Different Coalescence Sources of Light Nuclei Production in Au-Au Collisions at $\sqrt{s_{NN}}=3$ GeV

We study the production of light nuclei in the coalescence mechanism in Au-Au collisions at midrapidity at $\sqrt{s_{NN}}=3$ GeV. We derive analytic formulas of momentum distributions of two bodies, three bodies and four nucleons coalescing into light nuclei, respectively. We naturally explain the transverse momentum spectra of the deuteron ($d$), triton ($t$), helium-3 ($^3$He) and helium-4 ($^4$He). We reproduce the data of yield rapidity densities and averaged transverse momenta of $d$, $t$, $^3$He and $^4$He. We give proportions of contributions from different coalescence sources for $t$, $^3$He and $^4$He in their productions. We find that besides nucleon coalescence, nucleon$+$nucleus coalescence and nucleus$+$nucleus coalescence may play requisite roles in light nuclei production in Au-Au collisions at $\sqrt{s_{NN}}=3$ GeV.

The coalescence mechanism, in which light nuclei are usually assumed to be produced by the coalescence of the jacent nucleons in the phase space, possesses its unique characteristics.Plenty of current experimental observations at high RHIC and LHC energies favor the nucleon coalescence [18,19,22,23,[49][50][51].Recently the STAR collaboration has extended the beam energy scan program to lower collision energy and published the data of both hadrons and light nuclei in Au-Au collisions at √ s NN = 3 GeV [52][53][54][55].
These data show very different properties compared to those at high RHIC and LHC energies, such as the disappearance of partonic collectivity [52] and dominant baryonic interactions [53].At this low collision energy besides nucleons, light nuclei in particular of light d, t and 3 He have been more abundantly created [55] compared to higher collision energies [56].It is easier in physics for these light nuclei to capture nucleons or other light nuclei to form heavier composite objects.In fact clear depletions below unity of proton−d and * shaofl@mail.sdu.edu.cnd − d correlation functions measured at such low collision energy indicate the strong final state interaction and further support the possible coalescence of the d with the nucleon or other d [57].How much space is there on earth for other particle coalescence except nucleons, e.g., composite particles of less mass numbers coalescing into light nuclei of larger mass numbers or composite particles capturing nucleons to recombine into heavier light nuclei?
In this article, we extend the coalescence model which has been successfully used to explain the momentum dependence of yields and coalescence factors of different light nuclei at high RHIC and LHC energies [51,58], to include nucleon+nucleus coalescence and nucleus+nucleus coalescence besides nucleon coalescence.We apply the extended coalescence model to hadronic systems created in Au-Au collisions at midrapidity area at √ s NN = 3 GeV to study the momentum and centrality dependence of light nuclei production in the low-and intermediate-p T regions.We compute the transverse momentum (p T ) spectra, the yield rapidity densities (dN/dy) and the averaged transverse momenta (⟨p T ⟩) of d, t, 3 He and 4 He from central to peripheral collisions.We give proportions of contributions from different coalescence sources for t, 3 He and 4 He respectively in their productions.Our studies show that in 0 − 10%, 10 − 20% and 20 − 40% centralities, besides nucleon coalescence, nucleon+d coalescence plays an important role in t and 3 He production and nucleon+d (t, 3 He) coalescence as well as d + d coalescence occupy significant proportions in 4 He production.But in the peripheral 40 − 80% centrality, nucleon coalescence plays a dominant role, and nucleon+nucleus coalescence or nucleus+nucleus coalescence seems to disappear.
The rest of the paper is organized as follows.In Sec.II, we introduce the coalescence model.We present analytic formulas of momentum distributions of two bodies, three bodies, and four nucleons coalescing into light nuclei, respectively.In Sec.III, we apply the model to Au-Au collisions in different rapidity intervals at midrapidity area at √ s NN = 3 GeV to study momentum and centrality dependence of the production of various species of light nuclei in the low-and intermediate-p T regions.We give proportions of contributions from different coalescence sources for t, 3 He and 4 He arXiv:2210.10271v2[hep-ph] 16 Oct 2023 in their productions.In Sec.IV we summarize our work.

