Three body open flavor decays of higher charmonium and bottomonium

In the present work, we study the OZI-allowed three body open flavor decay properties of higher vector charmonium and bottomonium states with an extended quark pair creation model. For the bottomonium system, we get that (i) the $BB\pi$ and $B^*B^*\pi$ partial decay widths of the $\Upsilon(5S)$ state are consistent with the experiment, and the $BB^*\pi$ partial decay width of the $\Upsilon(5S)$ state is smaller but very close to the Belle's experiment. Meanwhile, (ii) the $BB^*\pi$ and $B^*B^*\pi$ decay widths of $\Upsilon(11020)$ can reachs $2\sim3$ MeV. In addition, (iii) for the most of higher vector charmonium states, the partial decay widths of the $DD^*\pi$ and $D^*D^*\pi$ modes can reach up to several MeV, which may be observed in future experiments.

Since the charmonium-like states with normal quantum numbers have similar masses compared to the normal charmonium, in order to understand the nature of the exotic states, it is necessary to have a better understanding of the normal charmonium spectroscopy. In Ref. [21], Li et al. investigated the spectrum of higher charmonium with screened potential, and found that the vector states Y(4008), Y(4260), Y(4320/4360), Y(4660) might be assigned as the ψ(3S ), ψ(4S ), ψ(3D), ψ(6S ) states respectively, while X(3940) and X(4160) might be the η c (3S ) and χ c0 (3P) states. However, according to the constituent quark model description by Segovia et al. [22], the mass of ψ(4040), ψ(4160), X(4360), ψ(4415), X(4630) and X(4660) are compatible with the ψ(3S ), ψ(2D), ψ(4S ), ψ(3D), ψ(5S ) and ψ(4D) states. Among the charmonium or charmonium-like states, the 1 −− states are of special interest because they can be easily produced in the e + e − annihilation. In Table I, we have listed the predicted masses of the vector charmonium states from various models.
In addition, the decay properties of charmonium play a piv- * E-mail: xzhweng@pku.edu.cn † E-mail: lyxiao@pku.edu.cn ‡ E-mail: zhusl@pku.edu.cn otal role in revealing the nature of charmonium. From Table I, we see that the masses of these states are well above the allowed two body open charm decay threshold, thus the decay widths mainly come from the strong decays. A widely used framework for the strong decay is the quark pair creation ( 3 P 0 ) model. In this model, the cc pair in the initial charmonium regroups with a qq pair created from the vacuum, which carries the vacuum quantum number J PC = 0 ++ , and then decays into the outgoing open charm mesons. About forty years ago, Le Yaouanc et al. [23,24] used this model to study the open charm strong decays of ψ(4040) and ψ(4415). In 2005, Barnes et al. performed a systematic study of the higher charmonium states just above 4.4 GeV, with the charmonium masses calculated in the GI model and a nonrelativistic potential model [25]. In 2012, Segovia et al. [22] studied the strong decays of the vector charmonium states. For ψ(3770), ψ(4040), ψ(4160) and X(4360) with ψ(1D), ψ(3S ), ψ(2D) and ψ(4S ) assignments, the calculated widths are compatible with the experimental values. While for the ψ(4415), X(4640) and X(4660) states, the difference between theoretical and experimental values of the total widths is larger. Recently, Gui et al. [26] studied the open-charm strong decays of higher charmonium states up to the 6P multiplet with their wave functions of charmonium states calculated in the linear potential and screened potential quark models. Moreover, the 3 P 0 model has also been used to study the strong decays of bottomonium states [27][28][29].
