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Systematic studies of charmonium-, bottomonium-,

and B

-like tetraquark states

Jing Wu

†

School of Science, Shandong Jianzhu University, Jinan 250101, China

Xiang Liu

‡

School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China

and Research Center for Hadron and CSR Physics, Lanzhou University

and Institute of Modern Physics of CAS, Lanzhou 730000, China

Yan-Rui Liu

School of Physics, Shandong University, Jinan 250100, China

Shi-Lin Zhu

School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,

Beijing 100871, China

and Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

and Center of High Energy Physics, Peking University, Beijing 100871, China

(Received 20 October 2018; published 28 January 2019)

We study the mass splittings of Q

(Q ¼ c, b, q ¼ u, d, s) tetraquark states with chromo-

magnetic interactions between their quark components. Assuming that Xð4140Þ is the lowest J

þþ

s tetraquark, we estimate the masses of the other tetraquark states. From the obtained masses and

defined measure reflecting effective quark interactions, we find the following assignments for several

exotic states: (1) both Xð3860Þ and the newly observed Z

ð4100Þ seem to be 0

þþ

n tetraquarks;

(2) Z

ð4200Þ is probably a 1

þ−

n tetraquark; (3) Z

ð3900Þ, Xð3940Þ, and Xð4160Þ are unlikely

compact tetraquarks; (4) Z

ð4020Þ is unlikely a compact tetraquark, but seems the hidden-charm

correspondence of Z

ð10650Þ with J

¼ 1

þ−

; and (5) Z

ð4250Þ can be a tetraquark candidate but

the quantum numbers cannot be assigned at present. We hope further studies may check the predictions and

assignments given here.

DOI: 10.1103/PhysRevD.99.014037

I. INTRODUCTION

A hot topic in hadron physics study is to identify multi-

quark states from the observed exotic structures. Through

explorations on their masses, productions, and decay proper-

ties, we may understand the problem how the strong

interaction forces nonobservable quarks and gluons to

form observable hadrons. Before 2003, the situation in

understanding hadron structures was simple because the

quark model gave a successful and satisfactory description

for hadron spectra [1], although there exist a few hadrons

difficult to understand. In 2003, experimentalists opened the

Pandora’s box for exotic states through the observation of

Xð3872Þ [2]. Since then, more and more unexpected XYZ

states were observed and the situation for hadron physics

study became complicated [3–12]. To understand a little more

of the above mentioned problem, the discussions in this work

aim at basic features of ground charmonium-like, bottomo-

nium-like, and B

-like tetraquark states with ev en P-parities.

As the first exotic charmonium-like state above the D

threshold, the Xð3872Þ motivated heated discussions on its

nature [5,10]. Its J

are determined to be 1

þþ

but the mass

is tens of MeV lower than the quark model prediction if it is

a charmonium. Since the meson is extremely close to the

0

threshold, it is widely regarded as a loosely bound



molecule. Discussions in the tetraquark picture and

Corresponding author.

yrliu@sdu.edu.cn

†

wujing18@sdjzu.edu.cn

‡

xiangliu@lzu.edu.cn

zhusl@pku.edu.cn

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW D 99, 014037 (2019)

2470-0010=2019=99(1)=014037(22) 014037-1 Published by the American Physical Society

hybrid picture are also performed. However, without a c

core, it is difficult to understand the measured ratios

BðXð3872Þ → Ψ ð2SÞγÞ∶BðXð3872Þ → J=ΨγÞ¼2.46 

0.64  0.29 by LHCb [13]. The Xð3872Þ seems to be a

charmonium affected significantly by the D



threshold

[14,15]. Anyway, one cannot identify this exotic meson as a

pure tetraquark state.

