1 Introduction
###### Abstract

In this article, we choose the type tetraquark current to study the hadronic coupling constants in the strong decays , , , , , , , , with the QCD sum rules based on solid quark-hadron quality. The predicted width is in excellent agreement with the experimental data from the Belle collaboration, which supports assigning the to be the type tetraquark state with . In calculations, we observe that the hadronic coupling constants , which is consistent with the observation of the in the mass spectrum, and favors the molecule assignment. It is important to search for the process to diagnose the nature of the , as the decay is greatly suppressed.

Strong decays of the as an vector tetraquark state in solid quark-hadron duality

[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 2007, the Belle collaboration observed the and in the invariant mass distribution with statistical significances and respectively in the precess between threshold and using of data collected with the Belle detector at KEKB [1]. In 2008, the Belle collaboration observed the in the invariant mass distribution with a significance of in the exclusive process with an integrated luminosity of at the KEKB [2].

In 2014, the Belle collaboration measured the cross section from to with the full data sample of the Belle experiment using the ISR technique, and determined the parameters of the and resonances and superseded previous Belle determination [3]. The masses and widths are shown explicitly in Table 1. Furthermore, the Belle collaboration studied the invariant mass distribution and observed that there are two clusters of events around the masses of the and corresponding to the and , respectively.

In the scenario of conventional two-quark states, the structures of the and in the ideal mixing limit can be symbolically written as,

 f0(500)=¯uu+¯dd√2,f0(980)=¯ss. (1)

While in the scenario of tetraquark states, the structures of the and in the ideal mixing limit can be symbolically written as [4, 5, 6],

 f0(500)=ud¯u¯d,f0(980)=us¯u¯s+ds¯d¯s√2. (2)

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. We determine the mixing angles, which indicate that the dominant components are the two-quark components. The maybe have constituent. The decay takes place, if the and are the same particle, the decay is Okubo-Zweig-Iizuka suppressed, there should be some rescattering mechanism to account for the decay.

The threshold of the is from the Particle Data Group [8], which is just above the mass from the Belle collaboration [3]. The can be assigned to be a molecular state [9, 10, 11] or a hadro-charmonium [12]. Other assignments, such as a 2P tetraquark state [13], a state [14], a state [15], a ground state P-wave tetraquark state [16, 17, 18, 19, 20, 21, 22] are also possible.

In Table 2, we list out the predictions of the masses based on the QCD sum rules [10, 11, 16, 17, 18, 19, 20, 21, 22], where the , , and denote the scalar (), pseudoscalar (), axialvector () and vector () diquark states. From the Table, we can see that it is not difficult to reproduce the experimental value of the mass of the with the QCD sum rules. However, the quantitative predications depend on the quark structures, the input parameters at the QCD side, the pole contributions of the ground states, and the truncations of the operator product expansion.

In the QCD sum rules for the hidden-charm (or hidden-bottom) tetraquark states and molecular states, the integrals

 ∫s04m2Q(μ)dsρQCD(s,μ)exp(−sT2), (3)

are sensitive to the energy scales , where the are the QCD spectral densities, the are the Borel parameters, the are the continuum thresholds parameters, the predicted masses depend heavily on the energy scales . In Refs.[19, 23, 24], we suggest an energy scale formula with the effective -quark mass to determine the ideal energy scales of the QCD spectral densities. The formula enhances the pole contributions remarkably, we obtain the pole contributions as large as , the largest pole contributions up to now. Compared to the old values obtained in Ref.[19], the new values based on detailed analysis with the updated parameters are preferred [20]. The energy scale formula also works well in the QCD sum rules for the hidden-charm pentaquark states [25].

For the correlation functions of the hidden-charm (or hidden-bottom) tetraquark currents, there are two heavy quark propagators and two light quark propagators, if each heavy quark line emits a gluon and each light quark line contributes a quark pair, we obtain a operator , which is of dimension , we should take into account the vacuum condensates at least up to dimension in the operator product expansion.

In Refs.[19, 20, 21, 26], we study the mass spectrum of the vector tetraquark states in a comprehensive way by carrying out the operator product expansion up to the vacuum condensates of dimension , and use the energy scale formula or modified energy scale formula to determine the ideal energy scales of the QCD spectral densities in a consistent way. In the scenario of tetraquark states, we observe that the preferred quark configurations are the and . In this article, we choose the quark configuration .

