Search for a $D \bar{D}$ bound state in the $\Lambda_b \rightarrow \Lambda D\bar{D}$ process

We have investigated the process of $\Lambda_b\to \Lambda D\bar{D}$, by taking into account the contributions from the $s$-wave $D\bar{D}$ interaction within the coupled-channel unitary approach, and the intermediate $\psi(3770)$ resonance. In addition to the peak of the $\psi(3770)$, an enhancement near the $D\bar{D}$ mass threshold is found in the $D\bar{D}$ invariant mass distributions, which should be the reflection of the $D\bar{D}$ bound state. We would like to encourage our experimental colleagues to measure the $D\bar{D}$ invariant mass distribution of the $\Lambda_b\to \Lambda D\bar{D}$ process, which is crucial to search for the $D\bar{D}$ bound state and to understand the heavy-hadron heavy-hadron interactions.


I. INTRODUCTION
Although the quark model was proposed by Gell-Mann and Zweig more than half century ago [1,2], it is still valid in classifying all known hadrons by now. Since the X(3872) was observed by the Belle Collaboration in 2003 [3], many charmonium-like states were reported experimentally [4], and most of them cannot be explained as the conventional mesons (qq) or baryons (qqq) [5,6].
There are many explanations about those states, such as tetraquark states, molecular states, the conventional cc mesons, or the mixing between different components [7][8][9][10][11]. However, it is surprising that many resonant structures are observed around thresholds of a pair of heavy hadrons, such as X(3872) and Z c (3900) ± around the DD * threshold, Z cs (3985) around theD s D * andD * s D thresholds, and X(3930) around D sDs threshold. As discussed in Ref. [12], such structures should appear at any threshold of a pair of heavy-quark and heavy-antiquark hadrons which have attractive interaction at threshold. Thus, the experimental information about the threshold structures is crucial to deeply understand the heavyhadron heavy-hadron interactions, and the internal structures of the hidden-charm states [13,14].
In Ref. [15], one new hidden charm resonance with mass around 3700 MeV (denoted as X(3700) in this article) is predicted within the coupled channel unitary approach involving the D + D − , D 0D0 , D sDs , K + K − , K 0K 0 , π + π − , π 0 π 0 , ηη, and π 0 η channels. Later it was suggested to search for this predicted DD bound state in several processes, such as B → DDK [16], ψ(3770) → γX(3700) → γηη , ψ(4040) → γX(3700) → γηη , and e + e − → J/ψX(3700) → J/ψηη [17]. According to the studies of Refs. [18,19], the experimental data of e + e − → J/ψDD measured by the Belle Collaboration [20,21] are compatible with the existence of such a DD bound state around 3700 MeV, though other possibilities cannot be discarded due to the present quality of the Belle data. In Ref. [24], we have performed a global fit to the data of γγ → DD [22,23] and the e + e − → J/ψDD [21], by taking into account the swave DD final state interactions. Our results are consistent with the experimental data considering the uncertainties of the fitted parameters, and the modulus squared of the amplitude |t DD→DD | 2 show peaks around 3710 ∼ 3740 MeV [24]. Recently, a DD bound state with binding energy B = 4.0 +5.0 −3.7 MeV was also predicted according to the Lattice calculation in Ref. [25]. Thus, it is crucial to search for the signal of this predicted state.
On the other hand, the decays of Λ b is one of the important tool to study the hidden charm resonances [26], such as the processes of Λ b → J/ψΛ, Λ b → ψ(2S)Λ [27][28][29]. The process Λ b → ΛX 0 c (X 0 c ≡ ccuū(dd), ccss) is also proposed to search for the XY Z states in Ref. [30]. In this work, we will propose to search for the signal of the DD bound state in the single-Cabibbo-suppressed process of Λ b → ΛDD, which has not been measured experimentally up to our knowledge. It should be pointed out that the Λ b → ΛDD process is expected to have a larger branching fraction than the double-Cabibbo-Suppressed process Λ b → ΛK + K − with the branching fraction B(Λ b → ΛK + K − ) = (15.9±1.2±1.2±2.0)×10 −6 measured by the LHCb Collaboration [31].
Since the predicted mass of the DD bound state is lower than the DD threshold, it will manifest itself as the enhancement near the DD threshold, the similar work is found in Refs. [16,32]. For instance, a peak observed in the φω threshold in the J/ψ → γφω reaction [33] was interpreted as the manifestation of the f 0 (1710) resonance below the φω threshold [34]. In Ref. [35] the BESIII Collaboration has seen a bump structure close to threshold in the K * 0K * 0 mass distribution of the J/ψ → ηK * 0K * 0 decay, which can be interpreted as a signal of the formation of an h 1 resonance [34,36]. We expect there will be an enhancement near the threshold in the DD invariant mass distribution. On the other hand, since the ψ(3770), with a mass close to the DD threshold, mainly decays into DD in p-wave, we will take into account the contribution from the ψ(3770).
The paper is organized as follows. In Sect. II, we introduce our model for the process Λ b → ΛDD. Numerical results for the DD invariant mass distribution and discussions are given in Sect. III, and a short summary is given in the last section.

