$Y(4320)$ and $Y(4390)$ as the candidate for $\psi(3^3D_1)$ charmonium

Inspired by the recent observations of $Y(4320)$ and $Y(4390)$, we evaluate the possibility of these two charmonium-like states as $\psi(3^3D_1)$ by investigating their open charm decays in quark-pair creation model. In the present work, our estimations indicate that the total width of $\psi(3^3D_1)$ is consistent with those of both $Y(4320)$ and $Y(4390)$. In addition, we find $D_1(2430) \bar{D}$ is one of dominant decay modes of $Y(4390)$, which indicates the process $Y(4390) \to \pi^+ D^0 D^{\ast -}$ may dominantly occur via the cascade decay $Y(4390) \to D^0 \bar{D}_1(2430) \to D^0 (\pi^+D^{\ast-})$. Moreover, the dominant decay channels of $\psi(3^3D_1)$ are predicted, which could be tested by further measurement at BES III and Belle II.

To date, the intrinsic structures of the charmonium-like states, Y(4220), Y(4320) and Y(4390) are still under debate. In our previous work [15,16,19], we categerized Y(4220) as ψ(4S ) and ψ(4415) as ψ(5S ), thus, there are no room for Y(4320) and Y(4390) in the S wave vector charmonium. However, in the D wave charmonium sector, the ψ(3770) and ψ(4160) are ψ(1 3 D 1 ) and ψ(2 3 D 1 ), respectively. The higher D wave vector charmonia have not been established. The mass of ψ(3 3 D 1 ) was predicted to be 4519 MeV by the relativistic quark model [27]. However, for the higher charmonia, the couple channel effects will shift their mass to the open charm threshold [28], thus the predicted mass of ψ(3 3 D 1 ) in Ref. [27] should be too large since the couple channel effects are not included. In Ref. [29], a screened potential model were employed to depict the couple channel effect in the charmonium and the mass of ψ(3 3 D 1 ) was predicted to be 4317 MeV. Considering the uncertainty of the quark model, both Y(4320) and Y(4390) could be the candidate for ψ(3 3 D 1 ) state. To further test this possibility, we estimate the open charm decays of Y(4320) and Y(4390) with the assignment of ψ(3 3 D 1 ) state, which is the main task of the present work.
This paper is organized as follows. After introduction, the formula of open charm decays of ψ( 3 D 1 ) states are presented in Section II. Our numerical results are given in Section III. Section IV is devoted to summary. Here, we adopt the quark pair creation (QPC) model (also named 3 P 0 model since the J PC quantum numbers of the quark pair created from the vacuum are 0 ++ ) to estimate the open charm decays of the vector D wave charmonia. The QPC model was first proposed by Micu [30][31][32][33] and then widely used to estimate the OZI allowed strong decay processes [30][31][32][33][34][35][36][37][38][39]. In the QPC model, the related S − matrix of A → BC process reads, where the transition operator T is, where Y 1m (k) = |k|Y 1m (θ, φ), χ 34 1,−m , ϕ 34 0 = (uū + dd + ss)/ √ 3 and ω 34 0 = δ α 3 α 4 are the space, spin, flavor and color parts of the wave functions, respectively. α 3 and α 4 are the color indexes of the created quark pair. In the QPC model, the parameter γ is introduced to represent the strength of the quarkantiquark pair creation from the vacuum and it could be fixed by fitting the decay data. In the present work, we take γ = 6.3 for the up/down quark pair and γ s = γ/ √ 3 for strange quark pair creation [34,35].
In the initial rest frame, the matrix element of the transition operator is where ϕ 13 34 1−m are the flavor matrix element and spin matrix element, respectively. While the color matrix element ω 13 B ω 24 C |ω 12 A ω 34 0 = 1/3 cancels out the factor 3 in the transition operator defined in Eq.
(2). The matrix element of the spatial part reads which reflect the overlap of the spatial wave functions of the initial state and final states. The amplitude of the decay process is By the Jacobi-Wick rotation, the amplitude can be transformed into partial wave amplitude, which is, In terms of the partial wave amplitude, the partial width is where B. open charm decays of ψ( 3 D 1 ) charmonium To test the reliability of the QPC model, we first investigate the open charm decays of the ground and first radial excited 3 D 1 charmonia, which are ψ(3770) and ψ(4190). These two D wave vector charmonia have been well established and we can compared the QPC estimations with the corresponding experimental data. And then apply the same model to study the open charm decays of ψ(3 3 D 1 ) state. The decay modes of ψ(3770), ψ(4160) and ψ(3 3 D 1 ) are listed in Table I.
For Y(4320) and Y(4390), their mass are above the threshold of D 1 (2430)D and D ′ 1 (2420)D. The charmed meson D 1 (2430) and D ′ 1 (2420) are the mixture of the 1 3 P 1 and 1 1 P 1 states and the mixing scheme is, where the mixing angle θ = −54.7 • , which is determined by the heavy quark limit [40][41][42]. The partial wave amplitude of the involved open charm decays are given in Table II.

