Constructing $J/\psi$ family with updated data of charmoniumlike $Y$ states

Based on the updated data of charmoniumlike state $Y(4220)$ reported in the hidden-charm %and open-charm channels of the $e^+e^-$ annihilation, we propose a $4S$-$3D$ mixing scheme to categorize $Y(4220)$ into the $J/\psi$ family. We find that the present experimental data can support this charmonium assignment to $Y(4220)$. Thus, $Y(4220)$ plays a role of a scaling point in constructing higher charmonia above 4 GeV. To further test this scenario, we provide more abundant information on the decay properties of $Y(4220)$, and predict its charmonium partner $\psi(4380)$, whose evidence is found by analyzing the $e^+e^-\to \psi(3686)\pi^+\pi^-$ data from BESIII. If $Y(4220)$ is indeed a charmonium, we must face how to settle the established charmonium $\psi(4415)$ in the $J/\psi$ family. In this work, we may introduce a $5S$-$4D$ mixing scheme, and obtain the information of the resonance parameters and partial open-charm decay widths of $\psi(4415)$, which do not contradict the present experimental data. Additionally, we predict a charmonium partner $\psi(4500)$ of $\psi(4415)$, which can be accessible at future experiments, especially, BESIII and BelleII. The studies presented in this work provide new insights to establish the higher charmonium spectrum.

Since 2003, abundant charmoniumlike XYZ states have been reported by experiments (see Refs. [40,41] for a review). As the first observed charmoniumlike state, X(3872) was announced by the Belle Collaboration in the J/ψπ + π − invariant mass spectrum from the B meson decay [42]. Since the mass of X(3872) is lower than that of χ c1 (2P) state predicted by quenched quark models like the GI model [34] and is close to the threshold of DD * channel, there were extensive discussions of exotic hadron assignments like the DD * molecular state [43,44] or tetraquark state [45,46]. Theorists have not given up the effort to categorize X(3872) into the charmonium family. According to lessons from studying Λ(1405) [47,48], D s0 (2317) [49,51], and D s1 (2460) [50,51], the importance of a coupled-channel effect was realized. If considering the coupled-channel effect, the low mass puzzle of X(3872) can be understood [52][53][54], which means that X(3872) as χ c1 (2P) state becomes possible by an unquenched quark model.
The study experience of exploring X(3872) tells us that the coupled-channel effect should be considered seriously, especially for the higher radial and orbital states. When checking the charmonium spectrum, we notice that the DD channel is open for ψ(3770). More open-charm decay channels are open for higher states ψ(4040), ψ(4160), and ψ(4415). In 2014, the Lanzhou group once indicated that it is not suitable to assign ψ(4415) as ψ(4S ) state. Due to the similarity between the charmonium and bottomonium families [55], ψ(4S ) is roughly predicted to be 4263 MeV by a mass gap estimate, which is also consistent with mass of ψ(4S ) predicted by potential models [54,56] with a color-screening effect. Here, we need to emphasize that there exists some equivalence between the screening potential model and coupled-channel model [57], which is a reason why our mass value of the ψ(4S ) is totally different from quenched quark models.
Frankly speaking, in the past 40 years, the charmonium spectrum above 4.16 GeV was not established, which also reflects how poorly we understand the nonperturbative behavior of Quantum Chromodynamics. This situation stimulates our interest in hunting the evidence of missing higher charmonia by combining with the updated experimental information of charmoniumlike Y states in the e + e − annihilations.
In this work, we indicate Y(4220) may play a role of the scaling point when constructing the whole charmonium family, especially, higher charmonium above 4 GeV. We need to face several key points: (1) The observed charmonia below 4.2 GeV should be well described even assigning Y(4220) to a charmonium. (2) There must exist a charmonium partner of Y(4220), which is still missing in experiment and whose properties should be predicted. The search for this predicted charmonium partner can be applied to test our scenario. (3) It is also crucial how to settle ψ(4415) in the J/ψ family since ψ(4415) is an established charmonium by different experiments.
To quantitatively illustrate these three key points, we adopt an unquenched potential model to study charmonium mass spectrum, which will be introduced in the next section. Associated with the study of mass spectrum, we further investigate the open-charm decay channels, where the quark pair creation (QPC) model is employed. Thus, their total and partial decay widths can be obtained, which makes us possible to compare with the experimental data and to provide the crucial information to experimental investigation.
Usually, the mixture happens between nS -wave and (n − 1)D-wave states. A typical example in the charmonium family is ψ(3686) and ψ(3770), which can be considered as 2S -1D wave mixing states. Since masses of ψ(4S ) and ψ(3D) are close to each other, the S -D wave mixing scheme should be considered. This inspires us to consider 4S -3D mixing scheme for Y(4220). Our study supports Y(4220) as a 4S -3D mixing state existing in the J/ψ family since our result is consistent with the present experimental data. In addition, we provide more abundant open-charm decay information, which can be applied to test this explanation of Y(4220).
