→ D 0 ¯ D 0 K 0 , B + → D 0 ¯ D 0 K + and the scalar D ¯ D bound state

We study the B 0 decay to D 0 ¯ D 0 K 0 based on the chiral unitary model that generates the X(3720) resonance, and make predictions for the D 0 ¯ D 0 invariant mass distribution. From the shape of the distribution, the existence of the resonance below threshold could be induced. We also predict the rate of production of the X(3720) resonance to the D 0 ¯ D 0 mass distribution with no free parameters.


I. INTRODUCTION
The weak decay of heavy hadrons has brought a wealth of information concerning the interaction of mesons, has reported on many mesonic, as well as baryonic, resonances and, together with devoted theoretical work (see the recent review [1]), has brought new insight into the nature of many resonances which have been the subject of continuous debate [2,3]. The meson scalar sector has been emblematic and the chiral unitary approach, unitarizing in coupled channels the information contained in the chiral Lagrangians [4], has shown that the f 0 (500), f 0 (980), a 0 (980) resonances appear as a consequence of the interaction of pseudoscalar mesons and respond to a kind of molecular structure of these components [5][6][7][8][9], diverting from the standard qq nature of most mesons. The case of the σ (f 0 (500)) has been thoroughly discussed in a recent review [10] and the situation has been much clarified. In this picture, the f 0 (980) stands as a bound KK state, with a small component of ππ that provides the decay channel of this state. Much before the advent of the chiral unitary approach, the KK molecular nature of the f 0 (980) had already been claimed [11]. A perspective into these "extraordinary states" was also recently given in the Hadron2015 Conference by Jaffe [12].
The chiral Lagrangians can be obtained from a more general framework, which includes vector mesons, the local hidden gauge approach [13][14][15][16]. In this picture the chiral Lagrangians are obtained by exchanging vector mesons between the pseudoscalar mesons. This picture is most welcome because it allows us to extend the dynamics of the chiral Lagrangians to the heavy quark sector, and the interaction of DD, for instance, would be given by the exchange of light vector mesons. Heavy vector mesons could also be exchanged, but their large mass makes the contributions of these terms subdominant, and the dominant terms, where the heavy quarks act as spectators [17,18], automatically satisfy the rules of heavy quark spin symmetry [19,20]. It is then not surprising that, in analogy to the KK interaction, which generates the f 0 (980), the DD interaction also gives rise to a bound state, which was studied in [21]. This state was also predicted in [22,23] using effective field theory that implements Heavy Quark Spin Symmetry. In [24] the results of the e + e − → J/ψDD reaction close to threshold [25] were analyzed. A bump around the DD threshold was observed and the fit to the data was compatible with a state below threshold at 3720 MeV (we shall call this state X(3720) from now on).
On the other hand, the study of B and D weak decays looking at resonances in the final state, or threshold behavior of invariant mass distributions, has shown that these reactions have a potential to tell us about the existence of "hidden" resonances and their nature. In this sense, in [26] a natural explanation was given, in terms of the f 0 (500), f 0 (980) as dynamically generated resonances [5,10], for the experimental facts that in the B 0 s → J/ψπ + π − reaction the f 0 (980) was clearly observed and no trace of the f 0 (500) was seen [27], while in the case of the B 0 decay, the f 0 (500) was seen and only a minor fraction of the f 0 (980) was observed [28]. These results were complemented by the study of the D weak decays, were the production of f 0 (500), f 0 (980) and a 0 (980) were studied [29].
The idea in this paper will be to make predictions for the DD invariant mass distribution in the decay of B 0 . In this sense the work of [30], where the B 0 s → D − s (KD) + was studied, showed that from the spectrum of the KD invariant mass one could determine the existence of the D * ± s0 (2317) below threshold, and the amount of the KD component in its wave function, using the compositeness sum rule of [31][32][33][34]. A similar work was also done in [35] where the reactionsB 0 →K * 0 X(Y Z) andB 0 s → φX(Y Z) with X(4160), Y (3940), Z(3930) were studied. It was found there that from the study of D * D * and D * sD * s mass distributions close to threshold, the existence of resonances below threshold could be induced.
In the present paper we will study the B 0 decay to D 0D0 K 0 with a model based on the work of [21] that generates the X(3720) resonance, and will make predictions for the D 0D0 invariant mass distribution. The theory predicts the shape of the distribution close to threshold, but not the absolute normalization. However, from the shape of the distribution the existence of the resonance below threshold could be induced. Additionally, we shall also evaluate the rate of production of the X(3720) resonance, irrelevant of its decay channel, and will show that the ratio of this rate to the D 0D0 mass distribution is then predicted with no free parameters, under the assumption that the X(3720) resonance is dynamically generated. The implementation of the experiment would provide a boost in the search of this elusive state, which we think really exists. This experiment and related ones are currently under investigation by the LHCb collaboration [36] and this gives us a motivation to perform the calculations at the present time. So far, the related experiment B + → D 0D0 K + has already been done [37]. The D 0D0 invariant mass is measured, but with very small statistics close to threshold. A sharp peak is identified, that corresponds to the excitation of the ψ(3770) charmonium state, which decays in p-wave in D 0D0 . The X(3720) state is a scalar meson and it decays into D 0D0 in s-wave. In this sense, testing the invariant mass predicted here should require to separate the s-wave from the p-wave part of the spectrum, something which is already currently been done by the partial wave analysis of the LHCb collaboration, where the contribution of the ρ and f 0 (500) are separated in the B 0 decay to J/ψπ + π − [28]. In any case, in the work of [37] the contribution of the ψ(3770) is separated and this allows us to make a comparison of our results with this distribution. With present errors we find a good agreement with the data, thus getting extra support for the X(3720) state. However, our study indicates that the B 0 → D 0D0 K 0 reactions is better suited than the B + → D 0D0 K + one to give information on that state.

