Exploring open-charm decay mode $\Lambda_c\bar{\Lambda}_c$ of charmonium-like state $Y(4630)$

The newly observed $X, Y, Z$ exotic states are definitely not in the standard $Q\bar Q'$ structures, thus their existence composes a challenge to our understanding on the fundamental principles of hadron physics. Therefore the studies on their decay patterns which are determined by the non-perturbative QCD will definitely shed light on the concerned physics. Generally the four-quark states might be in a molecular state or tetraquark or their mixture. In this work, we adopt the suggestion that $Y(4630)$ is a charmonium-like tetraquark made of a diquark and an anti-diquark. If it is true, its favorable decay mode should be $Y(4630)$ decaying into an open-charm baryon pair, since such a transition occurs via strong interaction and is super-OZI-allowed. In this work, we calculate the decay width of $Y(4630)\to\Lambda_c\bar{\Lambda}_c$ in the framework of the quark pair creation (QPC) model. Our numerical results on the partial width computed in the tetraquark configuration coincide with the Belle data within a certain error tolerance.


I. INTRODUCTION
In 2007, the Belle collaboration reported that a J PC = 1 −− resonance peak Y(4630) with mass M = 4634 +9 −11 MeV and width Γ = 92 +41 −32 MeV appeared at the invariant mass spectra of the e + e − → Λ + c Λ − c channel [1]. Besides an interpretation that the observed Y(4630) is the 5 3 S 1 charmonium state [2,3], there are many alternative suggestions for the observed peak, for example, Y(4630) was considered to be induced by a threshold effect instead of being a genuine resonance [4], then it was also interpreted as a molecular state made of ψ(2S ) and f 0 (980) by another theoretical physics group [5]. Among those proposals, the suggestion that Y(4630) is a tetraquark state is more favorable [6,7]. In Ref. [6], the Y(4630) is identified as the ground state with its orbital angular momentum L = 1. It is noted that the mass and width of Y(4630) are consistent within errors with those for the Y(4660) state (M = 4652 ± 10 ± 8MeV, Γ = 68 ± 11 ± 1MeV), which is found in the invariant mass spectrum of ψ(2S )π + π − by the Belle collaboration [8,9]. By analyzing the Λ cΛc and ψ(2S )π + π − spectra, Cotugno et al. suggested that the Y(4630) and Y(4660) could be the same tetraquark state, and is the first radial excitation of the Y(4360) with L = 1 [7].
However, there definitely may exist other decay modes beside of the Λ cΛc pair, such as DD, DD * , D * D * , J/ψη, ψ(2S )η, etc. Such processes occur via color rearrangements which in principle can be depicted by hadronic loops even though the propagators in the loops do not correspond to real colorsinglet particles (see in text), so they suffer from a loop suppression. Even though the most promising tetraquark candidate Z(4430) + decays into the [cq][cq] mode ψ(2S )π + [10][11][12][13] with a broad width Γ = 172 ± 13 MeV, this case is very different from Y(4630). Since its mass is below the Λ cΛc threshold it would overwhelmingly decay into open-charmed mesons. For Y(4630) case, as its mass is above the doublebaryon threshold, the strong decay of such tetraquark state is OZI-super allowed. Therefore, following the suggestions given by other groups here we will assume that the decay mode Y(4630) → Λ cΛc would be dominant, namely this partial width could be at the same order of the total width.
A tetraquark is assumed to be made of the diquarkantidiquark [cq][cq], where q is a light quark either u or d and [cq] resides in a color anti-triplet whereas [cq] is in a color triplet (in later calculations we do not distinguish between u and d at all). In this work, we suppose that Y(4630) is a tetraquark in the dynamic picture suggested by Brodsky et al. [14]. In the tetraquark a diquark and an anti-diquark are bound together via the QCD confinement, but are separated by a substantial distance once they are created. Thus the Y(4630) state can be considered as a two-body meson-like state. The picture we adopt in this work is slightly different from that proposed by Maiani et al. [6,15], where the authors studied the tetraquark states by means of their spin structure of a Hamiltonian formalism [16], in fact, the two pictures are in principle consistent. Under this assignment, we study the strong decay of Y(4630) by computing the width of Y(4630) → Λ cΛc in the quark pair creation (QPC) model. The corresponding reaction mechanism is that first the diquark-antidiquark bound state is dissociated into a "free" diquark-antidiquark system and a light quark-antiquark pair is created from the vacuum, then the quark and anti-quark would join the diquark and antidiquark respectively to constitute a baryon-anti-baryon pair. Indeed, this association can be viewed as that due to soft gluon emission a light-quark pair is created and the soft gluons tear off the diquark-antidiquark bound state, then by absorbing light quark and antiquark respectively they transit into color singlet baryons. Moreover, M Λ c + MΛ c is only slightly below 4630 MeV, so that a suppression induced by matching different momenta as appearing at similar hadronic processes, does not exist. Surely the whole dissociation process is governed by non-perturbative QCD, so that one needs to introduce a few phenomenological factors which can only be obtained by fitting available data.
