1 Introduction
###### Abstract

In this article, we study the axialvector-diquark-axialvector-antidiquark type scalar, axialvector, tensor and vector tetraquark states with the QCD sum rules. The predicted mass for the axialvector tetraquark state is in excellent agreement with the experimental value from the BESIII collaboration and supports assigning the new state to be a tetraquark state with . The predicted mass disfavors assigning the or to be the vector partner of the new state. As a byproduct, we obtain the masses of the corresponding tetraquark states. The light tetraquark states lie in the region about rather than .

Light tetraquark state candidates

[2mm] Zhi-Gang Wang 111E-mail: zgwang@aliyun.com.

Department of Physics, North China Electric Power University, Baoding 071003, P. R. China

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark state, QCD sum rules

## 1 Introduction

Recently, the BESIII collaboration studied the process and observed a structure in the mass spectrum [1]. The fitted mass and width are and respectively with assumption of the spin-parity , the corresponding significance is ; while the fitted mass and width are and respectively with assumption of the spin-parity , the corresponding significance is . The state was observed in the decay model rather than in the decay model, they maybe contain a large component, in other words, it maybe have a large tetraquark component. In Ref.[2], Wang, Luo and Liu assign the state to be the second radial excitation of the . In Ref.[3], Cui et al assign the to be the partner of the tetraquark state with the .

We usually assign the lowest scalar nonet mesons to be tetraquark states, and assign the higher scalar nonet mesons to be the conventional quark-antiquark states [4, 5, 6]. In Ref.[7], we take the nonet scalar mesons below as the two-quark-tetraquark mixed states and study their masses and pole residues with the QCD sum rules in details, and observe that the dominant components of the nonet scalar mesons below are conventional two-quark states. The light tetraquark states maybe lie in the region about rather than lie in the region about .

In this article, we take the axialvector diquark operators as the basic constituents to construct the tetraquark current operators to study the scalar (), axialvector (), tensor () and vector () tetraquark states with the QCD sum rules, explore the possible assignments of the new state. We take the axialvector diquark operators as the basic constituents because the favored configurations from the QCD sum rules are the scalar and axialvector diquark states [8, 9], the current operators or quark structures chosen in the present work differ from that in Ref.[3] completely.

The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.

## 2 QCD sum rules for the ss¯s¯s tetraquark states

We write down the two-point correlation functions and firstly,

 Πμναβ(p) = i∫d4xeip⋅x⟨0|T{Jμν(x)J†αβ(0)}|0⟩, (1) Π(p) = i∫d4xeip⋅x⟨0|T{J0(x)J†0(0)}|0⟩, (2)

where , ,

 J2,μν(x) = εijkεimn√2{sTj(x)Cγμsk(x)¯sm(x)γνC¯sTn(x)+sTj(x)Cγνsk(x)¯sm(x)γμC¯sTn(x)}, J1,μν(x) = εijkεimn√2{sTj(x)Cγμsk(x)¯sm(x)γνC¯sTn(x)−sTj(x)Cγνsk(x)¯sm(x)γμC¯sTn(x)}, J0(x) = εijkεimnsTj(x)Cγμsk(x)¯sm(x)γμC¯sTn(x), (3)

where the , , , , are color indexes, the is the charge conjugation matrix. Under charge conjugation transform , the currents and have the properties,

 ˆCJ2,μν(x)ˆC−1 = +J2,μν(x), ˆCJ1,μν(x)ˆC−1 = −J1,μν(x), ˆCJ0(x)ˆC−1 = +J0(x). (4)

At the hadronic side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators and into the correlation functions and to obtain the hadronic representation [10, 11]. After isolating the ground state contributions of the scalar, axialvector, vector and tensor tetraquark states, we get the results,

 Π2,μναβ(p) = λ2XTm2XT−p2(˜gμα˜gνβ+˜gμβ˜gνα2−˜gμν˜gαβ3)+⋯ (5) = Π2+(p)(˜gμα˜gνβ+˜gμβ˜gνα2−˜gμν˜gαβ3)+⋯,
 Π1,μναβ(p) = ˜λ2XAm2XA−p2(p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ) (6) = Π1+(p2)(p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ) +Π1−(p2)(−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ),
 Π(p) = Π0+(p2)=λ2XSm2XS−p2+⋯, (7)

where , the subscripts , , and denote the spin-parity of the corresponding tetraquark states. The pole residues and are defined by

