Charmonium transport in the high- Bmedium Jiaxing Zhao1 2and Baoyi Chen3 1Department of Physics Tsinghua University Beijing 100084 China_2

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Charmonium transport in the high-µBmedium

Jiaxing Zhao1, 2 and Baoyi Chen3, ∗

1Department of Physics, Tsinghua University, Beijing 100084, China

2SUBATECH, Universit´e de Nantes, IMT Atlantique,

IN2P3/CNRS, 4 rue Alfred Kastler, 44307 Nantes cedex 3, France

3Department of Physics, Tianjin University, Tianjin 300354, China

(Dated: May 23, 2023)

We employ the transport model coupled with hydrodynamic equations to study the charmonium

dissociation and regeneration in the quark-gluon plasma (QGP) with the large baryon chemical

potential in Au-Au collisions at the energies of √sNN = (39, 14.5, 7.7) GeV. The baryon chemical

potential µBis encoded in both Debye mass characterizing the heavy-quark potential and also the

equation of state (EoS) of the bulk medium respectively. After considering µB-corrections in both

heavy quarkonium and the QGP medium, we calculate the nuclear modiﬁcation factor RAA of

charmonium. And ﬁnd the µBinﬂuence on charmonium production at √sNN = 39 and 14.5 GeV

is negligible, while the RAA of charmonium is reduced considering µBinﬂuence at √sNN = 7.7

GeV Au-Au collisions. It is crucial for studying charmonium production in low energy and also

ﬁxed-target heavy-ion collisions.

I. INTRODUCTION

In relativistic heavy-ion collisions, a hot deconﬁned

medium consisting of quarks and gluons, called “Quark-

Gluon Plasma” (QGP) is believed to be created. Heavy

quarkonia which are produced in the initial parton hard

scatterings have been regarded as clean probes of the

QGP production [1]. Heavy quarkonium is the bound

state of the heavy quark and its antiquark forced by

an attractive potential. In the hot medium, this attrac-

tive potential is screened by thermal partons, which re-

sults in the dissociation of heavy quarkonium and sup-

pression of its production in relativistic heavy-ion col-

lisions [2]. The nuclear modiﬁcation factor RAA is an

observable proposed to characterize such suppression.

And various theoretical models have been developed to

study the heavy quarkonium evolution and suppression

in the hot medium, such as the statistic hadronization

model [3], coalescence hadronization model [4,5], trans-

port model [6–11], open quantum system [12–15], time-

dependent Schr¨odinger equation [16–18], and newly ex-

tended Remler equation [19]. The heavy quarkonia not

only can be used to probe the QGP properties but also be

widely investigated to probe the early state tilted energy

deposition and ﬂuctuation of heavy-ion collisions [20,21],

and strong electromagnetic and vorticity ﬁelds created in

non-central heavy-ion collisions [22–26].

At low energy collisions such as the Beam Energy Scan

(BES) program at RHIC [27], NA60+ at SPS [28], and

Compressed Baryonic Matter (CBM) at FAIR [29], the

temperatures of the created medium are much lower than

that in heavy-ion collisions at top RHIC and LHC en-

ergies. While the baryon chemical potential µBin the

central regions of low energy collisions enhanced dramat-

ically. It can be much larger than the temperature of

∗Electronic address: baoyi.chen@tju.edu.cn

the QGP medium, such as the µB∼420 MeV at Au-Au

collisions with √sNN = 7 GeV. How such large baryon

chemical potential aﬀects heavy quarkonia evolution and

production is an interesting and worth-studying ques-

tion. The previous studies based on Hard Thermal and

Dense Loop (HTL/HDL) theory show the Debye mass

and heavy quark potential are changed in the baryon-rich

medium [30–32]. Recently, based on the time-dependent

Schr¨odinger equation, the charmonium dissociation is

studied with baryon chemical potential corrected poten-

tial [33]. In this paper, we will establish a more realistic

model to study carefully the quarkonium dissociation and

production in a high baryon density QGP medium which

is created in Au-Au collisions at the energies of √sNN =

(39, 14.5, 7.7) GeV.

We ﬁrst employ the µB- and T-dependent heavy quark

potential to calculate the binding energies and aver-

aged radius of charmonium via the two-body Schr¨odinger

equation in Section II. The binding energies are used to

estimate the charmonium dissociation in high µB-QGP.

The evolution of the QGP is described by the hydrody-

namic model. µBcontribution is encoded in the equation

of state of hot medium. While the evolution of charmo-

nium is controlled by the Boltzmann equation. Including

µBcontributions in both heavy quarkonium and the bulk

medium, we present the charmonium yield and suppres-

sion in section III. A summary is given in section IV.