II. THE COALESCENCE MODEL
In this section we introduce the coalescence model which is used to deal with the light nuclei production.The starting point of the model is a hadronic system produced at the late stage of the evolution of high energy collision.The hadronic system consists of different species of primordial mesons and baryons.In the first step of the model all primordial nucleons are allowed to form d, t, 3 He and 4 He via the nucleon coalescence.Then in the second step the formed d, t and 3 He capture the remanent primordial nucleons, i.e., those excluding consumed ones in the nucleon coalescence process, or other light nuclei to recombine into nuclei with larger mass numbers.In this model only d, t, 3 He and 4 He are included, and those light nuclei with mass number larger than 4 are abandoned.
In the following we present the deduction of the formalism of the production of various species of light nuclei via different coalescence processes, respectively.First we give analytic results of two bodies coalescing into light nuclei, which can be applied to processes such as p + n → d, n + d → t, p + d → 3 He, p + t → 4 He, n+ 3 He → 4 He and d + d → 4 He.Then we show analytic results of three bodies coalescing into light nuclei, which can be used to describe these processes, e.g., n + n + p → t, p + p + n → 3 He and p + n + d → 4 He.Finally, we give the analytic result of four nucleons coalescing into 4 He, i.e., p + p + n + n → 4 He.

A. Formalism of two bodies coalescing into light nuclei
We begin with a hadronic system produced at the final stage of the evolution of high energy collision and suppose light nuclei L j are formed via the coalescence of two hadronic bodies h 1 and h 2 .The three-dimensional momentum distribution of the produced light nuclei f L j ( p) is given by where Here and from now on we use bold symbols to denote three-dimensional coordinate or momentum vectors.
In terms of the normalized joint coordinate-momentum distribution denoted by the superscript '(n)', we have N h 1 h 2 is the number of all possible h 1 h 2 -pairs, and it is equal to ) is the number of the hadrons h i in the considered hadronic system.
The kernel function R L j (x 1 , x 2 ; p 1 , p 2 , p) denotes the probability density for h 1 , h 2 with momenta p 1 and p 2 at x 1 and x 2 to recombine into a L j of momentum p.It carries the kinetic and dynamical information of h 1 and h 2 recombining into light nuclei, and its precise expression should be constrained by such as the momentum conservation, constraints due to intrinsic quantum numbers e.g.spin, and so on [51,58,59].To take these constraints into account explicitly, we rewrite the kernel function in the following form where the spin degeneracy factor g L j = (2J ]. J L j is the spin of the produced L j and J h i is that of the primordial hadron h i .The Dirac δ function guarantees the momentum conservation in the coalescence.The remaining R (x,p) L j (x 1 , x 2 ; p 1 , p 2 ) can be solved from the Wigner transformation once the wave function of L j is given with the instantaneous coalescence approximation.It is as follows as we adopt the wave function of a spherical harmonic oscillator as in Refs.[60,61].The superscript ' ′ ' in the coordinate or momentum variable denotes the hadronic coordinate or momentum in the rest frame of the h 1 h 2 -pair.m 1 and m 2 are the rest mass of hadron h 1 and that of hadron h 2 .The width parameter σ = 2(m 1 +m 2 ) 2 3(m 2 1 +m 2 2 ) R L j , where R L j is the rootmean-square radius of L j and its values for different light nuclei can be found in Ref. [62].The factor ℏc comes from the used GeV•fm unit, and it is 0.197 GeV•fm.
