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Eur. Phys. J. C (2016) 76:142

DOI 10.1140/epjc/s10052-016-3983-2

Regular Article - Theoretical Physics

Analysis of the

pentaquark states in the

diquark–diquark–antiquark model with QCD sum rules

Zhi-Gang Wang

Department of Physics, North China Electric Power University, Baoding 071003, People’s Republic of China

Received: 25 December 2015 / Accepted: 25 February 2016 / Published online: 14 March 2016

Abstract In this article, we construct both the axial

vector-diquark–axialvector-diquark–antiquark type and the

axialvector-diquark–scalar-diquark–antiquark type interpo-

lating currents, then calculate the contributions of the vac-

uum condensates up to dimension 10 in the operator prod-

uct expansion, and we study the masses and pole residues

of the J

hidden-charm pentaquark states with the

QCD sum rules in a systematic way. In calculations, we use

the formula μ =



− (2M

)

to determine the energy

scales of the QCD spectral densities. We take into account

the SU(3) breaking effects of the light quarks, and we obtain

the masses of the hidden-charm pentaquark states with the

strangeness S = 0, −1, −2, −3, which can be confronted

with the experimental data in the future.

1 Introduction

Recently, the LHCb collaboration studied the 

→

J/ψ K

−

p decays, and one observed two pentaquark can-

didates P

(4380) and P

(4450) in the J/ψ p mass spectrum

with the signiﬁcances of more than 9 σ [1]. The measured

masses and widths are M

(4380)

= 4380 ± 8 ± 29 MeV,

(4450)

= 4449.8 ± 1.7 ± 2.5MeV, 

(4380)

= 205 ±

18±86 MeV, and 

(4450)

= 39±5±19 MeV, respectively.

The decays P

(4380) → J/ψ p take place through the rel-

ative S-wave channel, while the decays P

(4450) → J /ψ p

take place through the relative P-wave channel, the decays

(4380) → J /ψ p are kinematically favored and P

(4380)

has larger width. The preferred spin-parities of P

(4380) and

(4450) are J

−

and

, respectively. There have

been several attempted assignments, such as 

∗

, 

∗

p, J/ψ N (1440), and the J /ψ N(1520) molecule-like

pentaquark states [2–11] (or not the molecular pentaquark

states [12]), the diquark–diquark–antiquark type pentaquark

e-mail: zgwang@aliyun.com

states [13–22], the diquark–triquark type pentaquark states

[23], re-scattering effects [24–27], etc.

The quarks have color SU(3) symmetry; we can con-

struct the pentaquark conﬁgurations according to the routine

quark → diquark → pentaquark,

(3 ⊗ 3) ⊗ (3 ⊗ 3) ⊗

= (

3 ⊕ 6) ⊗ (3 ⊕ 6) ⊗ 3 = 3 ⊗ 3 ⊗ 3 ⊕···=1 ⊕···,

(1)

where 1, 3 (

3), and 6 denote the color singlet, triplet

(antitriplet), and sextet, respectively. The diquarks q

Cq



have ﬁve structures in Dirac spinor space, where C = Cγ

C, Cγ

, Cγ

, and Cσ

μν

for the scalar, pseudoscalar, vec-

tor, axialvector, and tensor diquarks, respectively, and the j

and k are color indices. The attractive interactions of one-

gluon exchange favor formation of the diquarks in color

antitriplet

, ﬂavor antitriplet 3

, and spin singlet 1

[28,29],

while the favored conﬁgurations are the scalar (Cγ

) and

axialvector (Cγ

) diquark states [30–32]. The calculations

based on the QCD sum rules indicate that the heavy-light

scalar and axialvector diquark states have almost degener-

ate masses [30,31], while the masses of the light axialvector

diquark states lie (150–200) MeV above that of the light

scalar-diquark states [32], if they have the same quark con-

stituents.