II. FORMALISM
If we followed the steps of [26,38], a possible way for the B 0 → D 0D0 K 0 to proceed would be the following: in the first place we would produce a cc together with a sd pair as shown in Fig.1. The next step would consist in introducing a new qq state with the quantum numbers of the vacuum,ūu +dd +ss +cc in between the created cc pair, and see which combinations of mesons appear. This is depicted in Fig.2. This would lead to some DDK 0 components. The hadronization is now done including a qq scalar pair inside the sc pair, as shown in Fig. 4. This way chosen to hadronize the quarks follows the path of [26,29,35,38], based on the topology of the internal emission [40,42]. However, in the present case, we also have external emisson, which is color favored and, thus,B we adhere to this other mechanism, which is depicted in Fig. 3 at the quark level. The hadronization is now done including a qq scalar pair inside the sc pair, as shown in Fig. 4, and technically this is done as follow: An easy way to see which mesons are produced in the hadronization of sc is to introduce the qq matrix uū ud us uc dū dd ds dc sū sd ss sc cū cd cs cc In order to get the pair of mesons it is convenient to write the qq in terms of pseudoscalar mesons and then the M matrix has its equivalent matrix in φ given by (2) which incorporates the standard η, η mixing [39]. Now we see that (see Refs. [30,41]) Thus, in terms of mesons the hadronized sc pair will be given by We have the meson-meson components of Eq. (4), together with a D + of Fig. 3. If we want to have DD 0 and K 0 at the end, we must take theK 0 D − component of Eq.
(4) and let the D + D − interact to have DD 0 . Similarly, if we want to produce the scalar resonance X(3720), which has zero charge, it is also theK 0 D − component of Eq.
(4) that we must take, and we will let the D + D − interact to give the resonance X(3720) at the end. Hence, diagramatically this latter process is depicted in Fig. 5. For the case of D 0D0 production the mechanism is depicted in Fig. 6. Analytically we will now have where G M1M2 is the loop function of the two intermediate meson propagators [5] and g R,M1M2 is the coupling of the resonance to the M 1 M 2 meson pair. Taking into account that, with the doublets (D + , −D 0 ) and (D 0 , D − ), the isospin I = 0 state of DD is eq. (6) can be rewritten as The factor V P englobes the weak amplitudes plus the hadronization factors. We take it as a constant since we are only concerned about a restricted range of invariant masses, and the factor prior to the final state interaction is smooth in that range [43,44] (see section 3.3 of Ref. [1], for a more complete discussion on this issue and other approaches). The partial decay width ofB 0 →K 0 R decay will be where pK0 is theK 0 momentum in the rest frame of thē B 0 .

