Study of the B − → K − ηη c decay due to the D ¯ D bound state

We study the B − → K − ηη c decay by taking into account the S -wave contributions from the pseudoscalar meson–pseudoscalar meson interactions within the unitary coupled-channel approach, where the D ¯ D bound state is dynamically generated. In addition, the contribution from the intermediate resonance K ∗ 0 (1430) − , with K ∗ 0 (1430) − → K − η , is also considered. Our results show that there is a clear peak around 3720 MeV in the ηη c invariant mass distribution, which could be associated with the D ¯ D bound state. The future precise measurements of the B − → K − ηη c process at the Belle II and LHCb experiments could be, therefore, used to check the existence of the D ¯ D bound state, and to deepen our understanding of the hadron-hadron interactions.


I. INTRODUCTION
Since the discovery of X(3872) by the Belle Collaboration in 2003 [1], many exotic states, which do not fit into the expectations of conventional quark models, have been observed experimentally during the past two decades [2].Many of these exotic states, especially the ones observed in the charmonium sector, are observed around the threshold of a pair of heavy hadrons; some of them, such as X(3872) [3], Z c (3900) [4] and X(4160) [5], can be explained as the hadronic molecules.However, the hadronic molecular states with mass near the D D threshold have not yet been observed experimentally, and further detailed studies are therefore required both theoretically and experimentally [6].
Although the D D bound state X(3700) couples mainly to the D D and D s Ds channels, it is not easy to search for any signals of the state in these systems.This is due to the fact that, since its mass is a little bit lower than the D D threshold, the X(3700) state would manifest itself as a near-threshold enhancement in the D D invariant mass distribution, which may be difficult to identify due to the low detection efficiencies near the threshold [27,30].On the other hand, the X(3700) state has also a sizeable coupling to the ηη c channel, as observed in Refs.[7,8].Since the ηη c threshold is about 200 MeV lower than the predicted mass of X(3700), one expects that, if the D D bound state exists, a clear peak near the D D threshold would appear in the ηη c invariant mass distribution of some processes with large phase space.
As is well known, the three-body weak decays of the B mesons involve much more complicated dynamics than do the two-body decays and can, therefore, provide a wealth of information about the meson-meson interactions and the hadron resonances [31][32][33][34][35] (see e.g., Ref. [36] for a recent review).For instance, the B → K + X/Y /Z decay is an ideal process to produce the charmoniumlike hadronic molecular states [11,[37][38][39][40], and many exotic states have been observed experimentally through the B-meson weak decays during the past few years, such as Z cs (4000), Z cs (4220) [41] and X(4140) [42,43] in B + → J/ψϕK + , as well as X 0 (2900) and X 1 (2900) in B + → D + D − K + decay [44,45].In this paper, we propose to search for the D D bound state X(3700) in the B − → K − ηη c decay.It is worth mentioning that the Belle Collaboration has already searched for the process in 2015 based on 772 × 10 6 B B pairs collected at the Υ(4S) resonance [46], but no significant signal of the D D bound state was observed due to insufficient statistics.However, the Belle II Collaboration will accumulate about 50 times the Belle dataset [47,48], and is expected to make further precise measurements of the B − → K − ηη c decay, which will shed more light on the existence of the D D bound state in this process.In addition, the authors of Ref. [49] have suggested to search for the D D bound state in the ηη c mass distribution of the B + → K + ηη c decay, and predicted the branching ratio of B(B + → X q q (→ η c η)K + ) = (0.9 ∼ 6.7) × 10 −4 .
In this paper, motivated by the observations made above, we will study the B − → K − ηη c decay by taking into account the pseudoscalar meson-pseudoscalar meson interactions within the chiral unitary approach, from where the D D bound state is generated dynamically.On the other hand, the B − → K − ηη c decay can also proceed through the subsequent decay of the intermediate resonance K * 0 (1430), i.e.K * 0 (1430) → Kη, whose contribution will be considered in this paper too.We will demonstrate that, besides a peak of K * 0 (1430) in the K − η invariant mass distribution, there is a clear peak around 3720 MeV in the ηη c invariant mass distribution, which could be associated with the D D bound state.Therefore, future precise measurements of the B − → K − ηη c decay at the Belle II and LHCb experiments could be used to check the existence of the D D bound state, and to deepen our understanding of the hadron-hadron interactions.
This paper is organized as follows.In Sec.II, we will firstly introduce our formalism for the B − → K − ηη c decay.Our numerical results and discussions are then presented in Sec.III.In Sec.IV, we give our final conclusion.

