Light tetraquark state candidates

In this article, we study the axialvector-diquark-axialvector-antidiquark type scalar, axialvector, tensor and vector sss̄s̄ tetraquark states with the QCD sum rules. The predicted mass mX = 2.08±0.12GeV for the axialvector tetraquark state is in excellent agreement with the experimental value (2062.8± 13.1± 4.2)MeV from the BESIII collaboration and supports assigning the new X state to be a sss̄s̄ tetraquark state with J = 1. The predicted mass mX = 3.08± 0.11GeV disfavors assigning the φ(2170) or Y (2175) to be the vector partner of the new X state. As a byproduct, we obtain the masses of the corresponding qqq̄q̄ tetraquark states. The light tetraquark states lie in the region about 2GeV rather than 1GeV. PACS number: 12.39.Mk, 12.38.Lg


Introduction
Recently, the BESIII collaboration studied the process J/ψ → φηη ′ and observed a structure X in the φη ′ mass spectrum [1]. The fitted mass and width are m X = (2002.1 ± 27.5 ± 15.0) MeV and Γ X = (129 ± 17 ± 7) MeV respectively with assumption of the spin-parity J P = 1 − , the corresponding significance is 5.3σ; while the fitted mass and width are m X = ((2062.8 ± 13.1 ± 4.2) MeV and Γ X = (177 ± 36 ± 20) MeV respectively with assumption of the spin-parity J P = 1 + , the corresponding significance is 4.9σ. The X state was observed in the φη ′ decay model rather than in the φη decay model, they maybe contain a large ssss component, in other words, it maybe have a large tetraquark component. In Ref. [2], Wang, Luo and Liu assign the X state to be the second radial excitation of the h 1 (1380). In Ref. [3], Cui et al assign the X to be the partner of the tetraquark state Y (2175) with the J P C = 1 +− .
We usually assign the lowest scalar nonet mesons {f 0 (500), a 0 (980), κ 0 (800), f 0 (980)} to be tetraquark states, and assign the higher scalar nonet mesons {f 0 (1370), a 0 (1450), K * 0 (1430), f 0 (1500)} to be the conventional 3 P 0 quark-antiquark states [4,5,6]. In Ref. [7], we take the nonet scalar mesons below 1 GeV 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 1 GeV are conventional two-quark states. The light tetraquark states maybe lie in the region about 2 GeV rather than lie in the region about 1 GeV.
In this article, we take the axialvector diquark operators as the basic constituents to construct the tetraquark current operators to study the scalar (S), axialvector (A), tensor (T ) and vector (V ) tetraquark states with the QCD sum rules, explore the possible assignments of the new X 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 ssss tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.

QCD sum rules for the ssss tetraquark states
We write down the two-point correlation functions Π µναβ (p) and Π(p) firstly, where J µν (x) = J 2,µν (x), J 1,µν (x), where the i, j, k, m, n are color indexes, the C is the charge conjugation matrix. Under charge conjugation transform C, the currents J µν (x) and J 0 (x) have the properties, At the hadronic side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µν (x) and J 0 (x) into the correlation functions Π µναβ (p) and Π(p) 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, where g µν = g µν − pµpν p 2 , the subscripts 2 + , 1 + , 1 − and 0 + denote the spin-parity J P of the corresponding tetraquark states. The pole residues λ X and λ X are defined by where the ε µν and ε µ are the polarization vectors of the tetraquark states. Now we contract the s quarks in the correlation functions with Wick theorem, there are four s-quark propagators, if two s-quark lines emit a gluon by itself and the other two s-quark lines contribute a quark pair by itself, we obtain a operator GGssss, which is of order O(α k s ) with k = 1 and of dimension 10. In this article, we take into account the vacuum condensates up to dimension 10 and k ≤ 1 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 quarkhadron duality below the continuum thresholds s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: where and λ X A/V = m X A/V λ X A/V . We derive Eq.(9) with respect to τ = 1 T 2 , then obtain the QCD sum rules for the masses of the tetraquark states through a fraction, 3 Numerical results and discussions  at the energy scale µ = 1 GeV [10,11,13], and choose the M S mass m s (µ = 2 GeV) = 0.095 ± 0.005 GeV from the Particle Data Group [14], and evolve the s-quark mass to the energy scale µ = 1 GeV with the renormalization group equation, furthermore, we neglect the small u and d quark masses. We choose suitable Borel parameters and continuum threshold parameters to warrant the pole contributions (PC) are larger than 40%, i.e.
and convergence of the operator product expansion. The contributions of the vacuum condensates D(n) in the operator product expansion are defined by, where the subscript n in the QCD spectral density ρ n (s) denotes the dimension of the vacuum condensates. We choose the values |D(10)| ∼ 1% 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 10. 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 ssss 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 0.6 GeV [15]. In Fig.1, we plot the masses of the scalar, axialvector, tensor and vector ssss 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 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,

Conclusion
In this article, we construct the axialvector-diquark-axialvector-antidiquark type currents to interpolate the scalar, axialvector, tensor and vector ssss 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 m X = 2.08 ± 0.12 GeV for the axialvector tetraquark state is in excellent agreement with the experimental value, m X = (2062.8 ± 13.1 ± 4.2) MeV, from the BESIII collaboration and supports assigning the new X state to be an axialvector-diquark-axialvector-antidiquark type ssss tetraquark state. The predicted mass m X = 3.08 ± 0.11 GeV for the vector tetraquark state lies above the experimental value of the mass of the φ(2170), m φ = 2188 ± 10 MeV, from the Particle Data Group, and disfavors assigning the φ(2170) to be the vector partner of the new X state. As a byproduct, we also obtain the masses and pole residues of the corresponding qqqq tetraquark states. The present predictions can be confronted to the experimental data in the future.