The paper is organized as follows: after this introduction, we calculate the rate of Y(4630) decaying into the Λ cΛc pair in section. II A&II B and perform a numerical analysis in Sec. II C. The other decays of Y(4630) are discussed in Sec. III. Sec. IV is devoted to our summary.
In this work, we use the two-body wave function for the diquark-antidiquark bound system Y(4630), since the constituents (diquark and antidiquark) are treated as two pointlike color sources. In this structure, the diquark Qq of coloranti-triplet in the tetraquark is in analog to a heavyQ residing in a common meson QQ whileQq is similar to Q by the same color configuration.
In the following, we present all the details of calculating Y(4630) → Λ cΛc in the QPC model.
As indicated in the introduction, we suppose Y(4630) as a tetraquark in the diquark-antidiquark structure, thus in our case, the decay of Y(4630) is a dissociation process where the diquark and antidiquark bound state is loosened by a quarkantiquark pair which is created in vacuum. Concretely, the quark and antiquark of the pair excited out from the vacuum would join the diquark and antidiquark respectively to compose a Λ cΛc pair, and the process is graphically shown in Fig. 1.
The quantum number of the created quark pair is 0 ++ [17,18]. In the non-relativistic limit, the transition operator is expressed as where i and j are the SU(3)-color indices of the created quark and anti-quark. ϕ 56 0 = (uū + dd + ss)/ √ 3 and ω 56 0 = δ i j are for flavor and color singlets, respectively. χ 56 1,−m is a spin triplet. Here the indices 5 and 6 distinguish between the quark and antiquark respectively as shown in Fig. 1. Y m (k) ≡ |k| Y m (θ k , φ k ) denotes the th solid harmonic polynomial. γ is a dimensionless constant for the strength of quark pair creation from vacuum and is fixed by fitting data.
we label the quark c and antiquarkc with subscripts 2 and 4, as well understood. q stands for the light quark u/d.
In the dynamical picture of tetraquark, the (anti)diquark is considered to be a point-like color source, then the twobody wave function(meson-like) should be a good approximation to describe the inner structure of Y(4630). Including the color (ω [12][34] Y ), spin (χ [12][34] Y ), flavor (ϕ [12][34] Y ) and the spatial (Ψ n Y L Y M L Y (p 1 , p 2 )) parts, the wave function is written as where we use the (super)subscript 1∼4 to mark the (anti)quark in the tetraquark as clearly shown in We also consider the diquark-quark picture [38][39][40][41][42] for the Λ c baryon in where the internal degrees of freedom of the diquark are neglected as in the tetraquarks, then we have where the (super)subscripts in the expressions correspond to the constituent quark and the diquark, and K Λ c is the 3-momentum of Λ c , p 1 (k 3 ) is the 3-momentum of the diquark(quark). The quantum numbers of Λ c are known as J P = 1 2 + and L = 0, so we only use M S Λc (= M J Λc ) to label the spin projection state. The wave functions respect the normalization conditions For Y(4630) → Λ c +Λ c process, the transition hadronic matrix element is written as In the center of the mass frame of Y(4630), Λc ϕ [ The expressions of Eq. (6) for Y 1,2,3,4 states are explicitly written out in terms of I M L Y ,m (K) as listed in the Appendix A. The spatial integral I M L Y ,m (K) manifests an overlap between the spacial parts of the initial state (including the created light quark pair) and the final state, and is expressed as I M L Y ,m (K) = dp 1 dp 2 dk 5 dk 6 where µ = m The decay width of Y(4630) → Λ cΛc is then where |K|, as aforementioned, is the 3-momentum of the final states in the center of mass frame.