In Ref.[27], we assign the to be the diquark-antidiquark type axialvector tetraquark state, study the hadronic coupling constants , , with the QCD sum rules by taking into account both the connected and disconnected Feynman diagrams in the operator product expansion. We pay special attentions to matching the hadron side of the correlation functions with the QCD side of the correlation functions to obtain solid duality. The routine works well in studying the decays [28].

In this article, we assign the to be the vector tetraquark state, and study the strong decays , , , , , , , , with the QCD sum rules based on the solid quark-hadron duality, and reexamine the assignment of the .

The article is arranged as follows: we illustrate how to calculate the hadronic coupling constants in the two-body strong decays of the tetraquark states with the QCD sum rules in section 2, in section 3, we obtain the QCD sum rules for the hadronic coupling constants , , , , , , , ; section 4 is reserved for our conclusion.

## 2 The hadronic coupling constants in the two-body strong decays of the tetraquark states

In this section, we illustrate how to calculate the hadronic coupling constants in the two-body strong decays of the tetraquark states with the QCD sum rules. We write down the three-point correlation functions ,

 Π(p,q) = i2∫d4xd4yeipxeiqy⟨0|T{JB(x)JC(y)J†A(0)}|0⟩, (4)

where the currents interpolate the tetraquark states , the and interpolate the conventional mesons and respectively,

 ⟨0|JA(0)|A(p′)⟩ = λA, ⟨0|JB(0)|B(p)⟩ = λB, ⟨0|JC(0)|C(q)⟩ = λC, (5)

the , and are the pole residues or the decay constants.

At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators , , into the three-point correlation functions and isolate the ground state contributions to obtain the result [29, 30],

 Π(p,q) = λAλBλCGABC(m2A−p′2)(m2B−p2)(m2C−q2)+1(m2A−p′2)(m2B−p2)∫∞s0CdtρAC′(p′2,p2,t)t−q2 (6) +1(m2A−p′2)(m2C−q2)∫∞s0BdtρAB′(p′2,t,q2)t−p2 +1(m2B−p2)(m2C−q2)∫∞s0AdtρA′B(t,p2,q2)+ρA′C(t,p2,q2)t−p′2+⋯ = Π(p′2,p2,q2),

where , the are the hadronic coupling constants defined by

 ⟨B(p)C(q)|A(p′)⟩ = iGABC, (7)

the four functions , , and have complex dependence on the transitions between the ground states and the higher resonances or the continuum states.

We rewrite the correlation functions at the hadron side as

through dispersion relation, where the are the hadronic spectral densities,

 ρH(s′,s,u) = limϵ3→0limϵ2→0limϵ1→0Ims′ImsImuΠH(s′+iϵ3,s+iϵ2,u+iϵ1)π3, (9)

where the and are the thresholds, the , , are the continuum thresholds.

Now we carry out the operator product expansion at the QCD side, and write the correlation functions as

 ΠQCD(p′2,p2,q2) = ∫s0BΔ2sds∫u0CΔ2uduρQCD(p′2,s,u)(s−p2)(u−q2)+⋯, (10)

through dispersion relation, where the are the QCD spectral densities,

 ρQCD(p′2,s,u) = limϵ2→0limϵ1→0ImsImuΠQCD(p′2,s+iϵ2,u+iϵ1)π2. (11)

However, the QCD spectral densities do not exist,

 ρQCD(s′,s,u) = limϵ3→0limϵ2→0limϵ1→0Ims′ImsImuΠQCD(s′+iϵ3,s+iϵ2,u+iϵ1)π3 (12) = 0,

because

 limϵ3→0Ims′ΠQCD(s′+iϵ3,p2,q2)π = 0 (13)

Thereafter we will write the QCD spectral densities as for simplicity.

We math the hadron side of the correlation functions with the QCD side of the correlation functions, and carry out the integral over firstly to obtain the solid duality [27],

 ∫s0BΔ2sds∫u0CΔ2uduρQCD(s,u)(s−p2)(u−q2) = ∫s0BΔ2sds∫u0CΔ2udu1(s−p2)(u−q2)[∫∞Δ2ds′ρH(s′,s,u)s′−p′2],

the denotes the thresholds . Now we write down the quark-hadron duality explicitly,

 ∫s0BΔ2cds∫u0CΔ2uduρQCD(s,u)(s−p2)(u−q2) = ∫s0BΔ2cds∫u0CΔ2udu∫∞(mB+mC)2ds′ρH(s′,s,u)(s′−p′2)(s−p2)(u−q2) = λAλBλCGABC(m2A−p′2)(m2B−p2)(m2C−q2)+CA′B+CA′C(m2B−p2)(m2C−q2).

