Collision centrality and system size dependences of light nuclei production via dynamical coalescence mechanism

A multi-phase transport model  Lin et al. (2005) was used to provide the phase-space of nucleons in this work. The model is composed of four parts: the HIJING model Wang and Gyulassy (1991); Gyulassy and Wang (1994) is used to simulate the initial conditions, the Zhang’s Parton Cascade (ZPC) model Zhang (1998) is employed to describe partonic interaction, the Lund string fragmentation or coalescence model is used for the hadronization process, and A Relativistic Transport (ART) model Li and Ko (1995) is applied to describe the hadronic rescattering process. As an event generator used in this work, the AMPT model outputs the phase-space distribution at the final stage in the hadronic rescattering process (ART model Li and Ko (1995)) with considering baryon-baryon, baryon-meson, and meson-meson elastic and inelastic scatterings, as well as resonance decay or week decay. In Refs. Li and Ko (1995); Lin et al. (2005) the interaction cross section was presented and extended. The hadronic rescattering time would affect light nuclei spectra and yield which are based on the phase-space information of nucleons from the AMPT model. Refs. Lin et al. (2005); Ma and Lin (2016) suggest the maximum hadronic rescattering time ($t_{max,h}$), which means to cease a hadron interacting with others if it still dose not reach freeze-out state at that time, 30 $fm/c$ for the RHIC energy region and 200 $fm/c$ for the LHC energy region. Here the $p_{T}$ spectra of $p$, $d$ and $t$ with the cutoff of the maximum hadronic rescattering time of 30 $fm/c$ and 100 $fm/c$ are checked. Fig. 1 shows the $p_{T}$ spectra of proton, deuteron and triton of ¹⁹⁷Au + ¹⁹⁷Au collisions at mid-rapidity $(|y|<0.5)$ for different centralities at $\sqrt{s_{NN}}=39$ GeV, and Fig. 2 presents the $p_{T}$ results for the 0$-$10% central collisions of ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, ⁴⁰Ca + ⁴⁰Ca at $\sqrt{s_{NN}}=39$ GeV and mid-rapidity $(|y|<0.5)$. We can find that these two cases are very close to each other. Afterwords we choose the case of $t_{max,h}$ = 100 $fm/c$ for the following calculations. We would mention that the AMPT model has been successfully used to simulate physics in heavy-ion collisions at the RHIC and LHC energies Lin et al. (2005); Ma and Lin (2016); Lin and Zheng (2021) and the detailed parameter configurations can be found therein.

In the coalescence model Csernai and Kapusta (1986), the invariant yields of light nuclei with charge number $Z$ and atomic mass number $A$ can be described by the yields of cluster constituents (protons and neutrons) multiplying by a coalescence parameter $B_{A}$,

where $p_{p}$ and $p_{n}$ are the momenta of proton and neutron, respectively, and $p_{A}$ is the momentum of the nucleus with the mass number $A$ which is approximate $A$ times of proton momentum, i.e. $Ap_{p}$, assuming that the distributions of neutrons and protons are the same. The coalescence parameter $B_{A}$ related to the local nucleon density reflects the probability of nucleon coalescence. The coalescence parameter $B_{A}$ is also related to the effective volume of the nuclear matter at the time of coalescence of nucleons into light nuclei, called nucleon correlation volume $V_{eff}$ Csernai and Kapusta (1986), i.e. $B_{A}\propto 1/V_{eff}^{A-1}$.

The dynamical coalescence model can give the probability of light nuclei ($M$-nucleon cluster) by the overlap of the cluster Wigner phase-space density with the nucleon phase-space distributions at an equal time in the $M$-nucleon rest frame at the freeze-out stage Chen et al. (2003). The momentum distribution of a cluster in a system containing $A$ nucleons can be expressed by,

where $M$ and $Z$ are the number of the nucleon and proton of the cluster, respectively; $f_{n}$ and $f_{p}$ are the neutron and proton phase-space distribution functions at freeze-out, respectively; $\rho^{W}$ is the Wigner density function; $\vec{r}_{i_{1}},\cdots,\vec{r}_{i_{M-1}}$ and $\vec{k}_{i_{1}},\cdots,\vec{k}_{i_{M-1}}$ are the relative coordinates and momentum in the $M$-nucleon rest frame; the spin-isospin statistical factor $G$ is 3/8 for deuteron and 1/3 for triton Chen et al. (2003), note whether to consider the isospin effect is still an unresolved problem, neglecting the isospin effect can be found in Zhao et al. (2018); Sun et al. (2021). While the neutron and proton phase-space distribution comes from the transport model simulations, the multiplicity of a $M$-nucleon cluster is then given by,

where the $\left<\cdots\right>$ denotes the event averaging.