II. FORMALISM
In analogy to Refs. [37][38][39][40][41], the mechanism of the decay Λ b → ΛDD (DD ≡ D 0D0 , D + D − ) can happen via three steps: the weak decay, hadronization, and the final state interaction. In the first step as depicted in Fig. 1, the b quark of the initial Λ b weakly decays into a c quark and a W − boson, followed by the W − boson decaying into ā cs quark pair, where we take the flavor wave functions In order to give rise to the final state D 0D0 Λ (or D + D − Λ), the quark c and antiquarkc need to hadronize together with theqq (≡ūu +dd +ss) created from the vacuum with J P C = 0 ++ , which could be expressed as the mechanisms of the internal W − emission and external W − emission, respectively shown in Figs. 2(a) and 2(b). Thus, we have, for the internal W − emission mechanism of Fig. 2(a) , and for the external W − emission mechanism of Fig. 2(b).
Here the color factor C accounts for the relative weight of the external W − emission with respect to the internal W − emission, and we take C = 3 in the case of color number N c = 3 [42][43][44].
The final states can also undergo the interactions of the DD and ΛD, which may generate dynamically the resonances. The interaction of the coupled channels including ΛD was studied within a unitary coupled-channel approach which incorporates heavy-quark spin symmetry, and two resonances Ξ c (2790) and Ξ c (2815) are identified as the dynamically generated resonances [45]. Since their masses are about 150 ∼ 200 MeV below the ΛD threshold, their contributions do not affect the structure close to the DD threshold, which can be easily understood from the Dalitz plot of Fig. 3. Thus, we neglect the ΛD interaction in this work, because only the DD invariant mass distribution near the threshold is relevant for the DD bound state.
The next step is to consider the final state interaction of these channels to give D 0D0 (or D + D − ) at the end. We can have the final states of D 0D0 (or D + D − ) through the direct production in the Λ b decay, or the re-scattering of the primarily produced channels D 0D0 , spectively. Apart from the three coupled channels D 0D0 , D + D − , and D + s D − s , we only consider one light channel ηη to account for the width of the DD bound state, as in Refs. [16][17][18]24]. Then, the total amplitudes for the Λ b → ΛD 0D0 and Λ b → ΛD + D − can be expressed as, where G l is the loop function for the two-meson propagator in the l-th channel, with the subtraction constant α l = −1.3 (l = 1, 2, 3, 4 correspond to the channels D 0D0 , D + D − , D + s D − s , and ηη, respectively) and µ = 1500 MeV as Ref. [15]. P ≡ √ s = M DD is the invariant mass of the two mesons in the l-th channel. m 1 and m 2 are the masses of the two mesons in the l-th channel. p is the three-momentum of the meson in the center of mass frame of the mesonmeson system, with the Källen function λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx.
With the isospin doublets (D + , −D 0 ), (D 0 , D − ), we have, Taking the averaged mass of D meson in Eqs.(4) and (5), it is easy to find that only the isospin I = 0 component of the DD has the contribution to the Λ b → ΛDD process, The scattering matrices t i→j in Eqs. (4) and (5) are obtained by solving the Bethe-Salpeter equation in coupled channels, where the elements of the diagonal matrix G is the loop function of Eq. (6), and the matrix element V i,j are the transition potential of the i-th channel to the j-channel. The transition potentials V i,j (i, j = D 0D0 , D + D − , D + s D − s ) are tabulated in the Appendix A of Ref. [15]. We introduce the potentials of ηη → D 0D0 and ηη → D + D − with a dimensionless strength a = 50 to give the width of the DD bound state, and the transition potentials of ηη → ηη and ηη → D + s D − s are not relevant and are taken as zero [16][17][18]24]. Both the G l and t i→j in Eqs. (4) and (5)  In addition, we also take into account the decays Λ b → ΛD 0D0 and Λ b → ΛD + D − via the intermediate resonance ψ(3770), which is depicted in Fig. 6. The amplitude can be written as, where the normalization factor V p is the same as the one in Eqs. (4) and (5), and we introduce the parameter β to account for the relative weight of the ψ(3770) strength with respect to the s-wave contribution of Eqs. (4) and (5).p D is the momentum of the D 0 (or D + ) in the rest frame of the D 0D0 (or D + D − ) system, We take the width for ψ(3770) energy dependent, which is given by, With the amplitudes of Eqs. (4), (5) and (13), we can write the differential decay width for the decays Λ b → ΛD 0D0 and Λ b → ΛD + D − , with

