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

The European Physical Journal C, Jan 2018

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 \(Z_c^+(3900)\rightarrow J/\psi \pi ^+, \eta _c\rho ^+, D^+ \bar{D}^{*0}, \bar{D}^0 D^{*+}\) 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.

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

Eur. Phys. J. C The decay width of the Zc(3900) as an axialvector tetraquark state in solid quark-hadron duality Zhi-Gang Wang 0 Jun-Xia Zhang 0 0 Department of Physics, North China Electric Power University , Baoding 071003 , People's Republic of China In this article, we tentatively assign the Zc±(3900) to be the diquark-antidiquark type axialvector tetraquark state, study the hadronic coupling constants G Zc J/ψπ , G Zcηcρ , G Zc D 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 Zc+(3900) → J /ψ π +, ηcρ+, D+ D¯ ∗0, D¯ 0 D∗+ and obtain the total width of the Zc±(3900). The numerical results support assigning the Zc±(3900) to be the diquark-antidiquark type axialvector tetraquark state, and assigning the Zc±(3885) to be the meson-meson type axialvector molecular state. 1 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 Zc(3900) in the π ± J /ψ mass spectrum [ 1 ]. Then the structure Zc(3900) was confirmed by the Belle and CLEO Collaborations [ 2,3 ]. Also in 2013, the BESIII Collaboration studied the process e+e− → π D D¯ ∗, and observed a distinct charged structure Zc(3885) in the (D D¯ ∗)± mass spectrum [4]. The angular distribution of the π Zc(3885) system favors a J P = 1+ assignment [ 4 ]. Furthermore, the BESIII Collaboration measured the ratio Rex p [ 4 ], Rex p = (Zc(3885) → D D¯ ∗) (Zc(3900) → J /ψ π ) = 6.2 ± 1.1 ± 2.7. (1) In 2015, the BESIII Collaboration observed the neutral parter Zc0(3900) with a significance of 10.4 σ in the process e+e− → π 0π 0 J /ψ [ 5 ]. Recently, the BESIII Collaboration determined the spin and parity of the Zc±(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. Zc±(3900) : M = 3899.0 ± 3.6 ± 4.9 MeV, = 46 ± 10 ± 20 MeV, BESIII [ 1 ], Zc±(3900) : M = 3894.5 ± 6.6 ± 4.5 MeV, Zc±(3900) : M = 3886 ± 4 ± 2 MeV, = 63 ± 24 ± 26 MeV, Belle [ 2 ], = 37 ± 4 ± 8 MeV, CLEO [ 3 ], Zc±(3885) : M = 3883.9 ± 1.5 ± 4.2 MeV, Zc0(3900) : M = 3894.8 ± 2.3 ± 3.2 MeV, = 24.8 ± 3.3 ± 11.0 MeV, BESIII [ 4 ], = 29.6 ± 8.2 ± 8.2 MeV, BESIII [ 5 ]. (2) 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 Zc(3900) and Zc(3885) may be the same particle according to the mass, spin and parity. Faccini et al. tentatively assign the Zc(3900) to be the negative charge conjunction partner of the X (3872) [ 7 ]. There have been several possible assignments, such as tetraquark state [ 8–13 ], molecular state [ 14–20 ], hadro-charmonium [21], rescattering effect [22–24]. In Ref. [ 12 ], we study the masses and pole residues of the J PC = 1+± hidden charm tetraquark states with the QCD sum rules by calculating the contributions of the vacuum 6.2 ± 1.1 ± 2.7, if the Zc(3900) and Zc(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 Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗ in Sect. 2; in Sect. 3, we present the numerical results and discussions; and Sect. 4 is reserved for our conclusion. 