No approximation is needed, we do not need the continuum threshold parameter in the channel. The channel and channel are quite different, we can not set the continuum threshold parameters in the channel as , i.e. we can not set in the present case, where the denote the , , , , because the contaminations from the excited states , , , are out of control.

We can introduce the parameters , , and to parameterize the net effects,

 CAC′ = ∫∞s0CdtρAC′(p′2,p2,t)t−q2, CAB′ = ∫∞s0BdtρAB′(p′2,t,q2)t−p2, CA′B = ∫∞s0AdtρA′B(t,p2,q2)t−p′2, CA′C = ∫∞s0AdtρA′C(t,p2,q2)t−p′2. (16)

In numerical calculations, we take the relevant functions and as free parameters, and choose suitable values to eliminate the contaminations from the higher resonances and continuum states to obtain the stable QCD sum rules with the variations of the Borel parameters.

If the are charmonium or bottomnium states, we set and perform the double Borel transform with respect to the variables and , respectively to obtain the QCD sum rules,

 λAλBλCGABCm2A−m2B[exp(−m2BT21)−exp(−m2AT21)]exp(−m2CT22)+ (CA′B+CA′C)exp(−m2BT21−m2CT22)=∫s0BΔ2sds∫u0CΔ2uduρQCD(s,u)exp(−sT21−uT22), (17)

where the and are the Borel parameters. If the are open-charm or open-bottom mesons, we set and perform the double Borel transform with respect to the variables and , respectively to obtain the QCD sum rules,

 λAλBλCGABC4(˜m2A−m2B)[exp(−m2BT21)−exp(−˜m2AT21)]exp(−m2CT22)+ (CA′B+CA′C)exp(−m2BT21−m2CT22)=∫s0BΔ2sds∫u0CΔ2uduρQCD(s,u)exp(−sT21−uT22), (18)

where .

## 3 The width of the Y(4660) as vector tetraquark state

Now we write down the three-point correlation functions for the strong decays , , , , , , , , , respectively, and apply the method presented in previous section to obtain the QCD sum rules for the hadronic coupling constants , , , , , , , .

For the two-body strong decays , , the correlation function is

 Πμν(p,q) = i2∫d4xd4yeipxeiqy⟨0|T{JJ/ψ,μ(x)Jf0(y)J†ν(0)}|0⟩, (19)

where

 JJ/ψ,μ(x) = ¯c(x)γμc(x), Jf0(y) = ¯s(y)s(y), Jν(0) = εijkεimn√2[sTj(0)Cck(0)¯sm(0)γνC¯cTn(0)−sTj(0)Cγνck(0)¯sm(0)C¯cTn(0)]. (20)

For the two-body strong decay , the correlation function is

 Πμν(p,q) = i2∫d4xd4yeipxeiqy⟨0|T{Jηc(x)Jϕ,μ(y)J†ν(0)}|0⟩, (21)

where

 Jηc(x) = ¯c(x)iγ5c(x), Jϕ,μ(y) = ¯s(y)γμs(y). (22)

For the two-body strong decay , the correlation function is

 Πμν(p,q) = i2∫d4xd4yeipxeiqy⟨0|T{Jχc0(x)Jϕ,μ(y)J†ν(0)}|0⟩, (23)

where

 Jχc0(x) = ¯c(x)c(x). (24)

For the two-body strong decay , the correlation function is

 Πν(p,q) = i2∫d4xd4yeipxeiqy⟨0|T{J†Ds(x)JDs(y)J†ν(0)}|0⟩, (25)

where

 JDs(y) = ¯s(y)iγ5c(y). (26)

For the two-body strong decay , the correlation function is

 Παβν(p,q) = (27)

where

 JD∗s,β(y) = ¯s(y)γβc(y). (28)

For the two-body strong decay , the correlation function is

 Πμν(p,q) = i2∫d4xd4yeipxeiqy⟨0|T{J†D∗s,μ(x)JDs(y)J†ν(0)}|0⟩. (29)

For the two-body strong decay , the correlation function is

 Παβν(p,q) = i2∫d4xd4yeipxeiqy⟨0|T{JJ/ψ,α(x)Jϕ,β(y)J†ν(0)}|0⟩. (30)

At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators into the three-point correlation functions and isolate the ground state contributions to obtain the hadron representation [29, 30],

For the decays , ,

 Πμν(p,q) = fJ/ψmJ/ψff0mf0λYGYJ/ψf0(p′2−m2Y)(p