In Refs. [19,20], we choose the light scalar diquark

and heavy axialvector diquark (or heavy scalar diquark) as

the basic constituents, construct both the scalar-diquark–

axialvector-diquark–antiquark type and scalar-diquark–

scalar-diquark–antiquark type interpolating currents, which

are supposed to couple potentially to the lowest pentaquark

states according to the light scalar-diquark constituent [32],

then calculate the contributions of the vacuum condensates

up to dimension 10 in the operator product expansion and

study the masses and pole residues of J

−

, and

hidden-charm pentaquark states with the QCD sum rules. The

numerical results favor assigning P

(4380) and P

(4450)

123

142 Page 2 of 21 Eur. Phys. J. C (2016) 76 :142

to be the

−

and

diquark–diquark–antiquark type pen-

taquark states, respectively [19]. In Ref. [19], we take the

energy-scale formula μ =



− (2M

)

to determine the

energy scales of the QCD spectral densities, the resulting pole

contributions are about (40–60) %, and the contributions of

the vacuum condensates of dimension 10 are less than 5 %,

the two criteria (pole dominance at the phenomenological

side and convergence of the operator product expansion) of

the conventional QCD sum rules can be satisﬁed. Now we

extend our previous work to study the

hidden-charm pen-

taquark states in a systematic way, where the energy-scale

formula μ =



− (2M

)

serves as an additional con-

straint on the predicted masses. All the predictions can be

confronted with the experimental data in the future, and the

assignments of P

(4380) and P

(4450) in the scenario of

the diquark–diquark–antiquark type pentaquark states can

be testiﬁed.

In this article, we take the light axialvector diquarks and

heavy axialvector diquarks (and heavy scalar diquarks) as

the basic constituents, and study the axialvector-diquark–

axialvector-diquark–antiquark type and axialvector-diquark–

scalar-diquark–antiquark type pentaquark conﬁgurations.

Now we illustrate how to construct the pentaquark conﬁg-

urations in the diquark–diquark–antiquark model according

to the spin-parity J

⊗ 0

⊗

−

¯c

−

c ¯c

⊕

−

c ¯c

, (2)

⊗ 1

⊗

−

¯c



⊕ 1

⊕ 2



⊗

−

¯c

−

c ¯c

⊕



−

c ¯c

⊕

−

c ¯c



⊕



−

c ¯c

⊕

−

c ¯c



, (3)

⊗ 0

⊗



−

⊗

−

¯c



= 1

⊗ 0

⊗



¯c

⊕

¯c





c ¯c

⊕

c ¯c



⊕



c ¯c

⊕

c ¯c

⊕

c ¯c



, (4)

⊗ 1

⊗



−

⊗

−

¯c





⊕ 1

⊕ 2



⊗



¯c

⊕

¯c



c ¯c

⊕



c ¯c

⊕

c ¯c



⊕



c ¯c

⊕

c ¯c



⊕

c ¯c

⊕



c ¯c

⊕

c ¯c

⊕

c ¯c



⊕



c ¯c

⊕

c ¯c

⊕

c ¯c

⊕

c ¯c



(5)

where 1

−

denotes the contribution of the additional P-wave

to the spin-parity, the subscripts q

, q

c, ... denote the

quark constituents. The quark and antiquark have opposite

parity, we usually take it for granted that the quarks have

positive parity, while the antiquarks have negative parity, so

the ¯c-quark has J

−

In this article, we study the pentaquark states with the

underlined spin-parity

c ¯c

, which are supposed to

be the lowest pentaquark states with the light axialvec-

tor diquarks. We construct both the axialvector-diquark–

axialvector-diquark–antiquark type and the axialvector-

diquark–scalar-diquark–antiquark type currents J (x) =

(x) according to Eqs. (2)–(3), where the superscripts

and j

denote the spins of the light diquark and heavy

diquark, respectively, the subscripts q

, q

denote the light

quark constituents. We calculate the vacuum condensate up

to dimension 10 in the operator product expansion, and study

the masses and pole residues of the lowest pentaquark states

in a systematic way.

The article is arranged as follows: we derive the QCD sum

rules for the masses and pole residues of the

pentaquark

states in Sect. 2; in Sect. 3, we present the numerical results

and discussions; and Sect. 4 is reserved for our conclusion.

2 QCD sum rules for the

pentaquark states

We write down the two-point correlation functions 

( p)

in the QCD sum rules,



( p) = i



ip·x

0|T {J

(x)

(0)}|0,

(6)

where

uuu

(x ) = ε

ila

ijk

lmn

(x )Cγ

(x )u

(x )

×Cγ

(x )C ¯c

(x ),

uud

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )d

(x )Cγ

(x )

+2u

(x )Cγ

(x )u

(x )Cγ

(x )]C ¯c

(x ),

123

Eur. Phys. J. C (2016) 76 :142 Page 3 of 21 142

udd

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )u

(x )Cγ

(x )

+2d

(x )Cγ

(x )d

(x )Cγ

(x )]C ¯c

(x ),

ddd

(x ) = ε

ila

ijk

lmn

(x )Cγ

(x )d

(x )