2 The width of the Zc(3900) as an axialvector tetraquark state We study the two-body strong decays Zc+(3900) → J /ψ π +, ηcρ+, D+ D¯ ∗0, D¯ 0 D∗+ with the following three-point correlation functions 1μν ( p, q), 2μν ( p, q) and 3μν ( p, q), respectively, 1μν ( p, q) = i2 2μν ( p, q) = i2 3μν ( p, q) = i2 d4xd4 yeipx eiqy 0|T {JμJ/ψ (x)J5π (y)Jν (0)}|0 , (3) d4xd4 yeipx eiqy 0|T {J5ηc (x)Jμρ (y)Jν (0)}|0 , (4) d4xd4 yeipx eiqy 0|T {JμD∗ (x)J5D(y)Jν (0)}|0 , (5) where the currents J J/ψ (x ) = c¯(x )γμc(x ), μ J5π (y) = u¯(y)i γ5d(y), J5ηc (x ) = c¯(x )i γ5c(x ), Jμρ (y) = u¯(y)γμd(y), JμD∗ (x ) = u¯(x )γμc(x ), 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 MX = 3.87+−00..0099 GeV and MZ = 3.91+−00..1019 GeV support assigning the X (3872) and Zc(3900) to be the 1++ and 1+− diquark–antidiquark type tetraquark states, respectively. In Ref. [ 20 ], 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), Zc(3900), Zb(10610) to be the color singletsinglet type molecular states with J PC = 1++, 1+−, 1+−, respectively. We can reproduce the experimental value of the mass of the Zc(3900) based on the QCD sum rules both in the scenario of tetraquark states and in the scenario of molecule states [ 12,20 ]. Additional theoretical works on the width are still needed to identify the Zc(3900). In Ref. [ 13 ], Dias et al identify the Zc±(3900) as the charged partner of the X (3872) state, and study the twobody strong decays Zc+(3900) → J /ψ π +, ηcρ+, D+ D¯ ∗0, D0 D¯ ∗+ with the QCD sum rules by evaluating the threepoint correlation functions and take into account only the connected Feynman diagrams, and they obtain the width Zc = 63.0 ± 18.1 MeV. In Ref. [25], Agaev et al study the two-body strong decays Zc+(3900) → J /ψ π +, ηcρ+ with the light-cone QCD sum rules by taking into account both the connected and disconnected Feynman diagrams, and obtain the width Zc = (Zc+(3900) → J /ψ π +) + (Zc+(3900) → ηcρ+) = 65.7 ± 10.6 MeV. It is interesting to know that the connected Feynman diagrams alone or the connected plus disconnected Feynman diagrams lead to the same result [ 13,25 ]. 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 [26] or the connected plus disconnected Feynman diagrams [27,28]. We should prove that the contributions of the disconnected Feynman diagrams can be neglected safely. In this article, we assign the Zc(3900) to be the diquark– antidiquark type tetraquark state with J PC = 1+−, study the hadronic coupling constants G Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗ 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 Zc+(3900) → J /ψ π +, ηcρ+, D+ D¯ ∗0, D0 D¯ ∗+, and diagnose the nature of the Zc±(3900) based