Besides putting Y(4220) into the J/ψ family under this 4S -3D mixing scheme, what is more important is the prediction of its charmonium partner ψ(4380). Under this mixing scheme, an interesting phenomenon appears, i.e., the predicted ψ(4380) mainly decays into DD * 2 (2460) and has a weak coupling to DD. Thus, we discuss the possible evidence of ψ(4380) existing in the reported open-charm decay channels [74,75]. According to our studies, we strongly suggest to search for the charmonium partner of Y(4220) via the DD 1 (2430), D * D * , and DD * 2 (2460) channel, which will give a good chance for BESIII and Belle II to observe.
When categorizing Y(4220) in the J/ψ family, we have to face how to settle the well established ψ(4415) in the J/ψ family. In this work, we continue to propose 5S -4D mixing scheme for ψ(4415) and find that the obtained result strongly suggests this possibility. To give a definite conclusion, we need more precise measurements of ψ(4415) like the resonance parameters, and the partial widths of open-charm and hidden-charm decays. Under this mixing scheme, we naturally predict a charmonium partner of ψ(4415), which is also still missing. In this work, its mass, width and partial decay behavior are obtained. The search for it will be an interesting research issue, and this 5S -4D mixing scheme assignment to ψ(4415) can be tested in future.
Stimulated by the existence of Y(4220) in the e + e − → ψ(3686)π + π − process [64], we consider whether the predicted charmonium partner of Y(4220) may exist in the present experimental data of e + e − → ψ(3686)π + π − . In this work, we reanalyze the data of e + e − → ψ(3686)π + π − by introducing the Fano interference picture [72,73] proposed by us, and find the evidence of the charmonium partner of Y(4220).
This paper is organized as follows. After Introduction, we will give the description of the charmonium spectrum when setting Y(4220) as a charmoium state (see Sec. II) and predicting its charmonium partner. In addition, we discuss how to settle ψ(4415) in the J/ψ family, where 5S -4D mixing scheme is proposed and the corresponding charmonium partner of ψ(4415) is predicted. In Sec. III, we continue to analyze the recent e + e − → ψ(3686)π + π − data, and find there should exist the evidence of the predicted charmonium partner of Y(4220). Finally, the paper ends with a summary in Section IV.

A. A concise introduction of the methods adopted
To provide the description of the charmonium spectrum, in this work, we adopt an unquenched potential model, which has been applied to study heavy-light meson systems [76,77], kaon family [78], and bottomonium zoo [79].
The interaction between charm quark and anti-charm quark can be expressed by the Hamiltonian [34] where m c and mc are the masses of charm quark and anticharm quark, respectively.Ṽ eff (p, r) contains a short range γ µ ⊗ γ µ interaction of one-gluon-exchange and a long range 1⊗1 linear color confining interaction [34]. In the nonrelativistic limit,Ṽ eff (p, r) can be translated into a familiar non- Here, the first term is a spin-independent potential including the linear confinement and Coulomb-type potential, and the second term denotes the color-hyperfine interaction composed of the tensor and contact terms, and the third term is from the spin-orbit interaction including the color-magnetic term H SO(cm) qq and the Thomas-precession term H SO(tp) qq [34]. There are two aspects reflecting the relativistic corrections [34], i.e., smearing transformation and momentum-dependent factors. By introducing smearing function the confining potential S (r) = br + c and one-gluon exchange potential G(r) = −4α s (r)/(3r) are smeared out tõ This smearing treatment actually takes into account the nonlocality property of interaction between quark and antiquark. Besides, a general relativistic form of the potential should be dependent on momenta of interacting quarks in the center-ofmass system, so a smeared potentialṼ i (r) could be modified according tõ where E c = (p 2 + m 2 c ) 1/2 and Ec = (p 2 + m 2 c ) 1/2 and a parameter ε i corresponds to a different type of interaction, such as contact, vector spin-orbit, etc [34].
In order to include the unquenched effect in this potential model, we need to consider the screening effect, which can be achieved by modifying a linear confining br + c as A similar smearing transformation and momentum-dependent factor for the S scr (r) are also performed, and the more detailed descriptions of this unquenched potential model can be found in Ref. [76]. To some extent, this unquenched potential model by considering a screening effect is partly equivalent to the coupled channel effect [57,76]. The screening effect has also been supported by unquenched lattice QCD calculations [80,81]. When we get the charmonium mass spectrum, the numerical spatial wave functions are obtained by this unquenched potential model, which can be applied to calculate the opencharm decays of the discussed charmonia. To quantitatively study their decay behaviors, we will employ the quark pair creation (QPC) model [82,83], which is a successful phenomenological method to deal with the Okubo-Zweig-Iizuka (OZI)-allowed strong decays of hadrons. In the following, we concisely introduce it.