×Cγ

(x )C ¯c

(x ), (7)

uus

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )s

(x )Cγ

(x )

+2u

(x )Cγ

(x )u

(x )Cγ

(x )]C ¯c

(x ),

uds

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )s

(x )Cγ

(x )

(x )Cγ

(x )d

(x )Cγ

(x )

(x )Cγ

(x )u

(x )Cγ

(x )]C ¯c

(x ),

dds

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )s

(x )Cγ

(x )

+2d

(x )Cγ

(x )d

(x )Cγ

(x )]C ¯c

(x ),

(8)

uss

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )u

(x )Cγ

(x )

+2s

(x )Cγ

(x )s

(x )Cγ

(x )]C ¯c

(x ),

dss

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )d

(x )Cγ

(x )

+2s

(x )Cγ

(x )s

(x )Cγ

(x )]C ¯c

(x ),

(9)

sss

(x) = ε

ila

ijk

lmn

(x)Cγ

(x)s

(x)

×Cγ

(x)C ¯c

(x), (10)

uuu

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )

×u

(x )Cγ

(x )γ

C ¯c

(x ),

uud

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )d

(x )Cγ

(x )

+2u

(x )Cγ

(x )u

(x )Cγ

(x )]γ

C ¯c

(x ),

udd

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )u

(x )Cγ

(x )

+2d

(x )Cγ

(x )d

(x )Cγ

(x )]γ

C ¯c

(x ),

ddd

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )d

(x )

×Cγ

(x )γ

C ¯c

(x ), (11)

uus

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )s

(x )Cγ

(x )

+2u

(x )Cγ

(x )u

(x )Cγ

(x )]γ

C ¯c

(x ),

uds

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )s

(x )Cγ

(x )

(x )Cγ

(x )d

(x )Cγ

(x )

(x )Cγ

(x )u

(x )Cγ

(x )]γ

C ¯c

(x ),

dds

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )s

(x )Cγ

(x )

+2d

(x )Cγ

(x )d

(x )Cγ

(x )]γ

C ¯c

(x ),

(12)

uss

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )u

(x )Cγ

(x )

+2s

(x )Cγ

(x )s

(x )Cγ

(x )]γ

C ¯c

(x ),

dss

(x ) =

ila

ijk

lmn

√

(x )Cγ

(x )d

(x )Cγ

(x )

+2s

(x )Cγ

(x )s

(x )Cγ

(x )]γ

C ¯c

(x ),

(13)

sss

(x) =

ila

ijk

lmn

√

(x)Cγ

(x)s

(x)

×Cγ

(x)γ

C ¯c

(x), (14)

i, j, k, l, m, n, and a are color indices, C is the charge

conjugation matrix. In the currents J

(x), the light axi-

alvector diquark combines with the heavy scalar diquark to

form a tetraquark with J

= 1

in color triplet according to

Eq. (2), while in the currents J

(x), the light axialvector

diquark combines with the heavy axialvector diquark to form

a tetraquark with J

= 0

in color triplet according to Eq.

(3). Then they couple with the antiquark to form pentaquark

states with J

−

in color singlet. In this article, we

take the isospin limit, and classify the currents couple to the

pentaquark states with degenerate masses into the following

eight types:

uuu,μ

(x), J

uud,μ

(x), J

udd ,μ

(x), J

ddd,μ

(x) ;

uus,μ

(x), J

uds,μ

(x), J

dds,μ

(x) ;

uss,μ

(x), J

dss,μ

(x) ;

sss,μ

(x) ;

uuu,μ

(x), J

uud,μ

(x), J

udd ,μ

(x), J

ddd,μ

(x) ;

uus,μ

(x), J

uds,μ

(x), J

dds,μ

(x) ;

uss,μ

(x), J

dss,μ

(x) ;

sss,μ

(x). (15)

In calculations, we choose the ﬁrst current in each type.

The currents J

(0) have negative parity, and cou-

ple potentially to the

−

hidden-charm pentaquark states

123

142 Page 4 of 21 Eur. Phys. J. C (2016) 76 :142

(

−

), where the superscript

denotes the total angu-

lar momentum of the anti-c-quark ¯c,



0|J

(0)|P



−



( p)



= λ

−

( p, s), (16)

the λ

−

are the pole residues, the spinors U

−

( p, s) satisfy the

Dirac equations ( p − M

P,−

−

( p, s) = 0, the s are the

polarization vectors of the pentaquark states. On the other

hand, the currents J

(0) also couple potentially to the

hidden-charm pentaquark states P

(

) as multiplying

iγ

to the currents J

(x) change their parity,



0|J

(0)|P





( p)



= λ

iγ

( p, s), (17)

the spinors U

( p, s) (pole residues λ

) have analogous

properties [33–40]. We can study the J

hidden-

charm pentaquark states without introducing the additional

P-wave explicitly; see Eqs. (4)–(5).

