Strong decays of higher charmonium states into open-charm meson pairs

The open-charm strong decays of higher charmonium states up to the mass of the $6P$ multiplet are systematically studied in the $^3P_0$ model. The wave functions of the initial charmonium states are calculated in the linear potential (LP) and screened potential (SP) quark model. The decay widths for most of the well-established charmonium states above the open-charm thresholds can be reasonably described. By comparing our quark model calculations with the experimental observations we also discuss the nature of some of the newly observed charmonium-like states. It is found that (i) the $\psi(4415)$ may favor the $\psi(4S)$ or $\psi_1(3D)$ assignment. There may exist two highly overlapping vector charmonium states around 4.4 GeV; (ii) In the LP model the $J^{PC}=1^{--}$ $Y(4660)$ resonance and the $J^{PC}=0^{++}$ $X(4500)$ resonance may be assigned as the $\psi(5S)$ and $\chi_{c0}(4P)$, respectively; (iii) The newly observed state $X^*(3860)$ can be assigned as the $\chi_{c0}(2P)$ state with a narrow width of about $30$ MeV; (iv) It seems to be difficult to accommodate the $X(4140)$ and $X(4274)$ states in the same potential model as excited $\chi_{c1}$ states. (v) The $X(3940)$ resonance can be assigned as the $\eta_c(3S)$ state; (vi) The vector charmonium-like states $Y(4230/4260,4360)$ and scalar $X(4700)$ cannot be described by any conventional charmonium states self-consistently in our model.


I. INTRODUCTION
During the last decade, many new charmonium-like states above/near open-charm thresholds, such as X(3940), X(4140/4274), Y(4230/4260), Y(4360), Y(4660), X(4500) and X(4700), have been reported by the Belle, BaBar, LHCb, BESIII, CLEO Collaborations and so on [1]. Lately, a new charmonium-like state X * (3860) was observed in the e + e − → J/ψDD process by the Belle Collaboration [2]. These newly observed charmonium-like states have attracted a lot of attention from the hadron physics community. One obvious feature is that most of them have masses located around S -wave open-flavor thresholds and cannot be easily accommodated by the conventional quark model. Because of this, they have initiated tremendous interests and different ideas. Detailed reviews on the status of these charmonium-like states can be found in Refs. [3][4][5][6][7][8][9], where some of these states are categorized as exotic hadrons.
Although there has been progress made during the past years, there still exist many mysteries to be uncovered. As we have known that exotic states with normal quantum numbers can hardly be distinguished from the normal ones, in order to understand exotic candidates, one should also have a reasonable description of the normal hadron spectrum. In the charmonium sector the low-lying states can be very well described by non-relativistic potential quark model such as the Cornell model [10] and Godfrey-Isgur model [11][12][13]. How-ever, it is realized that the open channel effects would become essential for higher excited states and it is still challenging to include such effects in a coherent way [12,15].
In order to understand these recently observed charmonium-like states, a better understanding of the charmonium spectrum can be regarded as a prerequisite. In this work we take the strategy of studying systematically the charmonium open flavor decays within the widely used linear potential (LP) model [11][12][13] and screened potential (SP) model [15,16] such that most of the conventional charmonium states can be identified by comparing with the experimental measurements. Although it should be recognized that the SP model may not be sufficient for including the full open threshold effects, we anticipate that unusual phenomena arising from such a study would indicate signals for unconventional structures of some of those charmoniumlike states. In Ref. [14] the charmonium spectrum and their electromagnetic (EM) transitions have been studied within the LP and SP model. For the low-lying charmonium states with a mass of M < 4.0 GeV, both models give comparable predictions. However, for the higher charmonium states with a mass of M > 4.1 GeV, the SP model gives very different results from the LP model. For example, in the SP model, the J PC = 1 −− charmonium-like states Y(4260) and Y(4360) are good candidates for the ψ(4S ) and ψ 1 (3D) states, respectively, while the J PC = 1 ++ charmonium-like states X(4140) may be assigned as the χ c1 (3P) state. In contrast, there is no room for the charmonium-like states Y(4260), Y(4360) and X(4140) in the LP model, the well-established state ψ(4415) may be assigned to ψ(4S ) or ψ 1 (3D), and the charmonium-like states X(4274) seems to be a candidate of χ c1 (3P). Such a result has already shown different dynamic origins introduced by the color screening effects. To clarify the nature of the newly observed charmonium-like states, we continue to investigate the open-flavor strong decays in the SP and LP models in this work. The differences between these two models and their comparisons with experimental observations can provide valuable information on the internal structures of these charmonium-like states.
