Analysis of the mass and width of the $Y(4274)$ as axialvector molecule-like state

In this article, we assign the $Y(4274)$ to be the color octet-octet type axialvector molecule-like state with $J^{PC}=1^{++}$ tentatively, and construct the color octet-octet type axialvector current to study its mass and width with the QCD sum rules in details. The predicted mass favors assigning the $Y(4274)$ to be the color octet-octet type molecule-like state, but the predicted width disfavors assigning the $Y(4274)$ to be the color octet-octet type molecule-like state strongly. The $Y(4274)$ may be the conventional charmonium state $\chi_{c1}(\rm 3P)$, and it is important to observe the decay $Y(4274)\to J/\psi \omega$ to diagnose the nature of the $Y(4274)$.

In 2014, the Belle collaboration analyzed theB 0 → K − π + J/ψ decays and observed a resonance (Z c (4200)) in the J/ψπ + invariant mass distribution with a statistical significance of greater than 6.2 σ, the measured Breit-Wigner mass and width are M Zc(4200) = 4196 +31 −132 MeV, respectively [11,12]. The preferred spin-parity is J P = 1 + . In Ref. [13], we study the axialvector hidden charm and hidden bottom tetraquark states in details with the QCD sum rules and obtain the mass M cucd,J P =1 + = (4.44 ± 0.19) GeV for the diquark-antidiquark type tetraquark state. In Ref. [14], Chen and Zhu study the vector and axialvector charmonium-like tetraquark states with the QCD sum rules in a systematic way and obtain the mass M cucd,J P =1 + = (4.16 ± 0.10) GeV for the diquark-antidiquark type tetraquark state. In Ref. [13], we choose the input parameters m c (1 GeV), qq (1 GeV), qg s σGq (1 GeV), while in Ref. [14], Chen and Zhu choose the input parameters m c (m c ), qq (1 GeV), qg s σGq (1 GeV). The different predictions for the Cγ 5 ⊗ γ µ C type axialvector tetraquark state in Ref. [13] and Ref. [14] originate from the different choice of the c-quark mass. If we take different choice of the heavy quark masses as a source of uncertainties, the predicted mass is about M cucd,J P =1 + = (4.06 − 4.63) GeV.
In Ref. [15], we distinguish the charge conjugations of the interpolating currents, study the diquark-antidiquark type axialvector tetraquark states in a systematic way with the QCD sum rules by taking into account the energy scale dependence of the QCD spectral densities for the first time, and obtain the predictions M X(3872) = 3.87 +0.09 −0.09 GeV and M Zc(3900) = 3.91 +0.11 −0.09 GeV. In Ref. [16], R. Albuquerque et al take into account the next-to-leading order and next-to-nextto-leading order factorizable radiative corrections to the perturbative terms, and obtain the predication M cucd,J P =1 + = (3.888 ± 0.130) GeV, which also depends on special choice of the energy scale µ, in other words, the M S mass m c (µ). The non-factorizable radiative corrections are still needed to make precise predictions. In leading order approximation, M X(3872) = 3.87 +0.09 −0.09 GeV and M Zc(3900) = 3.91 +0. 11 −0.09 GeV [15], after taking into account the next-to-leading order and next-tonext-to-leading order factorizable radiative corrections to the perturbative terms, M cucd,J P =1 + = (3.888 ± 0.130) GeV [16], the predicted masses only change slightly. On the other hand, including the next-to-leading order and next-to-next-to-leading order factorizable radiative corrections to the perturbative terms leads to the value of the pole residue λ cucd,J P =1 + undergoes the replacement λ cucd,J P =1 + → 1.09 λ cucd,J P =1 + . According to Refs. [15,16], the masses of the ground state diquark-antidiquark type axialvector tetraquark states cucd are about 3.9 GeV.