The Wigner phase-space density of (anti)deuteron is assumed as Chen et al. (2003),

where the Gaussian fit coefficient $c_{i}$ and $w_{i}$ are given in Ref. Chen et al. (2003). $\vec{k}$ = ($\vec{k}_{1}$-$\vec{k}_{2}$)/2 is the relative momentum and $\vec{r}$ = ($\vec{r}_{1}$-$\vec{r}_{2}$) is the relative coordinate of proton and neutron inside deuteron.

The Wigner phase-space density of triton is obtained from a spherical harmonic oscillator Chen et al. (2003); Zhang et al. (2010); Sun and Chen (2015),

where $\rho$ and $\lambda$ are relative coordinates, $\vec{k}_{\rho}$ and $\vec{k}_{\lambda}$ are the relative momenta in the Jacobi coordinate, the parameter $b$ is obtained from the root-mean-square radius, 1.61 $fm$ for triton Chen et al. (2003).

In practice, the coalescence procedure by using Eq. (3) can not guarantee the energy conservation, such as for the formation of dueteron $p+n\rightarrow d$. If a proton and a neutron with momentum-energy $(\vec{k},E_{p})$ and $(-\vec{k},E_{n})$ coalesces a deuteron with $(\vec{0},m_{d})$, and then the lost energy is $\Delta E=\sqrt{k^{2}+m_{p}^{2}}+\sqrt{k^{2}+m_{n}^{2}}-m_{d}$. From Eq. (4), it can be seen that the lost energy is ignorable since the Wigner density is suppressed exponentially at the large relative momentum. For the three-body case, a similar derivation can be obtained. Actually, we made a numerical check for the effect of lost energy, it is found that it is negligible for the yield and spectra of the light nuclei production.

In this calculation, the AMPT model provides the phase-space of nucleons at the freeze-out stage in heavy-ion collisions and the followed coalescence model is coupled to give the transverse momentum $p_{T}$ spectra of deuterons ($d$) and tritons ($t$). Based on the obtained $p_{T}$ spectra, the yields of $d$ and $t$, as well as the coalescence parameters, are discussed.

Figure 4 presents the transverse momentum spectra of $p$ ($\bar{p}$), $d$ ($\bar{d}$) and $t$ ($\bar{t}$) calculated by the AMPT model coupling with the coalescence model in Au + Au collisions at $\sqrt{s_{NN}}=39$ GeV. The results are shown for the collision centrality classes of 0$-$5%, 5$-$10%, 10$-$20%, 20$-$30%, 30$-$40%, 40$-$50%, 50$-$60%, and 60$-$70% for $p$ ($\bar{p}$) in Fig. 4(a) and (d), 0$-$10%, 10$-$20%, 20$-$40%, 40$-$60%, and 60$-$80% for $d$ ($\bar{d}$) in Fig. 4 (b) and (e), 0$-$10%, 10$-$20%, 20$-$40%, and 40$-$80% for $t$ ($\bar{t}$) in Fig. 4(c) and (f). It is found that the results can well describe the experimental data for $p$ Adamczyk et al. (2017a), $d$ Adam et al. (2019) and $t$ Zhang (2021) spectra from the STAR collaboration, especially in central collisions. Besides, we compared the transverse momentum spectra of $p$ ($\bar{p}$), $d$ ($\bar{d}$) and $t$ with the results from the iEBE-MUSIC hybrid model plus coalescence model Zhao et al. (2020). Note that the isospin effect in the statistical factor in Eq. (2) can result in a constant factor among the results Zhao et al. (2020); Sun et al. (2021) and does not affect the shape of the spectra. It is interesting to see that two models are consistent, which implies that the phase-space of the two models have similar properties or distributions.

