Charmonium decays into ${\Lambda}_c\bar{\Lambda}_c$ pair governed by the hadronic loop mechanism

In this work, we investigate the open-charm decay process $\psi\to\Lambda_c\bar{\Lambda}_c$ via the hadronic loop mechanism for vector charmonia above $\Lambda_c\bar{\Lambda}_c$ threshold. The branching ratios of these vector charmonium states to $\Lambda_c\bar{\Lambda}_c$ are estimated. The charmonium explanation of the $Y(4630)$ observed in $e^+e^- \to \Lambda_c\bar{\Lambda}_c$ is tested. Furthermore, for the predicted higher vector charmonia above 4.7 GeV, the branching ratios $\mathcal{B}[\psi(nS)\to\Lambda_c\bar{\Lambda}_c]$ with $n=7,8,9$ are found to be of the order of magnitude of $10^{-4}-10^{-3}$ while $\mathcal{B}[\psi(mD)\to\Lambda_c\bar{\Lambda}_c]$ with $m=6,7,8$ are of the order of magnitude of $10^{-3}-10^{-2}$. The experimental signals of these missing charmonium states are discussed. The search for them may be an interesting topic in the future BESIII and Belle II experiments.


I. INTRODUCTION
As the important member of hadron spectroscopy, charmonium family has attracted extensive attention from both theorist and experimentalist since the first charmonium J/ψ was discovered in 1974 [1,2]. Since charmonium is a typical mesonic system composed of charm and anti-charm quarks, charmonium corresponding to low-energy particle physics can be treated as ideal platform to help us to deepen our understanding of non-perturbative behavior of quantum chromodynamics (QCD), which is full of challenge and opportunity.
At present, the number of charmonium states reported in experiment is constantly increasing, especially, with the observation of a series of charmoniumlike XYZ states in the past 18 years [3,4]. In Fig. 1, we summarize the present status of the charmonium family. We may find that there exist 8 charmonium states below the DD threshold, which results in the narrow widths of these charmonia due to the Okubo-Zweig-Iizuka (OZI) rule. Above the DD threshold, more charmonia can be found, where the OZI-allowed decay channels composed of charmed and anti-charmed mesons have dominant contribution to the corresponding total width of these involved charmonia. For these charmonia above the Λ cΛc threshold, a new type of OZI-allowed channel Λ cΛc is open. The Λ cΛc decay channel attracts less attention because all the established charmonia cannot decay into this channel.
The OZI-allowed strong decay of charmonium to DD was well described by the quark pair creation (QPC) model [23,24] which assumes the creation of one light quark pair with vacuum quantum number J PC = 0 ++ . However, the ψ → Λ cΛc needs the creation of two light quark pairs, which is beyond the naive QPC model. Xiao et al. [25] chose to extend the QPC model by directly assuming the same strength of two quark-antiquark pair creation vertexes. Simonov [26,27] developed a double string breaking model with Λ cΛc emission to depict this process. These models are tentative explorations to the mechanism of the ψ → Λ cΛc process.
As an available application of the hadronic loop mechanism in ψ → Λ cΛc , we examine the branching ratios of the Λ cΛc decays of charmonia ψ 6S −5D and ψ 6S −5D proposed to explain the Y(4630) in our previous work [22]. We find that the branching ratios of ψ 6S −5D → Λ cΛc and ψ 6S −5D → Λ cΛc extracted from the e + e − → Λ cΛc data are well reproduced within the hadronic loop mechanism. Furthermore, we also estimate the Λ cΛc branching ratios of more higher charmonia predicted in Ref. [22]. In order to arouse the interest of experimentalists, we try to reproduce the rough data of e + e − → Λ cΛc up to center-of-mass (CM) energy of 4.9 GeV based on our theoretical results of branching ratios. By this study, we want to show the possible evidence of higher vector charmonia with masses above 4.7 GeV existing in the present experimental data of the e + e − → Λ cΛc process.
This paper is organized as follow. Firstly, we introduce a hadronic loop mechanism for ψ → Λ cΛc in Sec. II. Then the applications of the hadronic loop mechanism to the decay of ψ 6S −5D , ψ 6S −5D and higher charmonia were presented in Sec. III. Finally, we conclude this paper in Sec. IV, where possible signals of higher charmonia above 4.7 GeV existing in the present experimental data are discussed.