To identify multiquark states, we may look for structures

according to several ideas. The easiest approach is to study

structures with explicitly exotic quantum numbers, e.g.,

charged charmonium-like or bottomonium-like states. The

quark content of the charged hidden-charm or hidden-

bottom mesons should be at least four if their nonresonance

interpretations are excluded. Up to now, experiments have

observed several charged structures, Z

ð4430Þ [16–18],

ð4050Þ [19], Z

ð4250Þ [19], Z

ð3900Þ [20–24],

ð3885Þ [25–27], Z

ð4020Þ [28,29], Z

ð4025Þ [30,31],

and so on. Very recently, LHCb found the evidence for a

charged charmonium-like resonance Z

−

ð4100Þin the decay

→ Z

−

ð4100ÞK

→ η

−

[32]. The measured mass

and width are 4096  20

þ18

−22

MeV and 152  58

þ60

−35

MeV,

respectively. Its possible quantum numbers are J

¼ 0

or 1

−

. They are certainly four-quark state candidates.

However, it is not easy to justify whether they are compact

tetraquarks or meson-meson molecules. In this paper, we

will try to understand whether parts of these charged states

are compact tetraquarks or just molecules.

It is also possible to identify a multiquark state from

its high mass that a conventional hadron cannot have.

The observed P

ð4380Þ and P

ð4450Þ by the LHC

Collaboration [33] are two such states. They look like

excited nucleons but can be identified as pentaquark states

because an orbital or radial excitation energy larger than

3 GeV for light quarks is an unnatural interpr etation for

the high masses while the creation of a c

c pair can

naturally explain. Ref. [34] predicted the existence of

hidden-charm p entaquarks with this idea. Similarly, one

may identify other high mass states looking like conven-

tional hadrons as multiquark states if experiments could

observe them. However, one still cannot easily distinguish

compact tetraquarks from molecules except the QQ

case [35–37] in this possibility.

If experiments could observe an exotic structure that the

molecule picture is not applicable, it is possible to identify

it as a compact tetraquark. In Refs. [38,39], the D0

Collaboration claimed an exotic B



state and named it

Xð5568Þ. This meson contains four different flavors. From

its low mass (∼200 MeV lower than the B

K threshold), the

Xð5568Þ is unlikely a molecule. If it really exists, it might

be a compact tetraquark. Unfortunately, the LHCb [40],

CMS [41], CDF [42], and ATLAS [43] Collaborations did

not confirm this state. The identification of compact

tetraquarks along this idea has not been achieved yet.

We have one more possibility to identify compact

multiquarks through number of states. The exotic structure

Xð4140Þ was first observed by the CDF Collaboration [44]

in the invariant mass distribution of J=ψϕ. In the latter

measurements with the same channel by various collabo-

rations [45–49], LHCb confirmed the Xð4140Þ, determined

its quantum numbers to be J

¼ 1

þþ

, established another

þþ

state Xð4274Þ, and observed two more 0

þþ

structures

Xð4500Þ and Xð4700Þ. The existence of two 1

þþ

states

does not support the molecule interpretations for them [49].

On the other hand, the cs

s tetraquark configuration can

account for such an observation [50]. This picture also

favors the assignment for the Belle Xð4350Þ [51] as their

þþ

tetraquark partner [52]. In this paper, we identify the

Xð4140Þ as the lowest 1

þþ

s tetraquark state and use its

mass as an input to estimate the masses of other charmo-

nium-, bottomonium-, and B

-like tetraquark states.

This paper is organized as follows. After the introductory

Sec. I, we present the theoretical formalism in Sec. II by

showing necessary wave functions and Hamiltonian matri-

ces. In Sec. III, we determine model parameters, present

strategy for the estimation of tetraquark masses, list

numerical results, analyze possible assignments for the

observed exotic mesons, and predict possible tetraquarks.

The last section is for discussions and summary.

II. FORMALISM

In this article, we use the notation Q

(Q ¼ c, b;

q ¼ n, s; n ¼ u, d) to generally denote the considered

system. If the system is truly neutral, Q

¼ Q

¼ Q, q

¼ q and the notation becomes Qq

q. From the SUð3Þ

symmetry, the tetraquarks belong to 8

and 1

representa-

tions. Since the flavor symmetry is broken, the isoscalar

states would mix with some angle. In principle, the

resulting flavor wave functions of the physical I ¼ 0 states

contain both Q

n and Q

s parts. At present, we

just consider the ideal mixing case, i.e., Q

n and

s do not mix.