Figure 5 shows the calculated transverse momentum spectra for $p$ ($\bar{p}$) ((a) and (b)), $d$ ($\bar{d}$) ((c) and (d)) and $t$ ($\bar{t}$) ((e) and (f)) in ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, ⁴⁰Ca + ⁴⁰Ca, and ¹⁹⁷Au + ¹⁹⁷Au in 0$-$10% central collisions at $\sqrt{s_{NN}}$ = 39 GeV. The $p_{T}$ spectra present an obvious collision system dependence in central collisions and drop with the decreasing of the collision system size Zhang et al. (2011).

The rapidity densities ($\langle dN/dy\rangle$) of $p$ ($\bar{p}$), $d$ ($\bar{d}$) and $t$ ($\bar{t}$) are calculated in mid-rapidity as a function of $\langle N_{ch}\rangle$ in ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, ⁴⁰Ca + ⁴⁰Ca, and ¹⁹⁷Au + ¹⁹⁷Au collisions at $\sqrt{s_{NN}}$ = 39 GeV, as shown in Fig. 6. It is found that $\langle dN/dy\rangle$ of $p$ as a function of $\langle N_{ch}\rangle$ (Fig. 6(a)) can well describe the data Adamczyk et al. (2017a) but underestimate $\bar{p}$ data in Au + Au collisions at $\sqrt{s_{NN}}$ = 39 GeV. For $d$ and $\bar{d}$ (Fig. 6(b)), it presents the similar description quality to the data Adam et al. (2019). $\langle dN/dy\rangle$ of $t$ and $\bar{t}$ as a function of $\langle N_{ch}\rangle$ is presented in Fig. 6(c). As shown in Fig. 6(a) and (b), $\langle dN/dy\rangle$ of $p$ and $d$ are comparable to those from the iEBE-MUSIC hybrid model plus coalescence model Zhao et al. (2020) in central collisions, and little difference for anti-matter partners. In addition, the yields of these light (anti)nuclei for the 0-10% central collisions of ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, and ⁴⁰Ca + ⁴⁰Ca systems at $\sqrt{s_{NN}}$ = 39 GeV are also shown in Fig. 6, and it seems that they follow the similar $\langle N_{ch}\rangle$ systematics. In general, it is reasonably speculated that $\langle dN/dy\rangle$ of (anti)proton, (anti)deuteron and triton present an increasing trend with $\langle N_{ch}\rangle$ (collision system size) in different collision centralities as well as collision systems.

Furthermore, we calculate the $\langle N_{ch}\rangle$ dependence of ratios of $d/p$ and $t/p$ by using a thermal model Braun-Munzinger et al. (2004),

where $\lambda_{i}(T,\vec{\mu})=\exp(\frac{B_{i}\mu_{B}+S_{i}\mu_{s}+Q_{i}\mu_{Q}}{T})$, $B_{i}$, $S_{i}$ and $Q_{i}$ are the baryon number, strangeness number and charge number, $\mu_{B}$, $\mu_{S}$ and $\mu_{Q}$, are their corresponding chemical potentials of particle $i$, $K_{2}$ is the modified Bessel function and the upper sign is for bosons and lower for fermions, $g_{i}$ is the spin$-$isospin degeneracy factor. We use the parameters such as the chemical freeze-out temperature as well as the baryon chemical potential from Ref. Adamczyk et al. (2017a). As shown in Fig. 6(d), the $d/p$ and $t/p$ ratios of AMPT + coalescence model are bigger than STAR data Adamczyk et al. (2017a); Adam et al. (2019); Zhang (2019). And the $d/p$ ratio from the thermal model can describe the STAR data Adamczyk et al. (2017a); Adam et al. (2019); Zhang (2019) but overestimates the $t/p$ ratio, which is consistent with the results in references Zhang (2019); Yu and Luo (2019); Vovchenko et al. (2020).