II. THEORETICAL FRAMEWORK

A. Heavy quark properties with high baryon

density

In a vacuum, the mass spectrum of heavy quarko-

nium is well described with the Cornell potential, V(r) =

−α/r+σr. At ﬁnite temperatures, heavy quark potential

is screened. The hot medium eﬀects can be absorbed in

the heavy quark potential via the Debye mass mD[34].

arXiv:2210.04661v2 [nucl-th] 21 May 2023

The in-medium potential can be parametrized as [30],

V(r, T ) = −αmD+e−mDr

r

+σ

mD2−(2 + mDr)e−mDr,(1)

where the parameter α= 0.4105 and the string strength

σ= 0.2 (GeV)2are ﬁxed with the charmonium masses

in vacuum [26]. At zero baryon chemical potential, the

Debye mass can be extracted by ﬁtting the lattice QCD

data [30],

mD(T) = g(Λ)TrNc

3+Nf

+NcT g2(Λ)

4πlog 1

g(Λ)rNc

3+Nf

+κ1T g2(Λ) + κ2T g3(Λ) + κ3T g5(Λ),(2)

where g(Λ) is the coupling constant depending on the

renormalization scale, which can be chosen as Λ = 2πT .

In this paper, we utilize the four-loop result given in [35]

with ΛQCD=0.2 GeV. The factors of color and ﬂavor are

taken as Nc=Nf= 3. At higher orders, the parameters

κ1=0.6, κ1=-0.23, and κ3=-0.007.

In the high baryon density and hot medium (T, µB

mqwith mqis light quark mass), the leading or-

der HTL/HDL calculations give the Debye screening

mass [31,32],

D(T, µB) = g2T2Nc

3+Nf

6+g2X

µ2

18π2,(3)

where µBis baryon chemical potential. Due to the lack

of lattice data on heavy quark potential at high baryon

density regions, we following the study [30] to consider

the baryon chemical potential through

D(T, µB) = m2

D(T) + g2Nf

µ2

18π2,(4)

where mD(T) is given by lattice results as shown in

Eq.(2). Besides, the renormalization scale in the cou-

pling constant is also modiﬁed to Λ = 2πpT2+µ2

B/π2.

The temperature-scaled Debye mass is plotted in Fig. 1

by taking diﬀerent values of µB. We can see the Debye

mass changes a lot considering the µB, especially at low

temperature regions. Taking mD(T, µB) into Eq.(1), we

obtain the baryon chemical potential related to heavy

quark potential.

With (T,µB)-dependent heavy quark potential, one

can calculate the in-medium binding energies of char-

monium. As the charm quark mass is relatively large

compared with the inner motion of charm quarks in the

bound state, we neglect the relativistic eﬀect and em-

ploy the Schr¨odinger equation to calculate their binding

energies via ψ(T, µB) = E−V(r=∞, T, µB). The po-

tential Eq.(1) is a central potential, the radial part of the

     















[]

��/�

●Lattice QCD

w/oμB

μB=0.5GeV

μB=0.3GeV

μB=0.1GeV

FIG. 1: The temperature-scaled Debye mass as a function

of temperature. Diﬀerent values of µBare taken to compare

with the data which is from lattice QCD results and without

the µB[30].

Schr¨odinger equation is separated to be,

1

2µ−d2

d2r−2

dr +l(l+ 1)

r2+V(r, T, µB)Rnl(r)

=ERnl(r),(5)

where Rnl(r) is the radial wave function labeled with the

radial and angular quantum number (n, l). r=|r2−r1|

is the distance between charm and anti-charm quarks lo-

cated at the positions of r1and r2. The reduced mass is

µ=mc/2 with charm mass mc= 1.5 GeV. Here we con-

sider three charmonium states (J/ψ, χc, ψ0). Their bind-

ing energies at diﬀerent µBand Tare plotted in Fig. 2.

As one can see, the binding energies of charmonium are

reduced considering the baryon chemical potential µB.

In the meantime, charmonia will suﬀer dynamic dis-

sociation in the QGP, such as gluo-dissociation and in-

elastic scattering with thermal partons. For the deeply

bounded charmonium state, the gluo-dissociation plays

an important role, which cross section can be obtained by

the operator-production-expansion (OPE) method [36,

37],

σ1s(ω) = A0

(x−1)3/2

x5,

σχc(ω)=4A0

(x−1)1p(9x2−20x+ 12)

x7,

σ2s(ω) = 16A0

(x−1)3/2(x−3)2

x7,(6)

where ωis the gluon energy. x≡ω/ψis the ratio of

gluon energy and the binding energy ψof charmonium

states. The coeﬃcient A0= 211π/(27pm3

cψ) is a con-

stant factor. At ﬁnite temperatures and baryon chemi-

cal potentials, heavy quark potential becomes weaker for

the charmonium states with larger geometry sizes. We

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Charmoniumtransportinthehigh-BmediumJiaxingZhao1,2andBaoyiChen3,1DepartmentofPhysics,TsinghuaUniversity,Beijing100084,China2SUBATECH,UniversitedeNantes,IMTAtlantique,IN2P3/CNRS,4rueAlfredKastler,44307Nantescedex3,France3DepartmentofPhysics,TianjinUniversity,Tianjin300354,China(Dated:May23,2023)We...

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Charmonium transport in the high- Bmedium Jiaxing Zhao1 2and Baoyi Chen3 1Department of Physics Tsinghua University Beijing 100084 China_2

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