In Ref. [17], Chen et al assign the Z c (4200) to be the ground state axialvector tetraquark state cucd, calculate its decay width with the QCD sum rules, and obtain the value Γ Zc(4200) = 435 ± 180 MeV. In Ref. [7], Chen et al assign the X(4140) to be the ground state axialvector tetraquark state cscs. If the Z c (4200) and X(4140) are the color triplet-triplet Cγ 5 ⊗ γ µ C type axialvector tetraquark states, it is more natural in the case that the X(4140) has larger mass than the Z c (4200).
In Ref. [18], we assign the Z c (4200) to be the color octet-octet type axialvector molecule-like stateūλ a ccλ a d, where λ a is the Gell-Mann matrix, and construct the color octet-octet type axialvector current to study its mass (width) with the QCD sum rules by calculating the vacuum condensates up to dimension 10 (5)  −132 MeV from the Belle collaboration [11,12], and favor assigning the Z c (4200) to be the color octet-octet type molecule-like state with J P C = 1 +− . Moreover, we study the energy scale dependance of the QCD spectral density of the molecule-like state in details and suggest an empirical energy scale formula to determine the ideal energy scale, in other words, to determine the ideal c-quark mass.
Also in Ref. [18], we discuss the possible assignments of the Z c (3900), Z c (4200) and Z(4430) as the ground state color triplet-triplet diquark-antidiquark type tetraquark states with J P C = 1 +− in details. The QCD sum rules support assigning the Z c (3900) and Z(4430) to be the ground state and the first radial excited state of the diquark-antidiquark type axialvector tetraquark states with J P C = 1 +− , respectively [15,19]. If we assign the Z c (4200) and Y (4274) to be the molecule-like states with J P C = 1 +− and 1 ++ , respectively, the mass difference M Y (4274) − M Zc(4200) ≈ 77 MeV. It is reasonable, as the SU (3) breaking effects are very small for the four-quark systems [13,20,21]. In this article, we assign the Y (4274) to be the color octet-octet type molecule-like state tentatively, study its mass and decay width with the QCD sum rules in details, where the meson-like states D a s and D a * s have the same quark constituents as the mesons D s and D * s respectively, but they are in the color octet representation, the a corresponds to the Gell-Mann matrix.
The article is arranged as follows: we derive the QCD sum rules for the mass and width of the color octet-octet type axialvector molecule-like state Y (4274) in section 2 and in section 3 respectively; section 4 is reserved for our conclusion.
2 The mass of the color octet-octet type axialvector moleculelike state In the following, we write down the two-point correlation function Π µν (p) in the QCD sum rules, where the λ a is the Gell-Mann matrix in the color space. We construct the color octet-octet type current J µ (x) to study the molecule-like state Y (4274). One can consult Refs. [18,22,23] for more literatures on the color octet-octet type currents. Under charge conjugation transform C, the current J µ (x) has the property, At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operator J µ (x) into the correlation function Π µν (p) to obtain the hadronic representation [24,25], and isolate the ground state contribution, where the pole residue λ Y (4274) is defined by 0|J µ (0)|Y (4274) = λ Y (4274) ε µ , the ε µ is the polarization vector of the axialvector meson Y (4274). In the following, we briefly outline the operator product expansion for the correlation function Π µν (p). We contract the quark fields s and c in the correlation function Π µν (p) with Wick theorem, and obtain the result, where and [25], then compute the integrals both in the coordinate space and in the momentum space, and obtain the correlation function Π µν (p), therefore the QCD spectral density through dispersion relation. For technical details, one can consult Ref. [15]. Now we take the quark-hadron duality below the continuum threshold s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rule: where The explicit expressions of the QCD spectral densities ρ 0 (s), ρ 3 (s), ρ 4 (s), ρ 5 (s), ρ 6 (s), ρ 7 (s), ρ 8 (s) and ρ 10 (s) are given in the Appendix. Even in the leading order approximation, the strong coupling constant g 2 s (µ) = 4πα s (µ) appears according to the equation of motion (D α G βα ) a = g s q=u,d,sq γ β t a q, see the terms g 2 s ss 2 in the spectral density ρ 6 (s). So we have to consider the energy scale dependence of the QCD sum rules, the preferred c-quark mass is the M S mass m c (µ).
