Different coalescence sources of light nuclei production in Au-Au collisions atsNN3GeV Rui-Qin Wang1Ji-Peng L u1Yan-Hao Li1Jun Song2and Feng-Lan Shao1 1School of Physics and Physical Engineering Qufu Normal University Shandong 273165 China_2

2025-05-06 0 0 509.94KB 16 页 10玖币

侵权投诉

Diﬀerent coalescence sources of light nuclei production in Au-Au collisions at √sNN =3GeV

Rui-Qin Wang,1Ji-Peng L¨

u,1Yan-Hao Li,1Jun Song,2and Feng-Lan Shao1, ∗

1School of Physics and Physical Engineering, Qufu Normal University, Shandong 273165, China

2School of Physical Science and Intelligent Engineering, Jining University, Shandong 273155, China

We study the production of light nuclei in the coalescence mechanism in Au-Au collisions at midrapidity

at √sNN =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 spec-

tra of the deuteron (d), triton (t), helium-3 (3He) and helium-4 (4He). We reproduce the data of yield rapidity

densities and averaged transverse momenta of d,t,3He and 4He. We give proportions of contributions from

diﬀerent coalescence sources for t,3He and 4He in their productions. We ﬁnd that besides nucleon coales-

cence, nucleon+nucleus coalescence and nucleus+nucleus coalescence may play requisite roles in light nuclei

production in Au-Au collisions at √sNN =3 GeV.

PACS numbers: 25.75.-q, 25.75.Dw, 27.10.+h

I. INTRODUCTION

As a speciﬁc group of observables in relativistic heavy

ion collisions [1–12], light nuclei such as the deuteron (d),

triton (t), helium-3 (3He) and helium-4 (4He) have always

been under active investigation in recent decades both in ex-

periment [13–23] and in theory [24–29]. The STAR experi-

ment at the BNL Relativistic Heavy Ion Collider (RHIC) and

the ALICE experiment at the CERN Large Hadron Collider

(LHC) have collected a wealth of data on light nuclei pro-

duction. These data exhibit some fascinating features, espe-

cially their non-trivial energy-dependent behaviors in a wide

collision energy range from GeV to TeV magnitude [17–

23]. Theoretical studies have also made signiﬁcant progress.

Two production mechanisms, the thermal production mech-

anism [29–33] and the coalescence mechanism [26,27,34–

42], have proved to be successful in describing light nuclei

formation. In addition, transport scenario [43–48] is em-

ployed to study how light nuclei evolve and survive during

the hadronic system evolution.

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 coales-

cence [18,19,22,23,49–51]. Recently the STAR collabo-

ration 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 √sNN =3 GeV [52–55].

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 interac-

tions [53]. At this low collision energy besides nucleons,

light nuclei in particular of light d,tand 3He have been more

abundantly created [55] compared to higher collision ener-

gies [56]. It is easier in physics for these light nuclei to cap-

ture nucleons or other light nuclei to form heavier composite

objects. In fact clear depletions below unity of proton−dand

∗shaoﬂ@mail.sdu.edu.cn

d−dcorrelation functions measured at such low collision

energy indicate the strong ﬁnal state interaction and further

support the possible coalescence of the dwith the nucleon or

other d[57]. How much space is there on earth for other par-

ticle 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 depen-

dence of yields and coalescence factors of diﬀerent light nu-

clei at high RHIC and LHC energies [51,58], to include

nucleon+nucleus coalescence and nucleus+nucleus coales-

cence besides nucleon coalescence. We apply the extended

coalescence model to hadronic systems created in Au-Au

collisions at midrapidity area at √sNN =3 GeV to study the

momentum and centrality dependence of light nuclei pro-

duction in the low- and intermediate-pTregions. We com-

pute the transverse momentum (pT) spectra, the yield rapid-

ity densities (dN/dy) and the averaged transverse momenta

(⟨pT⟩) of d,t,3He and 4He from central to peripheral col-

lisions. We give proportions of contributions from diﬀerent

coalescence sources for t,3He and 4He respectively in their

productions. Our studies show that in 0 −10%, 10 −20%

and 20 −40% centralities, besides nucleon coalescence,

nucleon+dcoalescence plays an important role in tand 3He

production and nucleon+d(t,3He) coalescence as well as

d+dcoalescence occupy signiﬁcant proportions in 4He pro-

duction. But in the peripheral 40 −80% centrality, nucleon

coalescence plays a dominant role, and nucleon+nucleus co-

alescence or nucleus+nucleus coalescence seems to disap-

pear.

