Analysis of the tensor–tensor type scalar tetraquark states with QCD sum rules
Eur. Phys. J. C
Analysis of the tensor-tensor type scalar tetraquark states with QCD sum rules
Zhi-Gang Wang 0
Jun-Xia 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 tensor-tensor type scalar hidden-charm 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 tensor-tensor type scalar hidden-charm tetraquark states, which can be confronted with the experimental data in the future.
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The attractive interaction induced by one-gluon 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,
axial-vector 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 axial-vector light diquark
states have been studied with the QCD sum rules [1–4], the
scalar and axial-vector heavy-light 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, hidden-charm or hidden-bottom 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 hidden-charm or hidden-bottom
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
heavy-light 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 hidden-charm and hidden-bottom 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 form-factors, coupling constants, etc. [21–
23]. In this article, we construct the C σαβ ⊗ σ αβ C -type
currents to study the scalar hidden-charm tetraquark states.
There exist some candidates for the scalar hidden-charm
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 hidden-charm 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 p (...truncated)