The on-shell quark propagator has no infrared divergences in perturbation theory, and this provides a perturbative definition of the quark mass [26]. But the pole mass cannot be used to arbitrarily high accuracy because of nonperturbative infrared effects in QCD. The pole mass m c and the M S mass m c (m c ) have the relation, The value m c (m c ) = 1.275 ± 0.025 GeV from the Particle Data Group corresponds to m c = 1.67 ± 0.07 GeV [27]. If we take the pole mass, then 2 m c > M J/ψ > M ηc , at the phenomenological side of the QCD sum rules for the J/ψ and η c , If we want to obtain nonzero values, we have to choose smaller pole mass, 2 m c < M ηc < M J/ψ [28]. For an observable particle such as the electron, the physical mass appears as the pole mass, irrespective of the leading order, next-to-leading order, next-to-next-to-leading order, · · · , radiative corrections are concerned. In the leading order approximation, m c = m c (m c ), however, the m c (m c ) originates from the radiative corrections and renormalization, which are beyond the leading order approximation. So in the leading order approximation, the definition of the pole mass m c is of arbitrary. Again, we can see that the preferred c-quark mass is the M S mass m c (µ). Moreover, the full quark propagator has no pole because the quarks are confined. The pole mass corresponds to a non-confined particle, at the QCD side of the QCD sum rules for J/ψ and η c , the heavy quarks c andc are confined particles. We derive Eq.(11) with respect to τ = 1 T 2 , then eliminate the pole residue λ Y (4274) to obtain the QCD sum rule for the mass, Now we choose the input parameters at the QCD side of the QCD sum rules. We take the vacuum condensates to be the standard values qq = −(0.24 ± 0.01 GeV) 3 , ss = (0.8 ± 0.1) qq , sg s σGs = m 2 0 ss , m 2 0 = (0.8 ± 0.1) GeV 2 , αsGG π = (0.33 GeV) 4 at the energy scale µ = 1 GeV [24,25,29], and take the M S masses m c (m c ) = (1.275 ± 0.025) GeV and m s (µ = 2 GeV) = (0.095 ± 0.005) GeV from the Particle Data Group [27]. Moreover, we take into account the energy-scale dependence of the quark condensate, mixed quark condensate and M S masses from the renormalization group equation [27,30], 12 25 , m s (µ) = m s (2GeV) α s (µ) α s (2GeV) 4 9 , where t = log µ 2 , Λ = 213 MeV, 296 MeV and 339 MeV for the flavors n f = 5, 4 and 3, respectively [27].
As the quark masses m c (µ), m s (µ), the quark condensate ss (µ), the mixed condensate sg s σGs (µ) all depend on the energy scale µ, the QCD spectral density ρ(s) depends on the energy scale µ, we have to determine the energy scales of the QCD sum rules for those moleculelike states in a consistent way.
The hidden charm (or bottom) four-quark systems qq ′ QQ can be described by a double-well potential. In the four-quark system qq ′ QQ, the heavy quark Q serves as a static well potential and combines with the light quark q to form a heavy diquark D i in color antitriplet, q + Q → D i [15,19,20,21,31], or combines with the light antiquarkq ′ to form a heavy meson in color singlet (meson-like state in color octet),q ′ +Q →q ′ Q (q ′ λ a Q) [18,23,32]; the heavy antiquarkQ serves as another static well potential and combines with the light antiquarkq ′ to form a heavy antidiquark D i in color triplet,q ′ +Q → D i [15,19,20,21,31], or combines with the light quark q to form a heavy meson in color singlet (meson-like state in color octet), q +Q →Qq (Qλ a q) [18,23,32], where the i is color index, the λ a is Gell-Mann matrix. Then the two heavy quarks Q andQ stabilize the four-quark systems qq ′ QQ, just as in the case of the (µ − e + )(µ + e − ) molecule in QED [33].