The rest of the paper is organized as follows. In Sec. II, we

introduce the coalescence model. We present analytic for-

mulas of momentum distributions of two bodies, three bod-

ies, and four nucleons coalescing into light nuclei, respec-

tively. In Sec. III, we apply the model to Au-Au collisions in

diﬀerent rapidity intervals at midrapidity area at √sN N =3

GeV to study momentum and centrality dependence of the

production of various species of light nuclei in the low- and

intermediate-pTregions. We give proportions of contribu-

tions from diﬀerent coalescence sources for t,3He and 4He

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 start-

ing 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 pri-

mordial nucleons are allowed to form d,t,3He and 4He via

the nucleon coalescence. Then in the second step the formed

d,tand 3He 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,3He and 4He

are included, and those light nuclei with mass number larger

than 4 are abandoned.

In the following we present the deduction of the formal-

ism 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→3He, p+t→4He, n+3He →4He and

d+d→4He. 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→3He and

p+n+d→4He. Finally, we give the analytic result of four

nucleons coalescing into 4He, i.e., p+p+n+n→4He.

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 sup-

pose light nuclei Ljare formed via the coalescence of two

hadronic bodies h1and h2. The three-dimensional momen-

tum distribution of the produced light nuclei fLj(p) is given

fLj(p)=Zdx1dx2dp1dp2fh1h2(x1,x2;p1,p2)

× RLj(x1,x2;p1,p2,p),(1)

where fh1h2(x1,x2;p1,p2) is two-hadron joint coordinate-

momentum distribution; RLj(x1,x2;p1,p2,p) is the kernel

function. 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

fLj(p)=Nh1h2Zdx1dx2dp1dp2f(n)

h1h2(x1,x2;p1,p2)

× RLj(x1,x2;p1,p2,p).(2)

Nh1h2is the number of all possible h1h2-pairs, and it is equal

to Nh1Nh2and Nh1(Nh1−1) for h1,h2and h1=h2, respec-

tively. Nhi(i=1,2) is the number of the hadrons hiin the

considered hadronic system.

The kernel function RLj(x1,x2;p1,p2,p) denotes the

probability density for h1,h2with momenta p1and p2at

x1and x2to recombine into a Ljof momentum p. It carries

the kinetic and dynamical information of h1and h2recom-

bining into light nuclei, and its precise expression should

be constrained by such as the momentum conservation, con-

straints due to intrinsic quantum numbers e.g. spin, and so

on [51,58,59]. To take these constraints into account explic-

itly, we rewrite the kernel function in the following form

RLj(x1,x2;p1,p2,p)=gLjR(x,p)

Lj(x1,x2;p1,p2)

×δ(

i=1

pi−p),(3)

where the spin degeneracy factor gLj=(2JLj+1)/[

i=1

(2Jhi+

1)]. JLjis the spin of the produced Ljand Jhiis that of the

primordial hadron hi. The Dirac δfunction guarantees the

momentum conservation in the coalescence. The remaining

R(x,p)

Lj(x1,x2;p1,p2) can be solved from the Wigner transfor-

mation once the wave function of Ljis given with the instan-

taneous coalescence approximation. It is as follows

R(x,p)

Lj(x1,x2;p1,p2)=8e−(x′

1−x′

2)2

2σ2e−2σ2(m2p′

1−m1p′

2)2

(m1+m2)2ℏ2c2,(4)

as we adopt the wave function of a spherical harmonic oscil-

lator as in Refs. [60,61]. The superscript ‘′’ in the coordi-

nate or momentum variable denotes the hadronic coordinate

or momentum in the rest frame of the h1h2-pair. m1and m2

are the rest mass of hadron h1and that of hadron h2. The

width parameter σ=q2(m1+m2)2

3(m2

1+m2

2)RLj, where RLjis the root-

mean-square radius of Ljand its values for diﬀerent light

nuclei can be found in Ref. [62]. The factor ℏccomes from

the used GeV·fm unit, and it is 0.197 GeV·fm.