Figure 7 presents a comparison between $\langle N_{ch}\rangle$ dependence of the fireball radius ($\sqrt{\left<r^{2}_{f}\right>}$) calculated directly by the coordinates from the AMPT model and that $r_{V}$ from the analytic coalescence model Sun and Chen (2017) in ¹⁹⁷Au + ¹⁹⁷Au collisions at $\sqrt{s_{NN}}=39$ GeV. In analytic coalescence model Sun and Chen (2017), a blast-wave-like parametrization is used for the phase-space configuration of constituent particles at freeze-out. We extract the effective volume by equation (25) in Ref. Sun and Chen (2017), then the fireball radius $r_{V}$ can be calculated by assuming a spherical fireball. We find that the both radii ($\sqrt{\left<r^{2}_{f}\right>}$ and $r_{V}$) present a similar $\langle N_{ch}\rangle$ dependence, i.e. increasing as $\langle N_{ch}\rangle$. Of course, we notice that the values of size have model or calculation method dependence.

Figure 8(a) shows the $\langle dN/dy\rangle$ of proton, deuteron and triton as a function of baryon number $B$ from the coalescence model in 0$-$10% ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, and ⁴⁰Ca + ⁴⁰Ca collisions, as well as the 0$-$10%, 10$-$20%, 20$-$40%, 40$-$60%, and 60$-$80% ¹⁹⁷Au + ¹⁹⁷Au collisions at $\sqrt{s_{NN}}=39$ GeV. The lines are the fits to the calculated results by a function of $N_{0}e^{-rB}$, here $N_{0}$ denotes amplitude, $B$ is the baryon number and $r$ is the reduction factor. It is found that the yields of proton, deuteron ,and triton in each collision system exhibit a decreasing exponential trend with the baryon number. The reduction factor Shah et al. (2016); Sun and Chen (2015) by fitting the yields of proton, deuteron and triton as a function of $\langle N_{ch}\rangle$ is shown in Fig. 8(b). While the system size is expressed by $\langle N_{ch}\rangle$, the reduction factor decreases sharply with the increasing of $\langle N_{ch}\rangle$ and then saturate at large $\langle N_{ch}\rangle$. This implies that light nuclei production becomes more difficult in small systems, especially for that with baryon number $B>$3 in the relativistic heavy-ion collisions.

To further characterize the system size dependence of light nuclei production, the coalescence probability of forming light clusters is investigated by the coalescence parameters $B_{A}$ ($A$ = 2 and 3) as defined in Eq. (1). In panel (a) and (b) of Fig. 9, the calculated coalescence parameter $B_{2}$ are compared with the data measured by the STAR collaboration Adam et al. (2019) in ¹⁹⁷Au + ¹⁹⁷Au collisions at RHIC energy of 39 GeV in 0$-$10%, 10$-$20%, 20$-$40%, and 40$-$60% (40$-$80% for triton) centralities. The calculated results present a similar trend with the experimental data，the coalescence parameters $B_{2}$ in panel (a), (b) and $B_{3}$ in panel (c) as a function of $p_{T}/A$ in different collision centralities always present an increasing trend, this might be due to the increasing correlation volume with the decreasing of $p_{T}$, leading to a higher coalescence probability for larger $p_{T}$ values. In addition, the values of $B_{2}$ and $B_{3}$ decrease with collision centrality (i.e. the more central collisions the less $B_{A}$), which suggests that source volume being larger in central collisions. From the viewpoint of the coalescence probability of nucleons to form these light clusters, it is reasonable to have a bigger coalescence probability while the distance between the protons and neutrons is smaller. On the other hand, we note that the values of $B_{2}$ for deuterons are systematically larger than those of anti-deuterons in the same centrality, it is consistent with the experimental observation Adam et al. (2019), indicating that the correlated volume of baryons is smaller than that of anti-baryons. Besides, the comparison of our results of $B_{2}$ and $B_{3}$ with the iEBE-MUSIC hybrid model plus coalescence model Zhao et al. (2020) is also shown in this figure, and the trend remains similar.

