1 Introduction
###### Abstract

In this article, we extend our previous work to study the mass spectrum of the ground state hidden-bottom tetraquark states with the QCD sum rules in a systematic way. The predicted hidden-bottom 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 hidden-bottom tetraquark mass spectrum with the QCD sum rules

[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

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], re-scattering 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 charmonium-like states (bottomonium-like states), their quark constituents must be or ( or ), irrespective of the diquark-antidiquark type or meson-meson 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 diquark-antidiquark type axialvector tetraquark states, and study their masses with the QCD sum rules in details. Furthermore, we explore the energy scale dependence of the hidden-charm and hidden-bottom tetraquark states for the first time [20], and suggest a formula,

 μ = √M2X/Y/Z−(2MQ)2, (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 two-body 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 diquark-antidiquark type axialvector tetraquark state [20]. In Ref.[6], we use the method proposed in Ref.[23] to study the two-body 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 diquark-antidiquark type axialvector hidden-bottom tetraquark states, there should exist a holonomic spectrum for the scalar, axialvector and tensor hidden-bottom tetraquark states without introducing an additional P-wave. Now we extend our previous work to study the mass spectrum of the hidden-bottom tetraquark states in a systematic way. Those hidden-bottom 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,

 ˆPεabcqTb(x)Cσμνγ5Qc(x)ˆP−1 = εabcqTb(~x)Cσμνγ5Qc(~x), ˆPεabcqTb(x)CσμνQc(x)ˆP−1 = −εabcqTb(~x)CσμνQc(~x), (2)

where and . The tensor diquark states have both and components, we introduce the four vector and project out the and components explicitly,

 ˆPεabcqTb(x)Cσtμνγ5Qc(x)ˆP−1 = +εabcqTb(~x)Cσtμνγ5Qc(~x), ˆPεabcqTb(x)CσvμνQc(x)ˆP−1 = +εabcqTb(~x)CσvμνQc(~x), ˆPεabcqTb(x)CσtμνQc(x)ˆP−1 = −εabcqTb(~x)CσtμνQc(~x), ˆPεabcqTb(x)Cσvμνγ5Qc(x)ˆP−1 = −εabcqTb(~x)Cσvμνγ5Qc(~x), (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 hidden-bottom 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 hidden-bottom 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 hidden-bottom tetraquark states

We write down the two-point correlation functions , and in the QCD sum rules,

 Π(p) = i∫d4xeip⋅x⟨0|T{J(x)J†(0)}|0⟩, Πμν(p) = i∫d4xeip⋅x⟨0|T{Jμ(x)J†ν(0)}|0⟩, Πμναβ(p) = i∫d4xeip⋅x⟨0|T{Jμν(x)J†αβ(0)}|0⟩, (4)

where the currents , , , , , , , , , , , ,

 JSS(x) = εijkεimnuTj(x)Cγ5bk(x)¯dm(x)γ5C¯bTn(x), JAA(x) = εijkεimnuTj(x)Cγμbk(x)¯dm(x)γμC¯bTn(x), J~A~A(x) = εijkεimnuTj(x)Cσvμνbk(x)¯dm(x)σμνvC¯bTn(x), J~V~V(x) = εijkεimnuTj(x)Cσtμνbk(x)¯dm(x)σμνtC¯bTn(x), JSA−,μ(x) = εijkεimn√2[uTj(x)Cγ5bk(x)¯dm(x)γμC¯bTn(x)−uTj(x)Cγμbk(x)¯dm(x)γ5C¯bTn(x)], JAA−,μν(x) = εijkεimn√2[uTj(x)Cγμbk(x)¯dm(x)γνC¯bTn(x)−uTj(x)Cγνbk(x)¯dm(x)γμC¯bTn(x)], J˜AA−,μ(x) = εijkεimn√2[uTj(x)Cσμνγ5bk(x)¯dm(x)γνC¯bTn(x)−uTj(x)Cγνbk(x)¯dm(x)γ5σμνC¯bTn(x)], JS˜A−,μν(x) = εijkεimn√2[uTj(x)Cγ5bk(x)¯dm(x)σμνC¯bTn(x)−uTj(x)Cσμνbk(x)¯dm(x)γ5C¯bTn(x)], JSA+,μ(x) = εijkεimn√2[uTj(x)Cγ5bk(x)¯dm(x)γμC¯bTn(x)+uTj(x)Cγμbk(x)¯dm(x)γ5C¯bTn(x)], J˜VV+,μ(x) = εijkεimn√2[uTj(x)Cσμνbk(x)¯dm(x)γ5γνC¯bTn(x)−uTj(x)Cγνγ5bk(x)¯dm(x)σμνC¯bTn(x)], J˜AA+,μ(x) = εijkεimn√2[uTj(x)Cσμνγ5bk(x)¯dm(x)γνC¯bTn(x)+uTj(x)Cγνbk(x)¯dm(x)γ5σμνC¯bTn(x)], JAA+,μν(x) = εijkεimn√2[uTj(x)Cγμbk(x)¯dm(x)γνC¯bTn(x)+uTj(x)Cγνbk(x)¯dm(x)γμC¯bTn(x)], (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 hidden-bottom 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.