By adopting the wavefunctions of the charmonium states calculated with the LP and SP models in our previous work [14] their strong decay amplitudes can be calculated by the widely used 3 P 0 model [17][18][19][20][21][22]. In this method, one assumes that a qq pair is produced from the vacuum with the vacuum quantum numbers, J PC = 0 ++ , and the decay of the charmonium state takes place by regrouping the new qq pair created from the vacuum and the cc in the initial state into the outgoing open-charm meson pair via a rearrangement process.
As an important topic in hadron physics, the open-charm strong decays of the charmonium states are often discussed in the literature [12,. Several pioneering works can be found in Refs. [20][21][22], where the open charm strong decays of ψ(3770), ψ(4040), ψ(4160) and ψ(4415) have been evaluated about forty years ago. Stimulated by first observed charmonium-like resonance X(3872) at Belle [45] and CDF [46], Eichten, Lane and Quigg analyzed the open charm strong decays of charmonium states near threshold in the Cornell coupled-channel model [35,36]. In 2007, Ding, Zhu and Yan considered the open flavor strong decays of Y(4360) and Y(4660) as 3 3 D 1 and 5 3 S 1 canonical charmonium in the simple harmonic oscillator wavefunction approximation in the framework of flux tube model [37]. In 2008, Segovia et al. calculated the open-flavor strong decays of the J PC = 1 −− charmonium states in the 3 P 0 model [23], where the new X(4360) state was considered to be the ψ(4S ) state and the ψ(4415) as the ψ 1 (3D) state, which differs from other assignments. In 2009, as conventional charmonium states, the open-flavor strong decays of the newly observed resonances X(3915) and X(4350) were studied by Liu et al. within the 3 P 0 model, the strong decay properties indicate that they may be assigned as χ c0 (2P) and χ c2 (3P), respectively [43]. Further studies of the open-flavor strong decays of P-wave charmonium states were also carried out within the 3 P 0 model by several groups in recent years [25][26][27][28]. It is found that X(3915) may be disfavored the assignment of χ c0 (2P) [25,26], the X(4140) may favor the χ c1 (3P) state [27], while the newly observed state X * (3860) can be a good candidate of χ c0 (2P) with a broad width [28]. Recently, the Bethe-Salpeter method was also extended to deal with the open-charm strong decays of several charmonium states [38][39][40]. However, as emphasized earlier, systematic studies of the full spectrum are essential for a better understanding of the underlying dynamics. Furthermore, how to properly treat the strong S -wave threshold interactions is a key issue for the description of near-threshold states [6]. The SP model can partially account for such an effect which makes the systematic comparison between the LP and SP model results interesting. Note that the most recent systematic study of the strong decays of higher charmonium states was carried out by Barnes, Godfrey and Swanson [12] quite long ago. It is necessary to re-investigate in a systematic way the strong decays of the higher charmonium states by combining the recent progress in theory and experiments.
The paper is organized as follows. In Sec. II, a brief introduction to the 3 P 0 strong decay model is presented. In Sec. III, we focus on the calculation results and discuss the phenomenological consequences in comparison with the experimental data. A summary is given in Sec. IV.
In this work, we used the 3 P 0 model to calculate the Okubo-Zweig-Iizuka (OZI) allowed strong decay widths for the charmonium states above DD threshold. The 3 P 0 model is a model that describes the quark pair creation mechanism of the OZI allowed strong decays based on the quark model. It is firstly proposed by Micu [17] and then extended by Le Yaouanc et al [18,19]. This model has been widely applied to deal with the open-charm strong decays of the charmonium states [12,[20][21][22][23][24][25][26][27][28][29][30][31][32][33]. In the 3 P 0 model, one assumes that a quark-antiquark pair is produced from the vacuum with the quantum number 0 ++ and the heavy meson decay takes place via the rearrangement of the four quarks. Such a process is empirically illustrated in Fig. 1. The quark pair creation process from vacuum can be described as, where γ is a dimensionless constant that denotes the strength of the quark-antiquark pair creation with momentum p 3 and p 4 from vacuum; b † 3i (p 3 ) and d † 4 j (p 4 ) are the creation operators for the quark and antiquark, respectively; the subscriptions, i and j, are the SU(3)-color indices of the created quark and anti-quark; φ 34 0 = (uū + dd + ss)/ √ 3 and ω 34 0 = 1 √ 3 δ i j correspond to flavor and color singlets, respectively; χ 34 1,−m is a spin triplet state; and Y ℓm (k) ≡ |k| ℓ Y ℓm (θ k , φ k ) is the ℓ-th solid harmonic polynomial. The factor (−3) is introduced for convenience, which will cancel the color factor.
