Analysis of the $Y(4220)$ and $Y(4390)$ as molecular states with QCD sum rules

In this article, we assign the $Y(4390)$ and $Y(4220)$ to be the vector molecular states $D\bar{D}_1(2420)$ and $D^*\bar{D}_0^*(2400)$, respectively, and study their masses and pole residues with the QCD sum rules in details. The present calculations only favor assigning the $Y(4390)$ to be the $D\bar{D}_1(1^{--})$ molecular state.


Introduction
In 2013, Yuan studied the cross sections of the process e + e − → π + π − h c at center-of-mass energies 3.90 − 4.42 GeV measured by the BESIII and the CLEO-c experiments, and observed evidence for two resonant structures, a narrow structure of mass (4216 ± 18) MeV and width (39 ± 32) MeV, and a possible wide structure of mass (4293 ± 9) MeV and width (222 ± 67) MeV [1].
In 2014, the BES collaboration searched for the production of e + e − → ωχ cJ with J = 0, 1, 2, based on data samples collected with the BESIII detector at center-of-mass energies from 4.21 − 4.42 GeV, and observed a resonance in the ωχ c0 cross section, the measured mass and width of the resonance Y (4230) are 4230 ± 8 ± 6 MeV and 38 ± 12 ± 2 MeV, respectively [2].
Recently, the BES collaboration measured the cross sections of the process e + e − → π + π − h c at center-of-mass energies 3.896 − 4.600 GeV using data samples collected with the BESIII detector, and observed two structures, the Y (4220) has the mass 4218.4 ± 4.0 ± 0.9 MeV and with 66.0 ± 9.0 ± 0.4 MeV respectively, and the Y (4390) has the mass 4391.6 ± 6.3 ± 1.0 MeV and width 139.5 ± 16.1 ± 0.6 MeV respectively [3]. The Y (4230) and Y (4220) may be the same particle. The Y (4230) has been assigned to be a vector-diquark-vector-antidiquark type vector tetraquark state [4,5] or a conventional meson ψ(4S) [6]. The near thresholds are M D + D1(2420) − = 4293 MeV, M D 0 D1(2420) 0 = 4285 MeV, M D * + D * 0 (2400) − = 4361 MeV, M D * 0 D * 0 (2400) 0 = 4325 MeV [7]. It is also possible to assign the Y (4220) and Y (4390) to be the DD 1 (2420) or D * D * 0 (2400) molecular states. Eleven years ago, the BaBar collaboration observed a broad resonance (Y (4260)) in the initialstate radiation process e + e − → Y (4260) → J/ψπ + π − in the invariant-mass spectrum of the J/ψπ + π − [8]. Later, the BaBar collaboration measured the mass and width of the Y (4260) in a more precise way [9]. The cross section rises rapidly below the peak of the Y (4260) and falls more slowly above the peak [7]. The BESIII experiment may be indicate that in fact the Y (4260) consist of two peaks, a narrow peak around 4.22 GeV and a wider peak around 4.39 GeV accounting for the asymmetry.
In Ref. [10], Zhang and Huang study the QqQ ′ q type scalar, vector and axialvector molecular states with the QCD sum rules systematically by calculating the operator product expansion up to the vacuum condensates of dimension 6. The predicted molecule masses M D * D * 0 = 4.26 ± 0.07 GeV and M DD1 = 4.34 ± 0.07 GeV are consistent with the Y (4220) and Y (4390), respectively. However, the charge conjugations of the molecular states are not distinguished and the higher dimensional vacuum condensates are neglected. In Ref. [11], Lee, Morita and Nielsen distinguish the charge conjugations of the interpolating currents, calculate the operator product expansion up to the vacuum condensates of dimension 6 including the vacuum condensates of dimension 8 partly. They obtain the mass of the DD 1 (2420) molecular state with J P C = 1 −+ , M DD1 = 4.19 ± 0.22 GeV, which differs from the prediction M DD1 = 4.34 ± 0.07 GeV significantly.
In Refs. [10,11], some higher dimensional vacuum condensates involving the gluon condensate, mixed condensate and four-quark condensate are neglected. The terms associate with 1 in the QCD spectral densities manifest themselves at small values of the Borel parameter T 2 , we have to choose large values of T 2 to warrant convergence of the operator product expansion. In the Borel windows, the higher dimensional vacuum condensates play a less important role. The higher dimensional vacuum condensates play an important role in determining the Borel windows therefore the ground state masses and pole residues, we should take them into account consistently.
In this article, we assign the Y (4390) and Y (4220) to be the vector molecular states DD 1 (2420) and D * D * 0 (2400), respectively, distinguish the charge conjugations, construct the color singletsinglet type currents to interpolate them. We calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion in a consistent way, and use the energy scale formula to determine the energy scales of the QCD spectral densities [12,13], which differs from the routines taken in Refs. [10,11] significantly, then study the masses and pole residues with the QCD sum rules in details.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the vector molecular states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.

