Analysis of the tensor–tensor type scalar tetraquark states with QCD sum rules
Eur. Phys. J. C
Analysis of the tensortensor type scalar tetraquark states with QCD sum rules
ZhiGang Wang 0
JunXia Zhang 0
0 Department of Physics, North China Electric Power University , Baoding 071003 , People's Republic of China
In this article, we study the ground states and the first radial excited states of the tensortensor type scalar hiddencharm tetraquark states with the QCD sum rules. We separate the ground state contributions from the first radial excited state contributions unambiguously, and obtain the QCD sum rules for the ground states and the first radial excited states, respectively. Then we search for the Borel parameters and continuum threshold parameters according to four criteria and obtain the masses of the tensortensor type scalar hiddencharm tetraquark states, which can be confronted with the experimental data in the future.

The attractive interaction induced by onegluon exchange
favors formation of diquark states in color antitriplet and
disfavors formation of diquark states in color sextet. The
antitriplet diquark states εi jk q Tj C qk have five Dirac
tensor structures, scalar C γ5, pseudoscalar C , vector C γμγ5,
axialvector C γμ and tensor C σμν . The structures C γμ and
C σμν are symmetric, while the structures C γ5, C and C γμγ5
are antisymmetric. The scalar and axialvector light diquark
states have been studied with the QCD sum rules [1–4], the
scalar and axialvector heavylight diquark states have also
been studied with the QCD sum rules [5,6]. The calculations
based on the QCD sum rules indicate that the scalar and
axialvector diquark states are more stable than the
corresponding pseudoscalar and vector diquark states, respectively. We
usually construct the C γ5 ⊗ γ5C type and C γμ ⊗ γ μC
type currents to study the lowest scalar light tetraquark
states, hiddencharm or hiddenbottom tetraquark states [7–
18], the corresponding C ⊗ C type and C γμγ5 ⊗ γ5γ μC
type scalar tetraquark states have much larger masses. The
C σαβ ⊗ σ αβ C type scalar hiddencharm or hiddenbottom
tetraquark states have not been studied with the QCD sum
rules, so it is interesting to study them with the QCD sum
rules.
The instantons play an important role in understanding
the UA(1) anomaly and in generating the spectrum of light
hadrons [19]. The calculations based on the random
instanton liquid model indicate that the most strongly correlated
diquarks exist in the scalar and tensor channels [20]. The
heavylight tensor diquark states, although they differ from
the light tensor diquark states due to the appearance of the
heavy quarks, maybe play an important role in understanding
the rich exotic hadron states, we should explore this
possibility, the lowest hiddencharm and hiddenbottom tetraquark
states maybe of the C γ5 ⊗ γ5C type, C γμ ⊗ γ μC type or
C σαβ ⊗ σ αβ C type.
The QCD sum rules method provides a powerful
theoretical tool in studying the hadronic properties, and it has been
applied extensively to the study of the masses, decay
constants, hadronic formfactors, coupling constants, etc. [21–
23]. In this article, we construct the C σαβ ⊗ σ αβ C type
currents to study the scalar hiddencharm tetraquark states.
There exist some candidates for the scalar hiddencharm
tetraquark states. In Ref. [24], Lebed and Polosa propose that
the X (3915) is the ground state scalar csc¯s¯ state based on
lacking of the observed D D¯ and D∗ D¯ ∗ decays, and attribute
the single known decay mode J /ψ ω to the ω–φ mixing
effect. Recently, the LHCb collaboration observed two new
particles X (4500) and X (4700) in the J /ψ φ mass spectrum
with statistical significances 6.1σ and 5.6σ , respectively,
and determined the quantum numbers to be J PC = 0++
with statistical significances 4.0σ and 4.5σ , respectively
[25,26]. The X (4500) and X (4700) are excellent candidates
for the csc¯s¯ tetraquark states. In Refs. [27,28], we study
the C γμ ⊗ γ μC type, C γμγ5 ⊗ γ5γ μC type, C γ5 ⊗ γ5C
type, and C ⊗ C type scalar csc¯s¯ tetraquark states with the
QCD sum rules. The numerical results support assigning the
X (3915) to be the 1S C γ5 ⊗ γ5C type or C γμ ⊗ γ μC
type csc¯s¯ tetraquark state, assigning the X (4500) to be the
2S C γμ ⊗ γ μC type csc¯s¯ tetraquark state, assigning the
× Tr σ αβ C n n(−x )σ μν C Sm mT (−x )C ,
d¯u ( p) = i εi jk εimnεi j k εi m n
where Si j (x ), Ui j (x ), Di j (x ) and Ci j (x ) are the full s, u, d
and c quark propagators, respectively,
i δi j x δi j ms
Si j (x ) = 2π 2x 4 − 4π 2x 2 −
X (4700) to be the 1S C γμγ5 ⊗ γ5γ μC type csc¯s¯ tetraquark
state. For other possible assignments of the X (4500) and
X (4700), one can consult Refs. [29–34]. In this article, we
study the C σαβ ⊗ σ αβ C type hiddencharm tetraquark states
with the QCD sum rules, and explore whether or not the
X (3915), X (4500) and X (4700) can be assigned to be the
C σαβ ⊗ σ αβ C type tetraquark states.