In the QPC model, the transition matrix of the process A → B + C can be written as BC|T |A = δ 3 (P B + P C )M M J A M J B M J C (P), where the transition operator T describes a quark-antiquark pair creation from the vacuum and reads as We introduce a dimensionless constant γ depicting the strength of the quark pair creation from the vacuum, which can be fixed by fitting the experimental data. Later, we discuss how to fix it by the present charmonium data. χ 34 1,−m is a spin-triplet state, and φ 34 0 and ω 34 0 denote S U(3) flavor and color singlets, respectively. Y m (p) ≡ |p| Y m (θ p , φ p ) denotes the -th solid harmonic polynomial. By the Jacob-Wick formula [84], the helicity amplitudes M M J A M J B M J C (P), which are extracted by the transition matrix element, could be related to the partial wave amplitudes, i.e., where J = J B + J C , and L is the orbital angular momentum between final states B and C. The general partial width of the A → BC reads as In the above expression, m A is the mass of the initial state A. By this adopted unquenched potential model, we get the numerical spatial wave functions of the involved charmonia and charmed/charm-strange mesons. It can eliminate the parameter dependence of theoretical results compared to the previous calculation [55]. In addition, the relevant mass values of involved mesons are taken from the PDG [9] while the masses of the discussed missing charmonia in this work are from our theoretical calculation. In our calculation, the constituent quark mass m c , m u = m d , and m s are taken as 1.65 GeV, 0.22 GeV and 0.419 GeV, respectively. A parameter γ = 5.84 for qq can be extracted by fitting the experimental data as shown in Table I, where χ 2 /d.o. f = 3.6/3 = 1.20 is obtained. And then, the strength for ss creation satisfies γ s = γ/ √ 3 which was suggested in Ref. [83].
B. Charmonium mass spectrum by scaling Y(4220) as ψ(4S ) In this subsection, the main task is to present the charmonium mass spectrum by the unquenched model when scaling Y(4220) as ψ(4S ). Until now, there are fourteen established charmonia [9] together with Y(4220), which can be employed to limit our potential model parameters. These parameters mainly include the charm quark mass, three screening confinement parameters, and four ε i related to the relativistic corrections of momentum factors. By getting a minimum of the where A T h (i) and A Exp (i) are theoretical and experimental values, respectively, and A Er (i) is an experimental error. In our fit, the experimental masses are taken from PDG [9]. However, we find the experimental errors are generally too small to be suitable as our fitting errors. Here, we will choose A Er (i) = 10 MeV for most of the states except for two S -wave ground charmonia, whose fitting errors are set to one MeV. Then, the fitting χ 2 /d.o. f = 16.7 is obtained. Based on the above parameters, the charmonium spectrum in our unquenched potential model are summarized in Table  II, where the experimental masses of observed charmonia are also given. From Table II, we can clearly see that the charmonium mass spectrum below 4.2 GeV is well described, especially for the S -wave ground states. In this Table, we identify Y(4220) as our ψ(4 3 S 1 ). Therefore, the first key point of constructing the whole charmonium spectrum by the updated Y(4220) has been achieved.
In this subsection, we firstly discuss the OZI-allowed strong decay behavior of ψ(4S ). We get the total decay width 27.2 MeV for ψ(4S ) when the input mass is chosen as 4274 MeV. The above results show that treating the charmoniumlike Y(4220) state [60,65,67] as ψ(4S ) state is reasonable since the resonance parameter of Y(4220) can be reproduced under the ψ(4S ) assignment. Our result also supports the conclusion of the ψ(4S ) as a narrow state in Ref. [55].
In the following, we further list the obtained branching ra-tios of the open-charm decay channels of ψ(4S ), i.e., where DD * and D s D * s represent DD * +DD * and D sD * s +D s D * s , respectively.  For ψ(4S ), the main decay modes are composed of six typical open-charm decays just shown in Eqs. (8)- (13). If identifying Y(4220) to be a ψ(4S ) state, this Y(4220) structure should be found in the corresponding open-charm decay channels. Especially, our result shows that D * D * is the dominant decay channel of ψ(4S ). In Fig. 3, we collect the experimental data of open-charm decay channels from the e + e − annihilation, which were released by the Belle Collaboration as early as 2007 [74,85]. There does not exist the evidence of enhancement structures around 4.2 GeV to support this scenario of Y(4220) as ψ(4S ) 1 . Here, we should point out that the Belle measurements of the open-charm decay channels are still rough since the bin size of energy is large, which is not enough to provide a definite test of this scenario, especially for a narrow charmonium. Thus, we should wait for more precise data from BESIII and Belle II.