The effective Hamiltonian in the adopted chromomag-

netic interaction (CMI) model reads,

H ¼

þ H

−

i<j

· σ

; ð1Þ

where

¼ λ

ð−λ



Þ for quarks (antiquarks). The involved

parameters are only effective coupling constants C

and effective masses m

containing various effects. This

Hamiltonian is reduced from a realistic model, which can

be found in Refs. [53,54]. Then the formula for the mass

estimation is

M ¼

i¼1

þhH

i: ð2Þ

In calculating the last term, we use the diquark-antidiquark

bases to express the wave functions for the S-wave

WU, LIU, LIU, and ZHU PHYS. REV. D 99, 014037 (2019)

014037-2

systems whose P-parities are always positive.

Here, the notation “diquark” just means two quarks and

does not mean a compact substructure. If one uses the

meson-meson bases, the same eigenvalues after diagonal-

ization will be obtained. In the present case, the Pauli

principle has no restriction on the total wave functions, but

one should notice the possible C-parity once a state is truly

neutral. The involved color (spin) wave functions ϕ

1;2

(χ

1;2;…;6

) are

¼j

; 3

; 1

i; ϕ

¼j6

;

; 1

¼j1

; 1

; 2

i; χ

¼j1

; 1

i; χ

¼j1

; 1

; 0

¼j1

; 0

; 1

i; χ

¼j0

; 1

i; χ

¼j0

; 0

ð3Þ

where the color representations (spins) in order in ϕ

(χ

)

are for diquark, antidiquark, and system, respectively. We

define the total wave function as

≡ ðQ

Þ ⊗ ϕ

⊗ χ

: ð4Þ

Compared with the cs

s case where a C-parity can be

given, the CMI matrices in the present cases are the

generalized ones in Ref. [52]. Now we have

¼2



ð2τ þ αÞ 2

ﬃﬃﬃ

ð5α − 2τÞ



; ð5Þ

¼0

ðτ − αÞ 4

ﬃﬃﬃ

ν −

ﬃﬃ

ν 2

ﬃﬃﬃ

−

ðτ þ5αÞ 2

ﬃﬃﬃ

α −

ﬃﬃ

−8τ 0

4τ

;

ð6Þ

and

¼1

ð2τ − αÞ

ﬃﬃ

β −

ﬃﬃ

μ 2

ﬃﬃﬃ

ν −4μ 4β

ð2θ − τÞ

ν −4μ 0 −2

ﬃﬃﬃ

−

ðτ þ2θÞ 4β −2

ﬃﬃﬃ

α 0

−

ð2τ þ 5αÞ

ﬃﬃ

β −

ﬃﬃ

ðτ − 2θÞ

ðτ þ2θÞ

; ð7Þ

where the defined variables are

τ ¼ C

þ C

; θ ¼ C

− C

;

α ¼ C

þ C

;

β ¼ C

− C

þ C

;

μ ¼ C

− C

þ C

− C

;

ν ¼ C

þ C

− C

ð8Þ

and the corresponding base vectors for the matrices are ðϕ

; ϕ

, ðϕ

; ϕ

, and ðϕ

; ϕ

;

; ϕ

, respectively. When the considered state is truly neutral, the matrices for the cases J

¼ 2

þþ

and

þþ

are the same as above, but that for the case J

¼ 1

þþ

i¼

−

ð4C

− C

þ 2C

Þ −2

ﬃﬃﬃ

ðC

þ C

þ 2C

ð4C

þ 5C

− 10C

ð9Þ

and that for the case J

¼ 1

þ−

SYSTEMATIC STUDIES OF CHARMONIUM-, … PHYS. REV. D 99, 014037 (2019)