 ⟨0|J2,μν(0)|XT(p)⟩ = λXTεμν, ⟨0|J1,μν(0)|XA(p)⟩ = ˜λXAεμναβεαpβ, ⟨0|J1,μν(0)|XV(p)⟩ = ˜λXV(εμpν−ενpμ), ⟨0|J0(0)|XS(p)⟩ = λXS, (8)

where the and are the polarization vectors of the tetraquark states.

Now we contract the quarks in the correlation functions with Wick theorem, there are four -quark propagators, if two -quark lines emit a gluon by itself and the other two -quark lines contribute a quark pair by itself, we obtain a operator , which is of order with and of dimension . In this article, we take into account the vacuum condensates up to dimension and in a consistent way. For the technical details, one can consult Refs.[7, 12]. Once the analytical expressions of the QCD spectral densities are obtained, we take the quark-hadron duality below the continuum thresholds and perform Borel transform with respect to the variable to obtain the QCD sum rules:

 λ2Xexp(−m2XT2)=∫s00dsρ(s)exp(−sT2), (9)

where , , and ,

 ρS(s) = s43840π6−13sms⟨¯sgsσGs⟩384π4+2s⟨¯ss⟩23π2−17⟨¯ss⟩⟨¯sgsσGs⟩48π2+s2192π4⟨αsGGπ⟩ (10) +19ms⟨¯ss⟩96π2⟨αsGGπ⟩−16ms⟨¯ss⟩33δ(s)+⟨¯sgsσGs⟩2192π2δ(s)−⟨¯ss⟩224⟨αsGGπ⟩δ(s),
 ρA(s) = s411520π6−s2ms⟨¯ss⟩12π4+sms⟨¯sgsσGs⟩9π4+4s⟨¯ss⟩29π2−5⟨¯ss⟩⟨¯sgsσGs⟩18π2 (11) −s22304π4⟨αsGGπ⟩+3ms⟨¯ss⟩64π2⟨αsGGπ⟩−32ms⟨¯ss⟩39δ(s)−2⟨¯ss⟩227⟨αsGGπ⟩δ(s),
 ρV(s) = s411520π6+s2ms⟨¯ss⟩12π4−7sms⟨¯sgsσGs⟩72π4−2s⟨¯ss⟩29π2+5⟨¯ss⟩⟨¯sgsσGs⟩18π2 (12) +s2768π4⟨αsGGπ⟩−79ms⟨¯ss⟩1728π2⟨αsGGπ⟩+16ms⟨¯ss⟩39δ(s) −2⟨¯ss⟩281⟨αsGGπ⟩δ(s)−⟨¯sgsσGs⟩218π2δ(s),
 ρT(s) = s45376π6−3s2ms⟨¯ss⟩20π4+29sms⟨¯sgsσGs⟩96π4+8s⟨¯ss⟩29π2−37⟨¯ss⟩⟨¯sgsσGs⟩48π2 −11s21920π4⟨αsGGπ⟩+43ms⟨¯ss⟩864π2⟨αsGGπ⟩−64ms⟨¯ss⟩39δ(s)−4⟨¯ss⟩227⟨αsGGπ⟩δ(s),

and .

We derive Eq.(9) with respect to , then obtain the QCD sum rules for the masses of the tetraquark states through a fraction,

 m2X = −∫s00dsddτρ(s)exp(−τs)∫s00dsρ(s)exp(−τs). (14)

## 3 Numerical results and discussions

We take the standard values of the vacuum condensates , , , , , at the energy scale [10, 11, 13], and choose the mass from the Particle Data Group [14], and evolve the -quark mass to the energy scale with the renormalization group equation, furthermore, we neglect the small and quark masses.