Furthermore, the coalescence parameter $B_{2}$ for (anti)deuterons as a function of $p_{T}/A$ is also calculated for ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, and ⁴⁰Ca + ⁴⁰Ca collisions at 0$-$10% centrality at $\sqrt{s_{NN}}$ = 39 GeV, and the results are presented in Fig. 10 (a) and (b). It is found that the coalescence parameter $B_{2}$ presents a system size dependence, i.e. $B_{2}$ decreases as the system size increases. This result is consistent with the centrality dependence in the same system such as Au + Au collisions. The $p_{T}$ dependence of $B_{2}$ also presents an upward trend as shown in Fig. 9. The coalescence parameter $B_{3}$ is presented as a function of $p_{T}/A$ for the 0$-$10% central collisions of ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, ⁴⁰Ca + ⁴⁰Ca, and ¹⁹⁷Au + ¹⁹⁷Au systems at $\sqrt{s_{NN}}$ = 39 GeV in Fig. 10(c), it shows the similar trend with $B_{2}$ even though the error remains larger.

Figure 11 shows the $\langle N_{ch}\rangle$ dependence of coalescence parameters $B_{2}$ and $B_{3}$ of $d$ ($\bar{d}$) (a, b), $t$ ($\bar{t}$) (c, d) in ¹⁹⁷Au+¹⁹⁷Au collisions at 0$-$10%, 10$-$20%, 20$-$40%, and 40$-$60% (40$-$80% for $t$) centralities as well as 0$-$10% central collisions of ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, ⁴⁰Ca + ⁴⁰Ca systems at $\sqrt{s_{NN}}=39$ GeV. It is observed that the coalescence parameters $B_{2}$ and $B_{3}$ present an obvious collision centrality dependence, the values of $B_{2}$ for deuteron and anti-deuteron decrease with the increasing of $\langle N_{ch}\rangle$. The $\langle N_{ch}\rangle$ dependence of $B_{3}$ for triton in ¹⁹⁷Au + ¹⁹⁷Au collisions at 0$-$10%, 10$-$20%, 20$-$40%, and 40$-$80% centralities at $\sqrt{s_{NN}}=39$ GeV is also shown in this figure. $B_{3}$ also shows a decreasing trend with $\langle N_{ch}\rangle$. Besides, it is observed that the values of $B_{2}$ and $B_{3}$ present an obvious collision system dependence in 0-10% central collisions of ¹⁰B + ¹⁰B, ¹⁶O + ¹⁶O, ⁴⁰Ca + ⁴⁰Ca, and ¹⁹⁷Au + ¹⁹⁷Au systems, the values of $B_{2}$ and $B_{3}$ for deuteron and anti-deuteron drop with the increasing of system size, the value of $B_{3}$ also shows a decreasing trend with $\langle N_{ch}\rangle$. Considering the properties of system size dependence from figure 3 as well as the relationship between coalescence parameter and nucleon correlation volume, i.e. $B_{A}\propto 1/V_{eff}^{A-1}$ Csernai and Kapusta (1986), we found that $B_{A}$ can be expressed by a simple function, $B_{A}\propto 1/\left(\left<N_{ch}\right>\right)^{(A-1)}$ (here $A$ = 2 or 3). From the viewpoint of light nuclei production by coalescence mechanism, it is concluded that the coalescence parameter $B_{A}$ can reflect the collision system size when the system is at kinetic freeze-out stage.

The thermal model has been successfully used to describe the multiplicities or particle ratios of hadrons and light nuclei Andronic et al. (2018) in relativistic heavy-ion collisions, while the coalescence model basing on phase space data is another useful tool to treat light nuclei production. In practice, the phase-space data can be generated from various models, such as blast-wave model Sun and Chen (2016), hydrodynamics Zhao et al. (2020), transport model Sombun et al. (2019) or pure analytical calculation Sun and Chen (2017). In our work, the coalescence model basing on the AMPT phase space data is used to study the light nuclei production at RHIC lower energy in the Beam Energy Scan project Aggarwal et al. (2010); Adamczyk et al. (2017b), the results are consistent with the previous calculations and provide a more comprehensive understanding of the experiment data. Therefore we argue that these models could approach an equivalent simulation of the production of light nuclei assuming the thermal or kinetic freeze-out properties of the collision systems, respectively.