III. NUMERICAL RESULTS AND DISCUSSIONS
With above preparations, we could investigate the open charm decays of the D wave vector charmonia. In Eq. (4), the spatial wave functions of the mesons are involved. In principle, these mesons wave functions could be estimated by the constitute quark model. However, as we discussed in the first section of the present work, there exist some uncertainties in the quark models. Thus, in the present work, we employ the simple harmonic oscillator wave function to simulate the spatial distribution of the quark-antiquark in meson. the detailed form of the spatial wave function is where n, ℓ and m ℓ are the radial, angular momentum and magnetic quantum numbers, respectively. F(−n, ν, x) and Y ℓm ℓ indicate the hypergeometric function and spherical harmonic function, respectively. In the spatial wave function, a parameter R is introduced. As for the lowest charmed mesons, the predictions of the relativistic quark model are well consistent with the experimental measurements. Thus, in the present work, the values of parameter R for the charmed and charmed-strange mesons are fixed such that it reproduces the root mean square radius estimated by the relativistic quark model [40]. In Ref. [35][36][37][38][39], the simple harmonic oscillator wave function with a parameter R has been used to investigate the decay behavior of mesons and the estimated results could well reproduce the corresponding experimental data, which proves such an approach is reliable to investigate the strong decays of the hadrons. As for the charmonia, the R values are quite different in different quark model, such as in the relativistic quark model, the R values for ψ(3770), ψ(4160) and ψ(3 3 D 1 ) are estimated to be 1.84, 2.09, and 2.24 GeV −1 , respectively, while in the screened potential model, these R values are 2.59, 3.12 and 3.59, respectively, thus in the present work, we vary the R values of the charmonia to check the R dependence of the decay widths. The masses and R values of the involved mesons are presented in Table III. In addition, the constituent quark masses for the charm, up/down, strange quarks are adopted to be 1.60, 0.22 and 0.419 GeV, respectively [34,35]. As for the ψ(3770), it is only about 40 MeV above the threshold of DD, thus, it dominantly decays into DD. The R dependence of the partial width of ψ(3770) → DD is presented in Fig. 2. The PDG average of the the branching ratio for ψ(3770) → DD is (93 +8 −9 )% and the width of the ψ(3770) is 27.2 ± 1 MeV. Thus, the measured partial width of ψ(3770) → DD is 22.7 ∼ 27.3 MeV. We find that the estimated partial width of ψ(3770) → DD with R = 1.56 ∼ 1.76 GeV −1 could well reproduce the experimental measurement.
The R dependent partial and total widths of ψ(4160) are presented in Fig. 3. The PDG average of the width of ψ(4160) is 70 ± 10 MeV and our estimations with R = 1.82 ∼ 1.97 could well consistent with the experimental data [43]. The R   These ratios are evaluated to be 0.46/0.01 and 0.2/0.05 by the QPC model with relativistic quark model and linear potential model, respectively [44,45]. In Ref. [46] , by using the Connell coupled-channel mode, the ratios are determined to be 0.08 and 0.16. On the experimental side, the BaBar collaboration performed a measurement of the exclusive production of DD, D * D and D * D * , the ratios were measured to be 0.02 ± 0.03 ± 0.02 and 0.34 ± 0.14 ± 0.05 [47], respectively, which is different from the QPC model estimations in the present work. It should be noticed that in Ref. [47], the data are fitted with three charmonia with fixed mass and width, which are ψ(4040), ψ(4160) and ψ(4415). From the current situation, there should exist more vector states in this energy range and thus the fitted results will be changed if more states are included. Moreover, in the analysis, the mass and width of ψ(4160) are fixed to be 4153 MeV and 103 MeV, respectively [48]. The values of the resonance parameters used in Ref. [47] are much different from latest PDG average, which are 4191 MeV and 70 MeV, respectively [43]. We expect the new precise measurement and analysis of the open charm decays of ψ(4160) at BESIII and BelleII could determine these ratios and test the results in the present work. Our estimations of ψ(3770) and ψ(4160) decays indicate that the QPC model is reliable to study the open charm decays of the charmomnia. Especially, the total width of these two established D wave charmonia could be well reproduced in a proper R range, which are R = 1.56 ∼ 1.76 GeV −1 and R = 1.82 ∼ 1.97 GeV −1 for ψ(3770) and ψ(4160), respectively. Thus, we could apply this model to study the decays of ψ(3 3 D 1 ) state to evaluate the possibility of Y(4320) and Y(4390) as ψ(3 3 D 1 ) state. Considering the R values of ψ(3770) and ψ(4160) determined by the total widths, we take R = 2.0 ∼ 3.0 GeV −1 for ψ(3 3 D 1 ) to study its open charm decays. The R dependent total and partial widths of the open charm decays of Y(4320) are presented in Fig. 4, where Y(4320) is assigned as ψ(3 3 D 1 ) charmonium. Our estimations indicate that when R = 2.49 ∼ 2.92 GeV −1 the evaluated total width are consistent with the measured one from the BESIII Collaboration [2]. This R value for ψ(3 3 D 1 ) is larger than the one of ψ(4160) and ψ(3770), which is consistent with our expectation. In this R range, ψ(  4390) reported in the e + e − → π + π − h c and e + e − → π + D 0 D * − by the BES III Collaboration [3,4] We present the R dependence of the total and partial widths of the open charm decay of Y(4390) in Fig. 5, where Y(4390) is assigned as ψ(3 3 D 1 ) charmonium. The width of Y(4390) reported in e + e − → π + π − h c is 139.5 +16.2 −20.6 ± 0.6 MeV and 181.7 ± 16.9 MeV in e + e − → π + D 0 D * − . The estimated total width of ψ(3 3 D 1 ) could overlap with one from both channels. In particular, the constrained R values from e + e − → π + π − h c and e + e − → π + D 0 D * − are R = 2.58 ∼ 2.78 GeV −1 and R = 2.83 ∼ 3.00 GeV −1 , respectively. In the range R = 2.58 ∼ 3.00 GeV −1 , Y(4390) also dominantly decays into DD, D * D , D * D * and DD 1 (2430)D. The ratios of these four channels are predicted to be In this R range, our estimations indicate the partial widths for D + s D − s , D * + s D − s , D * D 0 , and DD ′ 1 (2420) could be several MeV, while the D * + s D * − s and DD 2 decay modes could be ignored. From the results in the present work, one can find that D 1 (2430)D is one of dominant decay channels of Y(4390) and the width of D ′ 1 (2420)D is also about several MeV. Both D 1 (2430) and D 1 (2420) primaryly decay into D * π, which indicates the π + D 0 D * − may be dominantly from the cascade decay Y(4390) → D 0D 1 (2430), D 0D′ 1 (2420) → D 0 (π + D * − ), which could be test by BES III and Belle II.

IV. SUMMARY
The observations of the vector charmonium-like states in the e + e − annihilation processes make the states between 4.0 and 4.5 GeV overcrowed. Besides the higher excited state of J/ψ, these charmonium-like states could also be higher ψ( 3 D 1 ) states. In the present work, we evaluate the possibility of Y(4320) and Y(4390) as ψ(3 3 D 1 ) charmonium by investigate the open charm decays of ψ(3 3 D 1 ) in a QPC model.
The success in the study of open charm decays of the lowlying D wave vector charmonia indicate the reliable of the QPC model and encourage us to apply the same model to calculate the open charm decays of ψ(3 3 D 1 ) state. The mass of Y(4320) is well consistent with the one of ψ(3 3 D 1 ) charmonium predicted by the screened potential model and the total width estimated in the present work could overlap with the experimental measurement when R = 2.49 ∼ 2.92 GeV −1 . In the range R = 2.49 ∼ 2.92 GeV −1 , Y(4320) dominantly decays into DD, D * D and D * D * . As for Y(4390), the width reported from e + e − → π + D 0D * − is larger than the one reported from e + e − → π + π − h c . The total width estimated in the present work could overlap with the those from e + e − → π + π − h c and e + e − → π + D − D * − in the range R = 2.58 ∼ 2.78 GeV −1 and R = 2.83 ∼ 3.0 GeV −1 . In the range R = 2.58 ∼ 3.00 GeV −1 , the dominant decay modes of Y(4390) are also DD, D * D , D * D * and D 1 (2430)D. The observation of Y(4390) in the π + D 0 D * − could be well interpreted by the large decay width of D 1 (2430)D and D ′ 1D modes. To summarize, in present work, we evaluate the possibility of Y(4320) and Y(4390) as ψ(3 3 D 1 ) state by study the open charm decays in a QPC model. Our results indicate that the total width of ψ(3 3 D 1 ) could be consistent with the one of Y(4320) and moreover, the mass of ψ(3 3 D 1 ) predicted by the screened potential model is also in line with the one of Y(4320), thus Y(4320) could be a good candidate of ψ(3 3 D 1 ). As for Y(4390), the estimated width of ψ(3 3 D 1 ) could overlap with the reported width from e + e − → π + π − h c and e + e − → π + D 0 D * − in different R range. Our estimations also find that π + D 0 D * − may result from the large widths of D 1 (2430)D and D ′ 1 (2420)D. Thus, the possibility of Y(4390) as ψ(3 3 D 1 ) could not be rule out. To test the estimations in the present work, we expect the further precise searches for Y(4320) and Y(4390) in more channels, especially in the open charm channels, which could be performed by BES III and Belle II.