We choose suitable Borel parameters and continuum threshold parameters to warrant the pole contributions (PC) are larger than , i.e.

 PC = ∫s00dsρ(s)exp(−sT2)∫∞0dsρ(s)exp(−sT2)≥40% , (15)

and convergence of the operator product expansion. The contributions of the vacuum condensates in the operator product expansion are defined by,

 D(n) = ∫s00dsρn(s)exp(−sT2)∫s00dsρ(s)exp(−sT2) , (16)

where the subscript in the QCD spectral density denotes the dimension of the vacuum condensates. We choose the values to warrant the convergence of the operator product expansion. In Table 1, we present the ideal Borel parameters, continuum threshold parameters, pole contributions and contributions of the vacuum condensates of dimension . From the Table, we can see that the pole dominance is well satisfied and the operator product expansion is well convergent, we expect to make reliable predictions.

We take into account all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the tetraquark states, which are shown explicitly in Fig.1 and Table 1. In this article, we have assumed that the energy gaps between the ground state and the first radial state is about [15]. In Fig.1, we plot the masses of the scalar, axialvector, tensor and vector tetraquark states with variations of the Borel parameters at larger regions than the Borel windows shown in Table 1. From the figure, we can see that there appear platforms in the Borel windows.

The predicted mass for the axialvector tetraquark state is in excellent agreement with the experimental value from the BESIII collaboration [1], which supports assigning the new state to be an axialvector-diquark-axialvector-antidiquark type tetraquark state. The predicted mass for the vector tetraquark state lies above the experimental value of the mass of the or , , from the Particle Data Group, and disfavors assigning the or to be vector partner of the new state. If the have tetraquark component, it maybe have color octet-octet component [16]. As a byproduct, we obtain the masses and pole residues of the corresponding tetraquark states, which are shown in Table 1. The present predictions can be confronted to the experimental data in the future.

Now we perform Fierz rearrangement to the currents both in the color and Dirac-spinor spaces,

 J0 = 2¯ss¯ss+2¯siγ5s¯siγ5s+¯sγαs¯sγαs−¯sγαγ5s¯sγαγ5s, J1,μν = √2{i¯ss¯sσμνs−¯sσμνγ5s¯siγ5s+iεμναβ¯sγαγ5s¯sγβs}, J2,μν = 1√2{2¯sγμγ5s¯sγνγ5s−2¯sγμs¯sγνs+2gαβ¯sσμαs¯sσνβs+gμν(¯ss¯ss (17) +¯siγ5s¯siγ5s+¯sγαs¯sγαs−¯sγαγ5s¯sγαγ5s−12¯sσαβs¯sσαβs)}.

The diquark-antidiquark type currents can be re-arranged into currents as special superpositions of color singlet-singlet type currents, which couple potentially to the meson-meson pairs or molecular states, the diquark-antidiquark type tetraquark states can be taken as special superpositions of meson-meson pairs, and embodies the net effects. The decays to their components are Okubo-Zweig-Iizuka supper-allowed, we can search for those tetraquark states in the decays,

 XS → η′η′,f0(980)f0(980),ϕ(1020)ϕ(1020), XA/V → f0(980)h1(1380),ϕ(1020)η′,ϕ(1020)ϕ(1020), XT → η′η′,f0(980)f0(980),ϕ(1020)ϕ(1020). (18)

## 4 Conclusion

In this article, we construct the axialvector-diquark-axialvector-antidiquark type currents to interpolate the scalar, axialvector, tensor and vector tetraquark states, then calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and obtain the QCD sum rules for the masses and pole residues of those tetraquark states. The predicted mass for the axialvector tetraquark state is in excellent agreement with the experimental value, , from the BESIII collaboration and supports assigning the new state to be an axialvector-diquark-axialvector-antidiquark type tetraquark state. The predicted mass for the vector tetraquark state lies above the experimental value of the mass of the , , from the Particle Data Group, and disfavors assigning the to be the vector partner of the new state. As a byproduct, we also obtain the masses and pole residues of the corresponding tetraquark states. The present predictions can be confronted to the experimental data in the future.

## Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number 11775079.

## References

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