Strong decays of the Y(4660) as a vector tetraquark state in solid quark-hadron duality

In this article, we choose the [sc]P[s¯c¯]A-[sc]A[s¯c¯]P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[sc]_P[\bar{s}\bar{c}]_A-[sc]_A[\bar{s}\bar{c}]_P$$\end{document} type tetraquark current to study the hadronic coupling constants in the strong decays Y(4660)→J/ψf0(980)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y(4660)\rightarrow J/\psi f_0(980)$$\end{document}, ηcϕ(1020)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \eta _c \phi (1020)$$\end{document}, χc0ϕ(1020)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \chi _{c0}\phi (1020)$$\end{document}, DsD¯s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D_s \bar{D}_s$$\end{document}, Ds∗D¯s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D_s^* \bar{D}^*_s$$\end{document}, DsD¯s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D_s \bar{D}^*_s$$\end{document}, Ds∗D¯s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D_s^* \bar{D}_s$$\end{document}, ψ′π+π-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \psi ^\prime \pi ^+\pi ^-$$\end{document}, J/ψϕ(1020)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi \phi (1020)$$\end{document} with the QCD sum rules based on solid quark-hadron quality. The predicted width Γ(Y(4660))=74.2-19.2+29.2MeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma (Y(4660) )= 74.2^{+29.2}_{-19.2}\,\mathrm{{MeV}}$$\end{document} is in excellent agreement with the experimental data 68±11±1MeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$68\pm 11\pm 1 { \text{ MeV }}$$\end{document} from the Belle collaboration, which supports assigning the Y(4660) to be the [sc]P[s¯c¯]A-[sc]A[s¯c¯]P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[sc]_P[\bar{s}\bar{c}]_A-[sc]_A[\bar{s}\bar{c}]_P$$\end{document} type tetraquark state with JPC=1--\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{PC}=1^{--}$$\end{document}. In calculations, we observe that the hadronic coupling constants |GYψ′f0|≫|GYJ/ψf0|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |G_{Y\psi ^\prime f_0}|\gg |G_{Y J/\psi f_0}|$$\end{document}, which is consistent with the observation of the Y(4660) in the ψ′π+π-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^\prime \pi ^+\pi ^-$$\end{document} mass spectrum, and favors the ψ′f0(980)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^{\prime }f_0(980)$$\end{document} molecule assignment. It is important to search for the process Y(4660)→J/ψϕ(1020)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y(4660)\rightarrow J/\psi \phi (1020)$$\end{document} to diagnose the nature of the Y(4660), as the decay is greatly suppressed.


Introduction
In 2007, the Belle collaboration observed the Y (4360) and Y (4660) in the π + π − ψ invariant mass distribution with statistical significances 8.0σ and 5.8σ respectively in the precess e + e − → γ ISR π + π − ψ between threshold and √ s = 5.5 GeV using 673fb −1 of data collected with the Belle detector at KEKB [1]. In 2008, the Belle collaboration observed the Y (4630) in the + c − c invariant mass distribution with a significance of 8.2σ in the exclusive process e + e − → γ ISR + c − c with an integrated luminosity of 695fb −1 at the KEKB [2]. The values of the mass and width of the Y (4630) are consistent within errors with that of the new charmonium-like state Y (4660).