In the center-of-mass (c.m.) frame of the initial meson A, the helicity amplitude can be written as, with the integral in the momentum space, In A partial wave amplitude can be obtained by using the Jacob-Wick formula [47] where Then the strong decay width for a given decay mode of meson A is given by When calculating a decay width of a charmonium state, we adopt the numerical wave function for a charmonium state calculated by the LP and SP models from our previous work [14]. For the emitted charmed mesons in a decay process, such as D and D * , we use simple harmonic oscillator (SHO) wavefunc-tions as an approximation, where β is the universal harmonic oscillator parameter, and L L+1/2 n p 2 β 2 is an associated Laguerre polynomial. To partly remedy the inadequacy of the nonrelativistic wave function as the momentum P increases, a commonly used Lorentz boost factor γ f is introduced into the decay amplitudes [55][56][57][58] where γ f ≡ M B /E B . In most decays, the three momenta P carried by the final state mesons are relatively small, which means the nonrelativistic prescription is reasonable and corrections from the Lorentz boost are not drastic. In our calculations, we set m u = m d = 330 MeV, m s = 450 MeV and m c = 1483 MeV for the constituent quark masses. The masses of the well-established hadrons in the final states used in the calculations are adopted from the PDG [1]. In the present work, both β and the pair creation strength γ are considered as free parameters, which are determined by fitting the decay widths of the well-established charmonium states ψ(3770), ψ(4040), ψ(4160) and χ c2 (2P). If we adopt the wavefunction of a charmonium state calculated using the LP model, we obtain β = 0.38 GeV and γ = 0.234. And if we adopt the wavefunction of a charmonium state calculated using the SP model, we have β = 0.36 GeV and γ = 0.217, which are consistent with the LP results. With these parameters, the decay widths of ψ(3770), ψ(4040), ψ(4160) and χ c2 (2P) can be reasonably described in both LP and SP models (see Table I). The strong decay properties for the higher charmonium states up to the mass of the 6P multiplet have been listed in Tables II-VI.

III. RESULTS AND DISCUSSION
A. Well-established cc states We choose four states above the DD threshold to determine the parameters in our model, i.e. ψ(3770), ψ(4040), ψ(4160), and χ c2 (3927) which are broadly accepted as 1 3 D 1 , 3 3 S 1 , 2 3 D 1 , and 2 3 P 2 states, respectively. The first three states, ψ(3770), ψ(4040), and ψ(4160), have been well-established for a long time, while χ c2 (3927) was observed in experiment quite recently. A good understanding of their strong decay properties is the starting point for our study of the strong decay properties of other charmonium states.   The ψ(3770) is assigned to be the 1 3 D 1 charmonium state though a small S -wave component is allowed. This is the first D-wave vector charmonium state in the spectrum and located close to the DD threshold. In principle, the production of a Dwave state will be highly suppressed in e + e − annihilations due to the heavy quark spin symmetry (HQSS) constraint. However, as found by experiment, the production cross section for ψ(3770) is actually sizeable. It indicates quite large HQSS breakings in the charmonium sector mainly because the charm quark mass is not heavy enough. As the consequence, its non-DD branching ratio turns out to be much larger than a naive estimate based on the HQSS (see e.g. Refs. [48][49][50] for a modern view of this topical issue).
The dominant decay mode of ψ(3770) into DD is driven by the D-wave component in its wavefunction. By treating it as a pure 1 3 D 1 state as the leading approximation, its strong decay properties can be well understood within both LP and SP models. Note that the differences of the momentum transfers between the charged and neutral DD meson pairs will introduce isospin breaking effects to the ψ(3770) → DD couplings in the 3 P 0 model. Taking into account such isospin breaking effects, we obtain the partial width ratio between the two modes D 0D0 and D + D − , which is consistent with the world average value 1.26 ± 0.021 from the PDG [1].

ψ(4040)
The ψ(4040) resonance is assigned to be the 3 3 S 1 charmonium state in the potential quark model. Four open-charm decay modes DD, DD * + c.c., D * D * and D sDs have been seen in experiment [1]. For convenience, we apply as follows the abbreviations DD, DD * , D * D * and D s D s etc for the corresponding charmed and anti-charmed meson pairs in the final state. Its OZI allowed two-body strong decays in the LP and SP models are calculated and listed in Table V.