II. THE HADRONIC LOOP MECHANISM IN DECAY PROCESS OF ψ → Λ cΛc
The dominant decay channels of a charmonium states above charmed meson pair threshold are usually the two-body opencharm modes composed of charmed meson pairs. The decay of higher charmonia to Λ cΛc needs one more light quarkantiquark pair creations than the decay channel composed of charmed meson pairs. Here, we suppose the decay process ψ → Λ cΛc can proceed via a two-step way. Firstly, the ψ decays into their dominant final states D ( * ) To evaluate these processes, we adopt the effective Lagrangian approach. We need the Lagrangian depicting the interaction between a charmonium state and charmed/charmedstrange meson pairs and the Lagrangian of the couplings involving in charmed baryon Λ c and a charmed/charmedstrange meson together with a baryon N/Λ. Using the forms listed in Ref. [41], we can obtain the Lagrangian of the ψD ( * ) D ( * ) interaction, which respects the heavy quark symmetry [42] and reads as where ψ/ψ 1 is the S /D-wave vector charmonium field and D ( * ) is the charmed/charmed-strange meson S U(3) triplet (D 0( * ) , D +( * ) , D +( * ) s ). For the interaction Lagrangian of Λ c D ( * ) N, we use the following form [43]: where N denotes nucleon field. The above Lagrangian can be directly applied to the coupling of Λ c D s Λ, With these effective Lagrangians, the corresponding scattering amplitudes of ψ → D ( * ) The amplitudes for ψ(mD) → D ( * )D( * ) → Λ cΛc are where the dipole form factor F (q 2 ) is introduced to describe off-shell effect of the exchanged baryon in the rescattering process D ( * )D( * ) → Λ cΛc and avoid the divergence of the loop integral, which has the following form: Here, m E and q are the mass and four-momentum of the exchanged baryon, respectively. Λ QCD = 220 MeV and α is a phenomenological dimensionless parameter. The amplitudes s → Λ cΛc can be obtained by replacing g ψDD , g Λ c DN , m D , m N to the corresponding parameters in the D sDs case.
The total amplitude of ψ → Λ cΛc in the hadronic loop mechanism reads as where the factor of 2 in front of M q i comes from the sum over charmed meson isospin doublet (D 0( * ) , D +( * ) ), and M s i is the amplitude for intermediate D ( * ) sD ( * ) s case. Then, the branching ratio of ψ → Λ cΛc can be calculated by where the factor of 1 3 comes from spin average over an initial charmonium state. The loop integral in Eq. (5)-(12) are evaluated with the help of LoopTools package [44,45], by which both the real and imaginary parts are considered in our calculation.

III. TWO APPLICATIONS
In this section, we apply the hadronic loop mechanism discussed in Sec. II to calculate the branching ratios of the decay ψ → Λ cΛc . We follow our previous work [22], in which the spectrum and partial widths to charmed meson pairs for charmonia above Λ cΛc threshold were studied, and further explore ψ → Λ cΛc decay process for higher vector charmonia here. I: The second column is the product of the dilepton width and the decay branching ratio of Λ cΛc channel of ψ 6S −5D and ψ 6S −5D given in Ref. [22]. The corresponding ranges of α of ψ 6S −5D and ψ 6S −5D for different mixing angles are shown in the last two columns. A. Reproducing branching ratios of ψ 6S −5D → Λ cΛc and ψ 6S −5D → Λ cΛc The Y(4630) observed in e + e − → Λ cΛc has attracted much attention, where many explanations was proposed to explain this novel structure. Dai et al. [15,16] found that the Y(4630) may be treated as Y(4660) in the ψ f 0 (980) bound state picture when taking into account the Λ cΛc final state interaction. Cao et al. [19] found that the enhancement right above the Λ cΛc threshold was well explained by a virtual pole generated by Λ cΛc attractive final state interaction. Cotugno et al. [14] analyzed the Belle data of Y(4630) → Λ cΛc and Y(4630) → ψ(2S )π + π − and found that these two observations are likely to be due to the same state which are very likely to be a charmed baryonium constituted by four quarks. Besides, the charmonium explanation to the Y(4630) was also proposed [20,22]. Obviously, the present experimental data cannot exclude any possible explanations mentioned above. This situation motivate us to carry out further investigation around this puzzling phenomenon.