In 2018, the BESIII Collaboration released the measurements of the cross section of e + e − → D 0 D * − π + [86]. In their analysis, the e + e − → D * D * → D 0 D * − π + contribution was rejected, and the e + e − → D * 2 (2460) + D * − → D 0 D * − π + and e + e − → D 1 (2420) 0 D 0 → D 0 D * − π + are allowed [86]. A clear enhancement around 4.23 GeV was observed in the D 0 D * − π + invariant mass spectrum, which hints that this structure may have a strong coupling to the virtual D 1 (2420) 0 D 0 channel. Under our ψ(4S ) scenario, this phenomenon can be qualitatively understood. In Sec. II B, the sreening parameter µ reflects the importance of a screening effect, which means that the coupled-channel effect plays an important role to modulate a bare state of ψ(4S ) due to the partial equivalence between screening and coupled-channel effects [57]. A bare ψ(4S ) state associated with other channels like DD, DD * , and D * D * strongly couples to the nearby D 1 (2420)D channel, which shifts the original mass of ψ(4S ) to the present value 4274 MeV. Here, interaction between ψ(4S ) and D 1 (2420)D is a typical S -wave coupling while coupling of ψ(4S ) with DD, DD * , and D * D * occurs via P-wave interaction. Thus, D 1 (2420)D is one of the most important channels among the allowed coupled channels for ψ(4S ), which results in a chain reaction e + e − → ψ(4S ) → D 0 D * − π + via the virtual D 1 (2420) 0 D 0 as revealed by BESIII [86]. Although our theoretical results on ψ(4S ) are in good agreement with the experimental data of Y(4220), we cannot fully exclude a possibility of an extotic Y(4220), where a popular one is the charmonium hybrid state assignment to Y(4220) (a detailed discussion can be found in Ref. [41]). For the charmonium hybrid, the calculation of QCD sum rule [87,88] and flux tube model [89] suggest that the decay into two S -wave charmed mesons D ( * )D( * ) is suppressed. Instead, the modes of one S -wave and one P-wave charmed mesons are very important. Thus, an experimental study of the open-charm channel D ( * )D( * ) will provide a crucial test of different assignments to the Y(4220) since a charmonium has open-charm decay behavior different from a charmonium hybrid.
If explaining the charmoniumlike state Y(4220) as a ψ(4S ) state, we may expect an existence of its D-wave partner ψ(3D) state, which is still missing in experiment. Our calculation shows that the mass and total width of ψ(3D) are 4.334 GeV and 28.8 MeV, respectively. Similar to ψ(4S ), ψ(3D) is also a narrow charmonium.
The calculated branching ratios of the open-charm decays of ψ(3D) are Thus, DD channel is the dominant decay mode of the ψ(3D) state. The Belle data of e + e − → DD, however, does not show the evidence of ψ(3D) as presented in Fig. 3 (a). We try to find the evidence of ψ(3D) in the reported data of charmoniumlike states and notice the famous Y(4360) from the e + e − → ψ(3686)π + π − [9]. The mass of Y(4360) is close to that of ψ(3D), but the width of Y(4360) is broader than the predicted ψ(3D). This deviation should be faced when treating Y(4360) as ψ(3D). In addition, the e + e − annihilation decays of D-wave vector quarkonium states are generally one to three orders of magnitude smaller than those of corresponding Swave states [79]. Thus, it is not an easy task to observe this 3D state through the hidden charm decay channels from the electron-positron annihilation.
When further checking the early data of the open-charm process e + e − → DD * 2 → D 0 D − π + in Fig. 4 (a), a suspicious signal at 4.37 GeV is found. We may consider whether this enhancement structure is the predicted ψ(3D). However, our result indicates that ψ(3D) → DD * 2 (2460) has a tiny partial width (67.6 keV). It is obvious that this structure in e + e − → DD * 2 → D 0 D − π + cannot explain a ψ(3D) state. To understand this puzzling phenomenon, we need a new idea.