The article is arranged as follows: we derive the QCD sum
rules for the masses and pole residues of the ground state
C σαβ ⊗ σ αβ C type tetraquark states in Sect. 2; in Sect. 3, we
derive the QCD sum rules for the masses and pole residues
of the ground state and the first radial excited state of the
C σαβ ⊗ σ αβ C type tetraquark states; Sect. 4 is reserved for
our conclusion.
2 QCD sum rules for the Cσαβ ⊗ σ αβ Ctype tetraquark
states without including the first radial excited states
In the following, we write down the twopoint correlation
functions s¯s/d¯u ( p) in the QCD sum rules:
s¯s/d¯u ( p) = i
d4x ei p·x 0T Js¯s/d¯u (x ) Js¯†s/d¯u (0) 0 ,
where the i , j , k, m, n are color indices, C is the charge
conjugation matrix.
At the hadronic side, we insert a complete set of
intermediate hadronic states with the same quantum numbers as
the current operators Js¯s/d¯u (x ) into the correlation functions
s¯s/d¯u ( p) to obtain the hadronic representation [21–23].
After isolating the ground state contributions of the scalar
csc¯s¯/cuc¯d¯ tetraquark states Xs¯s/d¯u , we get the results,
s¯s/d¯u ( p) =
Xs¯s/d¯u ( p) = λs¯s/d¯u .
In the following, we briefly outline the operator product
expansion for the correlation functions s¯s/d¯u ( p) in
perturbative QCD. We contract the u, d, s and c quark fields in
the correlation functions s¯s/d¯u ( p) with Wick theorem, and
obtain the results:
and t n = λ2n , the λn is the GellMann matrix, Dα =
∂α − i gs Gnαt n [23]. Then we compute the integrals both in
the coordinate space and the momentum space, and obtain
the correlation functions s¯s/d¯u ( p) at the quark level,
therefore the QCD spectral densities through dispersion relation.
In this article, we calculate the contributions of the vacuum
condensates up to dimension 10 in a consistent way, for
technical details, one can consult Ref. [35].
Once the analytical QCD spectral densities are obtained,
we take the quark–hadron duality below the continuum
d4ke−ik·x
k − mc
(k2 − mc2)2
3(k2 − mc2)4
4(k2 − mc2)5
thresholds ss¯0s/d¯u and perform Borel transform with respect
to the variable P2 = − p2 to obtain the QCD sum rules:
1−y
1−y
yi zi
× 10s2 − 12smc2 + 3mc4
1−y
1−y
dz yz (1 − y − z)
dz yz s − mc2
2s − mc2
dz s − mc2 ,
1−y
× s − mc2
1−y
1−y
1−y
1−y
dz (1 − y − z)3
dz (1 − y − z)2
× (1 − y − z) s − mc2
× 2s − mc2 ,
1−y
1−y
1−y
dz yz(1−y−z)
dz yz s − mc2
dy y(1 − y) 3s − 2mc2
1−y
1−y
1−y
s2
+ 28(y + z) 2s − mc2 + 6 δ s − mc2
dy y(1 − y) 3s − 2mc2
1−y
dz (1 − y − z)
mc2 2 + s δ s − mc2
3s − 2mc2
s s2
× (1 − y − z) 1 + T 2 + 2T 4
1−y
1−y
1−y
1−y
1−y
dz (1 − y − z)
1−y
1−y
Table 1 The input parameters in the QCD sum rules, the values in the
bracket denote the energy scales μ = 1, 2 GeV and mc, respectively
−(0.24 ± 0.01 GeV)3 [21–23,36]
(0.8 ± 0.1) q¯q (1 GeV) [21–23,36]
m20 q¯q (1 GeV) [21–23,36]
m20 s¯s (1 GeV) [21–23,36]
(0.8 ± 0.1) GeV2 [21–23,36]
(0.33 GeV)4 [21–23,36]
(1.275 ± 0.025) GeV [37]
(0.095 ± 0.005) GeV [37]
dy s2 δ s − mc2 , (18)
y f =
mc2 = (y+yzz)mc2 , mc2 = y(1m−c2 y) , yyi f dy → 01 dy, z1i−y dz →
01−y dz, where the δ functions δ s − mc2 and δ s − mc2
appear.