014037-3

i¼



− 2C

−C

− C



ﬃﬃﬃ



þ C

q¯q

−2C

Q¯q



ðC

− C

q¯q

Þ −4

ﬃﬃﬃ

ðC

− C

q¯q

−



þ 10C

þ5C

þ 5C



−4

ﬃﬃﬃ

ðC

− C

ðC

− C

−



− 2C

þC

þ C



ﬃﬃﬃ



þ C

q¯q

þ2C

Q¯q





þ 10C

−5C

− 5C



: ð10Þ

Their corresponding base vectors are ðϕ

; ϕ

and

ðϕ

; ϕ

, respectively. Here ϕ

(ϕ

)

represents C-even (C-odd) wave function. The forms of

such wave functions are similar to those obtained in

Ref. [52]. Since we also consider the color structure

;

; 1

i for the tetraquarks, the above Eqs. (5)–(10)

can be actually thought of as generalizations of those for

; 3

; 1

i tetraquarks given in Ref. [55].

III. NUMERICAL ANALYSIS

A. Parameters and estimation strategy

The parameters in the CMI model are effective masses of

the quarks and coupling strengths between quark compo-

nents. We need 14 coupling strengths in the present study:

, C

c¯n

, C

c¯s

, C

b¯n

, C

b¯s

, C

c¯c

, C

n¯n

, and C

. Most of them can be extracted from the

measured masses [56] of the low-lying conventional

hadrons (see Table I), but the determination of C

s¯s

and C

needs approximations. We here assume C

¼ 10.5 MeV and adopt C

¼ 3.3 MeV

extracted fr om M



− M

¼ 70 MeV [1].Partsof

spectroscopic coupling parameters have been derived in

Ref. [55]. The values of our coupling parameters are

consistent with those in that paper, see discussions in

Refs. [52,57]. The effective quark masses we extracted are

¼ 361.7 MeV, m

¼ 54 0.3 MeV, m

¼ 1724.6 MeV,

and m

¼ 505 2.8 MeV, which are close to those obtained

in Ref. [58].

When one substitutes these parameters into the mass

formula (2), the tetraquark masses may be estimated.

However, if we check the numerical values for the masses

of the conventional hadrons with this formula and the above

parameters, deviations from experimental results are found

TABLE I. Chromomagnetic interactions for various hadrons and obtained effective coupling constants in units of

MeV.

Hadron hH

i Hadron hH

i C

N −8C

Δ 8C

¼ 18.4

−



¼ 12.4

0 8

ðC

− 4C

0 8

ðC

þ C

Ω 8C

¼ 6.5

Λ −8C

π −16C

¼ 29.8

K −16C



¼ 18.7

D −16C



¼ 6.7

−16C



¼ 6.7

B −16C



¼ 2.1

−16C



¼ 2.3

−16C

J=ψ

¼ 5.3

−16C

¼ 2.9

−



¼ 4.0

−



¼ 4.5

−



¼ 1.3

−



¼ 1.2

WU, LIU, LIU, and ZHU PHYS. REV. D 99, 014037 (2019)

014037-4

(see Table IVof Ref. [54]). Usually, the obtained masses are

larger than the measured values, which indicates that the

attractions between quark components are not sufficiently

considered in the simple model. The application of this

formula to multiquark states should also lead to higher

masses than those they should be. On the theoretical side,

such values can be treated as upper limits of the tetraquark

masses.

The reason for the overestimated masses is because of

the adopted assumption that the above extracted parameters

are applicable to every system. In principle, each system

has its own values of parameters. From the reduction

procedure for the model Hamiltonian and the fact that

the spacial wave functions are not the same for different

systems, this assumption certainly induces uncertainties.