As mentioned in Introduction, the established charmonium states ψ(3686) and ψ(3770) are admixtures with a small S -D mixing angle rather than a pure S -wave or D-wave state [36]. This lesson tells us that 4S -3D mixing scheme should be considered, which may shed light on the above puzzling phenomenon. In the next subsection, we pay more attention to this issue. In this subsection, we discuss the 4S -3D mixing scheme. Under this framework, we introduce to describe the 4S -3D mixing. Here, θ denotes the mixing angle. Then, the mass eigenvalues of ψ 4S −3D and ψ 4S −3D are determined by the masses of two basis vectors m 4S , m 3D and the mixing angle θ, i.e., As shown in Table II  Focusing on the interesting Y(4220), we discuss the treatment of Y(4220) as ψ 4S −3D . According to the experimental results for Y(4220) [60,62,65,67,86], we set the mass range of Y(4220) to be 4204 ∼ 4243 MeV, which is lower than the mass of ψ(4S ). In fact, this mass difference between Y(4220) and pure ψ(4S ) is also a main motivation to stimulate us to introduce the 4S -3D mixing scheme.
Using the mass range of Y(4220), we may predict the mass range (4364 ∼ 4400 MeV) for ψ 4S −3D given in Fig. 5. Since its central value is 4384 MeV, we tentatively name this ψ 4S −3D state as ψ(4380) in the following discussion, which is nothing but the partner of the discussed Y(4220). The corresponding mixing angle θ = ±(30 • ∼36 • ) is obtained.
In the 4S -3D mixing scheme, we need to illustrate the decay properties of Y(4220), and further give the decay behaviors of its partner ψ(4380), which are collected in Fig. 6. We notice that the decay behavior of Y(4220) under this 4S -3D mixing scheme is similar to that of Y(4220) as a pure ψ(4S ) state when taking a positive mixing angle. That is, the obtained total decay width is 26.0 MeV, and the obtained partial widths of the allowed strong decays of Y(4220) are listed in Table III, where a typical θ = +34 • , which corresponds to an average measured mass of 4222 MeV for Y(4220), is taken. When taking a negative mixing angle, the decay property of Y(4220) under this 4S -3D mixing scheme is different from that of Y(4220) as a pure ψ(4S ) state, where the total width of Y(4220) as a mixture of 4S and 3D states becomes smaller and the branching ratio of the DD * mode is larger than that of the DD mode. Considering the above two cases, we suggest to adopt a positive mixing angle θ = (30 • ∼36 • ) in the following discussion.
Next, we investigate the decay behaviors of ψ(4380) with the running of a mixing angle in Fig. 6. To our surprise, two main conclusions can be made for the above mixing scheme: 1. The total width of ψ(4380) has a significant enhancement, which shows that ψ(4380) should be a broad state since its total width is nearly three times larger than that of a pure ψ(3D) state (28.8 MeV). This conclusion can be understood as follows. Since the phase space from the decays of ψ(4380) into P-wave and S-wave charmed mesons is larger than that of a pure ψ(3D) state, the channels of ψ(4380) have large contributions to the total decay width of ψ(4380).

2.
The dominant decay channels of ψ(4380) are DD 1 (2430), D * D * , and DD * 2 (2460), especially sizable enhancement of B(ψ(4380) → DD * 2 (2460)). Additionally, the contribution of the DD mode to the total width becomes unimportant. Thus, the decay behavior of ψ(4380) is totally different from a pure ψ(3D) state. This result can be due to the change of the spatial wave function of ψ(4380) obtained in the 4S -3D mixing scheme.
The concrete values reflecting the decay behaviors of ψ(4380) are shown in Table III.
The above information indicates that the potential enhancement structure around 4.37 GeV existing in the e + e − → DD * 2 → D 0 D − π + process (see Fig. 4 (a)) can be the pre- the broad structure may be from the contributions of ψ(4415) and other resonances [86]. Depicting the experimental data shown in Fig. 4 (b), we can see two clear enhancements near 4.38 GeV and 4.42 GeV. The former one implies an unknown resonance and the latter one can be related to the established ψ(4415). From our theoretical point of view, the ψ(4380) state mainly decays to D 0 D * − π + through the most dominant mode D 0 D 1 (2430) 0 and an important channel D 0 D 1 (2420) 0 . Therefore, the recent BESIII experimental data can support our prediction of a missing charmonium ψ(4380). In general, the existence of ψ(4380) predicted in the present work does not contradict the announced experimental results. We strongly suggest that experimentalists carry out precise measurements on the e + e − → D 0 D * − π + and e + e − → D 0 D − π + processes, which will provide a crucial test to our predictions. This is an excellent opportunity for the upgraded Belle II and the running BESIII. We also calculate the e + e − annihilation width of Y(4220) and ψ(4380) by the formula with the first-order QCD radiative corrections given in Refs. [90,91], i.e., where e c = 2/3 and α 0 = 1/137 are the charm quark charge and fine structure constant, respectively, R 4S (0) is the radial 4S -wave function at the origin, R 3D (0) is the second derivative of the radial 3D-wave function at the origin, and C = (1− 16α s 3π ) corresponds to the first-order QCD radiative correction with α s = 0.26 [90]. With the above expression and taking our obtained numerical spatial wave functions as input, we estimate the typical widths of 0.290 keV and 0.257 keV for Y(4220) → e + e − and ψ(4380) → e + e − , respectively, by setting θ = +34 • .