We differentiate Eq. (9) with respect to T12 , then eliminate
the pole residues λs¯s/d¯u , and obtain the QCD sum rules for
the ground state masses Ms¯s/d¯u of the C σαβ ⊗ σ αβ C type
scalar hiddencharm tetraquark states,
ysy−mmc2 c2 ,
Ms¯2s/d¯u = −
The input parameters are shown explicitly in Table 1.
The quark condensates, mixed quark condensates, and M S
masses evolve with the renormalization group equation; we
take into account the energyscale dependence according to
the following equations:
q¯ gs σ Gq (μ) = q¯ gs σ Gq (Q)
s¯gs σ Gs (μ) = s¯gs σ Gs (Q)
2
where t = log μ2 , b0 = 331−22πn f , b1 = 1532−4π192n f , b2 =
2857− 50933 n f + 32275 n2f
128π3 , = 213, 296 and 339 MeV for the
flavors n f = 5, 4 and 3, respectively [37]. Furthermore, we set
mu = md = 0.
In the diquark–antidiquark type tetraquark system Qq Q¯ q¯ ,
the Qquark serves as a static well potential and combines
with the light quark q to form a heavy diquark D in color
antitriplet, while the Q¯ quark serves as another static well
potential and combines with the light antiquark q¯ to form
a heavy antidiquark D¯ in color triplet; the D and D¯
combine to form a compact tetraquark state [16–18,35,38,39].
For such diquark–antidiquark type tetraquark systems, we
suggest an energyscale formula μ = M X2/Y/Z − (2MQ )2
to determine the energy scales of the QCD spectral
densities, where the X , Y and Z are the hiddencharm or
hiddenbottom tetraquark states Qq Q¯ q¯ , the MQ are the effective
heavy quark masses. In this article, we choose the updated
value Mc = 1.82 GeV [40].
Now we search for the Borel parameters T 2 and
continuum threshold parameters ss¯0s/d¯u according to the four
criteria:
1. Pole dominance at the hadron side;
2. Convergence of the operator product expansion;
3. Appearance of the Borel platforms;
4. Satisfying the energy scale formula.
We cannot obtain reasonable Borel parameters T 2 and
continuum threshold parameters ss¯0s/d¯u , if the energy gap
between the ground state and the first radial excited state
is about 0.3 − 0.7 GeV.
3 QCD sum rules for the Cσαβ ⊗ σ αβ Ctype tetraquark
states including the first radial excited states
Now we take into account both the ground state contribution
and the first radial excited state contribution at the hadronic
side of the QCD sum rules [41]. First of all, we introduce the
notations τ = T12 , Dn = − ddτ n, and use the subscripts 1
and 2 to denote the ground state and the first radial excited
state of the C σαβ ⊗σ αβ C type tetraquark states, respectively.
Then the QCD sum rules can be written as
D2 − M 2j D
D3 − M 2j D2
b = DD32⊗⊗DD00−−DD2⊗⊗DD ,
D3 ⊗ D − D2 ⊗ D2
c = D2 ⊗ D0 − D ⊗ D ,
D j ⊗ Dk = D j QCD(τ ) Dk
The squared masses Mi2 satisfy the following equation:
λ21 exp −τ M12 + λ22 exp −τ M22
the subscript QCD denotes the QCD side of the correlation
functions s¯s/d¯u (τ ). We differentiate both sides of the QCD
sum rules in Eq. (21) with respect to τ and obtain
λ21 M12 exp −τ M12 + λ22 M22 exp −τ M22
= D
Then we solve the two equations and obtain the QCD sum
rules,
where i = j . We differentiate both sides of the QCD sum
rules in Eq. (23) with respect to τ and obtain
i = 1, 2, j, k = 0, 1, 2, 3. We solve Eq. (25) and obtain two
solutions,
1. Pole dominance at the hadron side;
2. Convergence of the operator product expansion;
3. Appearance of the Borel platforms;
4. Satisfying the energy scale formula.
The pole contributions are
the pole dominance condition is well satisfied, the criterion
1 is satisfied. The contributions come from the vacuum
condensates of dimension 10 D10 are
for the central values of the continuum threshold parameters,
the operator product expansion is convergent, the criterion 2
is satisfied.
Now we take into account the uncertainties of all the input
parameters, and obtain the masses and pole residues of the
C σαβ ⊗ σ αβ C type tetraquark states,
Md¯u,1S = 3.82 ± 0.16 GeV,
Ms¯s,1S = 3.84 ± 0.16 GeV,
λd¯u,1S = (5.20 ± 1.35) × 10−2 GeV5 ,
λs¯s,1S = (4.87 ± 1.25) × 10−2 GeV5 ,
Md¯u,2S = 4.38 ± 0.09 GeV,
λd¯u,2S = (2.12 ± 0.31) × 10−1 GeV5,
Ms¯s,2S = 4.40 ± 0.09 GeV,
λs¯s,2S = (2.14 ± 0.33) × 10−1 GeV5 ,
at the energy scale μ = 2.50 GeV. The central values of
the predicted masses satisfy the energyscale formula, the
criterion 4 is satisfied.