on the width and the ratio Rex p = (6) (7) (8) (9) J5D(y) = c¯(y)i γ5d(y), εi jk εimn Jν (0) = √2 −cn (0)C γ5um (0)c¯k (0)γν C d¯ j (0)}, {cn(0)C γν um (0)c¯k (0)γ5C d¯ j (0) interpolate the mesons J /ψ, π, ηc, ρ , D∗, D and Zc(3900), respectively. We insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators into the three-point correlation functions 1μν ( p, q), 2μν ( p, q) and 3μν ( p, q) [29–31], and isolate the ground state contributions to obtain the following results, 1μν ( p, q) = fπ Mπ2 f J/ψ MJ/ψ λZc G Zc J/ψπ mu + md 3μν ( p, q) = = = = = −i + (MZ2c − p 2)(MJ2/ψ − p2) sπ0 ∞ dt ρZcπ ( p 2, p2, t) t − q2 −i + (MZ2c − p 2)(Mπ2 − q2) s0J/ψ −i + (MJ2/ψ − p2)(Mπ2 − q2) ∞ dt ρZcψ ( p 2, t, q2) t − p2 ∞ dt ρZc J/ψ (t, p2, q2) + ρZcπ (t, p2, q2) × sZ0c t − p 2 × (gμν + · · · ) + · · · 1( p 2, p2, q2) gμν + · · · , −i × (MZ2c − p 2)(MD2∗ − p2)(MD2 − q2) −i + (MZ2c − p 2)(MD2 − q2) s0D∗ −i ∞ + (MD2∗ − p2)(MD2 − q2) sZ0c −i ∞ dt ρZc D ( p 2, p2, t) + (MZ2c − p 2)(MD2∗ − p2) s0D t − q2 ∞ dt ρZc D∗ ( p 2, t, q2) t − p2 dt × ρZc D∗ (t, p2, q2) + ρZc D(t, p2, q2) t − p 2 , 0| J5D(0)|D(q) = mc Zc( p )| Jν (0)|0 = λZc ζν∗ J /ψ ( p)π(q)|Zc( p ) = ξ ∗( p) · ζ ( p ) G Zc J/ψπ , ηc( p)ρ(q)|Zc( p ) = ε∗(q) · ζ ( p )G Zcηcρ , D∗( p)D(q)|Zc( p ) = ς ∗( p) · ζ ( p ) G Zc D D¯ ∗ , the ξ, ε, ς and ζ are polarization vectors of the J /ψ , ρ , D∗ and Zc(3900), respectively. The sπ0 , sJ/ψ , sZ0c , sη0c , sρ0, sD0∗ 0 and sD0 are the continuum threshold parameters. The 12 unknown functions ρZcπ ( p 2, p2, t ), ρZcψ ( p 2, t, q2), ρZcπ (t, p2, q2), ρZc J/ψ (t, p2, q2), ρZcρ ( p 2, p2, t ), ρZcηc ( p 2, t, q2), ρZcρ (t, p2, q2), ρZcηc (t, p2, q2), ρZc D∗ ( p 2, t, q2), ρZc D ( p 2, p2, t ), ρZc D∗ (t, p2, q2), ρZc D(t, p2, q2) 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 gμν to study the hadronic coupling constants G Zc J/ψπ , G Zcηcρ and G Zc D∗ D to avoid the contaminations from the corresponding scalar and pseudoscalar mesons, as the following current-meson couplings are non-vanishing, 0| JμJ/ψ (0)|χc0( p) = fχc0 pμ, 0| Jμρ (0)|a0(q) = fa0 qμ, 0| JμD∗ (0)|D0∗( p) = f D0∗ pμ, Zc0( p )| Jν (0)|0 = −i λZc0 pν , where the fχc0 , fa0 , f D0∗ , λZc0 are the decay constants of the χc0(3414), a0(980), D0∗(2400) and Zc( J P = 0−), respectively. The terms proportional to pμ pν in the 1μν ( p, q) and 3μν ( p, q) and the terms proportional to qμ pν in the 2μν ( p, q) have contaminations from the hadronic coupling constants G Zcχc0π , G Zc D0∗ D and G Zcηca0 , respectively. (12) (13) (14) (15) (16) (17) We introduce the notations CZcπ , CZcψ , CZcπ , CZc J/ψ , CZcρ , CZcηc , CZcρ , CZcηc , CZc D∗ , CZc D , CZc D∗ and CZc D to parameterize the net effects, CZcπ = CZcψ = CZcπ = CZc J/ψ = CZcρ = CZcηc = CZcρ = CZcηc = CZc D∗ = CZc D = CZc D∗ = CZc D = ∞ dt ρZcπ ( p 2, p2, t ) t − q2 s0 π ∞ dt ρZcψ ( p 2, t, q2) s0J/ψ t − p2 , , ρZcπ (t, p2, q2) t − p 2 , ρZc J/ψ (t, p2, q2) t − p 2 0 sZc ∞ dt ρZcρ ( p 2, p2, t ) t − q2 