E. Settlement of ψ(4415) in the J/ψ family Although ψ(4415) was firstly reported in 1976 [8], its inner structure of ψ(4415) is still waiting for being revealed. When categorizing Y(4220) into the J/ψ family, we must face how to settle ψ(4415) in the J/ψ family, which is one of the main tasks in this work.
The mass spectrum result in Table II shows that the mass of ψ(4415) is close to that of ψ(5S ). Thus, we propose 5S -4D mixing scheme to study ψ(4415), which also borrows the idea when dealing with Y(4220) in Sec. II D. To depict this mixing scheme, we have an expression where φ is a mixing angle. The masses of ψ(5S ) and ψ(4D) are taken from our calculations listed in Table II. Then, the dependence of the masses of ψ 5S −4D and ψ 5S −4D on φ is given in Fig. 7.
Under the 5S -4D mixing scheme, we illustrate the decay behaviors of ψ(4415) and ψ(4500) dependent on the mixing angle φ in Fig. 8. We notice that within the allowed range of φ, the theoretically obtained total decay width of ψ(4415) is smaller than the average width value (62±20 MeV) of ψ(4415) collected in PDG [9]. Although 43 years have passed, the resonance parameters were not established well since the results from different experimental groups are very different. This can be seen in Fig. 9, and most of the results are from inclusive processes of the e + e − annihilation. Thus, we cannot test the assignment of ψ(4415) as a 5S -4D mixing state by the present experimental width of ψ(4415). Considering this situation, we strongly suggest to carry out the precise measurement of the resonance parameters of ψ(4415) especially
The dominant decay modes of the ψ(4415) state are predicted to be D * D * , DD 1 (2420), and DD 1 (2430), and the cor-responding branching ratios as a function of a mixing angle are shown in Fig. 8. This means that the decay chains ψ(4415) → DD 1 (2420)/DD 1 (2430) → DD * π are allowed. The latest measurements of e + e − → D 0 D * − π + by BESIII [86] indeed indicate a possible signal near 4.42 GeV as shown in Fig. 4 (b). In 2009, the BaBar Collaboration released two ratios [95] Γ(ψ(4415) → DD) Γ(ψ(4415) → D * D * ) = 0.14 ± 0.12 ± 0.03, which show the decay width of ψ(4415) → D * D * is much larger than those of other two decay channels. This experimental result is consistent with our calculation for ψ(4415). Due to large errors, the above two experimental data cannot be applied to distinguish positive and negative mixing angles. Thus, more accurate measurements are still necessary. In Table III, we list the typical partial decay widths of the opencharm decay channels of ψ(4415) and ψ(4500) when taking typical φ = +30 • and φ = −30 • . Additionally, we need to point out that the ψ(4415) decays into a pair of S-wave charmed-strange mesons are not obvious, which means that it is not an easy task to find a ψ(4415) signal in the e + e − → D ( * ) s D ( * ) s processes. In Ref. [96,97],  Table III. The experimental resonance parameters and di-lepton width of ψ(4415) from measurements of MARK1 [8], DASP [7], RVUE [92,93], Belle [75], and BESII Collaboration [94].
We finally discuss the di-lepton width of ψ(4415), which is the last remaining and available experimental information. Theoretically, the lepton width of a charmonium is proportional to a value of a resonance wave function at the origin. Our estimate gives Γ(ψ(4415) → e + e − ) = 0.303 ∼ 0.344 or 0.147 ∼ 0.251 keV in the mixture scheme with negative and positive angles, respectively. Similar to the measured resonance parameters of ψ(4415), the experimental differences of the di-lepton widths can be easily seen in Fig. 9. Here, our di-lepton width can meet the measured values of 0.35 ± 0.12 and 0.44 ± 0.14 keV from BESII [94] and MARK1 [8] within the experimental error range, respectively. Therefore, more precise measurements on ψ(4415) are important to test our assignment of the 5S -4D mixing state.