In Fig. 1, we plot the predicted masses Md¯u/s¯s with
variations of the Borel parameters T 2. From the figure, we can
see that the plateaus are rather flat, the criterion 3 is satisfied.
The four criteria are all satisfied, we expect to make reliable
predictions.
The energy gaps between the ground states and the first
radial excited states are
Md¯u,2S − Md¯u,1S = 0.56 GeV ,
Ms¯s,2S − Ms¯s,1S = 0.56 GeV.
Z (4430) is assigned to be the first radial excitation of
Zc(3900) according to the analogous decays,
and the mass differences MZ(4430) − MZc(3900) = 576 MeV
and Mψ − MJ/ψ = 589 MeV [42–44]. The energy gaps
Md¯u,2S − Md¯u,1S, Ms¯s,2S − Ms¯s,1S, MZ(4430) − MZc(3900)
are compatible with each other. The widths Zc(3900) =
46 ± 10 ± 20 MeV [45] and Z(4430) = 172 ± 13+−3374 MeV
[46] are not broad, the QCD sum rules for the ground
state Zc(3900) alone or without including the first radial
excited state Z (4430) work well [35]. If both the ground
state Zc(3900) and the first radial excited state Z (4430)
are included in, the continuum threshold parameter √s0 =
4.8 ± 0.1 GeV = MZ(4430) + (0.2 ∼ 0.4) GeV, the lower
bound of the √s0 − MZ(4430) is about 0.2 GeV, which is
large enough to take into account the contribution of the
Z (4430) [44]. In the present case, the lower bound of the
√s0 − Md¯u/s¯s,2S are about 0.3 GeV, which indicates that the
widths of the first radial excited states of the C σαβ ⊗ σ αβ C
type tetraquark states are rather large. According to the
energy gaps between the ground states and the first radial
excited states, the continuum threshold parameters should be
chosen as large as √s0 − Md¯u/s¯s,1S = 0.5 ± 0.1 GeV without
including the first radial excited states Xd¯u/s¯s,2S explicitly,
however, for such large continuum thresholds, the
contributions of the Xd¯u/s¯s,2S are already included in due to their large
widths. So the QCD sum rules in which only the ground state
C σαβ ⊗ σ αβ C type tetraquark states are taken into account
cannot work.
The predicted mass Ms¯s,1S = 3.84 ± 0.16 GeV
overlaps with the experimental value MX (3915) = 3918.4 ±
1.9 MeV slightly [37], the X (3915) cannot be a pure
C σαβ ⊗σ αβ C type csc¯s¯ tetraquark state. The predicted mass
Ms¯s,2S = 4.40 ± 0.09 GeV overlaps with the experimental
value MX (4500) = 4506 ± 11+−1125 MeV slightly [25,26], the
X (4500) cannot be a pure C σαβ ⊗σ αβ C type csc¯s¯ tetraquark
state. As the central values of the Ms¯s,1S and Ms¯s,2S differ
from the central values of the MX (3915) and MX (4500)
significantly, it is difficult to assign the MX (3915) and MX (4500) to
be the C σαβ ⊗ σ αβ C type csc¯s¯ tetraquark states. The Xs¯s,1S
and Xs¯s,2S are new particles, the present predictions can be
confronted to the experimental data in the future.
4 Conclusion
In this article, we study the ground states and the first
radial excited states of the C σαβ ⊗ σ αβ C type hiddencharm
tetraquark states with the QCD sum rules by calculating the
contributions of the vacuum condensates up to dimension 10
in a consistent way. We separate the ground state
contributions from the first radial excited state contributions
unambiguously, and obtain the QCD sum rules for the ground
states and the first radial excited states, respectively. Then
we search for the Borel parameters and continuum threshold
parameters according to the four criteria: (1) pole dominance
at the hadron side; (2) convergence of the operator product
expansion; (3) appearance of the Borel platforms; (4)
satisfying the energyscale formula. Finally, we obtain the masses
and pole residues of the C σαβ ⊗ σ αβ C type hiddencharm
tetraquark states. The masses can be confronted to the
experimental data in the future, while the pole residues can be used
to study the relevant processes with the threepoint QCD sum
rules or the lightcone QCD sum rules.
Acknowledgements This work is supported by National Natural
Science Foundation, Grant Numbers 11375063, and Natural Science
Foundation of Hebei province, Grant Number A2014502017.
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
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