ρZcηc ( p 2, t, q2) t − p2 ρZcρ (t, p2, q2) t − p 2 ρZcηc (t, p2, q2) , , t − p 2 0 sZc ∞ dt ρZc D∗ ( p 2, t, q2) t − p2 0 sD∗ ∞ dt ρZc D ( p 2, p2, t ) t − q2 ∞ 0 sZc . , fπ Mπ2 f J/ψ MJ/ψ λZc G Zc J/ψπ mu + md −i × (MZ2c − p 2)(Mη2c − p2)(Mρ2 − q2) −i CZcρ −i CZcηc mc + (MZ2c − p 2)(Mη2c − p2) + (MZ2c − p 2)(Mρ2 − q2) −i CZcηc − i CZcρ + (Mη2c − p2)(Mρ2 − q2) + · · · , the CZcπ , CZcψ , CZcπ , CZc J/ψ , CZcρ , CZcηc , CZcρ , CZcηc , CZc D∗ , CZc D , CZc D∗ and CZc D on the momentums p 2, p2, q2, 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 1( p 2, p2, q2) and 2( p 2, p2, q2) can be written as i ∞ 1( p 2, p2, q2) = 32√2π 4 4mc2 ds 1 s − p2 0 ∞ 1 × u − q2 u s + 2mc2 i mc q¯q ∞ ds + 4√2π 2 4mc2 4mc2 ds 1 p 2 − s − q2 × s − p2 q4 i ∞ 2( p 2, p2, q2) = − 32√2π 4 4mc2 ds 1 where the last two terms originate from the Feynman diagrams where a quark pair q¯q absorbs a gluon emitted from other quark line. The term |m A→mc , (26) in above equations comes from the connected Feynman diagrams, if we set p 2 = p2, then it reduces to ∂ ∂ s ∂ ∞ ∞ ∞ It has no contribution after performing the double Borel transformation with respect to the variables P2 = − p2 4mc2 ds 4mc2 s ∞ s ds (27) and Q2 = −q2. It is more reasonable to performing the Borel transformation than taking the limit q2 → 0, as we carry out the operator product expansion at the large spacelike region Q2 = −q2 → ∞. So the connected Feynman diagrams have no contributions in the correlation functions 1/2( p 2, p2, q2), which are in contrary to Refs. [ 13,26 ], where only the connected Feynman diagrams have contributions and the limit Q2 → 0 is taken. For the correlation function 3( p 2, p2, q2), only the connected Feynman diagrams have contributions, we can set p 2 = 4 p2 according to the relation MZc(3900) ≈ 2MD∗ , the complex expression of the correlation function 3( p 2, p2, q2) can be reduced to a more simple form, 3(4 p2, p2, q2) = i mc q¯ gs σ Gq 96√2π 2 ∞ mc2 ρ → 0, M D2 → 0 and mc2 → 0, In the limit Mπ2 → 0, M 2 we maybe expect to choose Q2 = −q2 off-shell, and match the terms proportional to Q12 in the limit Q2 → 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 Zc J/ψπ (Q2), G Zcηcρ (Q2), G Zc D D¯ ∗ (Q2), then extract the values to the mass-shell Q2 = −Mπ2 , −Mρ2 or −M D2 to obtain the physical values [ 13 ]. However, the approximations Mρ2 → 0, M D2 → 0 and mc2 → 0 are rather crude, and we carry out the operator product expansion at the large space-like region Q2 = −q2 → ∞. We prefer taking the imaginary parts of the correlation functions 1/2/3( p 2, p2, q2) with respect to q2 + i through dispersion relation and obtain the physical hadronic spectral densities, then take the Borel transform with respect to the Q2 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 1/2/3( p 2, p2, q2), as there appears the variable p 2 = ( p + q)2. We rewrite the correlation functions 1H/2/3( p 2, p2, q2) on the hadron side into the following form