Light Nuclei Production in Au+Au Collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5-200 GeV from JAM model

Light nuclei production is sensitive to the baryon density fluctuations and can be used to probe the QCD phase transition in relativistic heavy-ion collisions. In this work, we studied the production of proton, deuteron, triton in central Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5, 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV from a transport model (JAM). Based on the coalescence production of light nuclei, we calculated the energy dependence of rapidity density $dN/dy$ and particle ratios ($d/p$, $t/p$, and $t/d$). More importantly, the yield ratio $N_{{t}} \times N_{{p}} / N_{{d}}^{2}$, which is sensitive to the neutron density fluctuations, shows a flat energy dependence and cannot describe the non-monotonic trend observed by the STAR experiment. Based on the nucleon coalescence, this work can provide constraint and reference to search for the QCD critical point and/or first order phase transition with light nuclei production in future heavy-ion collision experiments.


I. INTRODUCTION
Understanding the Quantum Chromodynamics (QCD) phase diagram of strongly interacting matter is of fundamental importance in nuclear physics. The QCD phase diagram can be displayed in the two dimensional phase diagram of temperature (T ) versus baryon chemical potential (µ B ). Lattice QCD calculations show that the transition from hadronic phase to quark-gluon plasma (QGP) is smooth crossover at small values of µ B [1]. While at finite µ B , it is of first order phase transition based on QCD model calculations [2]. If those predictions are true, by definition, there should be a QCD critical point as the end point of the first order phase boundary. However, there is still large uncertainties in determining the location and even the existence of the QCD critical point from theoretical side. Experimentally, relativistic heavy-ion collisions can provide us a useful and controllable way to explore the QCD phase structure, especially on finding the QCD critical point [3][4][5]. This is one of the main physics motivation of the Beam Energy Scan program (BES I & II: 2010-2021) at Relativistic Heavy-ion Collider (RHIC) [6].
Light nuclei, such as deuteron and triton, are loosely bounded objects with small binding energies (d with 2.2 MeV and t with 8.4 MeV). Those are formed via coalesence of nucleons in a very restricted phase-space volume [7,8]. Because of the small binding energies, their existence at high temperature enviroment created in heavy-ion collisions is called "snowball in hell" [7], which seems counter intuitive. For example, at LHC energies, the yield of deuteron, triton and even hypertrion in Pb+Pb collisions at √ s NN = 2.76 TeV measured by ALICE experiment [9] can be well described by thermal model, which is hard to understand. A recent hybrid model (hydrodynamics + hadronic afterburner) study of deuteron production at LHC [10] energy provide a possi- * xfluo@mail.ccnu.edu.cn ble explanation to the "snowball in hell" phenomena and the successful thermal description of ALICE data: the initially thermal produced light nuclei are dynamically destroyed and re-generated via detail balanced hadronic re-scattering process in heavy-ion collisions, which keeps the light nuclei yield almost unchanged. This indicates the hadronic re-scattering stage play a very important role for light nuclei production in heavy-ion collisions. At RHIC BES energies, the STAR experiment has collected the data of Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4, and 200 GeV and measured the light nuclei production (deuteron and triton) [11,12]. It was found that thermal model can describe the deuteron yield well, but overestimate the triton yield in central Au+Au collisions at √ s NN = 7.7 to 200 GeV measured by STAR experiment. This could be related to the different production mechanisms of light nuclei at different energies, which still remains an open question. On the other hand, due to increasing of the correlation length and formation of instability spinodal domain, both of the critical fluctuations and first order phase transition can induce large baryon density fluctuations. It is predicted that the production of light nuclei is sensitive to the baryon density fluctuations and thus can be used to probe the QCD phase transition in heavyion collisions [13][14][15]. For instance, the neutron density fluctuation (∆n = (δn) 2 / n 2 ) can be extracted from the yield ratio of proton, deuteron and triton in the following form N p × N t /N 2 d . Interestingly, it was observed that the neutron density fluctuation (∆n) extracting from yield ratio N p × N t /N 2 d in central Au+Au collisions measured by STAR experiment shows a clear non-monotonic energy dependence, with a peak around 20 GeV [12]. However, without dynamical modeling of the QCD phase transition in heavy-ion collisions, it is still not possible to give a definitive conclusion about the signature of the QCD critical point and/or first order phase transition.
In this paper, we studied the production of deuteron and triton in most central (b < 3f m) Au+Au collisions at √ s NN = 5, 7.7, 11.5, 14. GeV from JAM model. Our paper are organized as following: In Sec. II, we will give a brief introduction to the JAM model. In Sec. III, we show the relations between neutron density fluctuation and the yield ratio in relativistic heavy-ion collisions. In Sec. IV, we will present the mid-rapidity transverse momentum spectra for proton, deuteron and triton. Furthermore, we show the energy dependence of particle yield dN/dy, the particle ratios ( d/p, t/p and t/d), and the yield ratio N p × N t /N 2 d . Finally, the summary will be given in Sec. IV.

II. JAM MODEL
In relativistic heavy-ion collisions, the whole process from first N N collision stage to the final state interaction among produced particles is very complicated and involves a lot of dynamic evolution. To explore these evolutionary processes, many microscopic hadronic transport models are used to simulate the relativistic heavyion collisions, such as RQMD [22,23], UrQMD [24,25], ARC [26], ART [27] and AMPT [28]. JAM model [20,21] (Jet AA Microscopic Transportation Model) has been developed based on the resonance and string degrees of freedom. To describe nuclear collisions from low to high energy, in the JAM model, particles are produced via resonance or excitation of strings and their decay. In our analysis, the formation of light nucleus are taken into account. In the phase space, if the distance (∆R) and momentum (∆P ) [16] between two nucleons are both less than the given values (R 0 , P 0 ) [33], they are considered to be a nuclear cluster. In our study, we set the coalescence conditions R 0 = 3.0 fm and P 0 = 0.3 GeV/c.