QCD sum rules for the vector molecular states
In the isospin limit, the quark structures of the molecular states DD 1 (2420) and D * D * 0 (2400) can be symbolically written asū The isospin tripletūdcc,ū u−dd √ 2c c,ducc and isospin singletū u+dd √ 2c c have degenerate masses. In this article, we take the isospin limit and study the masses of the charged partners of the Y (4220) and Y (4390) for simplicity.
In the following, we write down the two-point correlation functions Π µν (p) in the QCD sum rules, where Under charge conjugation transform C, the currents J µ (x) have the properties, The charge conjugations of the molecular states Y (4220) and Y (4390) are unknown. If the decays take place through the charge conjugation is positive; on the other hand, if the decays take place through the charge conjugation is negative, where we assume that there is a relative S-wave between the intermediate mesons has been observed [2]. If the Y (4220) and Y (4230) are the same particle, the Y (4220) maybe have the quantum numbers J P C = 1 −− . At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µ (x) into the correlation functions Π µν (p) to obtain the hadronic representation [14,15]. After isolating the ground state contributions of the vector molecular states, we get the following results, where the pole residues λ Y are defined by 0|J µ (0)|Y (p) = λ Y ε µ , the ε µ are the polarization vectors of the vector molecular states.
In the following, we perform Fierz re-arrangement for the currents J µ both in the color space and Dirac-spinor space to obtain the results, The componentsūΓdcΓ ′ c andūΓλ a dcΓ ′ λ a c couple potentially to a series of charmonium-lightmeson pairs or charmonium-like molecular states or charmonium-like molecule-like states, where Γ, Γ ′ = 1, γ µ , γ µ γ 5 , iγ 5 , σ µβ , σ µβ γ 5 . For example, the current J 1 µ couples potentially to the meson pairs through its components, We cannot distinguish those contributions to study them exclusively, and assume that the currents uΓdcΓ ′ c andūΓλ a dcΓ ′ λ a c couples to a particular resonance Y , which is a special superposition of the scattering states, molecular states and molecule-like states, and embodies the net effects. Some meson pairs (in other words, its components) such as χ c0 ρ, J/ψa 0 (980), J/ψπ, · · · lie below the Y , the Y can decay to those meson pairs easily through fall-apart mechanism, although the rearrangements in the color space and Dirac-spinor space are highly non-trivial, the decays contribute a finite width to the Y .
In the following, we study the contributions of the intermediate meson-loops to the correlation function Π µν (p) for the current J 1 µ (x) as an example, the current J 1 µ (x) has nonvanishing couplings with the scattering states J/ψa 0 (980), χ c0 ρ, etc.