III. COMPLEMENTARY TEST OF THE MOLECULAR NATURE OF THE RESONANCES
In this section we make a test that is linked to the molecular nature of the resonances. We study the decaȳ B 0 →K 0 D 0D0 close to the DD threshold depicted in Fig. 6.
The production matrix will be given by We must evaluate the coupled channels s amplitudes which will contain I = 0 and I = 1, but close to the DD threshold they are dominated by I = 0. The meson-meson loop function G and the scattering matrix t i→j are evaluated following Ref. [21] as discussed later in the results section.
The differential cross section for production will be given by where pK0 is theK 0 momentum in theB 0 rest frame andp D the D momentum in the DD rest frame. By comparing this equation with Eq. (8) for the coalescence production of the resonance inB 0 →K 0 R, we find where we have divided the ratio of widths by the phase space factor pK0p D and multiplied by M 3 R to get a constant value at threshold and a dimensionless magnitude. We apply this method for the X(3720) resonance that couples strongly to DD.
The results obtained are easily translated to the B − → D 0D0 K − decay. The diagrams equivalent to Figs. 3,4,5,6, are now in Figs. 7,8,9,10. The situation is analogous to the former one, the hadronization of the sc pair proceeds in the same way as in Eq. (4), and the extra cū pair gives rise to a D 0 , unlike in the former case where the cū pair gave rise to a D + . Hence, from the (K −D0 +K 0 D − )D 0 contribution at the primary step, we can already produce K − D 0D0 at tree level and the DD rescattering will be done by the D 0D0 component.
The novelty is in Eq. (13) because now we can have K − D 0D0 production at tree level, and this is the term unity in Eq. (13).

IV. RESULTS
First, we use the scattering matrices based on the work of [21], and solve the Bethe-Salpeter equation in the coupled channels D + D − , D 0D0 , D + S D − s . With the |I = 0, DD wave function of Eq. (6), the I = 0 amplitude for DD is given by where 1,2,3 stand for the D + D − , D 0D0 , D + S D − s channels. The coupling g I=0 R,DD is obtained from this amplitude in the limit where M R is the energy where the pole of the bound DD state appears.
The resonance pole at √ s R = 3719.4 + i0 MeV is obtained with no width, using the same parameters as those in [21], α = −1.3, µ = 1500 MeV. In the following, from Eq. (10), we obtain the spectrum for the DD invariant mass distribution close to threshold in the decay ofB 0 . The differential cross section for the reactionB 0 →K 0 D 0D0 is given in Fig.  11, where the dashed line corresponds to a phase space distribution normalized to the same area in the range examined. We can see in the figure that the shape of the D 0D0 mass distribution is quite different from phase space, and this is due to the presence of the X(3720) resonance below threshold.
Next have evaluate Eq. (11) with the input of Eqs. (7) and (9), and the results are shown in Fig. (12). We observe that the ratio has some structure. There is a fall down of the ratio as a function of energy, as it would correspond to the tail of a resonance below the threshold of DD, the X(3720), since it basically is giving us the modulus squared of the t I=0 DD→DD amplitude.  Now we turn to the B − → D 0D0 K − reaction. We notice that the related experiment B + → D 0D0 K + has already been done [37], where the D 0D0 invariant mass is measured, but with very small statistics close to threshold. In the above experiment, a sharp peak is identified which should correspond to the excitation of the ψ(3770) charmonium state, which decays in p-wave in D 0D0 . Since the X(3720) state is a scalar meson, it couples to D 0D0 in s-wave. In any case, in the work of [37], the contribution of the ψ(3770) is separated and this allows us to make a comparison of our results with this distribution.
In order to compare our results with those of Ref. [37] for B + → K + D 0D0 , we subtract from the experimental data the explicit contribution of the Ψ(3770) and D s1 (2700) which give some contribution in the region of 100 MeV above DD threshold which we study.
We have taken a normalization such as to agree with that of the data and have We shall come back to this after the discussion of the next section.