The normalized two-hadron joint distribution

f(n)

h1h2(x1,x2;p1,p2) is generally coordinate and momentum

coupled, especially in central heavy-ion collisions with rela-

tively 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 ana-

lytically and present centrality and momentum dependence

of light nuclei more intuitively in Au-Au collisions at low

RHIC energy √sNN =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.,

f(n)

h1h2(x1,x2;p1,p2)=f(n)

h1h2(x1,x2)f(n)

h1h2(p1,p2).(5)

Substituting Equations (3-5) into Equation (2), we have

fLj(p)=Nh1h2gLjZdx1dx2f(n)

h1h2(x1,x2)8e−(x′

1−x′

2)2

2σ2

×Zdp1dp2f(n)

h1h2(p1,p2)e−2σ2(m2p′

1−m1p′

2)2

(m1+m2)2ℏ2c2δ(

i=1

pi−p)

=Nh1h2gLjALjMLj(p),(6)

where we use ALjto denote the coordinate integral part in

Equation (6) as

ALj=8Zdx1dx2f(n)

h1h2(x1,x2)e−(x′

1−x′

2)2

2σ2,(7)

and use MLj(p) to denote the momentum integral part as

MLj(p)=Zdp1dp2f(n)

h1h2(p1,p2)e−2σ2(m2p′

1−m1p′

2)2

(m1+m2)2ℏ2c2δ(

i=1

pi−p).

(8)

ALjstands for the probability of a h1h2-pair satisfying the

coordinate requirement to recombine into Lj, and MLj(p)

stands for the probability density of a h1h2-pair satisfying

the momentum requirement to recombine into Ljwith mo-

mentum p.

Changing integral variables in Equation (7) to be X=

x1+x2

√2and r=x1−x2

√2, we have

ALj=8ZdXdrf(n)

h1h2(X,r)e−r′2

σ2,(9)

and the normalizing condition

Zf(n)

h1h2(X,r)dXdr=1.(10)

We further assume the coordinate joint distribution is coor-

dinate variable factorized, i.e., f(n)

h1h2(X,r)=f(n)

h1h2(X)f(n)

h1h2(r).

Adopting f(n)

h1h2(r)=1

(πCwR2

f)3/2e−r2

CwR2

fas in Refs. [51,63], we

have

ALj=8

(πCwR2

f)3/2Zdre−r2

CwR2

fe−r′2

σ2.(11)

Here Rfis the eﬀective radius of the hadronic system at the

light nuclei freeze-out. Cwis 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 h1h2-pair, i.e., ∆t′=0, we get

r=r′+(γ−1) r′·β

β2β,(12)

where βis the three-dimensional velocity vector of the

center-of-mass frame of h1h2-pair in the laboratory frame

and the Lorentz contraction factor γ=1/p1−β2. Substi-

tuting Equation (12) into Equation (11) and integrating from

the relative coordinate variable, we can obtain

ALj=8σ3

(CwR2

f+σ2)pCw(Rf/γ)2+σ2.(13)

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

e−(p′

1−m1

m2p′

2)2

(1+m1

m2)2ℏ2c2

2σ2≈"ℏc

σ(1 +m1

)rπ

2#3

δ(p′

1−m1

p′

2).(14)

After integrating p1and p2from Equation (8) we can obtain

MLj(p)=(ℏc√π

√2σ)3γf(n)

h1h2(m1p

m1+m2

,m2p

m1+m2

),(15)

where γcomes from p′

1−m1

m2p′

2=1

γ(p1−m1

m2p2).

