR collaborations are fitted with Tsallis–Pareto distributions in Fig. 1 Fig. 1 . The fit results are in good agreement with the data points overall. The same Tsallis–Pareto fits were also applied to the transverse momentum distributions of charmed mesons in Pb-Pb collisions at 2.76 and 5.02 TeV. Figure 2 Figure 2 shows the measured charmed meson spectra and the fitted Tsallis–Pareto curves for different centrality bins in Pb-Pb collisions at 2.76 TeV ［ Fig. 2(a) Fig. 2(a) ］ and 5.02 TeV ［ Fig. 2(b)-(d) Fig. 2(b)-(d) ］. The non-extensive feature q ≠ 1 appears in Table 1 Table 1 . To obtain a more physical understanding of this result, the dependence of the parameter q on the size of the collisional system is represented diagrammatically in Fig. 5 Fig. 5 . In addition, the spectral temperature T obtained from the Tsallis–Pareto fit of the hadron spectrum is larger than the original temperature T thermal by a blue-shift factor owing to the presence of a radial flow [ 15 15 ]. The ordinate in Fig. 5 Fig. 5 shows the thermal temperature values obtained by flow correction of the spectral temperatures for different hadron species. The flow correction formalism is given as <math display="block" id="M46"><mrow><msub><mi>T</mi><mrow><mtext>thermal</mtext></mrow></msub><mo>=</mo><mi>T</mi><msqrt><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><msub><mi>β</mi><mtext>t</mtext></msub></mrow><mrow><mn>1</mn><mo>+</mo><msub><mi>β</mi><mtext>t</mtext></msub></mrow></mfrac></mrow></msqrt><mn>,</mn></mrow></math> where β t is given by Eq.(10) for π ± , K ± , and p ( <math id="M47"><mrow><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> ) and by Eq.(11) for charmed mesons. Figure 5 Figure 5 shows the T –( q -1) parameter space for π ± , K ± , p ( <math id="M48"><mrow><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> ), and charmed mesons in Pb-Pb ［pp( <math id="M49"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> )］ collisions at 2.76 and 5.02 TeV (200 GeV, 500 GeV, 1.96 TeV, and 7 TeV). To simplify this figure, the results for strange particles ( ϕ , Λ 0 , and Ξ ) are not shown. The open and solid symbols represent the Tsallis–Pareto parameter values at 2.76 and 5.02 TeV, respectively. In addition, 〈d N ch /d η 〉 = 2047 and 〈d N ch /d η 〉 = 19.5 represent the central and peripheral centrality bins for light hadrons, respectively. There are five points around T = 0.25 at q - 1 = 0. This result indicates that at such high energies, the light-flavor particles are completely thermalized in most central Pb-Pb collisions; thus, their q values do not deviate from 1 at high temperature. The dotted and solid lines show the fitting results from Eq.(6). The fits for charmed mesons were performed in the range 0.14 < q -1 < 0.21 for 2.76 and 5.02 TeV, and the fit parameters are tabulated in Table 2 Table 2 . The charmed meson results for pp( <math id="M50"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions at different energies are indicated by diamonds. The vertical shaded band marks the saturated value of q -1 = 0.142 ± 0.010 in pp( <math id="M51"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions with increasing energy. This value was obtained by constant fitting in the saturated region. To further investigate the T –( q -1) correlations between the pp( <math id="M52"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) and Pb-Pb systems, we extended the fit lines along the decreasing q -1 direction for charmed mesons in Pb-Pb collisions. 1 1 <math id="M54"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> (GeV) 1 1 Hadron 1 1 E (GeV) 1 1 ζ 2 1 1 χ 2 / ndf 1 1 Pb-Pb, 2760 1 1 charmed mesons 1 1 3.000± 2.225 1 1 0.226± 0.055 1 1 2.11/4 1 1 1 1 π ± 1 1 0.329± 0.022 1 1 0.374± 0.017 1 1 7.27/8 1 1 1 1 K ± 1 1 0.362± 0.014 1 1 0.470±0.016 1 1 15.91/8 1 1 1 1 p ( <math id="M55"><mrow><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> ) 1 1 0.901± 0.036 1 1 0.270± 0.011 1 1 11.04/8 1 1 Pb-Pb, 5020 1 1 charmed mesons 1 1 4.800± 1.460 1 1 0.221± 0.015 1 1 1.85/7 1 1 1 1 π ± 1 1 0.237± 0.027 1 1 0.478± 0.040 1 1 4.68/8 1 1 1 1 K ± 1 1 0.407± 0.024 1 1 0.441± 0.018 1 1 2.53/8 1 1 1 1 p ( <math id="M56"><mrow><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> ) 1 1 0.982± 0.049 1 1 0.271± 0.010 1 1 0.62/8 We find that the Tsallis–Pareto distributions can provide satisfactory descriptions in a wide range of transverse momentum dependence of charmed meson production in pp( <math id="M57"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) and AA collisions