III. NUMERICAL RESULTS AND DISCUSSION
In our model, we have three free parameters, the global normalization V p , the color factor C, and β. V p is a global factor and its value does not affect the shapes of the D 0D0 and D + D − invariant mass distributions. β represents the relative weight of the ψ(3770) strength with respect to the one of s-wave, and we take its value β = 0.15 to give the contributions from the s-wave DD interaction and the ψ(3770) with the same order of magnitude. Next, we first show the results with the color factor C = 3 and V p = 1, and will present the results for different values of C and β.
We show the D 0D0 and D + D − invariant mass distributions in Fig. 7. One can find a clear enhancement near the D 0D0 threshold in the D 0D0 invariant mass distribution of the Λ b → ΛD 0D0 , due to the presence of the X(3700) resonance below the DD threshold. The enhancement structure near the threshold is a little weaker for the D + D − invariant mass distribution of the Λ b → ΛD + D − , because the D + D − threshold is higher than the D 0D0 one and farther away from the peak of X(3700). In Fig. 8, we show the D 0D0 and D + D − invariant mass distributions with the different values of color factor C = 3.0, 2.5, 2.0. One can find that both mass distributions near the threshold do not change too much, since the value of color factor C only affects the contribution from the D + s D − s loop of Fig. 4(b), which is smaller than the contributions from the D + D − and D 0D0 .
We also present our results for the different values of β = 0.30, 0.15, 0.10 in Fig. 9. One can see that the enhancement near the threshold will be identified difficultly for the larger value of β. Indeed, the ψ(3770) would provide the dominant contribution for the Λ b → ΛDD process, however, it is still expected to find an enhancement near the DD threshold, especially the D 0D0 one, if the DD bound state do exist, as predicted in Refs. [15,25]. Furthermore, since the ψ(3770) state couples to DD in p-wave, the partial wave analysis of this reaction would be helpful to test the existence of the DD bound state. At present, the LHCb Collaboration has accumulated a large number of Λ b events, thus, we would like to call the attention of the experimentalists to measure the Λ b → ΛDD decay, which should be useful to confirm the existence of X(3700) and to understand its nature.

IV. CONCLUSIONS
The study of the charmonium-like states is crucial to understand the heavy-hadron heavy-hadron interactions, and also the internal structures of the hidden-charm states. One DD bound state around 3700 MeV was predicted within the coupled channel unitary approach [15], and also the lattice investigation of the DD and D sDs scattering [24]. Although our previous studies on the e + e − → J/ψDD and γγ → DD data support the existence of the DD bound state, the other possibilities cannot be discarded due to the present quality of the experimental data [18,24]. Investigating the processes involving the s-wave DD system could provide the information about the existence of the DD bound state. In this paper, we have investigated the processes Λ b → ΛD 0D0 and Λ b → ΛD + D − within the coupled channel unitary approach, by taking into account the s-wave meson-meson interactions and the contribution from the intermediate resonance ψ(3770). The D 0D0 and D + D − invariant mass distributions in the Λ b → ΛDD reaction are investigated, and our results show an enhancement structure near the DD threshold, which should be the reflection of the DD bound state. Therefore, we strongly encourage our experimental colleagues to measure the Λ b → ΛDD process, which would be crucial to confirm the existence the X(3700) resonance, and to understand the heavy-hadron heavy-hadron interactions.