The four-quark systems qq ′ QQ are characterized by the effective heavy quark mass M Q and [15,18,19,20,21,23,31,32]. It is natural to take the energy scale µ = V . The M Q is just an empirical parameter to determine the optimal energy scales of the QCD spectral densities, and has no relation to the pole massm Q or the M S mass m Q (µ).
In Refs. [15,18,19,20,21,23,31,32], we observe that there exist three universal values for the effective heavy quark masses M Q , which correspond to the compact tetraquark states, molecular states, molecule-like states, respectively. The empirical energy scale formula µ = We evolve all the input parameters in the QCD spectral density to the special energy scale determined by the empirical formula, In Ref. [18], we obtain the effective mass M c = 1.98 GeV for the molecule-like states. Then we rechecked the numerical calculations and corrected a small error concerning the mixed condensate, the updated value is M c = 2.01 GeV. From the empirical energy scale formula, we can obtain the energy scale µ = 1.45 GeV. After taking into account the SU (3) symmetry breaking effect m s − m u/d ≈ 0.1 GeV, we obtain the optimal energy scale µ = 1.25 GeV for the QCD spectral density ρ(s). If we neglect the SU (3) symmetry breaking effect, the effective c-quark mass M c can be taken as M c = 2.04 GeV. Now we search for the Borel parameter T 2 and continuum threshold parameter s 0 to satisfy the following three criteria: 1 · Pole dominance at the phenomenological side; 2 · Convergence of the operator product expansion; 3 · Appearance of the Borel platforms. The resulting Borel parameter and continuum threshold parameter are T 2 = (3.1 − 3.5) GeV 2 and √ s 0 = (4.8 ± 0.1) GeV, respectively. At the Borel window, the pole contribution is about (41 − 62)%, the contributions of the vacuum condensates of dimension 8 and 10 are about |D 8 | = (5 − 7)% and D 10 < 1%, respectively, the first two criteria are satisfied.
We take into account all uncertainties of the input parameters, and obtain the values of the mass and pole residue, which are shown explicitly in Fig.1, In Fig.1, we plot the mass and pole residue of the Y (4274) with variation of the Borel parameter T 2 at a larger interval than the Borel window. From the figure, we can see that there appear platforms, the criterion 3 is also satisfied. Now the three criteria are all satisfied, it is reliable to extract the ground state mass. The predicted mass M Y (4274) = (4.27 ± 0.09) GeV is consistent with the experimental value 4273.3 ± 8.3 +17.2 −3.6 MeV from the LHCb collaboration [5,6], which supports assigning the Y (4274) to be the color octet-octet typesλ a ccλ a s molecule-like state.
In Ref. [16], R. Albuquerque et al study the hidden-charm and hidden-bottom molecular states and tetraquark states by taking into account the next-to-leading order and next-to-next-to-leading order radiative corrections to the preturbative terms from the factorizable Feynman diagrams (without including the non-factorizable Feynman diagrams). The numerical results indicate that the predicted masses are slightly modified, while the decay constants (which relate to the pole residues) are modified significantly, the largest modification amounts to multiplying the decay constants by a factor 1.8. So we expect that the predication M Y (4274) = (4.27 ± 0.09) GeV survives approximately even if the next-to-leading order radiative corrections to the preturbative terms are taken into account. Moreover, at the present time, even the next-to-leading order factorizable contributions are not available for the color octet-octet type molecule-like states, it is a challenging work to calculate both the next-to-leading order factorizable and non-factorizable Feynman diagrams.
