Li-Lin Zhu · Bo Wang · Meng Wang · Hua Zheng

Abstract We study the energy and centrality dependence of deuteron and triton (helium-3) production in relativistic heavy-ion collisions at the BNL Relativistic Heavy Ion Collider(RHIC)and CERN Large Hadron Collider(LHC)using the Tsallis distribution,blast-wave (BW) model, and stationary Fokker–Planck (FP) solution. Our study shows that good agreement can be reached between the fitting results from the stationary FP solution and the experimental data for Au + Au collisions from the beam energy scan (BES) program of RHIC at =7.7, 11.5, 14.5,19.6,27,39,62.4,and 200 GeV and for Pb + Pb collisions at =2.76 TeV. The Tsallis distribution and BW model can reasonably describe the deuteron and triton(helium-3)transverse momentum spectra obtained at RHIC and LHC. A more comprehensive comparison among the three methods suggests that the stationary FP solution is a sensible method, which is able to describe the energy dependence of the light nuclei yield ratio NtNp/N2d and provide a coherent description of deuteron and triton (helium-3)production for all centralities and various colliding energies at RHIC and LHC.

Keywords Light nuclei production·Heavy-ion collisions·Tsallis distribution · Blast-wave model · Fokker–Planck solution

1 Introduction

Various scenarios and mechanisms have been proposed to describe the production of light nuclei. Three main approaches are typically used to describe light nuclei production. The first approach is the thermodynamic model[17–21],in which the yields of hadrons and light nuclei are described using a few parameters related to the chemical freeze-out conditions. The production of light nuclei can also be described by the coalescence model, in which it is assumed that light nuclei are formed through the coalescence of protons and neutrons with similar positions and velocities on the kinetic freeze-out surface [22–30]. The third approach is kinetic theory, in which light nuclei are formed and destroyed during the evolution of the collision system [31–33].

Fig. 1 Fitting results obt ained using theTsallisdistribution(Eq.1)for deuterons in Au +Au collisions at =19.6 and200 GeV and Pb + Pb collisionsat =2.76 TeV.Forbetter visualization,both the data and curves have been scaled by a constant,as indicated.The data/fit ratios are shown in the bottom panels. The data are obtained from the STAR [3] and ALICE [5] collaborations

The remainder of this paper is organized as follows. In Sect. 2–4, we show our fitting results for the energy and centrality dependence of deuteron and triton (helium-3)production in Au + Au collisions and Pb + Pb collisions obtained using the Tsallis distribution, BW model, and stationary FP solution, respectively. In Sect. 5, a detailed comparison among the three methods and a brief discussion are presented. Finally, a summary is given in Sect. 6.

2 Tsallis distribution

In our previous works [38–40], we demonstrated that several versions of the Tsallis distribution can describe the pTspectra of hadrons produced in both p + p and A + A collisions at RHIC and LHC equally well [38, 39]. The

Fig. 2 Same as Fig. 1 for tritons in Au + Au collisions at9 .6 and 200 GeV [4] and helium-3 in Pb + Pb collisio=2.76 TeV [5]

Fig. 3 Fitting results obtained using the BW model (Eq. 2) for deuterons in Au + Au collisions at=19.6 and 200 GeV and Pb + Pb collisions at =2.76 TeV. For better visualization,both the data and curves have been scaled by a constant,as indicated.The data/fit ratios are shown in the bottom panels. The data are obtained from the STAR [3] and ALICE [5] collaborations

Fig. 4 Same as Fig. 3 for tritons in Au + Au collisions at .6 and 200 GeV [4] and helium-3 in Pb + Pb collisio=2.76 TeV [5]

3 The blast-wave model

Besides the Tsallis distribution, the BW model is also commonly adopted by experimental collaborations [3–5].This model describes particle production under the assumption that the particles are thermally emitted from an expanding source. The functional form of this model is given by

4 The Fokker-Planck solution

5 Discussion

From the above results, we have learned that the three different methods, i.e., the Tsallis distribution, BW model,and stationary FP solution, can describe the experimental data in general.To explicitly compare the agreement of the fitting results obtained using the three approaches with the experimental data,we define the relative discrepancy as the ratio

Fig. 5 Fitting results obtained using the stationary FP solution(Eq. 4) for deuterons in Au + Au collisions at =7.7–200 GeV. For better visualization, both the data and curves have been scaled by a constant, as indicated. The data/fit ratios are shown in the bottom panels. The data are obtained from the STAR collaboration [3]

To perform a comprehensive evaluation, it is also necessary to check the relative discrepancies for tritons,which are shown with empty symbols in Fig. 8. The ratios for tritons obtained using Eqs. (1), (2), and (4) are generally smaller than those for deuterons at the three collision energies. In more detail, the relative discrepancies for Eq. (1) (empty black squares) are generally larger than those for Eq. (2) (empty red circles) and Eq. (4) (empty blue triangles). The results demonstrate that both the BW model and stationary FP solution are better than the Tsallis distribution for reproducing the triton spectra at various collision energies.

Based on the above-detailed comparisons, we can conclude that the stationary FP solution is the optimal method for describing the deuteron and triton(helium-3)transverse momentum spectra of central to peripheral collisions at

Fig. 6 Same as Fig. 5 for tritons with experimental data from the STAR collaboration [4]

Fig. 7 Fitting results obtained using the stationary FP solution(Eq. 4)for deuterons and helium-3 produced in Pb + Pb collisions at=2.76 TeV. For better visualization, both the data and curves have been scaled by a constant, as indicated. The data/fit ratios are shown at the bottom panels. The data are obtained from the ALICE collaboration [5]

Figure 9 shows the energy dependence of the light nuclei yield ratio NtNp/N2dat 0-10% centrality for Au + Au collisions. It has been suggested that this ratio can be used to probe the QCD phase diagram[12,14].The red solid circles are the preliminary results for BES energies obtained by the STAR collaboration [4]. The yield ratio exhibits a non-monotonic energy dependence. The yield ratios obtained using the Tsallis distribution, BW model, and the stationary FP solution are represented by the lines with blue empty triangles, magenta empty diamonds, and black full squares, respectively. It is obvious that the black line (the stationary FP solution) best reproduces the experimental data, which is consistent with the conclusion above. Our results indicate that transport or hydrodynamic models should reproduce the transverse momentum spectra of light nuclei with a high level of accuracy to quantitatively reproduce the NtNp/N2dratio.

Fig. 8 (Color online) Relative discrepancies of Eqs. (1), (2),and (4) for the pT spectra of deuterons and tritons(helium-3)in Au + Au central collisions at=19.6 and 200 GeV and in Pb + Pb collisions at=2.76 TeV based on the deuteron and triton data for 0–10% centrality [3, 4] and helium-3 data for 0–20%centrality. [5]

Fig. 9 (Color online) Light nuclei yield ratio NtNp/N2d as a function of the collision energy for central Au + Au collisions. The results for the Tsallis distribution, BW model, and stationary FP solution are denoted by their respective lines. The preliminary experimental data (red solid circles) were obtained from Ref. [4]

6 Summary

In this paper,we presented a detailed study of the Tsallis distribution, BW model, and stationary FP solution in which fitting was performed on the transverse momentum

Author’s contribution All authors contributed to the study conception and design.Material preparation,data collection and analysis were performed by Li-Lin Zhu, Bo Wang, Meng Wang, and Hua Zheng. The first draft of the manuscript was written by Li-Lin Zhu and Hua Zheng and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.