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 hidden-bottom tetraquark contributions,

 Π(p) = λ2Z+M2Z+−p2+⋯ = Π+(p2), Πμν(p) = λ2Z+M2Z+−p2(−gμν+pμpνp2)+⋯ = Π+(p2)(−gμν+pμpνp2)+⋯, ΠAA,−μναβ(p) = λ2Z+M2Z+(M2Z+−p2)(p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ) +λ2Z−M2Z−(M2Z−−p2)(−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ)+⋯ = ˜Π+(p2)(p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ) +˜Π−(p2)(−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ), ΠS˜A,−μναβ(p) = λ2Z−M2Z−(M2Z−−p2)(p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ) +λ2Z+M2Z+(M2Z+−p2)(−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ)+⋯ = ˜Π−(p2)(p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ) +˜Π+(p2)(−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ), ΠAA,+μναβ(p) = λ2Z+M2Z+−p2(˜gμα˜gνβ+˜gμβ˜gνα2−˜gμν˜gαβ3)+⋯, (6) = Π+(p2)(˜gμα˜gνβ+˜gμβ˜gνα2−˜gμν˜gαβ3)+⋯,

where , the superscripts (subscripts) in the hidden-bottom tetraquark states (components , ) denote the positive-parity and negative parity, respectively. The pole residues are defined by

 ⟨0|J(0)|Z+b(p)⟩ = λZ+, ⟨0|Jμ(0)|Z+b(p)⟩ = λZ+εμ, ⟨0|JS˜A−,μν(0)|Z−b(p)⟩ = λZ−MZ−εμναβεαpβ, ⟨0|JS˜A−,μν(0)|Z+b(p)⟩ = λZ+MZ+(εμpν−ενpμ), ⟨0|JAA−,μν(0)|Z+b(p)⟩ = λZ+MZ+εμναβεαpβ, ⟨0|JAA−,μν(0)|Z−b(p)⟩ = λZ−MZ−(εμpν−ενpμ), ⟨0|JAA+,μν(0)|Z+b(p)⟩ = λZ+εμν, (7)

the and are the polarization vectors of the hidden-bottom tetraquark states. In this article, we choose the components and to study the scalar, axialvector and tensor hidden-bottom 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 E-mail. 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 hidden-bottom tetraquark states through a ratio,

 M2Z+ = (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 energy-scale dependence of the input parameters at the QCD side,

 ⟨¯qq⟩(μ) = ⟨¯qq⟩(1GeV)[αs(1GeV)αs(μ)]1233−2nf, ⟨¯qgsσGq⟩(μ) = ⟨¯qgsσGq⟩(1GeV)[αs(1GeV)αs(μ)]233−2nf, mb(μ) = mb(mb)[αs(μ)αs(mb)]1233−2nf, αs(μ) = 1b0t[1−b1b20logtt+b21(log2t−logt−1)+b0b2b40t2], (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 hidden-charm tetraquark states [21, 31], hidden-charm pentaquark states [32] and hidden-bottom 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