The decay width of the $Z_c(3900)$ as an axialvector tetraquark state in solid quark-hadron duality

In this article, we tentatively assign the $Z_c^\pm(3900)$ to be the diquark-antidiquark type axialvector tetraquark state, study the hadronic coupling constants $G_{Z_cJ/\psi\pi}$, $G_{Z_c\eta_c\rho}$, $G_{Z_cD \bar{D}^{*}}$ with the QCD sum rules in details. We take into account both the connected and disconnected Feynman diagrams in carrying out the operator product expansion, as the connected Feynman diagrams alone cannot do the work. Special attentions are paid to matching the hadron side of the correlation functions with the QCD side of the correlation functions to obtain solid duality, the routine can be applied to study other hadronic couplings directly. We study the two-body strong decays $Z_c^+(3900)\to J/\psi\pi^+$, $\eta_c\rho^+$, $D^+ \bar{D}^{*0}$, $\bar{D}^0 D^{*+}$ and obtain the total width of the $Z_c^\pm(3900)$. The numerical results support assigning the $Z_c^\pm(3900)$ to be the diquark-antidiquark type axialvector tetraquark state, and assigning the $Z_c^\pm(3885)$ to be the meson-meson type axialvector molecular state.


Introduction
In 2013, the BESIII collaboration studied the process e + e − → π + π − J/ψ at a center-of-mass energy of 4.260 GeV using a 525 pb −1 data sample collected with the BESIII detector, and observed a structure Z c (3900) in the π ± J/ψ mass spectrum [1]. Then the structure Z c (3900) was confirmed by the Belle and CLEO collaborations [2,3]. Also in 2013, the BESIII collaboration studied the process e + e − → πDD * , and observed a distinct charged structure Z c (3885) in the (DD * ) ± mass spectrum [4]. The angular distribution of the πZ c (3885) system favors a J P = 1 + assignment [4]. Furthermore, the BESIII collaboration measured the ratio R exp [4], In 2015, the BESIII collaboration observed the neutral parter Z 0 c (3900) with a significance of 10.4 σ in the process e + e − → π 0 π 0 J/ψ [5]. Recently, the BESIII collaboration determined the spin and parity of the Z ± c (3900) state to be J P = 1 + with a statistical significance larger than 7σ over other quantum numbers in a partial wave analysis of the process e + e − → π + π − J/ψ [6]. Now we list out the mass and width from different measurements.
The values of the mass are consistent with each other from different measurements, while the values of the width differ from each other greatly. The Z c (3900) and Z c (3885) may be the same particle according to the mass, spin and parity.
In Ref. [9], we study the masses and pole residues of the J P C = 1 +± hidden charm tetraquark states with the QCD sum rules by calculating the contributions of the vacuum condensates up to dimension-10 in a consistent way in the operator product expansion, and explore the energy scale dependence in details for the first time. The predicted masses M X = 3.87 +0.09 −0.09 GeV and M Z = 3.91 +0. 11 −0.09 GeV support assigning the X(3872) and Z c (3900) to be the 1 ++ and 1 +− diquarkantidiquark type tetraquark states, respectively.
In Ref. [12], we study the axialvector hidden charm and hidden bottom molecular states with the QCD sum rules by calculating the vacuum condensates up to dimension-10 in the operator product expansion, and explore the energy scale dependence of the QCD sum rules for the heavy molecular states in details. The numerical results support assigning the X(3872), Z c (3900), Z b (10610) to be the color singlet-singlet type molecular states with J P C = 1 ++ , 1 +− , 1 +− , respectively.
We can reproduce the experimental value of the mass of the Z c (3900) based on the QCD sum rules both in the scenario of tetraquark states and in the scenario of molecule states [9,12]. Additional theoretical works on the width are still needed to identify the Z c (3900).
In Ref. [10], Dias et al identify the Z ± c (3900) as the charged partner of the X(3872) state, and study the two-body strong decays Z + c (3900) → J/ψπ + , η c ρ + , D +D * 0 , D 0D * + with the QCD sum rules by evaluating the three-point correlation functions and take into account only the connected Feynman Diagrams, and they obtain the width Γ Zc = 63.0 ± 18.1 MeV.
It is interesting to know that the connected Feynman Diagrams alone or the connected plus disconnected Feynman Diagrams lead to the same result [10,15]. As far as the X(5568) is concerned, if we take the scenario of tetraquark states, the width can also be reproduced based on the connected Feynman Diagrams alone [16] or the connected plus disconnected Feynman Diagrams [17,18]. We should prove that the contributions of the disconnected Feynman diagrams can be neglected safely.