The normalized two-hadron joint distribution f (n) h 1 h 2 (x 1 , x 2 ; p 1 , p 2 ) is generally coordinate and momentum coupled, especially in central heavy-ion collisions with relatively high collision energies where the collective expansion exists long.The coupling intensities and its specific forms are probably different at different phase spaces in different collision energies and different collision centralities.In this article, we try our best to derive production formulas analytically and present centrality and momentum dependence of light nuclei more intuitively in Au-Au collisions at low RHIC energy √ s NN = 3 GeV where the partonic collectivity disappears [52], so we consider a simple case that the joint distribution is coordinate and momentum factorized, i.e., Substituting Equations (3)(4)(5) into Equation (2), we have where we use A L j to denote the coordinate integral part in Equation ( 6) as and use M L j ( p) to denote the momentum integral part as A L j stands for the probability of a h 1 h 2 -pair satisfying the coordinate requirement to recombine into L j , and M L j ( p) stands for the probability density of a h 1 h 2 -pair satisfying the momentum requirement to recombine into L j with momentum p.
Changing integral variables in Equation ( 7) to be X = √ 2 , we have and the normalizing condition We further assume the coordinate joint distribution is coordinate variable factorized, i.e., f CwR 2 f as in Refs.[51,63], we have Here R f is the effective radius of the hadronic system at the light nuclei freeze-out.C w is a distribution width parameter and it is set to be 2, the same as that in Refs.[51,63].Considering instantaneous coalescence in the rest frame of h 1 h 2 -pair, i.e., ∆t ′ = 0, we get where β is the three-dimensional velocity vector of the center-of-mass frame of h 1 h 2 -pair in the laboratory frame and the Lorentz contraction factor γ = 1/ 1 − β 2 .Substituting Equation (12) into Equation (11) and integrating from the relative coordinate variable, we can obtain Noticing that ℏc/σ in Equation ( 8) has a small value of about 0.1, we can mathematically approximate the gaussian form of the momentum-dependent kernel function to be a δ function form as follows After integrating p 1 and p 2 from Equation (8) we can obtain ), (15) where . Substituting Equations ( 13) and ( 15) into Equation (6) and ignoring correlations between h 1 and h 2 hadrons, we have Denoting the Lorentz invariant momentum distribution d 2 N 2πp T d p T dy with f (inv) , we finally have where y is the rapidity.

B. Formalism of three bodies coalescing into light nuclei
For light nuclei L j formed via the coalescence of three hadronic bodies h 1 , h 2 and h 3 , the three-dimensional momentum distribution f L j ( p) is N h 1 h 2 h 3 is the number of all possible h 1 h 2 h 3 -clusters and it is equal to h 1 h 2 h 3 is the normalized three-hadron joint coordinate-momentum distribution.R L j is the kernel function.
We rewrite the kernel function as The spin degeneracy factor g L j = (2J The Dirac δ function guarantees the momentum conservation.R (x,p) L j (x 1 , x 2 , x 3 ; p 1 , p 2 , p 3 ) solving from the Wigner transformation [60,61] is The superscript ' ′ ' denotes the hadronic coordinate or momentum in the rest frame of the h 1 h 2 h 3 -cluster.The width With the coordinate and momentum factorization assumption of the joint distribution, we have Here we also use A L j to denote the coordinate integral part as f as in Refs.[51,63], we have Comparing relations of r 1 , r 2 with x 1 , x 2 , x 3 to that of r with x 1 , x 2 in Sec.II A, we see that C 1 is equal to C w and C 2 is 4C w /3 when ignoring the mass difference of m 1 and m 2 [51,63].Considering the Lorentz transformation and integrating from the relative coordinate variables in Equation ( 24), we obtain Approximating the gaussian form of the momentumdependent kernel function to be δ function form and integrating p 1 , p 2 and p 3 from Equation (23), we can obtain Substituting Equations ( 25) and ( 26) into Equation ( 21) and ignoring correlations between h 1 , h 2 and h 3 hadrons, we have Finally we have the Lorentz invariant momentum distribution C. Formalism of four nucleons coalescing into 4 He For 4 He formed via the coalescence of four nucleons, the three-dimensional momentum distribution is (29) where N ppnn = N p (N p − 1)N n (N n − 1) is the number of all possible ppnn-clusters; f (n)  ppnn is the normalized four-nucleon joint coordinate-momentum distribution; R4 He is the kernel function.
We rewrite the kernel function as where the spin degeneracy factor g4 He = 1/16, and Here σ4 He = 2 √ 2 3 R4 He , and R4 He = 1.6755 fm [62] is the root-mean-square radius of the 4 He.