Besides the two body decays, three body open flavor decay is also important access to dig into the properties of charmonium and bottomonium. In 2008, the Belle Collaboration [30] first measured the exclusive cross section for e + e − →D 0 D − π + over the center-of-mass energy range 4.0-5.0 GeV with ISR method and observed the decay ψ(4415)→D 0 D − π + . A detailed study found that the decay is dominated by ψ(4415)→DD * 2 (2460) and at 90% C.L. In 2009, they further measured the cross-section of e + e − →D 0 D * − π + + c.c. process and found no evidence of · · · · · · 4614 4642/4672 · · · 4463 4711/4472 ψ(6 3 S 1 ) · · · · · · · · · 4804/4828 · · · 4608 · · · ψ(2 3  · · · · · · 4641 4690/4707 · · · · · · · · · ψ(5 3 D 1 ) · · · · · · · · · 4840/4855 · · · · · · · · · Y(4260), Y(4360), ψ(4415), Y(4630) or Y(4660) with limited statistics [31]. Recently, the BESIII Collaboration found two resonances in the e + e − →D 0 D * − π + process [32,33]. The lower mass one is in good agreement with the Y(4220), and the other one might be ψ(4415). For the bottomonium state, the Belle Collaboration also measured Υ(5S ) decays into B mesons [34]. The measured fractions are The measured three-body fractions are significantly larger than the older predictions [35]. In Ref. [36], we extended the 3 P 0 model to study the Y(4660)→Λ cΛc process with two qq pairs created from the vacuum. In this paper, we will follow the extended 3 P 0 model to study the three body open flavor decays of higher charmonium and bottomonium states through a different rearrangement (see Fig. 1). In the framework of the extended 3 P 0 model, we find that (i) the BBπ and B * B * π partial decay widths of the Υ(5S ) state are consistent with the experiment (For simplicity, we abbreviate the BBπ, BB * π + B * B π and B * B * π to BBπ, BB * π and B * B * π respectively. A similar abbreviation is also used for the charmonium decays.). The BB * π partial decay width of the Υ(5S ) state is smaller but very close to the Belle's experiment. (ii) The partial decay widths of the DD * π and D * D * π modes can reach up to several MeV for the higher vector charmonium states. The three body open charm decay channels may be observed in the near future. This paper is organized as follows. In Sec. II the 3 P 0 model and its extension are briefly introduced. The numerical results are presented and discussed in Sec. III. Finally, a quick summary is given in Sec. IV.

II. THE 3 P 0 MODEL
The 3 P 0 model is widely used to calculate the Okubo-Zweig-Iizuka (OZI) allowed strong decays. It was first proposed by Micu [41] to study the strong decay properties of the P-wave mesons. Le Yaouanc et al. of the Orsay group further developed this model, and used it to study the open-flavor strong decays of mesons [23,24,42] and baryons [43,44]. Since then, this model has been widely used in the study of baryon strong decays [25,28,[45][46][47][48][49][50]. In the 3 P 0 model, a light qq pair is created with the vacuum quantum number J PC = 0 ++ (hence "the 3 P 0 model"), and then rearranged with the quarks within the initial meson to produce two final mesons. The decay matrix element can be described by the interaction Hamiltonian [25,47,48] where m f is the constituent quark mass, and ψ f is a Dirac field of quark. γ is a dimensionless constant standing for the qq pair creation strength, which can be extracted by fitting to data. In Ref. [36], we extended the 3 P 0 model to study the Y(4660)→Λ cΛc decay process which requires two light qq pairs to be created. Here, we go a further step to study the higher heavy quarkonium decaying into two heavy mesons plus a light meson, as shown in Fig. 1. The corresponding where m q is the mass of the created quark. Then Eq. (6) can be rewritten as The corresponding transition operator in the nonrelativistic limit reads [36] where p i is the momentum of the ith quark created from vacuum. ϕ 0 = (uū + dd + ss)/ √ 3 and ω 0 = δ i j stand for the flavor and color singlets, respectively. The solid harmonic polyno- (Ω p ) corresponds to the P-wave qq pair, and χ 1,−m(m ′ ) is the spin triplet state for the created qq pair. a † i b † j is the creation operator denoting the qq pair creation in the vacuum.