in a wide energy range. The values of the corresponding ζ 2 parameters in Table 2 Table 2 can be used to test the strength of the correlation between T and q . This correlation is inversely proportional to the hadron mass overall. The following conclusions can be drawn from Table 2 Table 2 and Fig. 5 Fig. 5 . (i) There is a significant linear relationship between the thermal temperature and Tsallis q parameter for π ± , K ± , p( <math id="M58"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ), and charmed mesons in Pb-Pb collisions at 2.76 TeV. In addition, the slope of the T –( q -1) parameter is positively correlated with the hadron mass. The same conclusion can be drawn for Pb-Pb collisions at 5.02 TeV, where it is better supported because of the larger statistics and more complete centrality bins. (ii) Charmed mesons have a significantly higher slope than light hadrons. The temperature of charmed mesons is found to be higher than that of light hadrons at the same q -1, indicating that heavy flavor requires a higher temperature to reach the same degree of non-extensivity as light flavors in heavy-ion collisions. The slope of a given hadron is smaller at 2.76 TeV than at 5.02 TeV. In Fig. 6 Fig. 6 , the slopes of the T –( q -1) correlations are plotted as a function of hadron mass for π ± , K ± , p( <math id="M59"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ), ϕ , Λ 0 , Ξ , and charmed mesons in Pb-Pb collisions at 2.76 and 5.02 TeV (open and solid circles, respectively). The data are fitted by a quadratic polynomial function, and a deeper theoretical explanation is needed. (iii) The charmed meson results for pp( <math id="M60"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions at different energies show that the thermal temperature increases with system energy, whereas the q parameter becomes saturated at the pp( <math id="M61"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) limit, q -1 = 0.142 ± 0.010. In addition, the results for most peripheral Pb-Pb collisions are found to approach the pp( <math id="M62"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) limit, which suggests that more peripheral heavy-ion collisions are less affected by the medium and more similar to pp( <math id="M63"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collisions. 4　Summary 1 We presented fits of the transverse momentum spectra of D 0 , D + , and D *+ mesons at mid-rapidity in Pb-Pb collisions at 2.76 and 5.02 TeV. A similar analysis with non-extensive statistics was performed to identify light hadron spectra for different centrality bins in Pb-Pb collisions at 2.76 TeV and 5.02 TeV after flow correction. Charmed meson production can be well described by the Tsallis–Pareto distributions. We observed that in the T –( q -1) parameter space, the slope has a positive dependence on hadron mass. In addition, the temperature of charmed mesons was found to be higher than that of light hadrons at the same q -1, indicating that heavy flavor requires a higher temperature to reach the same degree of non-extensivity as light flavors in heavy-ion collisions. In addition, the slope distribution of the T –( q -1) correlations ( Fig. 6 Fig. 6 ) and the anti-correlation between the thermal temperature and centrality for charmed mesons require a deeper theoretical explanation. For the pp( <math id="M64"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) collision system as a reference, we found that the thermal temperature increases with system energy, whereas the q parameter becomes saturated at the pp( <math id="M65"><mrow><mover accent="true"><mtext>p</mtext><mo stretchy="true">¯</mo></mover></mrow></math> ) limit, q -1 = 0.142 ± 0.010. 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Material preparation, data collection and analysis were performed by Yuan Su, Xiao-Long Chen, Yong-Jie Sun and Yi-Fei Zhang. The first draft of the manuscript was written by Yuan Su and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00945-4 This is supported by the National Key Research and Development Program of China (Nos. 2018YFE0205200 and 2018YFE0104700) , National Natural Science Foundation of China (Nos. 11890712 and 12061141008) , Strategic Priority Research Program of CAS (No. XDB34030000) , and Anhui Provincial Natural Science Foundation (No. 1808085J02). 2021-04-20 2021-08-26 2021-08-29 2021-10-05 2021-10-01 2021-11-29 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00945-4 http://dx.doi.org/10.1007/s41365-021-00945-4 . 2021 Nuclear Science and Techniques 10 32