Assuming that the normalized joint distribution is coordinate and momentum factorized, we have f4 He ( p) = N ppnn g4 He A4 He M4 He ( p). ( Here we use A4 He to denote the coordinate integral part in Equation ( 32) as 4 He e and use M4 He ( p) to denote the momentum integral part as We change integral variables in Equation ( 33) to be 4 He e C 1 , C 2 , C 3 are equal to be C w , 4C w /3 and 3C w /2, respectively [51,63].After the Lorentz transformation and integrating the relative coordinate variables from Equation (35), we obtain Approximating the gaussian form of the momentumdependent kernel function to be δ function form and after integrating p 1 , p 2 , p 3 and p 4 in Equation ( 34), we can obtain ). (37) Substituting Equations ( 36) and (37) into Equation ( 32), we have We finally have the Lorentz invariant momentum distribution where m is the nucleon mass.As a short summary of this section, we want to state that Equations (17,28,39) show the relationship of light nuclei with primordial hadronic bodies in momentum space in the laboratory frame.They can be directly used to calculate the yields and p T distributions of light nuclei formed via different coalescence channels as long as the primordial hadronic momentum distributions are given.In the case of ignoring the mass differences of primordial hadrons, Equations ( 17) and ( 28) return to our previous results for d, t and 3 He in Refs.[51,58] where only nucleon coalescence was considered.

III. RESULTS AND DISCUSSIONS
In this section, we apply the coalescence model in Sec.II to Au-Au collisions at √ s NN = 3 GeV to study the momentum and centrality dependence of the production of different light nuclei in the low-and intermediate-p T regions in different rapidity intervals at midrapidity area.We first introduce the p T spectra of the nucleons.We then present p T dependence of different coalescence sources for d, t, 3 He and 4 He in their productions.We finally give the yield rapidity densities dN/dy and the averaged transverse momenta ⟨p T ⟩ of different light nuclei.

A. p T spectra of nucleons
The invariant p T distributions at different rapidity intervals of primordial protons f (inv)  p,pri (p T , y) and neutrons f (inv)  n,pri (p T , y) are necessary inputs for computing p T distributions of light nuclei in our model.The relationship of primordial protons and those final-state ones is as follows The last three terms in the equation denote the invariant p T distributions of protons consumed in light nuclei production, those coming from hyperon weak decays and those finalstate ones, respectively.The feed-down contribution from the weak decays of hyperons to protons is about 1.5% [64] and that entering into light nuclei takes about 20% [55].
Considering that most of primordial protons, more than 80%, evolve to be final-state ones, we ignore the variation of the shape of the p T spectra of primordial protons and final-state ones.In this case, we can get f (inv) p,pri (p T , y) ≈ 1 81.5% f (inv) p,fin (p T , y).We here use the blast-wave model to get invariant p T distribution functions of final-state protons by fitting the proton experimental data in Ref. [55].The blast-wave function [65] is given as where r is the radial distance in the transverse plane and R is the radius of the fireball.m T is the transverse mass of the proton.I 0 and K 1 are the modified Bessel functions, and the velocity profile The surface velocity β s , the kinetic freeze-out temperature T kin and n are fitting parameters.
Figure 1 shows the invariant p T spectra of final-state protons in different rapidity intervals −0.1 < y < 0, −0.For the neutron, we assume the same normalized p T distribution as that of the proton in the same rapidity interval and the same collision centrality.For absolute yield density of the neutron, it is generally not equal to that of the proton due to the prominent influences of net nucleons from the colliding Au nuclei.We here use Z np to denote the extent of the yield density asymmetry of the neutron and the proton and take their relation as Z np = 1 corresponds to the complete isospin equilibration and Z np = 1.49 to isospin asymmetry in the whole Au nucleus.We here set Z np to be a free parameter, and its values in different centrality and rapidity windows are put in Table I.Values of Z np in central and semi-central 0 − 10%, 10 − 20%, 20 − 40% centralities are comparable and they are close to that evaluated in Ref. [66].Z np in 40 − 80% centrality becomes a little smaller.From the viewpoint of the effect of the neutron skin [67], Z np is expected to increase in peripheral collisions.But note that we here study light nuclei production in the midrapidity area, i.e., y < 0.5, in peripheral collisions the transparency of nucleons from the colliding nuclei become stronger due to smaller reaction area and they move to relative larger rapidity [55].The participant nucleons from colliding nuclei become less in midrapidity region, so the yield asymmetry extent due to the participant nucleons decreases.