In 2014, the Belle collaboration measured the e + e − → γ ISR π + π − ψ cross section from 4.0 to 5.5 GeV with the full data sample of the Belle experiment using the ISR (initial state radiation) technique, and determined the parameters of a e-mail: zgwang@aliyun.com the Y (4360) and Y (4660) resonances and superseded previous Belle determination [3]. The masses and widths are shown explicitly in Table 1. Furthermore, the Belle collaboration studied the π + π − invariant mass distribution and observed that there are two clusters of events around the masses of the f 0 (500) and f 0 (980) corresponding to the Y (4360) and Y (4660), respectively. The J PC quantum numbers of the final states accompanying the ISR photon(s) are restricted to J PC = 1 −− . According to potential model calculations [4][5][6], the 4 3 S 1 , 5 3 S 1 , 6 3 S 1 and 3 3 D 1 charmonium states are expected to be in the mass range close to the two resonances Y (4360) and Y (4660), however, there are no enough vector charmonium candidates which can match those new Y states consistently. Now, let us begin with discussing the nature of the f 0 (500) and f 0 (980) to explore the Y (4660). In the scenario of conventional two-quark states, the structures of the f 0 (500) and f 0 (980) in the ideal mixing limit can be symbolically written as, While in the scenario of tetraquark states, the structures of the f 0 (500) and f 0 (980) in the ideal mixing limit can be symbolically written as [7][8][9], In Ref. [10], 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. We determine the mixing angles, which indicate that the dominant components are the two-quark components. The Y (4660) maybe havess constituent. The decay Y (4630) →   [26] The threshold of the ψ f 0 (980) is 4676 MeV from the Particle Data Group [11], which is just above the mass m Y (4660) = 4652 ± 10 ± 8 MeV from the Belle collaboration [3]. The Y (4660) can be assigned to be a ψ f 0 (980) molecular state [12][13][14][15] or a ψ f 0 (980) hadro-charmonium [16]. Other assignments, such as a 2P [cq] S [cq] S tetraquark state [17,18], a ψ(6S) state [6], a ψ(5S) state [19], a ground state P-wave tetraquark state [20][21][22][23][24][25][26] are also possible.
In Table 2, we list out the predictions of the masses of the vector tetraquark (tetraquark molecule) states based on the QCD sum rules [14,15,[20][21][22][23][24][25][26], where the S, P, A and V denote the scalar (S), pseudoscalar (P), axialvector (A) and vector (V ) diquark states. From the Table, we can see that it is not difficult to reproduce the experimental value of the mass of the Y (4660) 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 hiddenbottom) tetraquark states and molecular states, the integrals are sensitive to the energy scales μ, where the ρ QC D (s, μ) are the QCD spectral densities, the T 2 are the Borel parameters, the s 0 are the continuum thresholds parameters, the predicted masses depend heavily on the energy scales μ. In Refs. [23,[27][28][29][30], we suggest an energy scale formula μ = 2 with the effective Q-quark mass M Q 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 (40−60)%, the largest pole contributions up to now. Compared to the old values obtained in Ref. [23], the new values based on detailed analysis with the updated parameters are preferred [24]. The energy scale formula also works well in the QCD sum rules for the hidden-charm pentaquark states [31][32][33].
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 GGqqqq, which is of dimension 10, we should take into account the vacuum condensates at least up to dimension 10 in the operator product expansion.
In Refs. [23][24][25]34,35], 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 10, and use the energy 2 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 for the Y ( In Ref. [36], we assign the Z ± c (3900) to be the diquarkantidiquark type axialvector tetraquark state, study the hadronic coupling constants G Z c J/ψπ , G Z c η c ρ , G Z c DD * 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 X (4140/4274) → J/ψφ(1020) [37,38].  (1020) with the QCD sum rules based on the solid quark-hadron duality, and reexamine the assignment of the Y (4660).
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 Sect. 2, in Sect. 3, we obtain the QCD sum rules for the hadronic coupling constants is reserved for our conclusion.