We further predict the decay properties of the charmonium partner ψ(4500) of ψ(4415), whose total and partial widths of open-charm channels by varying the mixing angle are shown in Fig. 8. In the positive and negative mixing schemes, the total widths of 36 ∼ 45 MeV and 30 ∼ 41 MeV for ψ(4500) are obtained, respectively. The dominant decay channels of ψ(4500) are DD 1 (2430), DD, D * D * , DD(2550), and DD 1 (2420). Additionally, the DD * 2 (2460) and DD * are also main decay modes when mixing angle φ > 0, while the D * D * 0 (2400) is not negligible in φ < 0. Their corresponding typical partial decay widths including both positive and negative mixing schemes are listed in Table III. As for ψ(4500), we can see that the decay modes DD 1 (2420), DD 1 (2430), D * D * 0 (2400), and DD(2550) begin to become important. In addition to the two-body decay modes DD and D * D * , the precise measurement of the three-body decay channel DD * π is also recommended in searching for the predicted ψ(4500) in future. The lepton annihilation width of ψ(4500) can also be predicted in 5S -4D mixing scheme, which is 2.25 × 10 −3 ∼ 0.0502 and 0.0913 ∼ 0.189 KeV for the negative and positive angles, respectively. A such small lepton width compared with ψ(4415), of course, causes the difficulty in searching for ψ(4500) in the electron-positron collider.

III. HINT OF THE PREDICTED ψ(4380) EXISTING IN
e + e − → ψ(3686)π + π − DATA It is interesting to notice that almost all the vector charmonium-like states observed in the electron-positron annihilation process were observed in the hidden charm channels, such as Y(4260), Y(4220), and Y(4320) in the π + π − J/ψ mode, Y(4360) an Y(4660) in the π + π − ψ(2S ) mode, since the final states of these hidden charm decay modes are easier to be detected or reconstructed. Therefore, a search for higher excited charmonia in the hidden charm processes will be interesting. However, as for higher excited charmonia, their mass splitting and their width are of the same order, thus the interferences between these resonances will become important. In Ref. [72], the authors suggest that the experimental cross sections for e + e − → J/ψπ + π − and e + e − → h c π + π − reported by the BESIII Collaboration can be reproduced by considering the contributions from three charmonium resonances ψ(4160), Y(4220), and ψ(4415) and interferences with a nonresonance background, which is a kind of the Fano-like interference. Since such an interference effect is a general quantum phenomenon, it has been applied to atomic and nuclear physics a long time ago to understand experimental data [98,99]. Specifically, the peak position of a genuine eigenstate is shifted by interferencing with the continuum via the Fano Hamiltonian and the corresponding Breit-Wigner distribution will be asymmetrically distorted [100]. The application of the Fano interference effect can explain why two well established charmonium ψ(4160) and ψ(4415) have no obvious signals in the cross sections for e + e − → π + π − J/ψ and e + e − → π + π − h c [65,67]. Similarly, we can extend such a kind of analysis to the cross sections for e + e − → ψ(3686)π + π − in the present work.
Recently, the BESIII Collaboration reported their precise measurements of the cross sections for e + e − → ψ(3686)π + π − process [64], which provides us a good chance to revise whether there are more potential structures other than ψ(4160) and ψ(4415) as in Ref. [73]. In addition, it may provide an evidence of the predicted ψ(4380) in the hidden-charm channel of ψ(3686)π + π − . In the Fano interference frame work, we firstly introduce an amplitude of a continuum background, which can be phenomenologically parameterized as, with u = √ s− f m f being the available kinetic energy, where f is the sum of masses of final states. In the nonresonance amplitude, two phenomenological parameters a and g are introduced, which are obviously related to non-perturbative QCD, and thus cannot be estimated from the first principle.
The genuine resonance contribution is described by a phase space corrected Breit-Wigner distribution, which is where Φ 2→3 denotes the phase space of e + e − → π + π − ψ(3686) and ψ is the intermediate vector charmonium. Here, the product of the electronic annihilation width Γ e + e − ψ and branching ratio B(ψ → π + π − ψ(3686)) is treated as a free parameter R ψ . The total amplitude is the coherent sum of the nonresonance and resonance amplitudes, which is where φ k is the phase angle between the continuum and the k-th intermediate resonance contribution.
It is worth mentioning that Y(4220) has been observed in the recent experimental data of e + e − → π + π − ψ(3686) from the BESIII Collaboration [64]. So, we first fit the cross sections for e + e − → π + π − ψ(3686) with a nonresonance continuum and three genuine resonances, which are ψ(4160), Y(4220), and ψ(4415). We set the masses and widths of all the involved resonances to be the average values of PDG [9]. The fitted results and corresponding parameters are shown in Fig. 10 (black dashed curve) and Table IV, respectively. It is interesting to notice that most of the experimental data can be reproduced in 3R scenario with χ 2 /d.o. f = 1.22. In particular, the enhancement signal of Y(4220) is very clear.