The other parameter in our model is R f , which is fixed by the data of the yield rapidity density of d [55].Values of R f in different rapidity intervals and different centralities in Au-Au collisions at √ s NN = 3 GeV are listed in Table I.For 0-10% centrality, our fixed values locate in the range evaluated by the linear dependence on the cube root of the rapidity density of charged particles, i.e., R f ∝ (dN ch /dy) 1/3 [58,68].For other collision centralities, R f cannot be evaluated by the relation R f ∝ (dN ch /dy) 1/3 due to the lack of the data of π ± and K ± currently.From Table I, one can see R f decreases very slightly as the increasing rapidity for the same centrality and it decreases from central to peripheral collisions.The smaller R f in more peripheral collisions leads to the stronger suppression of light nuclei production because of the non-negligible light nuclei sizes compared to R f as shown in Equations (17,28,39).This suppression effect of light nuclei production in small collision systems has been systematically studied in Ref. [69].We then study the invariant p T distributions of t, 3 He and 4 He in rapidity intervals −0.1 < y < 0, −0.But panel (d) in Figure 4 shows that 3 He production in peripheral Au-Au collisions favors p + p + n coalescence.This is similar to that of the triton.
Figure 5 shows the invariant p T spectra of 4 He.The spectra in different rapidity intervals are scaled by different factors for clarity as shown in the figure.Filled symbols with error bars are experimental data from the STAR collaboration [55].Short-dashed lines are the results of the nucleon coalescence, i.e., the contribution of the channel p + p + n + n → 4 He.Long-dashed lines are the results of the contributions from the channel p + t → 4 He, and largegap dotted lines are the results of the contributions from the channel n+ 3 He→ 4 He.Small-gap dotted lines are the results of the contributions from the channel d + d → 4 He.Dashed-dotted lines are the results of the contributions from the channel p + n + d → 4 He.Solid lines are the total results including the above five coalescence channels.From panels (a), (b) and (c) in Figure 5, one can see total results including the above five coalescence processes can describe the available data in central and semi-central Au-Au collisions at √ s NN = 3 GeV.But panel (d) in Figure 5 shows that 4 He production in peripheral Au-Au collisions favors nucleon coalescence, i.e., p + p + n + n coalescence.The other four coalescence cases involving nucleon+nucleus or nucleus+nucleus coalescence may not occur.When calculating contributions from different coalescence channels, we base on the hypothesis that the nucleon coalescence happens first and subsequently the formed lighter cluster captures other particle to form heavier cluster if they meet the coalescence requirements in the phase space.This coalescence time order is constrained to the local freeze-out instead of the whole phase space.Results in Figures 2, 3 and 4 show that our final results of p T spectra of d, t and 3 He can describe the experimental data in 0 − 10%, 10 − 20%, 20 − 40% centralities while in 40 − 80% centrality our results of nucleon coalescence itself can reproduce the available data.Results in Figure 5 show that our total results of nucleon coalescence plus nucleon+d (t, 3 He) co-alescence plus d + d coalescence can describe the data of p T spectra of 4 He in 0 − 10%, 10 − 20%, 20 − 40% centralities while in 40 − 80% centrality nucleon coalescence itself can reproduce the 4 He data.This indicates that besides nucleon coalescence, nucleon/nucleus+nucleus coalescence plays an important role in central and semicentral collisions.But in peripheral collisions nucleus coalescence seems to disappear.This is probably due to that the interactions between hadronic rescatterings become not so strong that the formed light nuclei can not capture other particles