II. FORMALISM
In analogy to the discussions made in Refs.[27,[50][51][52], the B − → K − ηη c decay proceeds via the following three steps: the weak decay, the hadronization, and the final-state interactions.Explicitly, the b quark of the B − meson firstly decays into a c quark and a virtual W − boson, and then the W − boson turns into a cs pair.In order to give rise to the K − ηη c final state, the ū antiquark of the initial B − meson and the cs pair from the W − subsequent decay have to hadronize together with the qq (≡ ūu+ dd+ ss) created from the vacuum with the quantum numbers J P C = 0 ++ .The relevant quark-level diagrams can be classified as the internal and external W − emission mechanisms, as depicted in Figs.1(a tion of q i , qj and qk q k are given by with the q q matrix defined as which could be expressed in terms of the physical pseu-doscalar mesons as [33], Thus, by isolating the meson K − , one could easily obtain the components of the meson systems for Figs.1(a) and 1(b) as follows: where V cb = 0.04182 and V * cs = 0.97349 are the CKM matrix elements, and V p encodes all the remaining factors arising from the production vertex.Then, the final-state interactions of D D, D s Ds , and η ′ η c will dynamically generate the D D bound state, which could decay into the ηη c system.Here we do not consider the component K − π 0 η c , since the isospin of the π 0 η c system is I = 1.
Similarly, we can write the hadron components for Figs.1(c) and 1(d) that could couple to the K − ηη c system as follows: where we have introduced the color factor C to account for the relative weight of the external W − emission mechanism with respect to the internal W − emission mechanism, and will take C = 3 in the case of color number N C = 3, as done in Refs.[53][54][55].According to the above discussions, the K − ηη c final state could not be produced directly through the treelevel diagrams of the B − decay, but can via the finalstate interactions of the coupled channels D 0 D0 , D + D − , D + s D − s , and η ′ η c , which could then generate the D D bound state, as shown in Fig. 2. The total amplitude of Fig. 2 can be expressed as where G l is the loop function for the two-meson propagation in the l-th channel, and its explicit expression is given by [7] with the subtraction constant s , and η ′ η c , and µ = 1500 MeV, being the same as used in Ref. [8].√ s = M ηηc is the invariant mass of the two mesons in the l-th channel, and m 1 and m 2 are the masses of these two mesons.P is the total four-momentum of the two mesons in the l-th channel, and p is the magnitude of the three-momentum of each meson in the meson-meson center of mass frame, with where λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx is the Källen function.The transition amplitudes in Eq. ( 8) are obtained by solving the Bethe-Salpeter equation in coupled channels [7,8], where the matrix V is the potential constructed at the tree level for each one of the possible channels.Here we take into account the channels of FIG. 3: The Dalitz plot for the B − → K − ηηc decay.The green dash-dotted line and band stand for the mass and width of X(3700), while the blue dashed line and band for the mass and width of the well-established resonance K * 0 (1430).
the channel K − η in S-wave with a branching fraction B(K * 0 (1430) → Kη) = (8.6 +2.7 −3.4 )% [2].Therefore, in this paper, we neglect all the other excited kaon mesons, and only take into account the contribution from the intermediate K * 0 (1430) resonance as shown by Fig. 4, whose amplitude can be expressed as where the parameter β accounts for the relative weight of the K * 0 (1430) contribution with respect to that of the D D bound state X(3700), and the phase factor e iφ is introduced to describe the interference between the amplitudes from the D D bound state and the K * 0 (1430) resonance.M K − η is the invariant mass of the K − η system.We will take as input M K * 0 (1430) = 1425 MeV and With the amplitudes given by Eqs. ( 8) and ( 12) at hand, the doubly differential decay width of the B − → K − ηη c process can be written as One could obtain the invariant mass distributions dΓ/dM ηηc , dΓ/dM K − η , and dΓ/dM K − ηc by integrating Eqs. ( 13) and ( 14) over each of the invariant mass variables.For instance, the differential decay width dΓ/dM ηηc can then be obtained by integrating Eq. ( 13) over the K − η invariant mass M K − η , with the final result given by Here the integration range is given by where E * K − and E * η are the energies of K − and η in the ηη c rest frame, respectively.Explicitly, we have Here all the meson masses involved are taken from Ref. [2].