We insert a complete set of intermediate pentaquark states

with the same quantum numbers as the current operators

J (x) and iγ

J (x) into the correlation functions 

( p)

to obtain the hadronic representation [41,42]. After isolat-

ing the pole terms of the lowest states of the hidden-charm

pentaquark states, we obtain the following results:



( p) = λ

−

 p + M

P,−

− p

+ λ

 p − M

P,+

− p

+···,

(18)

where the M

are the masses of the lowest pentaquark states

with the parity ±, respectively.

Now we obtain the hadronic spectral densities through the

dispersion relation,

Im

(s)

= p[λ

−

δ(s − M

P,−

) + λ

δ(s − M

P,+

)]

+[M

P,−

−

δ(s − M

P,−

) − M

P,+

δ(s − M

P,+

)],

= p ρ

(s) + ρ

(s), (19)

where the subscript index H denotes the hadron side, then

we introduce the weight function exp



−



to obtain the

QCD sum rules at the hadron side,





√

sρ

(s) + ρ

(s)



exp



−



= 2M

P,−

−

exp



−

P,−



, (20)





√

sρ

(s) − ρ

(s)



exp



−



= 2M

P,+

exp



−

P,+



, (21)

where the s

are the continuum threshold parameters and the

are the Borel parameters. We separate the contributions of

the negative-parity (positive-parity) pentaquark states from

the positive-parity (negative-parity) pentaquark states explic-

itly.

In the following, we brieﬂy outline the operator product

expansion for the correlation functions 

( p) in pertur-

bative QCD. First of all, we contract the u, s, and c quark

ﬁelds in the correlation functions 

( p) with Wick the-

orem, and obtain the results:



uuu

( p) =−i ε

ila

ijk

lmn





ip·x



(−x)C{2Tr[γ



(x)γ



(x)C]

×Tr[γ



(x)γ



(x)C]−4Tr[γ



(x)γ

×CU



(x)Cγ



(x)γ



(x)C]}, (22)



uus

( p) =−

ila

ijk

lmn





ip·x



(−x )C

×{2Tr[γ



(x )γ



(x )C] Tr[γ



(x )γ



(x )C]

+4Tr[γ



(x )γ



(x )C] Tr[γ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]},

(23)



uss

( p) =−

ila

ijk

lmn





ip·x



(−x )C

×{2Tr[γ



(x )γ



(x )C] Tr[γ



(x )γ



(x )C]

+4Tr[γ



(x )γ



(x )C] Tr[γ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]},

(24)



sss

( p) =−i ε

ila

ijk

lmn





ip·x



(−x )C

×{2Tr[γ



(x )γ



(x )C] Tr[γ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]},

(25)



uuu

( p) =−i ε

ila

ijk

lmn





ip·x



(−x )Cγ

×{2Tr[γ



(x )γ



(x )C]Tr[γ



(x )γ



(x )C]

−4Tr[γ



(x )γ



(x )Cγ



(x )γ



(x )C]}, (26)

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QCD和规则分析Diquark模型中作为五夸克态的Pc（4380）和Pc（4450）

从QCD和规则重新审视为0四夸克状态

胶子质量安萨兹在重味QCD中诱导的归一化夸克-反夸克哈密顿量

变形的AdS / QCD产生的夸克-反夸克电位

B介子中夸克-胶子三体分量的QCD和规则

具有QCD和规则的$$ Z_c（3900）$$ Zc（3900）的磁矩作为轴向矢量四夸克状态

关于QCD和SM中的b-夸克运行质量

固态夸克-强子对偶中Zc（3900）作为轴向矢量四夸克态的衰减宽度

隐身五夸克的QCD和规则研究

用QCD和规则分析标量，轴向矢量，矢量，张量双迷四方态

四夸克分子态的Landau方程和QCD和规则

Nambu–Jona-Lasinio模型中有限温度下的夸克物质和夸克星

多喷嘴最终状态下的夸克喷射速率和夸克-胶子判别

QCD拉普拉斯求和规则的轴向矢量cc和bb diquark质量

扰动QCD中弱磁场中夸克-胶子等离子体的剪切粘度：超前对数

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