to form heavier objects.To see contribution proportions of different coalescence sources of t, 3 He and 4 He in their production and depletion proportions of d, t and 3 He more clearly, we in this subsection study the yield rapidity densities dN/dy of light nuclei.After integrating over the p T , we can get dN/dy.Table II shows our results of d and Table III shows those of t and 3 He in different rapidity intervals and different centralities in Au-Au collisions at √ s NN = 3 GeV.Data with errors are from Ref. [55], and the errors denote the systematical un-  0.074 ± 0.015 0.074 0.031 0.002 0.103 0.047 ± 0.010 0.047 0.023 0.002 0.068 −0.2 < y < −0.1 0.086 ± 0.015 0.087 0.037 0.003 0.121 0.055 ± 0.007 0.057 0.027 0.002 0.082 −0.3 < y < −0.2 0.088 ± 0.006 0.089 0.038 0.003 0.124 0.064 ± 0.009 0.065 0.031 0.002 0.094 −0.4 < y < −0.3 0.116 ± 0.007 0.119 0.049 0.005 0.163 0.081 ± 0.005 0.083 0.038 0.003 0.118 −0.5 < y < −0.4 0.172 ± 0.010 0.188 0.075 0.008 0.255 0.120 ± 0.010 0.131 0.060 0.007 0.184 TABLE IV.Yield rapidity densities dN/dy of 4 He in different rapidity intervals and different centralities in Au-Au collisions at √ s NN = 3

Centrality
j=d,t, 3 He, 4 He Values of δ devi calculated with Theo fin for d, t, 3 He and Theo total for 4 He are put in the last column in Table IV, and those in the parentheses for the 40 − 80% centrality are calculated with the results only including nucleon coalescence.Our theoretical results in Tables II, III and IV clearly show contribution proportions of different production sources for d, t, 3 He and 4 He in their production in 0−10%, 10−20% and 20−40% centralities.The proportion of nucleon coalescence and that of nucleon+d coalescence in t and 3 He production take about 60% and 40%, respectively.The proportion of nucleon coalescence and those of p + n + d coalescence, p + t coalescence, n+ 3 He coalescence and d + d coalescence in 4 He production take about 20%, 15%, 30%, 25% and 10%, respectively.Tables II and III also show that the depletion of d takes about 7% ∼ 9% while the depletions of t and 3 He are both less than 3%.These results tell us that besides nucleon coalescence, other particle coalescences, e.g., composite particles of less mass numbers coalescing into light nuclei of larger mass numbers or composite particles capturing nucleons to recombine into heavier light nuclei, also play important roles in light nuclei production in central and semi-central collisions at relatively low collision energies.This provides a new possible window to cognize the underestimations of the yield densities of light nuclei in some specific models only including nucleon coalescence such as in Ref. [70].

D. Averaged transverse momenta of light nuclei
The averaged transverse momenta of different light nuclei reflect the collective motion and bulk properties of the hadronic matter at kinetic freezeout.In this subsection we study the averaged transverse momenta ⟨p T ⟩ of d, t, 3 He and 4 He in rapidity intervals −0.VI show that for t, 3 He and 4 He, the calculated ⟨p T ⟩ from different coalescence sources are almost the same.This is very different from dN/dy.Our theoretical results agree with the data, and the deviations are less than 10%.⟨p T ⟩ of d, t, 3 He and 4 He decreases gradually as the increasing rapidity from central to peripheral collisions.This further indicates the stronger transverse collective motion at midrapidity area in more central collisions.
At the end of Sec.III, we want to state that our results show the coalescence mechanism still works in describing light nuclei production in Au-Au collisions at √ s NN = 3 GeV. Compared to those at high RHIC and LHC energies in our previous works [51,58], relativistic heavy ion collisions at lower collision energies have some new characteristics in light nuclei production, e.g., isospin asymmetry from the colliding nuclei and the non-negligible nucleus+nucleon/nucleus coalescence.