The products of the branching ratio of Λ cΛc channel and the dilepton width of ψ 6S −5D and ψ 6S −5D have been extracted from experimental data in Ref. [22], which are listed in Table I. In general, the dilepton width of charmonium states are of the order of keV [46]. In this work, we take the dilepton width of these higher charmonia as 1 keV for the rough estimate, and get the branching ratios of ψ 6S −5D → Λ cΛc and ψ 6S −5D → Λ cΛc Before showing the numerical results, we introduce how to fix the values of the relevant coupling constants appearing in Eqs. (1)-(4). The Lagrangian L S in Eq. (1) can be applied to describe the interaction between ψ 6S −5D and charmed meson pairs, and the L D in Eq. (2) is used for the ψ 6S −5D case. The coupling constants in the ψD ( * ) D ( * ) coupling can be determined by theoretical partial decay widths of ψ → D ( * )D( * ) , which was given in Ref. [22]. Here, the corresponding partial widths and the extracted coupling constants are listed in Table  II. For the coupling constants involved in the Λ c D ( * ) N interaction, we take the values from the calculation of QCD lightcone sum rules [43], i.e., g Λ c DN = 13.8, g Λ c D * N = −7.9, and κ Λ c D * N = 4.7. The coupling constants g Λ c D ( * ) N can be directly related to the coupling constants in Λ c D ( * ) s Λ interaction under S U(3) symmetry: g Λ c D ( * ) N = − 3 2 g Λ c D ( * ) s Λ . After fixing all coupling constants, the only free parameter α is left, which is introduced in Eq. (13) to parameterize the cutoff Λ in the form factor F (q 2 ). Since the cutoff should not deviate far from the physical mass of the exchanged particle, α is expected to be the order of unity as indicated in Ref. [47]. In Fig. 3, we show the α dependence of the branching ratios of the decay ψ 6S −5D → Λ cΛc and ψ 6S −5D → Λ cΛc . Here, the results under taking two possible mixing angles are given since the sign of mixing angle θ was not determined [22]. For the purpose of comparison, we also show the extracted branching ratios in Fig. 3 as shadow regions. Indeed, the corresponding experimental data can be reproduced well. For the case of ψ 6S −5D , when α is taken as the range of 1.9 ∼ 2.3, the results with positive and negative mixing angles are similar to each other, where we get the branching ratio consistent with the extracted value. For the case of ψ 6S −5D , there exists big difference for the result under two mixing angles, where taking positive mixing angle results in a larger α value compared with the case of taking negative mixing angle. Thus, the negative mixing scheme is more preferred. In the following discussion, we take the negative mixing angle, where the extracted branching ratio of ψ 6S −5D → Λ cΛc decay can be well reproduced when α = 3.9 ∼ 4.3. In fact, these values of α are not far away from unity and can be seen to be reasonable.
In conclusion, by the above study of the Λ cΛc decay ψ 6S −5D and ψ 6S −5D , we find that the the charmonium explanation for the Y(4630) [22] can be tested.  . The relative minus sign in the couplings of ψ 6S −5D can be determined in the heavy quark limit [41].

Negative mixing scheme
Positive mixing scheme Partial width (MeV) Coupling constants Partial width (MeV) Coupling constants Final state for ψ(nS ) (n = 7, 8,9) and ψ(mD) (m = 6, 7, 8), which are converted by the corresponding partial widths calculated in Ref. [22]. In this subsection, we predict the branching ratios of the ψ → Λ cΛc decay for higher charmonia, where we focus on three ψ(nS ) (n = 7, 8,9) and three ψ(mD) (m = 6, 7, 8) which were predicted [22] to have the mass in the energy region between 4.7 and 4.9 GeV. Similarly, the ψD ( * ) D ( * ) coupling constants for these higher charmonia can also be fixed by the theoretically evaluated partial decay widths which are summarized in Table III. The α dependence of the calculated branching ratios of Λ cΛc channel by the hadronic loop mechanism are shown in Fig. 4   As shown in Fig. 4 and Fig. 5, the branching ratios of These features are understandable under the hadronic loop mechanism. For initial states with the same internal orbital angular momentum L, the dynamical difference of their decay behavior within hadronic loop mechanism comes from ψD ( * ) D ( * ) coupling vertexes and these couplings are determined by their partial decay widths to charmed meson pairs. In general, the amplitude for a two-body decay involves an overlap integral among three wave functions. The wave functions of highly radially excited charmonium states, as we consider here, highly oscillate and the overlap integrals have similar behavior for these charmonium states as indicated in Ref. [22].