3 The width of the color octet-octet type axialvector moleculelike state We can study the strong decay Y (4274) → J/ψφ with the three-point correlation function Π αµν (p, q), where the currents interpolate the mesons J/ψ and φ(1020) according to the current-hadron couplings, the f J/ψ and f φ are the decay constants, the ξ α and ζ µ are polarization vectors of the mesons J/ψ and φ(1020), respectively. At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J J/ψ α (x), J φ µ (y), J ν (0) into the three-point correlation function Π αµν (p, q) and isolate the ground state contributions to obtain the result, where p ′ = p + q, the G Y J/ψφ is the hadronic coupling constant, which is defined by In this article, we choose the tensor structure ε αµνλ p λ to study the coupling constant G Y J/ψφ . The two unknown functions ρ Y φ (p 2 , t, p ′2 ) and ρ Y J/ψ (t, q 2 , p ′2 ) parameterize transitions between the ground states and the higher resonances or the continuum states, the net effects can be parameterized by C Y φ and C Y J/ψ , In calculations, we take the C Y φ and C Y J/ψ as free parameters, and vary them to eliminate the contaminations to obtain Borel platforms [34]. We carry out the operator product expansion up to the vacuum condensates of dimension 5 and neglect the gluon condensate, which plays a minor important role. We obtain the QCD spectral density through dispersion relation, take the quark-hadron duality below the continuum thresholds, then set p ′2 = p 2 and take double Borel transform with respect to the variables P 2 = −p 2 and Q 2 = −q 2 respectively to obtain the QCD sum rule, where the s 0 Y and s 0 φ are the continuum threshold parameters for the Y (4274) and φ(1020), respectively. The If the radiative corrections to the perturbative term of the correlation function Π αµν (p, q) also amount to multiplying a factor about 1.8, as the color octet-octet type current J µ (x) is also presented, the value of the quantity f φ f J/ψ λ Y G Y J/ψφ at the hadronic side in the QCD sum rules in Eq.(26) changes according to the rule, In this article, we take the values f J/ψ = 0.418 GeV [35] and f φ = 0.253 GeV, which include nextto-leading order radiative corrections. The factors 1.8 come from the radiative corrections to the two-point correction function and three-point correlation function cancel out with each other, the net modification of the hadronic coupling constant G Y J/ψφ is estimated to be tiny, just like the hadronic coupling constants D * Dπ and B * Bπ, the net effects of the radiative corrections can be neglected [36]. Now it is easy to obtain the decay width, where p(a, b, c) = √ . It is difficult to assign the Y (4274) to be the color octetoctet type molecule-like statesλ a ccλ a s. In Ref. [20], we assign the Y (4140) to be the diquarkantidiquark type tetraquark state cscs with J P C = 1 ++ , and study the mass and pole residue with the QCD sum rules in details. The predicted mass disfavors assigning the Y (4140) to be the J P C = 1 ++ diquark-antidiquark type tetraquark state cscs. The Y (4140) and Y (4274) have the same quantum numbers except for the masses and widths, the QCD sum rules also disfavor assigning the Y (4274) to be the J P C = 1 ++ diquark-antidiquark type tetraquark state cscs.

Conclusion
In this article, we assign the Y (4274) to be the color octet-octet type axialvector molecule-like state with J P C = 1 ++ tentatively, and construct the color octet-octet type axialvector current to study its mass and width with the QCD sum rules in details. The predicted mass M Y (4274) = (4.27 ± 0.09) GeV is consistent with the experimental value 4273.3 ± 8.3 +17.2 −3.6 MeV from the LHCb collaboration, and favors assigning the Y (4274) to be the color octet-octet type molecule-like statesλ a ccλ a s. The predicted width Γ(Y (4274) → J/ψφ) = 1.8 GeV is much larger than the experimental value 56 ± 11 +8 −11 MeV from the LHCb collaboration and disfavors assigning the Y (4274) to be the color octet-octet type molecule-like state strongly. The Y (4274) may be the conventional charmonium state χ c1 (3P), and the preferred decays are Y (4274) → D * ρ 10 (s) = m 2 c sg s σGs 2 72π 2 T 6