We use the mock state [51] to define the meson (A) Here the p i (i = 1, 2) is the momentum of quarks in meson A.
Then the helicity amplitude in the center of mass frame can be written as where the factor (−3) 2 has been canceled by the color factor and I is the momentum space integration and more detailed calculations are shown in the Appendix A. Finally, the decay width Γ reads Following the literature in this field [23,25,28,[46][47][48][49][50], we adopt the simple harmonic oscilator (SHO) wave function to describe the momentum-space wave function of the meson where L l+1/2 n (p 2 /β 2 ) is an associated Laguerre polynomial.

A. Parameters
In the present work, we set m u = m d = 220 MeV, m s = 419 MeV, m c = 1628 MeV and m b = 4977 MeV for the constituent quark masses [38]. The masses of final state mesons are listed in Table II. For simplicity, we ignore the isospin breaking and obtain the meson masses by taking their isospin averages.
The harmonic oscillator strength β of light mesons takes the average value 400 MeV [36,50]. The parameter β's of heavylight mesons are taken from Refs. [28,50], which are obtained by comparing the rms radius of the SHO wave function to that of the wave functions calculated using the Godfrey-Isgur model (see Table II). We use β = 500 MeV for charmonium [25,52]. In Ref. [28], Godfrey et al. showed that the parameter β's are 638 MeV, 600 MeV and 578 MeV for Υ 4 3 S 1 , Υ 5 3 S 1 and Υ 6 3 S 1 respectively, thus we adopt the average value as 600 MeV for bottomonium states in this work.
For the qq pair creation strength, we use γ(cc) = 6.95 for charmonium decays, which is √ 96π times of that in Refs. [25,46] due to a different definition. However, in Ref. [28] it is found that this value underestimated the twobody strong decay widths of bottomonium, and the fitting of the open bottom decays of the Υ sector gives γ(bb) = 10.42.
Here we adopt the same value of γ for Υ sector as in Ref. [28]. The uncertainty of γ is about 30% [28,48,50,54], which may lead to a factor-of-3 change to the predicted three body decay widths, either smaller or bigger. Thus the uncertainty of our results may be quite large.
We present our results in Table III. According to our calculation, the partial decay width is which is quite small compared to the total decay width of ψ(4040). More precisely, the branching ratio is This ratio is smaller than that of the hidden charm decay modes of ψ(4040) by one order. Due to the narrow partial decay width, the DDπ decay mode might be not easy to be observed.
The DD * π mode is also available for ψ(4040). Since DD * π mode has little phase space, the partial width of ψ(4040) decaying into DD * π is about one magnitude smaller than the DDπ partial width. The partial decay width ratio is The mass of ψ(4160) is 4191 ± 5 MeV [53], which is about 150 MeV heavier than that of ψ(4040). According to the mass predictions in the quark model [25], this state is suggested to be the 2 3 D 1 cc state. Its two body open charm decays have been studied by many authors, which also support this assignment [23,25,26].
We analyze the three body decay properties of ψ(4160) as the 2 3 D 1 cc state, and collect its partial strong decay widths in Table III. We obtain the partial decay widths and The values are much bigger than the corresponding one of the ψ(4040). These widths seem not large compared to its total width (Γ tot. = 70 ± 10 MeV), but it is enough to be observed in those decay channels in experiments. Moreover, the branching ratios are predicted to be and which are comparable to the upper limit of hidden charm decays of ψ(4160). The partial decay width of D * D * π mode is This value is small and hard to be searched for at present.

Y(4360)
The state Y(4360) was first observed by the BaBar Collaboration in the e + e − → γ ISR π + π − ψ(2S ) process [12]. Later, the Belle Collaboration confirmed this state in the same process with a statistical significance of more than 8σ [13].