IV. SUMMARY
In the coalescence mechanism, we studied different coalescence sources of the production of various species of light nuclei in relativistic heavy ion collisions.We firstly extended the coalescence model to include two bodies, three bodies, and four nucleons coalescing into light nuclei, respectively.We used the assumption of the coordinatemomentum factorization of joint hadronic distributions.We adopted gaussian forms for the relative coordinate distributions.Based on these simplifications, we obtained analytic formulas of momentum distributions of light nuclei formed from different production sources which coalesced by different hadrons.
We then applied the extended coalescence model to Au-Au collisions at √ s NN = 3 GeV to simultaneously investigate the p T spectra of the d, t, 3 He and 4 He in different rapidity intervals at midrapidity area from central to peripheral collisions.We presented the p T dependence of different coalescence sources for d, t, 3 He and 4 He.We also studied yield rapidity densities dN/dy and averaged transverse momenta ⟨p T ⟩ of d, t, 3 He and 4 He.We gave proportions of yield densities from different coalescence sources for t, 3 He and 4 He in their production and those of depletions for d, t and 3 He.We found yield densities from different coalescence sources for a specific kind of light nuclei were very different, but averaged transverse momenta were almost unchanged.
Our results showed that (1) results of p + n coalescence minus those depleted in nucleus coalescence reproduced the available data of d well in central and semi-central collisions and the data in peripheral collisions favored p + n coalescence; (2) the nucleon coalescence plus nucleon+d coalescence reproduced the available data of t and 3 He in central and semi-central collisions (their depletions in forming 4 He are less than 3%) and the data in peripheral collisions favored only nucleon coalescence; (3) the nucleon coalescence plus nucleon+nucleus coalescence and nucleus+nucleus coalescence described the available data of 4 He in central and semi-central collisions and the data in peripheral collisions favored only p + p + n + n coalescence.
2 < y < −0.1, −0.3 < y < −0.2, −0.4 < y < −0.3, −0.5 < y < −0.4 in Au-Au collisions at √ s NN = 3 GeV in centralities 0−10%, 10−20%, 20−40%, 40−80%.The spectra in different rapidity intervals are scaled by different factors for clarity as shown in the figure.Filled symbols are experimental data from the STAR collaboration [55].Different lines are the results of the blast-wave model.Since we focus on testing the validity of the coalescence mechanism in describing the light nuclei production at low collision energy instead of predicting the momentum distributions of light nuclei, we only include the best fit from the blast-wave model for the proton, and do not consider the fitting errors.Here, the proton dN/dy and ⟨p T ⟩ obtained by these blast-wave results are just equal to central values of the corresponding data given by the STAR collaboration in Ref. [55].TABLE I. Values of Z np and R f in different rapidity intervals and different centralities in Au-Au collisions at √ s NN = 3 GeV. 2 < y < −0.1 1.33 3.21 −0.3 < y < −0.2 1.29 3.10 −0.4 < y < −0.3 1.34 3.09 −0.5 < y < −0.4 12 < y < −0.1 1.13 1.41 −0.3 < y < −0.2 1.01 1.40 −0.4 < y < −0.3 1.05 1.35 −0.5 < y < −0.4 1.05 1.34

B
. p T spectra of light nuclei With Equation (17), we first compute the invariant p T distributions of deuterons in rapidity intervals −0.1 < y < 0, −0.2 < y < −0.1, −0.3 < y < −0.2, −0.4 < y < −0.3, −0.5 < y < −0.4 in Au-Au collisions at √ s NN = 3 GeV in centralities 0 − 10%, 10 − 20%, 20 − 40%, 40 − 80%, respectively.Here, h 1 and h 2 in Equation (17) refer to the proton and the neutron.Different lines scaled by different factors for clarity in Figure 2 are our theoretical results for finalstate deuterons, i.e., those obtained by subtracting consumed ones in the nucleus coalescence from formed ones via the p + n coalescence.Filled symbols with error bars are experimental data from the STAR collaboration [55].From Figure 2, one can see our results can well reproduce the available data in different rapidity intervals at midrapidity area from central to peripheral Au-Au collisions at √ s NN = 3 GeV.