The uncertainties in coupling strengths affect the mass

splittings between the considered tetraquark states and the

effects should not be large. On the other hand, the

uncertainties in the effective quark masses affect the mass

shifts of the states, which may be significant. To reduce the

uncertainties in mass estimation, we adopt another method

by introducing a reference system and modifying the mass

formula to be

M ¼ðM

ref

− hH

ref

ÞþhH

i: ð11Þ

Here, M

ref

and hH

ref

are the physical mass of the

reference system and the corresponding CMI eigenvalue,

respectively. For M

ref

, one may use the mass of a reference

multiquark state or use the threshold of a reference hadron-

hadron system whose quark content is the same as the

considered multiquark states. With this method, the prob-

lem of using extracted quark masses from conventional

hadrons in multiquark systems [50] is evaded and part of

missed attractions between quark components is phenom-

enologically compensated. In previous studies [36,52,54,

57,59–62], we mainly adopted hadron-hadron thresholds.

One finds that the estimated multiquark masses with this

method are always lower than those with Eq. (2). Since the

number of thresholds may be more than 1, there is a

question which threshold leads to more reasonable masses.

As a multibody system, the size of a tetraquark state should

be larger than that of a conventional hadron and the

distance between two quark components in tetraquarks

may be larger than that in a conventional meson. The

resulting effect is that the attraction between quark com-

ponents should be weaker. Thus, although we cannot give a

definite answer, probably the meson-meson threshold

leading to higher masses gives more reasonable tetraquark

masses. In the present study, besides the possible hadron-

hadron thresholds, we may additionally turn to Xð4140Þ

by assuming it as the ground cs

s tetraquark state with

¼ 1

þþ

. It seems that using Xð4140Þ as an input is a

better approach than the adoption of meson-meson thresh-

olds. In Ref. [52], we have performed the exploration for

the cs

s states with this input and gotten higher masses

than with the D

threshold. This observation probably

indicates that the highest masses estimated with various

hadron-hadron thresholds are still lower than the tetraquark

masses. The discrepancy may be understood with the

additional kinetic energy [63]. From the comparison for

results in the current model [59] and in a dynamical study

[64], the calculated masses of heavy-full tetraquark states

are truly higher than the highest masses estimated with

meson-meson thresholds but lower than the theoretical

upper limits. In the following discussions, we use this

feature as a criterion for reasonable tetraquark masses. The

reasonability of the results may be tested in future studies.

B. Effective interactions and supplemental

results for the cs

s system

In Ref. [59], we have discussed the effects on the

tetraquark masses due to change of coupling parameters

and argued the stability of QQ

Q states by using the

effective color-spin interactions in the case that the mixing

of different color-spin structures is considered. In Ref. [57],

we further introduced a dimensionless measure to reflect

the effective color-spin interaction between the ith quark

component and the jth quark component,

ΔM

ΔC

→

∂M

∂C

: ð12Þ

With such measures, one may rewrite the multiquark

masses as

M ¼ M

i<j

: ð13Þ

When K

is a negative (positive) number, the effective

interaction between the ith and jth quark components is

attractive (repulsive). If K

and K

are negative but K

, K

, and K

are positive, the tetraquark state

is probably more stable than other cases. If

only K

or K

is negative, the state is probably less stable

than the mentioned case but more stable than other cases.

In the following parts, we qualitatively discuss the stability

of tetraquarks with such effective interactions.

In ours previous work [52], we considered the spectrum

of cs

s states. Here, we do not repeat the results given

there, but present the supplemental results about effective

interactions. The obtained K

’s of Eq. (13) are listed in

Table II. The order of states for each case of J

is the same

as the order of masses from high to low. From the results,

the highest 2

þþ

, the highest 1

þþ

, and the second highest

þþ

states are probably more stable than other states.

Although the Xð4274Þ as another 1

þþ

s state is higher

than the Xð4140Þ, its width can be narrower than that of

Xð4140Þ. This feature is not contradicted with the recent

LHCb measurement [49].

SYSTEMATIC STUDIES OF CHARMONIUM-, … PHYS. REV. D 99, 014037 (2019)

014037-5

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