We find that the total width Γ ∼ 60 MeV obtained in this work is slightly smaller than the world average value 80 ± 10 MeV. The ψ(4040) mainly decays into the D * D * channel. Within the LP model, the partial width ratio is consistent with the measured value 0.24 ± 0.05 ± 0.12 from the BaBar Collaboration [51]. However, our calculation of seems to be much larger than the measured value 0.18 ±0.14 ± 0.03 from the BaBar Collaboration [51]. In the SP model the partial width ratios are which is very different from the results of the LP model and the calculations of other works [12,36,42]. It should be noted that the decay channel of ψ(4040) → D * D * is quite sensitive to the kinematics due to the limited phase space. Furthermore, the partial width ratios extracted in various models seem not to agree with the data. Interestingly, the dominance of the D * D * decay channel is supported by the data for e + e − → D * D * from Belle [52], and the recent analyses of Ref. [53] by solving the Lippmann-Schwinger equation and lineshape studies in Ref. [54]. It shows that an improved measurement of the exclusive decays of ψ(4040) and reliable extraction of its resonance parameters are needed.

ψ(4160)
The ψ(4160) is assigned to be the 2 3 D 1 charmonium state in the quark model. Four open-charm decay modes DD, DD * , D * D * and D s D * s have been seen in experiments [1]. Its OZI allowed two-body strong decays using the wave functions calculated by the LP and SP models are evaluated, respectively, and the results are listed in Table V.
It shows that the measured width of ψ(4160), Γ ≃ 70 ± 10 MeV, can be well described by both the LP and SP models. The decay rate of ψ(4160) into D s D s is tiny, which can explain why the D s D s mode is not seen in experiment. In the main decay channels of ψ(4160), our calculation gives which is consistent with the result of Ref. [12]. Using the wave function calculated by the LP model, we find that which is similar to the ratios, 4 : 2 : 20, calculated by the SP model. However, these two ratios are very different from the measured ratios Γ(DD) : Γ(DD * ) : Γ(D * D * ) ≃ 1 : 17 : 50 from the BaBar Collaboration [51]. Notice that in these Pwave decay channels there exist obvious interfering effects between ψ(4040) and ψ(4160). A coherent partial wave analysis combining all these exclusive channels seems to be necessary for extracting the resonance parameters for these two states. It should be mentioned that the measured ratios cannot also be well understood in some existing models [36,42].
The X(3927) was observed in the γγ → DD process by the Belle [59] and BaBar [60] collaborations, and has been a good candidate for χ c2 (2P). We study its strong decay properties in both LP and SP models and the results are listed in Table V.
It shows that both models give similar strong decay properties for this state. The total width of χ c2 (2P) is predicted to be Γ ≃ 40 MeV, which is close to the upper limit of the measurements. The χ c2 (2P) dominantly decays into the DD channel, while the decay rate into the D * D channel is also sizeable. The branching fraction Br[χ c2 (2P) → D * D] can reach up to ∼ 40%. In the LP model, the partial width ratio is found to be which is slightly smaller than the ratio 0.68 obtained in the SP model. The D * D decay mode can be searched in e + e − → γD * D which is also the channel accessible for X(3872) as predicted in Ref. [61].  [1]. Some of them exhibit unusual properties that are very different from the expectations as conventional cc states. In addition, although ψ(4415) has been well-established in experiment, its structure and quark model assignment still need to be studied.
The Y(4260) state turns out to be a mysterious state from the very beginning. It was first reported by the BaBar Collab-oration in the initial state radiation e + e − → γ ISR J/ψπ + π − [62], and then confirmed by CLEO-c [63] and Belle [64] experiments in the same channel. However, its presence in open charm decay channels is not obvious at all, which has provoked a lot of theoretical interpretations in the literature. Comprehensive reviews can be found in several recent review articles [3,4,[6][7][8]. Recently, following the discovery of charged charmonium-like state Z c (3900) in e + e − → J/ψππ at the c.m. energy of 4.26 GeV [65], the BESIII Collaboration observed more detailed structures around the Y(4260) in several exclusive decay channels, namely, J/ψππ [66], h c ππ [67], ωχ c0 [68,69], and D 0 D * − π + + c.c. [70]. In particular, in Ref. [70] by treating these two structures around 4.23 and 4.29 GeV as from two Breit-Wigner states, a narrow resonance Y(4230) and a relatively broad resonance Y(4260) are extracted.