B. Baryon wavefunction
The charmed baryon Λ c is considered as the [cq]-q picture in our scenario, then a two body wavefunction, which can be gained by solving the Schrödinger equation, could be a reasonable approximation.
For our concrete calculation, we employ a non-relativistic Cornell-like potential where the concerned free parameters are fixed by fitting the mass spectra of charmed baryons. By solving the Schrödinger equation we obtain the wave function of Λ c . The general Hamiltonian of a diquark-quark system (i.e. a two body system) can be written as where the m [cq] (p [cq] ) and m q (p q ) are the masses(3-momenta) of the diquark [cq] and quark q respectively. It is worth of pointing out that in literature, the diquarkquark structure might be different, namely the two light quarks make a light diquark and the heavy quark stands as a color source. Instead the baryon still might be in [Qq]¯3q 3 structure [43], especially in our case the diquark (anti-diquark) does not have time to recombine into Q 3 [qq]¯3 by color rearrangement, namely the original diquark structure would remain to make a color singlet baryon by absorbing a light quark. The interaction potential is where −4/3 is the color factor specific to 3-3 attraction, b is the string tension and c is a global zero-point energy. Here we take the br κ + c part as the confinement which is slightly different form the usual Cornell br + c potential. α s is the phenomenological strong coupling constant. In this work, since only the wave function of Λ + c (2286) which is in S-wave is needed, the hyperfine interactions including the spin-spin interaction, the spin-orbit interaction and the color tensor interaction [30] are not included.
With the diquark mass m [cq] =1.86 GeV which is calculated by the QCD sum rules [44] and the light constituent quark mass m q =0.33 GeV, the parameters are fixed to be: α s = 0.45, b = 0.135 GeV 2 , κ = 0.84, c = 0.333 GeV. Here, as theoretical inputs, we ignore possible inaccuracies of the parameters.
The fitted spectra are presented in Table. I, and a comparison with the experimental data and other theoretical predictions in literature are also listed in the table. The radial wave function of Λ c (2286) is plotted in Fig. 2.

C. Numerical results
Following Ref. [48], in the numerical computations, we adopt the model parameter γ = 6.3 which is considered as universal in the QPC model. Meanwhile the R value for P-wave tetraquark in the SHO wave function, which represents the mean-square root (RMS) radius, can either not be determined from an underlying principle, so we perform a numerical analysis dependent on R Y with certain ranges, where R Y denote the R-value of the wave functions for Y(4630) in tetraquark structure. Since there still exists an ambiguity about the inner structure of Y(4630), we calculate the decay width for two possible cases: assuming (1) Y(4630) as the ground state with the radial quantum number n r = 1 and (2) the first radial excitation with n r = 2 assignments.  We first compute the decay width of Y(4630) → Λ cΛc with the n r = 1 assignment. The left panel of Fig. 3 shows the dependence of the calculated width Γ Y on R Y within a range (1 ∼ 6)GeV −1 . The colored curves correspond to the four spin states Y 1,2,3,4 which are marked on the figures. As discussed before, the decay mode Y(4630) → Λ cΛc should be dominant, so we compare this calculated partial width with the total width of Y(4630). In the plot one can find the predicted width for the Y 1,2,4 assignments do coincide with the data and the error band of 1σ (gray region) given by the Belle collaboration (Γ = 92 +41 −32 MeV). For the Y 3 case the figure shows that the values of the curves are obviously lower than the data Γ = 92 +41 −32 MeV. This suppression is caused by the relatively small overlap between the spin wave functions of initial and final states (one can see the appendix A for some details). Therefore it is concluded that the data do not favor Y(4630) to be a ground state with Y 3 spin structure.
Next, as Y(4630) being assigned as the first radial excitation state, our numerical results are shown at the right panel of Fig. 3 for all the four spin assignments. The results show that the Y 1,2 states can meet with the experimental data as long as R Y lies in a range of 1.5 ∼ 3 GeV −1 and/or around 5GeV −1 . The values correspond to the Y 4 state are slightly lower, however, they are still of the same order as the total width. Again, for Y 3 state, the situation is similar to that for n r = 1, the computed width are much below the data.
In a brief summary, our numerical results indicate that within certain regions of the parameter R Y , the partial width of Y(4630) → Λ cΛc can be comparable with the Belle data. Given the fact that the peak of Y(4630) has only been observed at the invariant spectrum of Λ cΛc , one is tempted to assume that the Λ cΛc mode dominates the decay of Y(4630) Moreover our calculation indicates that this predicted partial width is comparable with the total width of Y(4630). This consistency supports the assumption that the Y(4630) is a Pwave tetraquark in the diquark-antidiquark configuration and decays mainly into double charmed baryons. We will make more discussions on this issue in the next section.