through dispersion relation, 1H ( p 2, p2, q2) = 2H ( p 2, p2, q2) = 0 sZc (MJ/ψ +Mπ )2 ds 0 sJ/ψ ds 4mc2 0 u0π du ρ1H (s , s, u) × (s − p 2)(s − p2)(u − q2) + · · · , (29) s0Zc sη0c u0ρ (Mηc +Mρ )2 ds 4mc2 ds du 0 3H ( p 2, p2, q2) = ρ2H (s , s, u) × (s − p 2)(s − p2)(u − q2) + · · · , (30) sZ0c ds s0D∗ ds u0D du mc2 mc2 (MD∗ +MD)2 ρ3H (s , s, u) × (s − p 2)(s − p2)(u − q2) + · · · , (31) ρ1Q/C2/D3 ( p 2, s, u) = lim lim 2→0 1→0 × Ims Imu 1Q/C2/D3( p 2, s + i 2, u + i 1) π 2 , We math the hadron side of the correlation functions with the QCD side of the correlation functions, where the ρ1H/2/3(s , s, u) are the hadronic spectral densities, H lim lim ρ1/2/3(s , s, u) = 3li→m0 2→0 1→0 we add the superscript H to denote the hadron side. However, on the QCD side, the QCD spectral densities ρQ1/C2/D3(s , s, u) do not exist, ρQ1/C2/D3(s , s, u) = lim lim lim 3→0 2→0 1→0 Ims Ims Imu 1Q/C2/D3(s + i 3, s + i 2, u + i 1) π 3 × because lim 3→0 Ims 1Q/C2/D3(s + i 3, p2, q2) π = 0, we add the superscript QC D to denote the QCD side. On the QCD side, the correlation functions 1Q/C2/D3 ( p 2, p2, q2) can be written into the following form through dispersion relation, 1QC D( p 2, p2, q2) = 2QC D( p 2, p2, q2) = 3QC D( p 2, p2, q2) = 0 sJ/ψ 4mc2 ds du 0 u0π 4mc2 0 ρ1QC D( p 2, s, u) × (s − p2)(u − q2) + · · · , sη0c u0ρ ds du ρ2QC D( p 2, s, u) × (s − p2)(u − q2) + · · · , s0D∗ u0D ds du mc2 mc2 ρ3QC D( p 2, s, u) × (s − p2)(u − q2) + · · · , where the ρ1Q/C2/D3 ( p 2, s, u) are the QCD spectral densities, (38) (39) (40) (41) (32) = 0, (33) (34) (35) (36) (37) = 0 sηc 4mc2 = = mc2 = = 0 sJ/ψ 4mc2 = ds ∞ 0 u0π du ds ρ1QC D( p 2, s, u) (s − p2)(u − q2) s0J/ψ u0π ds du (MJ/ψ +Mπ )2 4mc2 0 ρ1H (s , s, u) × (s − p 2)(s − p2)(u − q2) where the integrals over ds are carried out firstly to obtain the solid duality, s0 2 s the s2 and 2u denote the thresholds 4mc2, mc2, 0, the 2 denotes the thresholds (MJ/ψ + Mπ )2, (Mηc + Mρ )2 and (MD∗ + MD)2. No approximation is needed, the continuum threshold parameter sZ0c in the s channel is also not needed. The present routine can be applied to study other hadronic rceoluaTpthiloiennngsfwudneicretsiceottnlysp. 2 1=/2( pp22, apn2d, qp2)2 an=d 4 p32( pin2,tph2e, qco2r)-, respectively, and perform the double Borel transformations with respect to the variables P2 = − p2 and Q2 = −q2, respectively to obtain the following QCD sum rules, (43) (44) where the s0J/ψ , u0π , sη0c , u0ρ , sD0∗ and u0D are the continuum threshold parameters, the T 2 and T22 are the Borel parameters. In the three QCD sum rules, the terms depend on T22 can be factorized out explicitly, (46) MD2∗ + CZc D∗ + CZc D exp − T 2 the dependence on the Borel parameter T22 is trivial, exp − u−T2M2ρ2 , exp − u−TM22D2 , u−Mπ2 , exp − T22 exp − mc2−T22MD2 , which differ from the QCD sum rules for the three-meson hadronic coupling constants greatly [ 32 ]. It is difficult to obtain T22 independent regions in the present three QCD