However, as studied in a series of works in Refs. [54,[71][72][73][74][75], the energy region of Y(4260) is close to the first narrow S -wave open charm threshold DD 1 (2420). Therefore, a strong near-threshold S -wave coupling can dynamically generate molecular state, which can then mix with nearby vector charmonium state and cause nontrivial near-threshold structures. In such a hadronic molecule scenario, the structures observed in these exclusive decay channels can be accounted for by the dynamics introduced by the DD 1 (2420) threshold.
The interesting phenomena arising from the 4.26 energy region is also a challenge for potential quark models in the study of the cc spectrum. In the LP model, it is almost impossible to accommodate the mass of Y(4230, 4260) in the spectrum. In contrast, in the SP model the mass of ψ(4S ) is found to be around 4.28 GeV. This might indicate the important role played by the S -wave threshold of DD 1 (2420) which can be partially accounted for by the screening effects. Therefore, if we assume that Y(4260) is dominated by the ψ(4S ) component in the wave function, we can investigate its decay properties as a charmonium state. As shown by the results listed in Table II, we find that Y(4260) should be a very narrow state with a width of ∼ 14 MeV, and its strong decays are dominated by the D * D * mode. Although such a result is consistent with that from Ref. [76] and several consequent works [77][78][79] with the assignment of Y(4260) as the ψ(4S ) state, the narrow width does not agree with the measured value of about 55 ± 19 MeV [1]. Note that such a solution also cannot be accommodated by the two-state fitting performed by Ref. [70]. Another issue is that if we assign Y(4260) to ψ(4S ), we will be unable to understand the decay properties of ψ(4415) in the SP model at all, which will be discussed later.

ψ(4415)
In the LP model the calculated masses of ψ(4S ) and ψ 1 (3D) are located around ∼ 4.4 GeV. Thus, the ψ(4415) might be a good candidate for ψ(4S ) or ψ 1 (3D) as broadly discussed in the literature. In contrast, in the SP model, the ψ(4415) is suggested to be a candidate of ψ(5S ). We discuss these possibilities below with details.
Assuming ψ(4415) as the ψ(4S ) state in the LP model, the to be mixed states via 3 P 1 -1 P 1 mixing as defined in Ref. [58]. While D 0 and D 2 stand for the states D 0 (2400) and D 2 (2460) listed in the PDG [1], respectively. ration [80]. The decay rates into D * D * , DD, D * D and D * s D s are relatively small with typical branching fractions O(1%). Our calculation result for the decay rate of ψ(4415) → D s D s is tiny, i.e., Br[ψ(4415) → D s D s ] < 10 −4 . The present data for e + e − → D * s D * s are still with large uncertainties [81] though some hints of enhancement around 4.415 seem to be present. Further improved measurement is strongly recommended.
In the LP model the partial width ratio between the DD and D * D * channel is which is close to the upper limit 0.29 measured by BaBar collaboration [51]. The ratio between D * D and D * D * , is also consistent with the measured value 0.17 ± 0.28 within uncertainties.
Assuming ψ(4415) as the ψ 1 (3D) state in the LP model, the calculation results are listed in Table VI is in the range of 0.14 ± 15 measured by BaBar Collaboration [51], while the partial width radio is also in the range of data 0.17 ± 0.28 [51]. Finally, we consider the possibility of ψ(4415) as the ψ(5S ) state in the SP models [14,15]. This is based on the assignment that in the SP model the mass of ψ(4S ) is found to be around 4.28 GeV as investigated in Subsection III B 1. With this hypothesis, the strong decay properties of ψ(4415) are calculated and listed in Table II. The results from such an assignment turns out to be inconsistent with the observations of ψ(4415).
In brief, it shows that the present data cannot distinguish the assignments of ψ(4415) as ψ(4S ) or ψ 1 (3D) in the LP model, while its assignment as the ψ(5S ) in the SP model cannot be supported. It should be mentioned that at the mass of 4.4 GeV, the nearby S -wave open-threshold may introduce coupled-channel effects of which if the interaction is strong enough, it can dynamically generate poles in a unitarized formulation and mix with the charmonium state. A recent study of the dynamic effects arising from the nearby D s0 (2317)D * s and D s1 (2460)D s thresholds and their impact on the property of ψ(4415) can be found in Ref. [82]. Taking into account that both the ψ(4S ) and ψ 1 (3D) states are likely located within this energy region, it is also possible that there may exist two highly overlapping vector charmonium states around 4.4 GeV. More accurate measurements and more observables are needed in order to understand the vector spectrum above 4 GeV in the future.