In this article, we assign the Z c (3900) to be the diquark-antidiquark type tetraquark state with J P C = 1 +− , study the hadronic coupling constants G ZcJ/ψπ , G Zcηcρ , G ZcDD * with the three-point QCD sum rules by including both the connected and disconnected Feynman diagrams, special attentions are paid to the hadronic spectral densities of the three-point correlation functions, then calculate the partial decay widths of the strong decays Z + c (3900) → J/ψπ + , η c ρ + , D +D * 0 , D 0D * + , and diagnose the nature of the Z ± c (3900) based on the width and the ratio R exp = 6.2 ± 1.1 ± 2.7, if the Z c (3900) and Z c (3885) are the same particle with the diquark-antidiquark type structure.
The article is arranged as follows: we derive the QCD sum rules for the hadronic coupling constants G ZcJ/ψπ , G Zcηcρ , G ZcDD * in section 2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion.
2 The width of the Z c (3900) as an axialvector tetraquark state We study the two-body strong decays Z + c (3900) → J/ψπ + , η c ρ + , D +D * 0 ,D 0 D * + with the following three-point correlation functions Π 1 µν (p, q), Π 2 µν (p, q) and Π 3 µν (p, q), respectively, where Zc are the decay constants of the mesons J/ψ, π, η c , ρ, D * , D and Z c (3900), respectively, the G ZcJ/ψπ , G Zcηcρ and G ZcDD * are the hadronic coupling constants, which are defined by the ξ, ε, ς and ζ are polarization vectors of the J/ψ, ρ, D * and Z c (3900), respectively. The s 0 have complex dependence on the transitions between the ground states and the high resonances or the continuum states.
In this article, we choose the tensor g µν to study the hadronic coupling constants G ZcJ/ψπ , G Zcηcρ and G ZcD * D to avoid the contaminations from the corresponding scalar and pseudoscalar mesons, as the following current-meson couplings are non-vanishing, where the f χc0 , f a0 , f D * 0 , λ Zc0 are the decay constants of the χ c0 (3414), a 0 (980), D * 0 (2400) and Z c (J P = 0 − ), respectively. The terms proportional to p µ p ′ ν in the Π 1 µν (p, q) and Π 3 µν (p, q) and the terms proportional to q µ p ′ ν in the Π 2 µν (p, q) have contaminations from the hadronic coupling constants G Zcχc0π , G ZcD * 0 D and G Zcηca0 , respectively. We introduce the notations Then the correlation functions on the phenomenological side can be written as In numerical calculations, we smear the dependencies of the C Zcπ ′ , on the momentums p ′2 , p 2 , q 2 , and take them as free parameters, and choose the suitable values to eliminate the contaminations from the high resonances and continuum states to obtain the stable QCD sum rules with the variations of the Borel parameters. We carry out the operator product expansion up to the vacuum condensates of dimension 5 and neglect the tiny contributions of the gluon condensate. On the QCD side, the correlation functions Π 1 (p ′2 , p 2 , q 2 ) and Π 2 (p ′2 , p 2 , q 2 ) can be written as where the last two terms originate from the Feynman diagrams where a quark pairqq absorbs a gluon emitted from other quark line. The term in above equations comes from the connected Feynman diagrams, if we set p ′2 = p 2 , then it reduces to It has no contribution after performing the double Borel transformation with respect to the variables P 2 = −p 2 and Q 2 = −q 2 . It is more reasonable to performing the Borel transformation than taking the limit q 2 → 0, as we carry out the operator product expansion at the large spacelike region Q 2 = −q 2 → ∞. So the connected Feynman diagrams have no contributions in the correlation functions Π 1/2 (p ′2 , p 2 , q 2 ), which are in contrary to Refs. [10,16], where only the connected Feynman diagrams have contributions and the limit Q 2 → 0 is taken. For the correlation function Π 3 (p ′2 , p 2 , q 2 ), only the connected Feynman diagrams have contributions, we can set p ′2 = 4p 2 according to the relation M Zc(3900) ≈ 2M D * , the complex expression of the correlation function Π 3 (p ′2 , p 2 , q 2 ) can be reduced to a more simple form, In the limit M 2 π → 0, M 2 ρ → 0, M 2 D → 0 and m 2 c → 0, we maybe expect to choose Q 2 = −q 2 off-shell, and match the terms proportional to 1 Q 2 in the limit Q 2 → 0 on the hadron side with the ones on the QCD side to obtain QCD sum rules for the momentum dependent hadronic coupling constants G ZcJ/ψπ (Q 2 ), G Zcηcρ (Q 2 ), G ZcDD * (Q 2 ), then extract the values to the massshell Q 2 = −M 2 π , −M 2 ρ or −M 2 D to obtain the physical values [10]. However, the approximations M 2 ρ → 0, M 2 D → 0 and m 2 c → 0 are rather crude, and we carry out the operator product expansion at the large space-like region Q 2 = −q 2 → ∞. We prefer taking the imaginary parts of the correlation functions Π 1/2/3 (p ′2 , p 2 , q 2 ) with respect to q 2 + iǫ through dispersion relation and obtain the physical hadronic spectral densities, then take the Borel transform with respect to the Q 2 to obtain the QCD sum rules for the physical hadronic coupling constants.