Substituting Equations (13) and (15) into Equation (6) and

ignoring correlations between h1and h2hadrons, we have

fLj(p)=(√2πℏc)3gLjγ

(CwR2

f+σ2)pCw(Rf/γ)2+σ2fh1(m1p

m1+m2

)

×fh2(m2p

m1+m2

).(16)

Denoting the Lorentz invariant momentum distribution

d2N

2πpTdpTdy with f(inv), we ﬁnally have

f(inv)

Lj(pT,y)=(√2πℏc)3gLj

(CwR2

f+σ2)pCw(Rf/γ)2+σ2

m1+m2

m1m2

×f(inv)

h1(m1pT

m1+m2

,y)f(inv)

h2(m2pT

m1+m2

,y),(17)

where yis the rapidity.

B. Formalism of three bodies coalescing into light nuclei

For light nuclei Ljformed via the coalescence of three

hadronic bodies h1,h2and h3, the three-dimensional mo-

mentum distribution fLj(p) is

fLj(p)=Nh1h2h3Zdx1dx2dx3dp1dp2dp3f(n)

h1h2h3(x1,x2,x3;

p1,p2,p3)RLj(x1,x2,x3;p1,p2,p3,p).(18)

Nh1h2h3is the number of all possible h1h2h3-clusters and it is

equal to Nh1Nh2Nh3,Nh1(Nh1−1)Nh3,Nh1(Nh1−1)(Nh1−2)

for h1,h2,h3,h1=h2,h3,h1=h2=h3,

respectively. f(n)

h1h2h3is the normalized three-hadron joint

coordinate-momentum distribution. RLjis the kernel func-

tion.

We rewrite the kernel function as

RLj(x1,x2,x3;p1,p2,p3,p)=gLjR(x,p)

Lj(x1,x2,x3;p1,p2,p3)

×δ(

i=1

pi−p).(19)

The spin degeneracy factor gLj=(2JLj+1)/[

i=1

(2Jhi+1)].

The Dirac δfunction guarantees the momentum conserva-

tion. R(x,p)

Lj(x1,x2,x3;p1,p2,p3) solving from the Wigner

transformation [60,61] is

R(x,p)

Lj(x1,x2,x3;p1,p2,p3)=82e−(x′

1−x′

2)2

2σ2

1e−2( m1x′

m1+m2+m2x′

m1+m2−x′

3)2

3σ2

×e−2σ2

1(m2p′

1−m1p′

2)2

(m1+m2)2ℏ2c2e−3σ2

2[m3p′

1+m3p′

2−(m1+m2)p′

3]2

2(m1+m2+m3)2ℏ2c2.(20)

The superscript ‘′’ denotes the hadronic coordinate or mo-

mentum in the rest frame of the h1h2h3-cluster. The width

parameter σ1=qm3(m1+m2)(m1+m2+m3)

m1m2(m1+m2)+m2m3(m2+m3)+m3m1(m3+m1)RLj,

and σ2=q4m1m2(m1+m2+m3)2

3(m1+m2)[m1m2(m1+m2)+m2m3(m2+m3)+m3m1(m3+m1)] RLj.

With the coordinate and momentum factorization assump-

tion of the joint distribution, we have

fLj(p)=Nh1h2h3gLjALjMLj(p).(21)

Here we also use ALjto denote the coordinate integral part

ALj=82Zdx1dx2dx3f(n)

h1h2h3(x1,x2,x3)e−(x′

1−x′

2)2

2σ2

×e−2( m1x′

m1+m2+m2x′

m1+m2−x′

3)2

3σ2

2,(22)

and use MLj(p) to denote the momentum integral part as

MLj(p)=Zdp1dp2dp3f(n)

h1h2h3(p1,p2,p3)δ(

i=1

pi−p)

×e−2σ2

1(m2p′

1−m1p′

2)2

(m1+m2)2ℏ2c2e−3σ2

2[m3p′

1+m3p′

2−(m1+m2)p′

3]2

2(m1+m2+m3)2ℏ2c2.

(23)

We change integral variables in Equation (22) to be Y=

(m1x1+m2x2+m3x3)/(m1+m2+m3), r1=(x1−x2)/√2

and r2=q2

3(m1x1

m1+m2+m2x2

m1+m2−x3), and further assume the co-

ordinate joint distribution is coordinate variable factorized,

i.e., 33/2f(n)

h1h2h3(Y,r1,r2)=f(n)

h1h2h3(Y)f(n)

h1h2h3(r1)f(n)

h1h2h3(r2).