The 3R scenario can reproduce most of the experimental data, and it should also be mentioned that the data from BE-SIII Collaboration obviously show the peak near 4.36 GeV in the fitted curve. This fact inspires us to propose an improved scheme, i.e., 4R fit scheme, where we consider an additional unknown Y state with free mass and width to interfere with the background and other resonances contributions. As shown in Fig. 10 (red solid curve), the experimental data can be perfectly reproduced in a 4R fit scheme, which is also reflected on an improved χ 2 /d.o. f = 0.748. The resonance parameters of the Y state are fitted to be m = 4374 ± 13 MeV, Γ = 106 ± 29 MeV, which are consistent with our predicted ψ(4380) state. The above results indicate a structure near 4.37 GeV should exist and it cannot be simply described by the interferences from three resonances ψ(4160), Y(4220), and ψ(4415) and continuum contribution. In other words, this conclusion shows a strong evidence of the existence of ψ(4380) dominated by the 3D-wave component in the hidden charm decay channel. At last, all of the puzzles are well resolved under our proposed theoretical picture, prompting us to have great confidence to believe that two longtime missing states ψ(4S ) and ψ(3D) in the vector charmonium family could be experimentally established in the near future.

IV. SUMMARY
The observation of J/ψ particle in 1974 opens a new era of particle physics [1,2]. Since then, more and more charmonia have been reported by experiments, which construct the main body of the present cc meson spectrum as listed in PDG [9]. Although the J/ψ family has become abundant with the effort made by experimentalists, the J/ψ family is far from being well established. In the past 15 years, the observations of a series of charmoniumlike states have brought us a new chance and challenge to study cc meson spectrum [40,41]. It is obvious that it is also a good opportunity for hadron physics.
In this work, we have focused on the updated data of charmoniumlike Y states from the e + e − annihilations, and have further revealed that Y(4220) observed in the e + e − → J/ψπ + π − processes is an important scaling point when constructing higher charmonia. Here, Y(4220) has been established as a charmonium under 4S -3D mixing scheme, and The fit to the cross section for the e + e − → ψ(2S )π + π − reaction in the Fano-like interference picture under 3R and 4R fit schemes. Here, the data of BaBar [101], Belle [62,102] and latest BESIII [64] are included. · · · 4374 ± 13 Γ Y (MeV) · · · 106 ± 29 R Y (eV) · · · 6.61 ± 2.91 φ 4 (rad) · · · 2.40 ± 0.46 χ 2 /d.o. f 1.22 0.748 further theoretical prediction of its decay behaviors has been given, which provides valuable information to test this scenario. What is more important is that we have also predicted the existence of the charmonium partner ψ(4380) of Y(4220). According to our calculation, we have obtained its resonance parameters and partial open-charm decay widths. Furthermore, we have also discussed how to identify the predicted ψ(4380) by the present data of open-charm and hidden-charm decay channels. Especially, we have analyzed the latest experimental data of e + e − → ψ(3686)π + π − measured by BESIII [64] by combining with the Fano interference picture, where the possible evidence of ψ(4380) has been found. Hence, we suggest future experiments like BESIII and Belle II to hunt for ψ(4380), which not only tests this charmonium assignment to Y(4220), but pushes experimental progress on charmonium or charmoniumlike states. When finishing the study, we have to face another crucial issue, i.e., how to settle the charmonium ψ(4415). In this work, we have investigated ψ(4415) under 5S -4D mixing scheme, and have found that the obtained results do not contradict with the experimental data of ψ(4415). If carefully checking the present experimental information listed in PDG [9], we notice that the precision of data is not enough since even the first observation of ψ(4415) has passed 42 years [8]. Therefore, further experimental studies on ψ(4415) are strongly encouraged, especially at BESIII and Belle II. As a charmonium partner of ψ(4415), a missing charmonium ψ(4500) has been predicted in this work. The search for it will be an interesting research issue.
We hope that our theoretical studies presented here can play an important role in constructing the J/ψ meson spectrum, especially higher charmonia. More experimental and theoretical joint efforts on this topic will be necessary in forthcoming years.

Acknowledgments
This work is partly supported by the China National Note added: When we are writing out the present work, we have noticed a recent result from BESIII [103]. By analyzing the data of the cross section of e + e − → ωχ c0 from √ s = 4.178 to 4.378 GeV, BESIIII has confirmed the existence of a narrow structure Y(4220) at 4.2 GeV. Especially, BESIII has also extracted the angular distribution of e + e − → ωχ c0 , which shows that there exists the evidence for a combination of S and D-wave contribution in the Y(4220) → ωχ c0 [103]. This updated measurement of e + e − → ωχ c0 supports our 4S -3D mixing scheme for Y(4220).