We have to be cautious in matching the QCD side with the hadron side of the correlation functions Π 1/2/3 (p ′2 , p 2 , q 2 ), as there appears the variable p ′2 = (p+q) 2 . We rewrite the correlation functions Π H 1/2/3 (p ′2 , p 2 , q 2 ) on the hadron side into the following form through dispersion relation, where the ρ H 1/2/3 (s ′ , s, u) are the hadronic spectral densities, we add the superscript H to denote the hadron side. However, on the QCD side, the QCD spectral densities ρ 1/2/3 QCD (s ′ , s, u) do not exist, because lim ǫ3→0 Im we add the superscript QCD to denote the QCD side.
On the QCD side, the correlation functions Π QCD 1/2/3 (p ′2 , p 2 , q 2 ) can be written into the following form through dispersion relation, where the ρ QCD 1/2/3 (p ′2 , s, u) are the QCD spectral densities, We math the hadron side of the correlation functions with the QCD side of the correlation functions, where the integrals over ds ′ are carried out firstly to obtain the solid duality, the ∆ 2  [24], M Zc = 3.899 GeV [1], and obtain the partial decay widths, and the total width, which is consistent with the experimental data considering the uncertainties [1,2,3,5]. If we take the central values of the hadronic coupling constants |G ZcJ/ψπ | = 3.63 GeV, G Zcηcρ = 4.38 GeV, |G ZcDD * | = 0.62 GeV, we can obtain the total width Γ Zc(3900) = 48.9 MeV, which happens to coincide with the central value of the experimental dada Γ = 46 ± 10 ± 20 MeV from the BESIII collaboration [1], while the predicted ratio from the BESIII collaboration [4]. It is difficult to assign the Z c (3900) and Z c (3885) to be the same diquark-antidiquark type axialvector tetraquark state. We can assign the Z c (3900) to be the diquark-antidiquark type axialvector tetraquark state, and assign the Z + c (3885) to be the molecular state D +D * 0 + D * +D0 according to the predicted mass 3.89 ± 0.09 GeV from the QCD sum rules [12]. If the Z c (3885) is the D +D * 0 + D * +D0 molecular state, the decays to D +D * 0 and D * +D0 take place through its component directly, it is easy to account for the large ratio R exp . Now we compare the present work with the work in Ref. [10] in details. In the two works, the same currents are chosen except for the currents to interpolate the π meson, the operator product expansion is carried out at the large space-like regions P 2 = −p 2 → ∞ and Q 2 = −q 2 → ∞. In the present work, we take into account both the connected and disconnected Feynman diagrams, and obtain the solid quark-hadron duality by getting the physical spectral densities through dispersion relation, then perform double Borel transforms with respect to the variables P 2 and Q 2 to obtain the QCD sum rules for the physical hadronic coupling constants directly. We pay special attention to the hadron spectral spectral densities, and present detailed discussions and subtract the continuum contaminations in a solid foundation. In Ref. [10], Dias et al take into account only the connected Feynman diagrams, and obtain the quark-hadron duality by taking the limit Q 2 → 0, M 2 π → 0, M 2 ρ → 0, M 2 D → 0 and m 2 c → 0 and choosing special tensor structures, then perform single Borel transform with respect to the variable P 2 to obtain the QCD sum rules for the momentum dependent hadronic coupling constants. They subtract the continuum contaminations by hand, then parameterize the momentum dependent hadronic coupling constants by some exponential functions with arbitrariness to extract the values to the mass-shell Q 2 = −M 2 π , −M 2 ρ or −M 2 D to obtain the physical hadronic coupling constants. Although the values of the width of the Z c (3900) obtained in the present work and in Ref. [10] are both compatible with the experimental data, the present predictions have much less theoretical uncertainties.

Conclusion
In this article, we tentatively assign the Z ± c (3900) to be the diquark-antidiquark type axialvector tetraquark state, study the hadronic coupling constants G ZcJ/ψπ , G Zcηcρ , G ZcDD * with the QCD sum rules in details. We introduce the three-point 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. In calculations, we take into account both the connected and disconnected Feynman diagrams, as the connected Feynman diagrams alone cannot do the work. Special attentions are paid to matching the hadron side of the correlation functions with the QCD side of the correlation functions to obtain solid duality, the routine can be applied to study other hadronic couplings directly. We study the two-body strong decays Z + c (3900) → J/ψπ + , η c ρ + , D +D * 0 , D 0 D * + and obtain the total width of the Z ± c (3900), which is consistent with the experimental data. The numerical results support assigning the Z ± c (3900) to be the diquark-antidiquark type axialvector tetraquark state, and assigning the Z ± c (3885) to be the meson-meson type axialvector molecular state.