TABLE III .
[55]d rapidity densities dN/dy of t and 3 He in different rapidity intervals and different centralities in Au-Au collisions at √ s NN = 3 GeV.Data are from Ref.[55], and the errors denote the systematical uncertainties.Theo nd Theo dep Theo fin Data Theo ppn Theo pd Theo dep Theo fin denotes the result of p + n coalescing into d.Theo nnp and Theo nd in the fourth and fifth columns in TableIIIdenote the result of n + n + p coalescing into t and that of n + d coalescing into t.Theo ppn and Theo pd in the ninth and tenth columns in Table III denote the result of p + p + n coalescing into 3 He and that of p + d coalescing into 3 He.Theo dep in the fifth column of Table II and in the sixth and eleventh columns of Table III denote the consumed d, t and 3 He in the nucleus coalescence process where they capture other particles to form objects with larger mass numbers.Theo fin in the sixth column of Table II and in the seventh and twelfth columns of Table III denote the final-state d, t and 3 He.From TablesII and III, one can see that our results Theo fin agree well with the experimental data in 0-10%, 10-20% and 20-40% centralities.But in the peripheral 40-80% centrality, our Theo fin of d underestimates the data and Theo fin of t and 3 He overestimates the data; our results only including nucleon coalescence Theo pn , Theo nnp and Theo ppn can describe the corresponding data much better.This further indicates that nucleon coalescence is the dominant production for light nuclei in peripheral 40 − 80% collisions, and other coalescence channels involving nucleon+nucleus and nucleus+nucleus may not occur.TableIVshows results of 4 He in different rapidity intervals and different centralities in Au-Au collisions at √ s NN = 3 GeV.Data with errors are from Ref. [55], and the errors denote the systematical uncertainties.Theo ppnn , Theo pnd , Theo pt , Theo n 3 He and Theo dd in the fifth, sixth, seventh, eighth and ninth columns denote the results of p + p + n + n, p + n + d, p + t, n+ 3 He and d + d coalescing into 4 He, respectively.Theo total in fourth column denote total results including all five coalescence sources for 4 He.Theo total in 0-10%, 10-20% and 20-40% centralities and Theo ppnn in the peripheral 40-80% centrality give about 20% ∼ 30% underestimations of the central values of the experimental data.
4 in Au-Au collisions at √ s NN = 3 GeV in centralities 0 − 10%, 10−20%, 20−40%, 40−80%, respectively.Table V and Table VI show the results.Data with errors are from Ref. [55], and the errors denote the systematical uncertainties.The ⟨p T ⟩ fin in the fourth, sixth and tenth columns in Table V denotes our theoretical results for final-state d, t, 3 He, respectively, and ⟨p T ⟩ total in the fourth column in Table VI denotes total results including all five coalescence sources for 4 He.⟨p T ⟩ nnp and ⟨p T ⟩ nd in the seventh and eighth columns in Table V denote the result of n + n + p coalescing into t and that of n + d coalescing into t.⟨p T ⟩ ppn and ⟨p T ⟩ pd in the eleventh and twelfth columns in Table V denote the result of p + p + n coalescing into 3 He and that of p + d coalescing into 3 He.⟨p T ⟩ ppnn , ⟨p T ⟩ pnd , ⟨p T ⟩ pt , ⟨p T ⟩ n 3 He and ⟨p T ⟩ dd in the fifth, sixth, seventh, eighth and ninth columns in Table VI denote the results of p + p + n + n, p + n + d, p + t, n+ 3 He and d + d coalescing into 4 He, respectively.Table V and Table

TABLE VI .
[55]aged transverse momenta ⟨p T ⟩ of4He in different rapidity intervals and different centralities in Au-Au collisions at √ s NN = 3 GeV.Data are from Ref.[55], and the errors denote the systematical uncertainties.⟩ total ⟨p T ⟩ ppnn ⟨p T ⟩ pnd ⟨p T ⟩ pt ⟨p T ⟩ n 3 He ⟨p T ⟩ dd