Adopting f(n)

h1h2h3(r1)=1

(πC1R2

f)3/2e−r2

C1R2

fand f(n)

h1h2h3(r2)=

(πC2R2

f)3/2e−r2

C2R2

fas in Refs. [51,63], we have

ALj=821

(πC1R2

f)3/2Zdr1e−r2

C1R2

fe−(r′

1)2

σ2

×1

(πC2R2

f)3/2Zdr2e−r2

C2R2

fe−(r′

2)2

σ2

2.(24)

Comparing relations of r1,r2with x1,x2,x3to that of rwith

x1,x2in Sec. II A, we see that C1is equal to Cwand C2is

4Cw/3 when ignoring the mass diﬀerence of m1and m2[51,

63]. Considering the Lorentz transformation and integrating

from the relative coordinate variables in Equation (24), we

obtain

ALj=82σ3

1σ3

(C1R2

f+σ2

1)qC1(Rf/γ)2+σ2

×1

(C2R2

f+σ2

2)qC2(Rf/γ)2+σ2

.(25)

Approximating the gaussian form of the momentum-

dependent kernel function to be δfunction form and inte-

grating p1,p2and p3from Equation (23), we can obtain

MLj(p)= πℏ2c2

√3σ1σ2!3

γ2×

f(n)

h1h2h3(m1p

m1+m2+m3

,m2p

m1+m2+m3

,m3p

m1+m2+m3

).(26)

Substituting Equations (25) and (26) into Equation (21)

and ignoring correlations between h1,h2and h3hadrons, we

have

fLj(p)=64π3ℏ6c6gLjγ2

3√3(C1R2

f+σ2

1)qC1(Rf/γ)2+σ2

×1

(C2R2

f+σ2

2)qC2(Rf/γ)2+σ2

fh1(m1p

m1+m2+m3

)

×fh2(m2p

m1+m2+m3

)fh3(m3p

m1+m2+m3

).(27)

Finally we have the Lorentz invariant momentum distribu-

tion

f(inv)

Lj(pT,y)=64π3ℏ6c6gLj

3√3(C1R2

f+σ2

1)qC1(Rf/γ)2+σ2

×1

(C2R2

f+σ2

2)qC2(Rf/γ)2+σ2

m1+m2+m3

m1m2m3

×f(inv)

h1(m1pT

m1+m2+m3

,y)f(inv)

h2(m2pT

m1+m2+m3

,y)

×f(inv)

h3(m3pT

m1+m2+m3

,y).(28)

C. Formalism of four nucleons coalescing into 4He

For 4He formed via the coalescence of four nucleons, the

three-dimensional momentum distribution is

f4He(p)=Nppnn Zdx1dx2dx3dx4dp1dp2dp3dp4

×f(n)

ppnn(x1,x2,x3,x4;p1,p2,p3,p4)

× R4He(x1,x2,x3,x4;p1,p2,p3,p4,p),(29)

where Nppnn =Np(Np−1)Nn(Nn−1) is the number of all

possible ppnn-clusters; f(n)

ppnn is the normalized four-nucleon

joint coordinate-momentum distribution; R4He is the kernel

function.

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DifferentcoalescencesourcesoflightnucleiproductioninAu-Aucollisionsat√sNN=3GeVRui-QinWang,1Ji-PengL¨u,1Yan-HaoLi,1JunSong,2andFeng-LanShao1,∗1SchoolofPhysicsandPhysicalEngineering,QufuNormalUniversity,Shandong273165,China2SchoolofPhysicalScienceandIntelligentEngineering,JiningUniversity,Shandong2731...

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Different coalescence sources of light nuclei production in Au-Au collisions atsNN3GeV Rui-Qin Wang1Ji-Peng L u1Yan-Hao Li1Jun Song2and Feng-Lan Shao1 1School of Physics and Physical Engineering Qufu Normal University Shandong 273165 China_2

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