Y(4360) was interpreted to be 3 3 D 1 state in the nonrela-tivistic screened potential model [21]. Ding et al. also interpreted Y(4360) as a 3 3 D 1 charmonium by evaluating its e + e − leptonic widths, E1 transitions, M1 transitions and the open flavor strong decays in the flux tube model. However, the possibility of the 4 3 S 1 assignment cannot be rule out [22]. As the possible assignments of Y(4360), it is crucial to study the decay properties of the ψ 4 3 S 1 and ψ 3 3 D 1 . The theoretical predictions are listed in Table IV.  Table IV, the dominant three body decay mode for both ψ 4 3 S 1 and ψ 3 3 D 1 is DD * π with a mass of M=4368 MeV, and the predicted partial decay widths are and Combing the measured width of Y(4360), we further get the branching ratios The sizeable branching ratios indicates that this state has a good potential to be observed in the DD * π decay channel if it indeed turns out to be either the state ψ 4 3 S 1 or ψ 3 3 D 1 .
Meanwhile, the partial decay widths of DDπ and D * D * π are sizable for the two assignments. If Y(4360) is the 4 3 S 1 state, we predict The DDη decay mode is also available kinetically. However, our calculation shows that its width [O (0.1 keV)] is too small to be observed because of its tiny phase space.
Unfortunately the three body decay properties of the two assignments ψ 4 3 S 1 and ψ 3 3 D 1 are very similar, which can't be used to distinguish these two states in future experiments.

ψ(4415)
The ψ(4415) state was discovered by SLAC and LBL in e + e − annihilation [56]. Later, it was confirmed by DASP Collaboration [57]. Its mass and width are (4421 ± 4) MeV and (62 ± 20) MeV [53], respectively. This state is the unique vector charmonium with experimental data of three body decays. The present study of the state ψ(4415) can not only provide an important test of our model but also let us obtain more information about the nature of ψ(4415).
In Ref. [24], Le Yaouanc et al. used the 3 P 0 model to calculate its open flavor decay and assigned it to be 4 3 S 1 state. Later, Barnes et al. confirmed this assignment by comparing the mass spectrum from GI model calculation. They calculated all ten open-charmed decay widths of ψ(4415) using the 3 P 0 model, and found that the total widths and the decay patterns were consistent with experiments [25]. Moreover, they predicted that DD 1 and DD * 2 were the major decay modes of ψ(4415), and the latter prediction was confirmed by Belle Collaboration [30]. Thus it is essential to study the three body decay properties of ψ 4 3 S 1 . Fixing the mass of ψ 4 3 S 1 at M = 4421 MeV, we calculated its partial decay widths and listed them in Table V. According to our calculation, its three body strong decay is governed by the DD * π channel with the branching ratio B[ψ 4 3 S 1 →DD * π] ∼ 3.2%, (29) which is less than the upper limit (< 11%) listed in PDG [53]. The role of the D * D * π channel is also important in the decays. The predicted partial width ratio between D * D * π and DD * π is Meanwhile, the partial decay width of the DDπ mode is predicted to be with the branching ratio This value is also less than the upper limit (2.2%) obtained by the Belle Collaboration [30]. To further confirm the nature of   body decay mode of ψ 5 3 S 1 and ψ 3 3 D 1 is DD * π. The partial decay widths are Γ[ψ 5 3 S 1 →DD * π] ∼ 0.96 MeV (33) and The above decay widths are large enough to be observed in future experiments. Moreover, which indicates the DD * π branching ratio of ψ 3 3 D 1 is larger that of ψ 5 3 S 1 .