Y(4360)
In the vector charmonium spectrum, the Y(4360) resonance was first reported by the BaBar Collaboration in e + e − → ψ(2S )π + π − [83]. Later, the Belle Collaboration confirmed this state in the same channel [84]. The interesting feature about Y(4360) is its presence in the hidden charm decay channel but seems to be absent from the open charm decays. This is very similar to Y(4260) when it was first observed in e + e − → J/ψππ. This mysterious state has also initiated many theoretical studies with different possible solutions [3,6,8] including possible open-charm effects which can mix and shift the nearby charmonium state.
In the SP model, the mass of ψ 1 (3D) is estimated to be 4.32 GeV. Considering only the mass position, Y(4360) can be a good candidate of the ψ 1 (3D). Our calculation results are listed in Table VI. It shows that the ψ 1 (3D) resonance should be a very narrow state with a width of Γ ≃ 20 MeV, and its decays should be dominated by the D * D * , DD and DD 1 modes. In contrast with the experimental value of the width, i.e. Γ = 102 ± 9 MeV, the calculated width of ψ 1 (3D) is too small. Moreover, it has not been observed in open-charm decay channels. Note that the SP model has partly included the open-charm effects, the mismatching of Y(4360) as the ψ 1 (3D) state with the experimental measurement has reflected some unusual properties of Y(4360).

Y(4660)
The Y(4660) was observed in association with Y(4360) by the Belle Collaboration in e + e − → ψ(2S )ππ [84]. Its assignment is still controversial though by filling the lower states with some of these observed enhancements, it leaves the ψ(5S ) as a possible option. However, it should be pointed out that the S -wave open-charm threshold D * D 2 is located nearby. Therefore, possible contributions from the open-channel effects or dynamically generated state cannot be ruled out [6].
In the SP model, the mass of ψ(5S ) is lower than 4.6 GeV and in Subsection III B 2 the assignment of ψ(4415) as the ψ(5S ) has been discussed. The results show that ψ(4415) does not favor such an assignment. In the LP model, the mass of ψ(5S ) is predicted to be 4711 MeV, which is about 50 MeV larger than the mass of Y(4660). As the nearest state we investigate its strong decay properties as the ψ(5S ), and the results are listed in Table II.
It shows that the calculated width ∼ 50 MeV is close to the measured value Γ = 70 ± 11 MeV. The main open-charm decay channels include DD ′ 1 , D * D 0 , D * D 1 , D * D ′ 1 and D * D 2 with the branching fractions at the order of 10−20%. As mentioned earlier, the Y(4660) was observed in e + e − → ψ(2S )ππ instead of open-charm decay channels. Therefore, further experimental studies confirming or denying its contributions to these open-charm decay channels should be essential for determining its nature.
C. Candidates of higher cc states with J PC = 1 ++ The higher cc states with J PC = 1 ++ , such as χ c1 (2P), χ c1 (3P) and χ c1 (4P), are still not established. During the past decade, several J PC = 1 ++ charmonium-like states, X(3872), X(4140) and X(4274), have been observed in experiments. They might be good candidates of the missing J PC = 1 ++ cc states, but associated by non-trivial dynamics.

X(3872)
The X(3872) resonance has the same quantum numbers as χ c1 (2P) (i.e., J PC = 1 ++ ) but with a much lighter mass than potential quark model predictions. Its mass is close to the DD * threshold and makes it an ideal candidate for the DD * hadronic molecule. Various scenarios have been discussed in the literature for which recent reviews can be found in Refs. [3,4,[6][7][8]. It is now broadly accepted that the X(3872) has both a long-ranged molecular wave function and a shortranged compact cc component [85]. It can be viewed as the mixture of the DD * + c.c. molecule and χ c1 (2P) combined by a unitarized strong S -wave interaction (see e.g. Ref. [6] for a detailed review).
We do not expect that the LP and SP model can explain the observed properties of X(3872) by treating it as the χ c1 (2P) state. But as a test of the potential model calculations we consider X(3872) as the χ c1 (2P) state and calculate its strong decays into D 0 D * 0 . It shows that a strong coupling can be extracted and the dominance of the partial widths of X(3872) → D 0 D * 0 (if the input mass is higher than the threshold) is consistent with the experimental observations. We mention that the radiative transitions of X(3872) → ψ(2S )γ and J/ψγ were studied in Ref. [14], where the X(3872) was also treated as the χ c1 (2P) state. It also shows that the radiative decay properties are consistent with the observations from the BaBar [86] and LHCb [87]. This feature was regarded as evidence for X(3872) being the χ c1 (2P) state. However, as studied by Refs. [88,89] the radiative decays of X(3872) are shown to be driven by the short-ranged component of the wavefunction. Therefore, our results actually can be regarded as a support of such a view that the short-ranged wavefunction is from the χ c1 (2P) state instead of concluding that it is a χ c1 (2P) state. Since the physical state of X(3872) is apparently different from a conventional charmonium state, and there have been tremendous works discussing its dynamics, we do not want to over-interpret it based on our approach.