sum rules, as no other terms to stabilize the QCD sum rules. We can take the local limit T22 → ∞, which is so called local-duality limit (the local QCD sum rules are reproduced from the original QCD sum rules in infinite Borel parameter limit) [ 33– 35 ], then exp − Tu22 = exp − mT22c2 = exp − MT22π2 = exp − MT22ρ2 = exp − MT22D2 = 1, the three QCD sum rules are greatly simplified. Now we write down the simplified QCD sum rules explicitly, + 3 Numerical results and discussions The input parameters on the QCD side are taken to be the standard values q¯q = −(0.24 ± 0.01 GeV)3, q¯ gs σ Gq = m20 q¯q , m02 = (0.8 ± 0.1) GeV2 at the energy scale μ = 1 GeV [ 29–31,36 ], mc(mc) = (1.28 ± 0.03) GeV from the Particle Data Group [ 37 ]. Furthermore, we set mu = md = 0 due to the small current quark masses. We take into account the energy-scale dependence of the input parameters from the renormalization group equation, q¯ gs σ Gq (μ) = q¯ gs σ Gq (Q) 2 αs (Q) 25 , αs (μ) 12 αs (μ) 25 αs (mc) , mc(μ) = mc(mc) 1 αs (μ) = b0t b1 log t 1 − b2 t 0 (49) + b12(log2 t − log t − 1) + b0b2 b4t 2 0 , (52) where t = log μ22 , b0 = 331−22πn f , b1 = 1532−4π192n f , b2 = 2857− 51093238nπf3+ 32275 n2f , = 210, 292 and 332 MeV for the flavors n f = 5, 4 and 3, respectively [ 37 ], and evolve all the input parameters to the optimal energy scale μ = 1.4 GeV to extract hadronic coupling constants [ 12,38 ]. The hadronic parameters are taken as Mπ = 0.13957 GeV, Mρ = 0.77526 GeV, MJ/ψ = 3.0969 GeV, Mηc = 2.9834 GeV [ 37 ], fπ = 0.130 GeV, fρ = 0.215 GeV, s0 π = 0.85 GeV, sρ0 = 1.3 GeV [ 36 ], MD = 1.87 GeV, f D = 208 MeV, u0D = 6.2 GeV2, MD∗ = 2.01 GeV, f D∗ = 263 MeV, sD0∗ = 6.4 GeV2 [ 39,40 ], f J/ψ = 0.418 GeV, fηc = 0.387 GeV [41], s0J/ψ = 3.6 GeV, sη0c = 3.5 GeV, MZc = 3.899 GeV, λZc = 2.1 × 10−2 GeV5 [ 12,38 ], fπ Mπ2 /(mu + md ) = −2 q¯q / fπ from the Gell– Mann–Oakes–Renner relation. In the scenario of tetraquark states, the QCD sum rules indicate that the Zc(3900) and Z (4430) can be tentatively assigned to be the ground state and the first radial excited state of the axialvector tetraquark states, respectively [ 42 ], the coupling of the current Jν (0) to the excited state Z (4430) is rather large, so the unknown parameters cannot be neglected. The unknown parameters are fitted to be CZc J/ψ + CZcπ = 0.001 GeV8, CZcηc + CZcρ = 0.0046 GeV8 and CZc D∗ + CZc D = 0.00013 GeV8 to obtain platforms in the Borel windows T 2 = (1.9 − 2.6) GeV2, (1.9 − 2.5) GeV2 and (1.5 − 2.1) GeV2 for the hadronic coupling constants G Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗ , respectively. Then it is easy to obtain the values of the hadronic coupling constants, |G Zc J/ψπ | = 3.63 ± 0.70 GeV, G Zcηcρ = 4.38 ± 1.86 GeV, |G Zc D D¯ ∗ | = 0.62 ± 0.09 GeV, (53) which are shown explicitly in Fig. 1 We choose the masses Mπ = 0.13957 GeV, Mρ = 0.77526 GeV, MJ/ψ = 3.0969 GeV, Mηc = 2.9834 GeV, MD+ = 1.8695 GeV, MD∗0 = 2.00685 GeV, MD0 = 