Yuan Su· Yong-Jie Sun · Yi-Fei Zhang· Xiao-Long Chen

Abstract The mid-rapidity transverse momentum spectra of charmed mesons in Pb-Pb and pp(p) 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 parameterin Pb-Pb collisions at=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(p)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(p) limit, q-1 = 0.142 ±0.010. In addition, the results of most peripheral Pb-Pb collisions are found to approach the pp(p) limit, which suggests that more peripheral heavy-ion collisions are less affected by the medium and more similar to pp(p)collisions.

Keywords Charmed meson · Non-extensive statistics ·Tsallis-Pareto distribution · QGP · Heavy-ion collision

1 Introduction

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,mc ～1.3 GeV/c2and mb ～4.8 GeV/c2, are larger than those of light quarks and the quantum chromodynamics(QCD) energy scale (ΛQCD). 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, 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 heavyflavor quarks and the medium by measuring the nuclear modification factor, flow, and production yield of charmed mesons [3-12].

2 Tsallis-Pareto distribution and its fit to charmed meson spectra

Much work on high-energy particle collisions has focused on the study of the transverse momentum distributions of outgoing particles. In the low-pTregime 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

where A is a normalizing factor and can reflect the production yield of the hadron pTspectrum. Tqis the temperature in the non-extensive statistical theory; the subscript q is omitted for brevity in the following discussion. Note that Tqcan differ from the temperature T in Eq. (1),but its physical meaning should be the same in the limiting case q →1.

The correlations between the parameters T and q have been presented in earlier studies [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 ～O(10 ) [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 [39-43]. This study focuses on the relationship between the parameters of heavy-flavor hadrons in the T-(q-1) parameter space using the nonextensive statistics described above. For simplicity, the fluctuations in the number of produced particles can be explained in a one-dimensional relativistic gas model[44],and the Tsallis parameters under consideration are given as

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]:

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

Fig. 1 Transverse momentum distributions dN/dpT of D0/D* in pp(p) collisions at 200 GeV, 500 GeV, 1.96 TeV, and 7 TeV, from bottom to top.Solid curves are results of Tsallis-Pareto fit.Error bars are quadratic sums of statistical and systematic uncertainties,and data are scaled by factors of 10n for better visibility

Table 1 Values of parameters from Tsallis-Pareto fit to charmed meson spectra in Pb-Pb [pp(p)] collisions. The uncertainties are from the fit

Fig. 2 Transverse momentum distributions dN/dpT of D0, D+, and D*+ for different centrality bins in Pb-Pb collisions at 2.76 TeV(a)and 5.02 TeV(b),(c),(d),where the production yields are scaled by various factors for visibility. Vertical bars represent quadratic sums of statistical and systematic errors; symbols are placed at the center of the bin. Detailed descriptions are presented in Sect. 2.1

Figure 2(b), (c), and (d) shows the transverse momentum distributions dN/dpT of D0mesons(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]. 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 D0and 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 pT = 1.5 GeV/c, we applied a D+/D0and D*+/D0ratio of approximately 0.5 from PYTHIA and performed the fit. The ratio obtained from PYTHIA is consistent with the experimental data[50]. The fitted T, q, and A parameters and χ2/ndf values are listed in Table1. The T and q parameters after transverse flow correction are shown in Fig. 5. In addition, we also applied the same Tsallis-Pareto fits to the transverse momentum spectra of π±, K±, and p(p) 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, 52]. The fit parameters after transverse flow correction are also shown in Fig. 5. We studied φ, Λ0, 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

The phenomenological model can describe almost all hadronic spectra by beginning with thermalization and collective flow as basic assumptions [15]. The mass dependence of the effective temperature T has been described by introducing a Gaussian parameterization[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, 57, 58].Many models have been used to investigate the radial flow [59]; a radial flow model [15, 58] we use in this analysis is written as

Fig.3 Effective temperature as a function of hadron mass for 0-10%(solid circles) and 30-50% (open circles) centrality in Pb-Pb collisions at 2.76 TeV. The lines are fits from Eq. (7)

Fig. 4 Average radial flow velocity from Eq.(8) as a function of average event multiplicity. Solid circles represent φ, Λ0, Ξ, Ω, and D0, which are combined to extract 〈ut〉 at 0-10% and 30-50%centrality. The results for light hadrons at 2.76 and 5.02 TeV are indicated by open squares and open diamonds, respectively. The fit result is shown at the bottom of the panel

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]: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〈βt〉and〈dNch/dη〉as solid circles in Fig. 4.We finally obtained the thermal temperatures after flow correction of the spectral temperatures for charmed mesons, as shown in Sect. 3.

3 Results and discussion

Fig.5 Thermal temperature T versus q-1 and values of parameters from the Tsallis-Pareto fit of the identified particle spectra at different centralities in Pb-Pb [pp(p)] collisions at 2.76 and 5.02 TeV (200 GeV, 500 GeV, 1.96 TeV, and 7 TeV) after transverse flow correction. Shaded vertical band marks the saturated value of q-1=0.142±0.010 in pp(p)collisions with increasing energy.The solid and dotted lines are from Eq.(6); the parameters are listed in Table 2

Table 2 Values of parameters from linear fit by Eq. (6) of the T-(q-1) correlations for π±,K±, p(p), and charmed mesons in Pb-Pb collisions. Quoted uncertainties are the errors of the fit

Fig. 6 Slope of T-(q-1) correlations as a function of hadron mass in Pb-Pb collisions at 2.76 TeV (open circles) and 5.02 TeV (solid circles). The curves are quadratic polynomial fits

4 Summary

We presented fits of the transverse momentum spectra of D0, D+, and D*+mesons at mid-rapidity in Pb-Pb collisions at 2.76 and 5.02 TeV. A similar analysis with nonextensive 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) and the anti-correlation between the thermal temperature and centrality for charmed mesons require a deeper theoretical explanation.

For the pp(p) collision system as a reference, we found that the thermal temperature increases with system energy,whereas the q parameter becomes saturated at the pp(p)limit,q-1=0.142±0.010.Moreover,the results of most peripheral Pb-Pb collisions were found to approach the pp(p)limit,which suggests that more peripheral heavy-ion collisions are less affected by the medium and more similar to pp(p) collisions. In addition, uniform descriptions of both small and large systems over a wide range of collision energies and hadron transverse momenta were presented.

Author Contributions All authors contributed to the study conception and design. 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.