X(4140) and X(4274)
In the SP model, the mass of χ c1 (3P) is predicted to be ∼ 4.19 MeV. Thus, the charmonium-like state X(4140) can be a candidate for the χ c1 (3P) state. In this scenario its total width is found to be ∼ 14 MeV, which is in agreement with the world average data 19 +8 −7 MeV [1]. If X(4140) corresponds to the χ c1 (3P) indeed, it may mainly decay into DD * , D * D * and D s D * s channels with comparable decay rates. The partial width ratios between these decay modes are predicted to be which can be tested in future experiment.
In contrast with the LP model calculations, the mass of χ c1 (3P) is predicted to be ∼ 4.28 MeV which is close to X(4274). Assigning X(4274) as the χ c1 (3P) state, we calculate its strong decays in the LP model. It shows that the LP model produces a narrow width of about 21 MeV which is close to the lower limit of the measured width 56 ± 11 +8 −11 MeV [90]. The ratios between different partial widths, i.e. DD * , D * D * and D s D * s , are predicted to be It is interesting to note that the recent analysis of the B + → J/ψφK + process [92], and the study of the masses of cscs tetraquark states [93] seem to favor that the X(4274) may be the conventional χ c1 (3P) state. If X(4274) is assigned as the χ c1 (3P) state, then one immediate question is how to understand X(4140). In Ref. [92] the analysis suggests that the X(4140) structure may not be a genuine resonance. There are possibilities for a non-resonance interpretation for Y(4140), such as the D s D * s CUSP [90,91], or D s D * s rescatterings via the open-charmed meson loops [92]. To better understand X(4140) and X(4274), experimental studies of their opencharm decay modes, i.e. DD * , D * D * and D s D * s , are strongly recommended.
Considering X * (3860) as the χ c0 (2P) state we study its strong decays into DD in the LP and SP models. Our results are listed in Table III. Both models give a similar value of Γ ≃ 22 ∼ 28 MeV, which is rather narrow. Our results are consistent with that from Ref. [12]. Note that the present experimental width is not well determined. In contrast, results from different theoretical calculations are still controversial. For instance, the recent analysis of Ref. [94] found a small width about 11 MeV for the χ c0 (2P) state [94], while the analysis of Ref. [28] in a 3 P 0 model obtained a large width of 110 ∼ 180 MeV. This suggests that further precise measurement of the X * (3860) and more theoretical studies of the χ c0 (2P) state are necessary. We mention that tetraquark interpretations were also proposed for the nature of X * (3860) in the literature [95,96].
In the LP model the predicted mass of χ c0 (4P), M ≃ 4544 MeV, is close to the mass of X(4500), 4506±11 +12 −15 MeV. Considering the X(4500) as the χ c0 (4P) state, we study its strong decay properties and the results are listed in Table IV. It is shows that the calculated total width, Γ ∼ 50 MeV, is compatible with the measured value of X(4500) within its uncertainties. In Ref. [101], the X(4500) was also considered as the  conventional χ c0 (4P) state in the coupled-channel approach. If X(4500) is the χ c0 (4P) state indeed, it may dominantly decay into the DD 1 , DD ′ 1 , D * D 0 and D * D 2 channels, and the branching fractions are about O(10%). In contrast, the decay rate into DD channel is relatively small, which is is predicted to be Br[X(4500) → DD] ∼ 10 −3 . Such a small branching ratio may be difficult to observe in experiment.
In the SP model the predicted mass of χ c0 (5P) is 4537 MeV, which also makes X(4500) a good candidate. Assigning X(4500) as the χ c0 (4P) state, the strong decay properties of X(4500) are studied and the results are listed in Table IV. It shows that the χ c0 (4P) state should be very narrow in the SP model with a width of Γ ≃ 15 MeV. Its dominant decay modes are the DD 1 and DD ′ 1 channels. One notices that the total width in the SP model is about a factor of 4 smaller than the observed width of X(4500).