1.86484 GeV, MD∗+ = 2.01026 GeV [ 37 ], MZc = 3.899 GeV [ 1 ], and obtain the partial decay widths, (Zc+(3900) → J /ψ π +) = 25.8 ± 9.6 MeV, (Zc+(3900) → ηcρ+) = 27.9 ± 20.1 MeV, (Zc+(3900) → D+ D¯ ∗0) = 0.22 ± 0.07 MeV, (Zc+(3900) → D¯ 0 D∗+) = 0.23 ± 0.07 MeV, and the total width, which is consistent with the experimental data considering the uncertainties [ 1–3,5 ]. If we take the central values of the hadronic coupling constants |G Zc J/ψπ | = 3.63 GeV, G Zcηcρ = 4.38 GeV, |G Zc D D¯ ∗ | = 0.62 GeV, we can obtain the total width Zc(3900) = 48.9 MeV, which hap(54) (55) pens to coincide with the central value of the experimental dada = 46 ± 10 ± 20 MeV from the BESIII Collaboration [1], while the predicted ratio (56) R = = (Zc(3900) → D D¯ ∗) (Zc(3900) → J /ψ π ) = 0.02 Rex p (Zc(3885) → D D¯ ∗) (Zc(3900) → J /ψ π ) = 6.2 ± 1.1 ± 2.7, from the BESIII Collaboration [ 4 ]. It is difficult to assign the Zc(3900) and Zc(3885) to be the same diquark–antidiquark type axialvector tetraquark state. We can assign the Zc(3900) to be the diquark–antidiquark type axialvector tetraquark state, and assign the Zc+(3885) to be the molecular state D+ D¯ ∗0 + D∗+ D¯ 0 according to the predicted mass 3.89 ± 0.09 GeV from the QCD sum rules [ 20 ]. If the Zc(3885) is the D+ D¯ ∗0 + D∗+ D¯ 0 molecular state, the decays to D+ D¯ ∗0 and D∗+ D¯ 0 take place through its component directly, it is easy to account for the large ratio Rex p. Now we compare the present work with the work in Ref. [ 13 ] 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 spacelike regions P2 = − p2 → ∞ and Q2 = −q2 → ∞. 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 P2 and Q2 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. [ 13 ], Dias et al take into account only the connected Feynman diagrams, and obtain the quark–hadron duality by taking the limit Q2 → 0, Mπ2 → 0, Mρ2 → 0, M D2 → 0 and mc2 → 0 and choosing special tensor structures, then perform single Borel transform with respect to the variable P2 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 Q2 = −Mπ2 , −Mρ2 or −M D2 to obtain the physical hadronic coupling constants. Although the values of the width of the Zc(3900) obtained in the present work and in Ref. [ 13 ] are both compatible with the experimental data, the present predictions have much less theoretical uncertainties. 4 Conclusion In this article, we tentatively assign the Zc±(3900) to be the diquark–antidiquark type axialvector tetraquark state, study the hadronic coupling constants G Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗ with the QCD sum rules in details. We introduce the threepoint 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. 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Zhi-Gang Wang, Jun-Xia Zhang. The decay width of the \(Z_c(3900)\) as an axialvector tetraquark state in solid quark–hadron duality, The European Physical Journal C, 2018, 14, DOI: 10.1140/epjc/s10052-017-5514-1