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 ( p, q) firstly, where the currents J A (0) interpolate the tetraquark states A, the J B (x) and J C (y) interpolate the conventional mesons B and C, respectively, the λ A , λ B and λ C are the pole residues or decay constants. At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J A (0), J B (x), J C (y) into the threepoint correlation functions ( p, q) and isolate the ground state contributions to obtain the result [39][40][41], where p = p + q, the G ABC are the hadronic coupling constants defined by the four functions ρ AC ( have complex dependence on the transitions between the ground states and the higher resonances or the continuum states. We rewrite the correlation functions H ( p 2 , p 2 , q 2 ) at the hadron side as through dispersion relation, where the ρ H (s , s, u) are the hadronic spectral densities, where the 2 s and 2 u are the thresholds, the s 0 A , s 0 B , u 0 C are the continuum thresholds. Now we carry out the operator product expansion at the QCD side, and write the correlation functions through dispersion relation, where the ρ QC D ( p 2 , s, u) are the QCD spectral densities, However, the QCD spectral densities ρ QC D (s , s, u) do not exist, because lim 3 →0 Thereafter we will write the QCD spectral densities ρ QC D ( p 2 , s, u) as ρ QC D (s, u) 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 ds firstly to obtain the solid duality [36], the 2 denotes the thresholds (m B + m C ) 2 . Now we write down the quark-hadron duality explicitly, No approximation is needed, we do not need the continuum threshold parameter s 0 A in the s channel. The s channel and s channel are quite different, we can not set the continuum threshold parameters in the s channel as s 0 2 in the present case, where the B denotes the J/ψ, η c ,D s ,D * s , because the contaminations from the excited states ψ , η c ,D s ,D * s are out of control. We can introduce the parameters C AC , C AB , C A B and C A C to parameterize the net effects, In numerical calculations, we take the relevant functions C A B and C A C 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 B are charmonium or bottomnium states, we set p 2 = p 2 and perform the double Borel transform with respect to the variables P 2 = −p 2 and Q 2 = −q 2 , respectively to obtain the QCD sum rules, where the T 2 1 and T 2 2 are the Borel parameters. If the B are open-charm or open-bottom mesons, we set p 2 = 4 p 2 and perform the double Borel transform with respect to the variables P 2 = −p 2 and Q 2 = −q 2 , respectively to obtain the QCD sum rules, where

The width of the Y (4660) as a vector tetraquark state
Now we write down the three-point correlation functions for the strong decays Y (4660) → J/ψ f 0 (980), η c φ(1020), χ c0 φ(1020), D sDs , D * sD * s , D sD * s , D * sD s , ψ π + π − , J/ψφ(1020), respectively, and apply the method presented in previous section to obtain the QCD sum rules for the hadronic coupling constants where For the two-body strong decay Y (4660) → η c φ(1020), the correlation function is where For the two-body strong decay Y (4660) → χ c0 φ(1020), the correlation function is where For the two-body strong decay Y (4660) → D sDs , the correlation function is where J D s (y) =s(y)iγ 5 c(y) .
For the two-body strong decay Y (4660) → D * sD * s , the correlation function is where For the two-body strong decay Y (4660) → D sD * s , the correlation function is For the two-body strong decay Y (4660) → J/ψ φ(1020), the correlation function is 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 [39][40][41].
For the decays Y (4660) → J/ψ f 0 (980), ψ f 0 (980) * , the correlation function can be written as For the decay Y (4660) → η c φ(1020), the correlation function can be written as For the decay Y (4660) → χ c0 φ(1020), the correlation function can be written as For the decay Y (4660) → D sDs , the correlation function can be written as For the decay Y (4660) → D * sD * s , the correlation function can be written as For the decay Y (4660) → D sD * s , the correlation function can be written as For the decay Y (4660) → J/ψ φ(1020), the correlation function can be written as In calculations, we observe that the hadronic coupling constant G Y J/ψφ is zero at the leading order approximation, and we will neglect the process Y (4660) → J/ψ φ(1020).
In Eqs. (31)-(37), we have used the following definitions for the decay constants and hadronic coupling constants, where the ξ , G Y J/ψφ are the hadronic coupling constants. We study the components ( p 2 , p 2 , q 2 ) of the correlation functions, and carry out the operator product expansion up to the vacuum condensates of dimension 5 and neglect the tiny contributions of the gluon condensate. Then we obtain the QCD spectral densities through dispersion relation and use Eqs. (17) and (18) to obtain the QCD sum rules for the hadronic coupling constants,   The decay Y (4660) → ψ f 0 (980) is kinematically forbidden, but the decay Y (4660) → ψ π + π − can take place through a virtual intermediate f 0 (980) * , the partial decay width can be written as, with the observation of the Y (4660) in the ψ π + π − mass spectrum, and favors the ψ f 0 (980) molecule assignment, as there may be appear some ψ f 0 (980) component due to the strong coupling. The decay Y (4660) → J/ψφ(1020) is greatly suppressed and can take place only through rescattering mechanism. It is important to search for the process Y (4660) → J/ψφ(1020) to diagnose the nature of the Y (4660).