In the SP model the predicted mass of χ c0 (6P) is about 4669 MeV, which is very close to that of X(4700). Considering X(4700) as the χ c0 (5P) state, we find that the predicted width is about 15 MeV, which is too narrow to be comparable with the observed width 120±31 +42 −33 MeV of X(4700). Although the mass seems to fit the experimental observation, the significant discrepancy in the width raises questions on the structure of X(4700). In Ref. [92], the X(4700) is explained as the ψ ′ φ rescattering via the ψ ′ K 1 loops.
In brief, we find that the mass and width of the X(4500) can be understood in the LP model. In contrast, although the masses of both X(4500) and X(4700) can be described by the SP model, their widths appear to be difficult to understand. E. Candidates of higher cc states with J PC = 2 ++ The X(4350) found by Belle [102] in the φJ/ψ mass spectrum has been a good candidate for the χ c2 (3P) state with J PC = 2 ++ . The extracted mass and width are 4350.6 +4.6 −5.1 ± 0.7 MeV and 13 +18 −9 ± 4 MeV, respectively. In Ref. [43] such a possible assignment was also considered.
In the LP model the calculated mass of χ c2 (3P) is about 4310 MeV, which is very close to the mass of X(4350). Assigning the X(4350) as the χ c2 (3P), we study its strong decays and the results are listed in Table. III. Its width is found to be about 90 MeV, which is much larger than the observed value. This indeed raises questions on such an assignment. Note that the present experimental information is still very rough. Therefore, future experimental search for its decays into DD * , DD, D * D * and D * s D * s are strongly recommended.
F. Candidates of higher cc states with J PC = 0 −+ The X(3940) was first observed by the Belle Collaboration in e + e − → J/ψ + X [103]. This state is also established in the invariant mass spectrum of D * D in e + e − → J/ψD * D process [104]. The updated mass and width of X(3940) are M = 3942 +7 −6 ± 6 MeV and Γ = 37 +26 −15 ± 8 MeV, respectively. Its decay into DD * but not DD suggests that it has unnatural parity. The most likely interpretation of X(3940) is that it is the η c (3S ) state with J PC = 0 −+ [9], although the predicted mass in potential models appears to be higher than the observations.
Considering X(3940) as the η c (3S ) state, we analyze its strong decays in both the LP and SP models and the results are listed in Table II. Due to the limited phase space, it shows that the DD * channel is the only open-charm decay for X(3940). Furthermore, in the SP model the decay width is predicted to be about 40 MeV, which is in good agreement with the measurements. We also mention that the width is consistent with the calculation result from the Bethe-Salpeter method [38]. So far, the X(3940) turns out to be a good candidate for η c (3S ) according to its strong decay properties.

IV. SUMMARY
In this work we carry out a systematical study of the opencharm strong decays of the higher charmonium states up to the mass of the 6P multiplet within the 3 P 0 model. The wavefunctions of the initial charmonium states are adopted from the calculations by the LP and SP models in our previous work.
Several key results from this study can be learned here: • The decay widths for the well-established charmonium states ψ(3770), ψ(4040), ψ(4160) and χ c2 (2P) can be reasonably described in the LP and SP models, although for ψ(4040) and ψ(4160) some partial width ratios between the open-charm decay modes appear to have large discrepancies with the data.
• In the LP model, the ψ(4415) favors the ψ(4S ) assignment, while the possibility of ψ(4415) as the ψ 1 (3D) state cannot be excluded. There may exist two highly overlap J PC = 1 −− charmonium states around 4.4 GeV.
• The newly observed state X * (3860) seems to be too broad if it is classified as the χ c0 (2P) state. Our calculation shows that the χ c0 (2P) state should be a narrow state with Γ ≃ 30 MeV.
• The X(3940) favors the assignment as the η c (3S ) state, although the predicted mass in the potential models is somehow higher than the experimental data.
• The X(4140) resonance may be a good candidate of the χ c1 (3P) state in the SP model, while in the LP model X(4274) seems to favor the χ c1 (3P) state. However, since it is difficult to accommodate these two states in the same model, future studies of these two states are needed for a better understanding of their nature.
• The vector charmonium-like states, Y(4230/4260) and Y(4360), and the scalar charmonium state X(4700) cannot be accommodated by the conventional charmonium spectrum.
In brief, we show that a systematic study of the charmonium spectrum in the LP and SP models is useful for identifying unusual features arising from some of those higher charmoniumlike states recently observed in experiment. We anticipate that future experimental measurements of some of those opencharm decay channels should be helpful for pinning down both conventional and unconventional charmonium states and provide more insights into the underlying dynamics in the charmonium mass regime.