+70
−132 MeV, can be safely absorbed into the pole residues λ X/Zc [16]. In this article, we take the zero width approximation, and expect that the predicted masses are reasonable.
We carry out the operator product expansion in a consistent way, and obtain the QCD spectral densities through dispersion relation, then we take the quark-hadron duality below the continuum thresholds s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the following QCD sum rules, where ρ(s) = ρ 1 (s), ρ 2 (s), ρ 3 (s), ρ 4 (s), the explicit expressions of the QCD spectral densities ρ(s, r) are given in the Appendix. In this article, we carry out the operator product expansion to the vacuum condensates up to dimension-10 and assume vacuum saturation for the higher dimension vacuum condensates. The condensates s ) respectively, and discarded. We take the truncations n ≤ 10 and k ≤ 1 in a consistent way, the operators of the orders O(α k s ) with k > 1 are discarded [12,13,17]. Furthermore, the numerical values of the condensates g 3 s GGG , αsGG π 2 , αsGG π qg s σGq are very small, and they are neglected safely. We derive Eq.(13) with respect to τ = 1 T 2 , and eliminate the pole residues λ Y to obtain the QCD sum rules for the masses, 3 Numerical results and discussions where t = log µ 2 The hidden charm (or hidden bottom) four-quark systems QqQq ′ could be described by a double-well potential in the heavy quark limit. The heavy quark Q serves as one static well potential and combines with the light antiquarkq ′ to form a heavy meson-like state or correlation (not a physical meson) in color singlet. The heavy antiquarkQ serves as the other static well potential and combines with the light quark state q to form another heavy meson-like state or correlation (not a physical meson) in color singlet. The two meson-like states (not two physical mesons) combine together to form a physical molecular state. Then the double heavy molecular state Y is characterized by the effective heavy quark mass M Q and the virtuality V = M 2 Y − (2M Q ) 2 [12,13]. It is natural to choose the energy scales of the QCD spectral densities as µ = V , which works well in the QCD sum rules for the molecular states. In Ref. [12], we obtain the optimal value M c = 1.84 GeV. Recently, we re-checked the numerical calculations and corrected a small error involving the mixed condensates. After the small error was corrected, the Borel windows are modified slightly and the predictions are also improved slightly, but the conclusions survive. In this article, we choose the updated value M c = 1.85 GeV.
In the scenario of molecular states, we study the color singlet-singlet type and octet-octet type scalar, axial-vector and tensor hadronic molecular states with the QCD sum rules in a systematic way [12,13], and assign the X(3872), Z c (3900/3885), Y (4140), Z c (4020/4025) and Z b (10610/10650) to be the molecular states tentatively, Now we search for the Borel parameters T 2 and continuum threshold parameters s 0 to satisfy the following four criteria: • Pole dominance at the phenomenological side; • Convergence of the operator product expansion; • Appearance of the Borel platforms; • Satisfying the energy scale formula. The resulting Borel parameters, continuum threshold parameters, pole contributions and energy scales are shown explicitly in Table 1. From the Table, we can see that the central values of the pole contributions are larger than 50%, the pole dominance condition can be satisfied. In the Borel windows, the contributions come from the vacuum condensates D i of dimension i are where i = 0, 3, 4, 5, 6, 7, 8, 10. The operator product expansion is well convergent. In the QCD sum rules for the hidden charm tetraquark states and molecular states, the operator product expansion converges slowly, we have to postpone the Borel parameters to large values. Larger Borel parameters lead to smaller pole contributions at the hadron side. So in the QCD sum rules for the hidden charm tetraquark states and molecular states, the Borel windows are rather small, T 2 max −T 2 min ≈ 0.4 GeV 2 , while the low bounds of the pole contributions are about (40−45)%. From Table 1, we can see that the threshold parameters and the predicted masses satisfy the relation √ s 0 = M Y + (0.4 ∼ 0.6) GeV. Naively, we expect that the energy gap between the ground state and the first radial excited state is about 0.4 ∼ 0.6 GeV, the present predictions are reasonable. Although the low bounds of the pole contributions are less than 50%, the contaminations of the radial excited states and continuum states are expected to be excluded by the continuum threshold parameters s 0 .
We take into account all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the vector molecular states, which are shown explicitly in Figs.1-2     In this article, we distinguish the charge conjugations of the currents, calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion in a consistent way, the intervals of the vacuum condensates are much larger than the ones in Refs. [10,11]. Moreover, we use the energy scale formula to determine the energy scales of the QCD spectral densities, which works well in our previous works [12,13]. We obtain the predications M DD1(1 −− ) = 4.36±0.08 GeV, M DD1(1 −+ ) = 4.60 ± 0.08 GeV, M D * D * 0 (1 −− ) = 4.78 ± 0.07 GeV and M D * D * 0 (1 −+ ) = 4.73 ± 0.07 GeV, which differ from the ones in Refs. [10,11] significantly, the conclusion is changed.

Conclusion
In this article, we assign the Y (4390) and Y (4220) to be the vector molecular states DD 1 (2420) and D * D * 0 (2400), respectively, distinguish the charge conjugations, construct the color singlet-singlet type currents to interpolate them. We calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion in a consistent way, use the energy scale formula to determine the energy scales of the QCD spectral densities, and study the masses and pole residues with the QCD sum rules in details. The present predictions only favor assigning the Y (4390) to be the DD 1 (1 −− ) molecular state. −r