III. NEUTRON DENSITY FLUCTUATION
Based on the coalescence model, the light nuclei is formed via nucleon coalescence and the neutron density fluctuation (∆n = (δn) 2 / n 2 ) at kinetic freeze out can be encoded in the yield ratio of light nuclei (N p ×N t /N 2 d ). According to Ref. [13,18], the number of deuteron can be expressed approximately as Similarly, the number of triton, which is formed via the coalescence of one proton and two neutrons can be expressed as where N p and N n are the number of protons and neutrons, respectively. The degeneracy factor g d = 3/4, and g t = 1/4, V is system volume. In COAL-SH model [17] for cluster production, the oscillator frequency of the cluster's internal wave function, ω, is much smaller than the effective temperature T eff at kinetic freeze-out, thus the suppression factor (x = (2T eff /ω) 1/2 ) will be much larger than unity [19]. In this coalescence picture, uniform distributions of nucleons in space are assumed. One can also neglect the mass difference between neutron and proton and set m p = m n = m 0 . Then, the Eq. 1 and 2 can be simplified as If we ignore the binding energies of deuteron and triton, the results obtained in Eq. 3 and Eq. 4 are consistent with thermal model [29][30][31]. Based on Eq. 3 and Eq. 4, we introduce neutron density fluctuations, i.e n( r) = 1 V n( r)d r + δn( r) = n + δn( r), here δn( r) = 0 with uniform distribution, we can rewrite Eq. 3 and Eq. 4 as where α is correlation coefficient between neutron and proton density, and ∆n = (δn) 2 / n 2 is the neutron density fluctuation. Finally, the neutron density fluctuations can be derived from the yield ratio as where g = 0.29. If we assume the correlation between density of protons and neutrons are small (α ≈ 0), then the Eq. 7 can be approximated as In Eq. 8, we can see that the neutron density fluctuation can be obtained by measuring the yield ratio of light nuclei in heavy-ion collisions. The light nuclei production can provide us an useful tool to study the QCD phase transition.

IV. RESULTS
In this section, we will present the transverse momentum spectra, dN/dy and yield ratios in Au+Au collisions at √ s NN = 5, 7.7, 11.5, 14.    Fig. 2 shows the energy dependence of dN/dy of p, d, t andp,d, 3 He. Comparing the results of particle and anti-particle [10,34], we find the particle yields decrease with the increasing energy, while the yield of anti-particles increase. This is due to the interplay of baryon stopping and pair production of nucleons at different energies: baryon stopping dominated at low energies, while the pair production dominated at high energies [3,35], which make the yields of particle and anti-particle get closer at higher energies. We also found the yields of triton and helium-3 are consistent within uncertainties, as they are both coalesced from three nucleons.

B. Light Nuclei Yield Ratios
Using the integrated yields, one can obtain the particle ratio of light nuclei (deuteron, triton) to proton (d/p, t/p, t/d) as a function of collision energy. As shown in Fig. 3, the lines calculated by JAM model are compared to the preliminary results from central Au+Au collisions at RHIC BES energies ( √ s NN = 7.7 to 200 GeV) measured by the STAR experiment [11,12]. For t/d ratio, we find the results from JAM model are in agreement with the data measured by STAR. However, the energy dependence of d/p and t/p ratios measured by STAR cannot be described and are slightly larger than the results from JAM model. In addition, the ratio decreases with increasing of collision energy, which can be explained by the lower baryon density at higher energy.
As discussed in section III, the yield ratio N p × N t /N 2 d is related to the neutron density fluctuation and can be used to study the QCD phase transition in heavyion collisions. The STAR experiment has measured the deuteron and triton production in Au+Au collisions at √ s NN = 7.7 to 200 GeV [11,12]. Fig. 4 shows the energy dependence of the yield ratio N p ×N t /N 2 d calculated from JAM model and its comparison with the experimental data measured by STAR experiment. The model results were scaled by a factor of two. We found that the experimental data shows a clear non-monotonic energy dependence with a peak around 20 GeV, while the model results are flat as a function of collision energy. Although the values of model calculations are much larger than the experimental data, but their energy dependence trends are very different. This non-monotonic energy dependence of light nuclei yield ratio could be related to the large baryon density fluctuation near the critical point or first order phase transition. To provide a definite physics conclusion on this non-monotonic behavior, we still need dynamical modeling of the heavy-ion collisions with more realistic equation of state. Our model study, which is without the physics of QCD phase transition, can be used as a baseline for future studies in heavy-ion collisions.

V. SUMMARY
We presented the light nuclei production in most central Au+Au collisions at √ s NN = 5, 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4, and 200 GeV within JAM model. We showed the transverse momentum spectra for proton, deuteron and triton, and the energy dependence of their yields and yield ratios. It was found that the particle ratio of deuteron to proton is in good agreement with STAR data, while the values of triton to proton ratios are larger than the experimental data. Due to the relations between the light nuclei yield ratio N p ×N t /N 2 d and the neutron density fluctuation (Eq.8), the energy dependence of the light nuclei yield ratio N p × N t /N 2 d from JAM model was shown and compared with the experimental data measured by STAR experiment. We found that the yield ratio N p × N t /N 2 d from JAM model shows an almost constant value as a function of collision energy and cannot describe the non-monotonic energy dependence trend observed by STAR. One should note that there is only hadronic degree of freedom in JAM model and the physics of QCD phase transition are not implemented. Thus, our studies could provide a baseline to perform further studies on searching for the QCD critical point and/or the first order phase transition in relativis-tic heavy-ion collisions at high baryon density region.