V. FURTHER CONSIDERATIONS
So far we have obtained the X(3720) as a bound DD state with no width. In practice, this resonance decays into lighter pseudoscalar-pseudoscalar channels as discussed in Refs. [21,45]. Actually, in Ref. [45] it has found that the width of the X(3720) state decaying to these channels was Γ = 36 MeV and the most important decay channel was ηη. In the present work, we do not want to go through all the coupled channels of [45] but just wish to have an idea of the effect of considering the width of the X(3720) state. For this reason, we work now with four coupled channels, adding ηη to the former ones, D + D − , D 0D0 , D + S D − s , and introduce the transition potential ηη → D + D − , ηη → D 0D0 with a strength a (dimensionless), similar to that of the model of [45] but tuned to give Γ = 36 MeV just from the ηη decay channel (this is accomplished with a = 42).
We perform the same calculations as before, using the new t DD→DD amplitude and the results are shown in Figs. 13, 14. We can see that the features of the mass distribution are very similar to those obtained in the case of zero width for the X(3720) state. Only the t DD→DD matrix becomes wider in terms of the DD invariant mass and hence the enhancement of the mass distribution close to the DD threshold is not as strong as before, but clearly is different from a phase space distribution. As a consequence, the ratio of Eq. (11) is a bit softer than before, but the fall down in the invariant mass is still clear.
Finally we want to compare with the B + → K + D 0D0 data) and we show the results in Figs. 15,16,17. In this case we only calculate the case with a width for the X(3720). In Fig. 15 we show dΓ/dM inv as a function of the invariant mass. We observe in this case that there is practically no enhancement close to threshold and the distribution is closer to phase space. The reason is the term unity in Eq. (13), in contrast to Eq. (9) where only the t DD matrix appears. As a consequence of this, in Fig. 16, we do not see the fall down of R Γ that we see in Fig. 14. In Fig. 17 we compare the mass distribution with the data of [37]. As mentioned at the end of section IV, we have removed the contributions of ψ(3770) and D s1 (2700). Since these contributions come from the analysis of the data of [37], we have also taken the results for the total distribution from the same analysis, instead of the raw data. We normalize the results of Fig. 15 to the number of events in Fig. 17 and observe that, within errors, the agreement with the data is good. However, we should note that the errors are large and the bins of 40 MeV too broad. It would be most helpful to have these results improved with more statistics and better resolution. However, the message of the work is that the B 0 → D 0D0 K 0 decay is better suited to determine the bound state below threshold, because in this reaction we find and justify the presence of an enhancement of the D 0D0 mass distribution close to threshold, which is due to the DD The comparison is made here with the limited experimental information available. Further comparison of these results with coming LHCb measurements will be very valuable to make progress in our understanding of the meson-meson interaction and the nature of the scalar meson X(3720).

VI. CONCLUSIONS
In the present paper we have studied the B 0 decay to D 0D0 K 0 based on the chiral unitary model that generates the X(3720) resonance, and have made predictions for the D 0D0 invariant mass distribution. From the shape of the distribution, the existence of the resonance below threshold could be induced. Additionally, we have also predicted the rate of production of the X(3720) resonance to the D 0D0 mass distribution with no free parameters, under the assumption that the X(3720) resonance  is dynamically generated. So far, the related experiment B + → D 0D0 K + has already been done, and the D 0D0 invariant mass is measured, but with very small statistics close to threshold, including the s-wave and p-wave parts of the spectrum. The X(3720) state is a scalar meson and it decays into D 0D0 in s-wave. In this sense, testing the invariant mass predicted here should require to separate the s-wave from the p-wave part of the spectrum. In any case, the contribution of the ψ(3770) is separated and this allows us to make a comparison of our results with this distribution.
With present errors we find a good agreement with the data. However, we found that the B 0 decay to D 0D0 K 0 is better suited to study the X(3720) resonance, since there is no tree level D 0D0 production in this decay and this forces the D + D − → D 0D0 transition to intervene to make D 0D0 at the end. The implementation of the experiment in the future would be very helpful in the search of this elusive state, and as a further test of the nature of the X(3720) resonance.