In addition, considering the uncertainty of the predicted masses in various models, we plot the partial decay widths of the 4 3 S 1 , 5 3
According to the quark model calculation, the mass of ψ 5 3 S 1 is very close to Y(4660). Ding et al. also suggested that the Y(4660) is a 5 3 S 1 charmonium after studying its e + e − leptonic widths, E1 transitions, M1 transitions and the open flavor strong decays in the flux-tube model [59]. As a possible assignment, we first study the decay property of the ψ 5 3 S 1 and list the corresponding results in Table VI. From the table, we find that the partial decay widths of DD * π and D * D * π modes are quite large, which read and respectively. The values are large enough to be observed in experiment. Meanwhile, the partial decay width of Γ[ψ 5 3 S 1 →DDπ] is considerable. The partial decay width ratio is Besides ψ 5 3 S 1 , the possibility that Y(4660) is a ψ 4 3 S 1 or ψ 6 3 S 1 state cannot be excluded completely. Thus we also calculate the partial decay widths of the ψ(4 3 S 1 ) and ψ(6 3 S 1 ) states. Similarly, we fix the mass of ψ(4 3 S 1 ) and ψ(6 3 S 1 ) at M = 4643 MeV, and collect their partial decay widths in Table VI.
As listed in Table VI, we obtain that the partial decay widths of DD * π and D * D * π for ψ(4 3 S 1 ) are the largest compared to those for ψ(5 3 S 1 ) and ψ(6 3 S 1 ). The predicted branching ratios are However, the ψ 6 3 S 1 state gives the smallest branching ratios, which are B[ψ 6 3 S 1 →DD * π] ∼ 1.5%, Furthermore, we study the decay properties of the ψ 3 3 D 1 , ψ 4 3 D 1 and ψ 5 3 D 1 states, and list their decay properties in Table VI as well. Combined with the total width of Y(4660), we obtain the branching ratios of the ψ 3 3 D 1 , ψ 4 3 D 1 and ψ 5 3 D 1 states as follows: These branching ratios are comparable to those of the S -wave states. If Y(4660) is a D-wave state, it is possible to be observed in the DD * π and D * D * π channels as well.
In addition to the DDπ, DD * π and D * D * π channels, Y(4660) can also decay into DDρ, DDω, DDη, DD * η, D * D * η, D s D s η and D s D * s η channels. In the same way, we fix the mass of the states ψ(4S ), ψ(5S ), ψ(6S ), ψ(3D), ψ(4D) and ψ(5D) at M = 4643 MeV, and calculate their widths of decaying into these channels. The results are collected in Table VI. The partial decay widths of these channels are relatively smaller. Among them, the partial decay widths of the DDρ, DDω and DD * η modes are around several tenths of MeV. If Y(4660) is one of the above states, it is still possible to observe these channels.
The mass spectrum predicted by various quark models bears a large uncertainty, and may have effect on the partial decay widths. To investigate this effect, we vary the mass of the states ψ(4S ), ψ(5S ), ψ(6S ), ψ(3D), ψ(4D) and ψ(5D) from 4300 MeV to 4700 MeV, and calculate their corresponding decay widths. Here, we just plot the results for the DDπ, DD * π and D * D * π channels in Figs. 2-4, and omit the theoretical predictions of other channels since their decay widths are relatively smaller.

C. Bottomonim
For the bottomonium system, there are three bb states above the open bottom threshold, namely Υ(4S ), Υ(10860) and Υ(11020). A number of literatures are available on the study of their strong decays with the 3 P 0 model [27,28,60] and other models [35,61]. Most of them focus on the two body strong decays. However, the Υ(10860) and Υ(11020) states can also decay into two bottomed mesons plus a π meson. Furthermore, these channels for Υ(10860) state have recently been observed by the Belle Collaboration [34]. We will investigate the three body decays of Υ(10860) and Υ(11020) with the extended 3 P 0 model. The Υ(10860) and Υ(11020) were discovered by the CLEO Collaboration in the e + e − annihilation [62]. Their masses and They are usually assigned to be the 5 3 S 1 and 6 3 S 1 bb states in the quark model. We will discuss the three body decays of the Υ(10860) and Υ(11020) states with this assignment.