Also Y(4630) could be in either the radial ground state with n r = 1 or the first excited state with n r = 2. In other words, the present data cannot rule out any of the two possible configurations. So definitely it needs to be studied with more experimental information in the future to decide the more accurate nature of Y(4630), so as the spin structures.
For the tetraquark structure, this decay mode requires a quark-antiquark rearrangement which is also a color exchange process. In the process a quark and an antiquark which belong to different clusters are switched round to produce the final states.
In the figure 4, tracing the diquark (antiquark) flow lines, one can draw an effective hadronic Feynman diagram as a diquark (scalar or vector) which brings a color-content (colortriplet 3 or anti-triplet (3) is exchanged between the diquark [cq] and antidiquark [cq], and results in the final state to be in color-singlet. In figure, such processes occur via a hadronic loop, therefore is suffering from a loop suppression. This Feynman diagram is similar to the final state interaction where all lines corresponding to (no matter inside the loop or outside finally produced hadrons) color-singlet hadrons, thus only difference between the quark rearrangement and final state interaction is their color configurations. But both of them are suppressed. In our another paper, we estimated the rates of Y(4630) → Λ cΛc → pp, DD, DD * , ππ, K + K − etc. through hadronic rescattering and found that such as final states could be observed by much more accurate measurements [49]. Similarly, we may conjecture that the color-re-arrangement which proceeds along similar way should have comparable rates.
In fact, such quark exchange mechanism was investigated by some authors for meson decays [50][51][52], but since it is completely induced by the non-perturbative QCD effect, the estimate in terms of the present theories cannot be accurate, or at the best can be valid to the order of magnitude if one can find an appropriate model to carry out numerical computations.

IV. SUMMARY
To evaluate the hadronic matrix elements which are governed by the non-perturbative QCD, phenomenological models are needed. For the OZI-allowed strong decays, the QPC model, flux tube model, QCD sum rules and lattice QCD, etc. have been successfully used to estimate the decay rates, even though except the lattice calculation none of them can be directly derived from quantum field theory. We are assured that all of those models have certain reasonability and they are in parallel somehow. In this work, we employed the QPC model to study the strong decay of Y(4630) → Λ cΛc .
First we assume that Y(4630) is a tetraquark which is a bound state of a diquark and an anti-diquark. As its mass is slightly above the threshold of two charmed baryons, it would favorably decay into Λ cΛc pair, therefore the fact that Y(4630) is only observed at the invariant spectrum of Λ cΛc , is understandable.
There could be different quantum structures for the diquarkanti-diquark bound state, and we try to assign it with various radial quantum numbers and spin assignments and then calculate the decay width of Y(4630) → Λ cΛc in all possible cases.
The numerical results show that, within certain parameter range of R Y , one can gain proper decay width Γ Y that agrees with the experimental data if we assign Y(4630) as either the radial ground state n r = 1 or the first radially excited state n r = 2. Whereas for the case of Y 3 , the obtained partial width are suppressed by the small overlap between the spin wave functions, so the Y 3 spin state is ruled out. Our analysis provides a strong support to the postulate that Y(4630) is the diquark-antidiquark bound state whose mainly decay channel should be Y(4630) → Λ cΛc .
We are looking forward to getting more information from the Belle-II, LHCb experiments, especially we will pay more attention to, such as DD, DD * , D * D * , ψ(2S )π + π − , ψ(2S )η etc, decay modes, which may shed more light on the structure of Y(4630). In particular, we suspect if there is a mixing between the tetraquark and molecular states to result in Y(4630) and Y(4660), it would be an interesting picture. Indeed in the near future, with the accumulated data at various accelerators, our understanding on the XYZ states will be improved and the observations of new states are expected.
Note added. When we make changes to our manuscript, we notice that another work [53] which suggests to use Y(4630) as a window to the landscape of tetraquarks appears, by J. Sonnenshein and D. Weissman, and we cite it at the end of this modified manuscript. In this work, we employ the SHO wave functions for Y(4630) as the input wave functions. For the decay channels of interest, we need a P-wave two-body wave function for the Y(4630).