In this work, we investigated the decay mechanism of ψ → Λ cΛc by introducing the hadronic loop mechanism. In this mechanism, a charmonium state firstly decays into a charmed meson pair D As an application of the hadronic loop mechanism, we have examined the decay behaviors of charmonium states ψ 6S −5D and ψ 6S −5D to Λ cΛc , where the ψ 6S −5D and ψ 6S −5D are the mixture of ψ(6S ) and ψ(5D) predicted in Ref. [22]. We found that the calculated branching ratios of ψ 6S −5D → Λ cΛc and ψ 6S −5D → Λ cΛc can match the corresponding extracted values from the fit to the experimental data of e + e − → Λ cΛc in Ref. [22], where the Y(4630) structure was reproduced by the contribution of ψ 6S −5D and ψ 6S −5D [22]. Thus, by the study of ψ / 6S −5D → Λ cΛc , the charmonium explanation to the Y(4630) is tested in the present work.
Furthermore, the branching ratios of the Λ cΛc decay mode of higher vector charmonium states up to 4.9 GeV were also explored. The branching ratios for the Λ cΛc decays of Swave charmonia are found to be of the order of magnitude of 10 −4 − 10 −3 and those of the corresponding D-wave charmonium partners are of the order of magnitude of 10 −3 − 10 −2 . The branching ratios of B[ψ(nS )/ψ(mD) → Λ cΛc ] with n = 7, 8, 9 and m = 6, 7, 8 are comparable with those of ψ 6S −5D /ψ 6S −5D .
Since the decay of these higher charmonia to Λ cΛc have sizable branching ratios, before ending this article, we discuss the possibility of finding out these missing charmonia in e + e − → Λ cΛc . In the following, we try to mimic the corresponding cross section by combining with our the present study. If considering the intermediate charmonium contribution to e + e − → Λ cΛc , a phase space corrected Breit-Wigner distribution reads as where ψ denotes the intermediate charmonium resonance and Φ(s) is the phase space. Additionally, we define a free param- Additionally, a non-resonance contribution is parameterized as [22] where g NoR , a, and b are free parameters. The total amplitude of the process e + e − → Λ cΛc can be written as where φ i is the phase angles between the i-th resonance amplitude and non-resonance term.
In this analysis, the masses and widths of the involved charmonium states are taken to be theoretical values from the screened potential model [22]. Because a similar fit was performed in Ref. [22] to extract the branching ratio of ψ 6S −5D and ψ 6S −5D , so we choose the same parameters g NoR , a, b, and R ψ 6S −5D as those in Ref. [22]. Here, the R ψ 6S −5D is considered as a free parameter in our present fit because the fitted width of ψ 6S −5D in Ref. [22] is almost three times larger than our theoretical estimation and we argued that this inconsistency may be due to the influence from possible ψ(7S ). Finally, other free parameters in the fit are the relative phases associated with various charmonium resonances.
The fitted parameters are listed in Table IV and the fitted result is shown in Fig. 6, where the χ 2 /d.o. f = 1.465 is obtained. Similar to the Y(4630) structure, one can see that the interference of adjacent ψ(nS ) and ψ((n − 1)D) really shows several obvious enhancements in the energy region between 4.7 and 4.9 GeV in Fig. 6. Unfortunately, the uncertainties of the Belle data are too large to draw any solid conclusions from the present fit. However, it is worth noting that the respective contributions of these higher charmonium states to the cross section of e + e − → Λ cΛc are also shown in Fig. 6, which are directly calculated from the predicted branching ratios in Eq. (17) and are independent of the fit schemes. Thus, it should provide some interesting evidences for the existence of these higher charmonia in this channel. We are hopeful for more precise experimental measurements to clearly confirm these local enhancements. We notice that the BESIII Collaboration has recently released their white paper on the future physics The fitted parameters to e + e − → Λ cΛc from Belle [6] and BESIII [48].  program [49]. At present, the BEPCII is going to take data in center-of-mass energy region between 4.6 and 4.9 GeV and the data set corresponding to 15 fb −1 of total integrated luminosity is expected. It is interesting to test the property of dense charmonium spectrum above 4.6 GeV, and the future BESIII and upcoming Belle II will provide a good platform to search for these charmonium states.