The partial decay widths of the Υ(10860) state are listed in Table VII. According to our calculation, we obtain The BB * π decay width is the largest one. Combining with the total width of Υ(10860), we obtain the branching ratios as follows: The predicted branching ratios of the BBπ and B * B * π decay modes are within the ranges of experimental values measured by the Belle Collaboration [53]. For the BB * π decay mode, our result is slightly smaller than the experiment data.
The branching ratios of the BB * π and B * B * π channels are quite large. Thus these two channels may be observed by the BelleII Collaboration in the near future.

D. The effect of β
We have investigated the three body open flavor decays of five charmonium-like states with various assignments and two bottomonium states. In this work, we carried out the calculation by fixing the harmonic oscillator parameter β to be 500 MeV (600 MeV) for charmonium (bottomonium) states. However, the parameter β of the initial states is not determined precisely, which may bring in uncertainty to our results. To estimate this effect, we carry out the preceding calculation by varying the parameter β of the initial states by 50 MeV. We investigate the decay properties with two different β values, β = 450 MeV and β = 550 MeV for charmonium states and β = 550 MeV and β = 650 MeV for bottomonium states. The numerical results are presented in Table III, Figs. 2-4 for charmonium states and Table VII for bottomonium states. In most cases, a larger β value leads to a larger decay width. Within a reasonable range of the parameter β, our main predictions and conclusions hold.

IV. CONCLUSIONS
In the present work, we have investigated the OZI-allowed three body open flavor decays of excited vector charmoniumlike states and bottomonium states in the framework of the extended 3 P 0 model. It is the first attempt along this direction in literatures to study this type of decay modes by considering the creation of two light qq pairs from vacuum. Our main results are summarized as follows.
For the well-established states ψ(4040) and ψ(4160), we estimate their three body open flavor decay properties with the assignments ψ(3 3 S 1 ) and ψ(2 3 D 1 ), respectively. The partial decay widths of ψ(4040) should be fairly small (about several tens keV), and those of ψ(4160) are a little larger, which are about 0.1 MeV for the DDπ and DD * π modes.
We also discuss the decay properties of Y(4360) as a candidate of ψ(4 3 S 1 ) or ψ(3 3 D 1 ). From our calculation, the partial decays width of DD * π mode can reach up to 1 MeV in both cases. Thus if Y(4360) is one of these states, it may be observed in the DD * π channel.
With the ψ(4 3 S 1 ) assignment, the DD * π and D * D * π decay widths of ψ(4415) are larger than 1 MeV. Meanwhile, the DDπ decay mode is sizable with a width of ∼ 0.38 MeV. Our predictions for the branching ratios of the DDπ and DD * π channels are within the upper limits measured by the Belle Collaboration [30,31]. However, assigning ψ(4415) to be ψ(5 3 S 1 ) or ψ(5 3 S 1 ) state, we obtain similar decay properties. Thus, to further determine the inner structure of ψ(4415), more precise experimental data are needed.
We calculated the three body open flavor decay widths of Y(4660) with various assignments, ψ(4S , 5S , 6S ) and ψ(3D, 4D, 5D). In both cases, its three body decays are dominated by the DD * π and D * D * π channels, and the partial decay widths can reach up to several MeV. Meanwhile, we notice that the DDρ and DDω decay widths of the D-wave states are larger than those of the S -wave states. If Y(4660) turns out to be ψ(3D), its DDρ decay width even reaches up to 1.86 MeV.
We have also investigated the three body open flavor decays of Υ(10860) and Υ(11020). The branching ratios of Υ(10860) decaying into BBπ and B * B * π are consistent with the experimental data, while the BB * π braching ratio is smaller but very close to the Belle's measurement. For Υ(11020), the BBπ, BB * π and B * B * π decay widths are 0.34 MeV, 3.17 MeV and 2.69 MeV, respectively. Hopefully the BB * π and B * B * π decay modes of the Υ(11020) state will be observed by BelleII Collaboration in the very near future.