Abstract
In this article, we extend our previous work to study the mass spectrum of the ground state hiddenbottom tetraquark states with the QCD sum rules in a systematic way. The predicted hiddenbottom tetraquark masses can be confronted to the experimental data in the future to diagnose the nature of the states. In calculations, we observe that the scalar diquark states, the axialvector diquark states and the axialvector components of the tensor diquark state are all good diquarks in building the lowest tetraquark states.
Analysis of the hiddenbottom tetraquark mass spectrum with the QCD sum rules
[2mm] ZhiGang Wang ^{1}^{1}1Email: 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
In 2011, the Belle collaboration observed the and in the and mass spectrum for the first time, the favored quantum numbers (Isospin, parity, Spin, Parity) are [1]. Later, the Belle collaboration updated the values of the masses and widths , , and [2], which are adopted in the Review of Particle Physics by the Particle Data Group now [3]. The possible assignments of the and are the tetraquark states [4, 5, 6], molecular states [7, 8, 9, 10, 11], threshold cusps [12], rescattering effects [13], etc. For more literatures on the states, one can consult the old review [14] and the recent review [15].
In 2013, the BESIII collaboration (also the Belle collaboration) observed the in the mass spectrum [16, 17]. Later, the BESIII collaboration observed the near the threshold [18], and the in the mass spectrum [19]. Now the and are taken to be the same particle, and are denoted as the in the Review of Particle Physics [3].
The and ( and ) are charged charmoniumlike states (bottomoniumlike states), their quark constituents must be or ( or ), irrespective of the diquarkantidiquark type or mesonmeson type substructures. The , , and are observed in the analogous decays to the final states , , , , and should have analogous structures. In Refs.[6, 20, 21], we assign the , , and to be the diquarkantidiquark type axialvector tetraquark states, and study their masses with the QCD sum rules in details. Furthermore, we explore the energy scale dependence of the hiddencharm and hiddenbottom tetraquark states for the first time [20], and suggest a formula,
(1) 
with the effective heavy mass to determine the optimal energy scales of the QCD spectral densities [6, 21]. The experimental values of the masses can be well reproduced. In Ref.[22], we study the twobody strong decays , , with the QCD sum rules in details. We take into account both the connected and disconnected Feynman diagrams, and pay special attentions to matching the hadron side with the QCD side of the correlation functions to obtain solid duality, the predicted width is consistent with the experimental data [16, 17], and supports assigning the to be the diquarkantidiquark type axialvector tetraquark state [20]. In Ref.[6], we use the method proposed in Ref.[23] to study the twobody strong decays , with the QCD sum rules by taking into account only the connected Feynman diagrams. Although the predictions are good, the subtractions of the higher resonances and continuum states are introduced by hand, the contaminations cannot be subtracted completely. The widths of the and should be studied in a consistent way to make the assignments more robust. A updated analysis of the masses and widths with the QCD sum rules is needed.
If the and are the diquarkantidiquark type axialvector hiddenbottom tetraquark states, there should exist a holonomic spectrum for the scalar, axialvector and tensor hiddenbottom tetraquark states without introducing an additional Pwave. Now we extend our previous work to study the mass spectrum of the hiddenbottom tetraquark states in a systematic way. Those hiddenbottom tetraquark states may be observed at the LHCb, Belle II, CEPC (Circular Electron Positron Collider), FCC (Future Circular Collider), ILC (International Linear Collider) in the future, and shed light on the nature of the exotic , , particles.
We usually take the diquarks (in color antitriplet) and antidiquarks (in color triplet) as the basic building blocks to construct the tetraquark states. The diquarks (or diquark operators) have five structures in Dirac spinor space, where , , , and (or ) for the scalar (), pseudoscalar (), vector (), axialvector () and tensor () diquarks, respectively, the , , are color indexes. The scalar, pseudoscalar, vector and axialvector diquark states have been studied with the QCD sum rules in details, the good diquark correlations in building the lowest tetraquark states are the scalar and axialvector diquark states [24, 25], the axialvector diquark states are not bad diquark states.
Under parity transform , the tensor diquark operators have the properties,
(2) 
where and . The tensor diquark states have both and components, we introduce the four vector and project out the and components explicitly,
(3) 
where , , , [26]. Thereafter, we will denote the axialvector diquark operators , as , and the vector diquark operators , as .
In this article, we take the scalar (), axialvector (, ), vector (, ) diquark operators and antidiquark operators as the basic building blocks to construct the tetraquark operators to study the mass spectrum of the hiddenbottom tetraquark states with the QCD sum rules in a systematic way.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the hiddenbottom 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 hiddenbottom tetraquark states
We write down the twopoint correlation functions , and in the QCD sum rules,
(4) 
where the currents , , , , , , , , , , , ,
(5) 
the , , , , are color indexes, the is the charge conjugation matrix, the subscripts denote the positive charge conjugation and negative charge conjugation, respectively. The diquark operators and have both positive parity and negative parity components, we project out the and components unambiguously with suitable diquark operators to obtain the current operators , and with the . The current operators and couple potentially to both the positive parity and negative parity tetraquark states, we separate those contributions explicitly to obtain reliable QCD sum rules. In Table 1, we present the quark structures and corresponding interpolating currents for the hiddenbottom tetraquark states. The four vector breaks down Lorentz covariance, the currents , , , , and are not Lorentz covariant, it is the shortcoming of the present method, the calculations can be understood as carried out at a particular (or given) coordinate system, which cannot impair the predictive ability.
Currents  
























At the hadron side, we 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 [27, 28], and isolate the ground state hiddenbottom tetraquark contributions,
(6)  
where , the superscripts (subscripts) in the hiddenbottom tetraquark states (components , ) denote the positiveparity and negative parity, respectively. The pole residues are defined by
(7) 
the and are the polarization vectors of the hiddenbottom tetraquark states. In this article, we choose the components and to study the scalar, axialvector and tensor hiddenbottom tetraquark states with the QCD sum rules.
At the QCD side, we carry out the operator product expansion for the correlation functions , and up to the vacuum condensates of dimension in a consistent way, and obtain the QCD spectral densities through dispersion relation. We match the hadron side with the QCD side of the correlation functions below the continuum threshold and perform Borel transform with respect to to obtain the QCD sum rules:
(8) 
The explicit expressions of the QCD spectral densities are available upon request by contacting me via Email. For the technical details, one can consult Refs.[20, 21].
We derive Eq.(8) with respect to , and obtain the QCD sum rules for the masses of the scalar, axialvector and tensor hiddenbottom tetraquark states through a ratio,
(9) 
3 Numerical results and discussions
We take the standard values of the vacuum condensates , , , at the energy scale [27, 28, 29], and take the mass from the Particle Data Group [3], and set . Furthermore, we take into account the energyscale dependence of the input parameters at the QCD side,
(10) 
where , , , , , and for the flavors , and , respectively [3, 30], and evolve all the input parameters to the optimal energy scales with the flavor to extract the tetraquark masses.
In all the QCD sum rules for the hiddencharm tetraquark states [21, 31], hiddencharm pentaquark states [32] and hiddenbottom tetraquark states [6], we search for the optimal Borel parameters and continuum threshold parameters to satisfy the four criteria:
Pole dominance at the hadron side;
Convergence of the operator product expansion at the QCD side;
Appearance of the Borel platforms;
Satisfying the energy scale formula,
via try and error.
The pole contributions (PC) or ground state tetraquark contributions are defined by