III. RESULTS AND DISCUSSION
In our model, we have three free parameters, V p , β and φ.The parameter V p is a global factor and its value does not affect the shapes of the ηη c , K − η, and K − η c invariant mass distributions, and thus we take V p = 1 for simplicity.The parameter β represents the relative weight of the K * 0 (1430) contribution with respect to that of X(3700), and the parameter φ is the relative phase between these two amplitudes.0.0 5.0×10 −8   1.0×10 −7   1.5×10 −7   2.0×10 −7   3600 3800 4000 4200 4400 4600 4800 8.0×10 −8   1.2×10 −7   1.6×10 −7   1000 1200 1400 1600 1800 2000 2200 6.0×10 −8   9.0×10 −8   1.2×10 −7   3600 3800 4000 4200 4400 4600 4800 As indicated by the current data on the branching fractions of B-meson decays [2], the branching fractions of the processes B 0 → K * 0 (1430) 0 η c and B 0 → K 0 D D are of the same order of magnitude.Thus, the contributions from the D D bound state and the K * 0 (1430) resonance are expected to be of similar magnitudes.By integrating the differential decay width over the corresponding invariant mass, one can estimate the partial decay widths Γ(B − → K * 0 (1430 ) are of the same order of magnitude.Therefore, in this work, we take the parameter β = 0.012 and also discuss our results with different values of β later.
Firstly, we show in Fig. 5 the ηη c , K − η, and K − η c invariant mass distributions with β = 0.012 and φ = 0.One can see a clear peak around 3720 MeV in the ηη c invariant mass distribution, which should be associated with the D D bound state X(3700).At the same time, a cusp structure appears around 3930 MeV in the same invariant mass distribution, which is due to the strong coupling of the D D bound state to the D s Ds channel.In addition, a K * 0 (1430) signal appears in the K − η invariant mass distribution, but gives rise to a smooth shape in the ηη c invariant mass distribution and thus does not affect the peak structure of the X(3700) significantly.It should be stressed that the line shape of the X(3700) in the ηη c invariant mass distribution is different from that of a Breit-Wigner form, which is a typical feature of the D D molecular state.On the other hand, one bump structure appears around 4400 MeV in the K − η c invariant mass distribution, which is due to the D D interaction and hence should not be associated with any resonance.
We also show in Fig. 6 the doubly differential decay width d 2 Γ/ dM ηηc dM K − η for the ) plane, where one can see two clear bands corresponding to the X(3700) and K * 0 (1430) resonances, respectively.
The parameters β and φ are unknown in our model, and their values could be determined if the precise experimental measurements of the B − → K − ηη c decay are available in the future.In order to study the dependence of our results on β and φ, we have calculated the ηη c , K − η, and K − η c invariant mass distributions with different values of β and φ, which are shown in Figs.7 and  8, respectively.From Fig. 7, one can see that the peak of the K * 0 (1430) resonance in the K − η invariant mass distribution becomes more significant when the value of β increases.From Fig. 8, on the other hand, the peak of K * 0 (1430) moves a little bit for different values of φ.However, the peak of the D D bound state X(3700) is always clear in the ηη c invariant mass distribution.
Finally, we should note that the value of the color factor C, which represents the relative weight of the external W − emission mechanism with respect to the internal W − emission mechanism, could vary around 3 in order to account for the potential nonfactorizable contributions [65].To this end, we show in Fig. 9 the ηη c , K − η, and K − η c invariant mass distributions of the B − → K − ηη c decay by taking three different values of C = 3.0, 2.5, 2.0.One can see that, although the peak of the X(3700) state in the ηη c invariant mass distribution becomes weaker when the value of C decreases, its signal is still clear and can be easily distinguished from the background contribution.Meanwhile, the peak of the K * 0 (1430) resonance in the K − η invariant mass distribution has little changes for these three different values of C, because the contribution from the D D bound state is smooth around the peak of K * 0 (1430) in the K − η invariant mass distribution, as observed already in Fig. 5.
From the above analyses, one can conclude that, within the variation ranges of the three free parameters, there is always a clear peak around 3720 MeV in the ηη c invariant mass distribution, which corresponds to the D D bound state.Thus, we strongly suggest our experimental colleagues to perform more precise measurements of the B − → K − ηη c decay at the Belle II and LHCb experiments in the future, which is very important for confirming the existence of the predicted D D bound state.

IV. CONCLUSIONS
In this paper, motivated by the theoretical predictions for the D D bound state X(3700), we propose to search for this state in the B − → K − ηη c decay.To this end, we have investigated the process within the unitary coupled-channel approach, by taking into account the contributions from the S-wave pseudoscalar meson-pseudoscalar meson interactions, which can dynamically generate the D D bound state X(3700).We have also taken into account the contribution from the intermediate resonance K * 0 (1430), since it couples to the Kη channel in S wave with a branching fraction of B(K * 0 (1430) → Kη) = (8.6 +2.7 −3.4 )%.Our results show that a clear peak appears around 3720 MeV in the ηη c invariant mass distribution, which should be associated with the D D bound state.It should be stressed that the line shape of the D D bound state is significantly different from that of a Breit-Winger form, which is a typical feature of the D D molecular state.On the other hand, one can also find the peak of the resonance K * 0 (1430) in the K − η invariant mass distribution, and the resonance gives a smooth contribution in the ηη c invariant mass distribution.
In this paper, we take into account the coupled channels π , and η ′ η c , and present the transition potential V ij in Table I.
continued on next page -continued from previous page Channel Potential

TABLE I :
The transition potentials Vij among different channels, where s, t and u are the Mandelstam variables.