Molecular picture for the $X_0(2866)$ as a $D^* \bar{K}^*$ $J^P=0^+$ state and related $1^+,2^+$ states

We recall the predictions made ten years ago about a bound state of $J^P=0^+$ in $I=0 $ of the $D^* \bar{K}^*$ system, which is manifestly exotic, and we associate it to the $X_0(2866)$ state reported in the recent LHCb experiment. Fine tuning the parameters to reproduce exactly the mass and width of the $X_0(2866)$ state, we report two more states stemming from the same interaction, one with $1^+$ and the other with $2^+$. For reasons of parity, the $1^+$ state cannot be observed in $D\bar{K}$ decay, and we suggest to observe it in the $D^*\bar{K}$ spectrum. On the other hand, the $2^+$ state can be observed in $D \bar{K}$ decay but the present experiment has too small statistics in the region of its mass to make any claim. We note that measurements of the $D^*\bar{K}$ spectrum and of the $D \bar{K}$ with more statistics should bring important information concerning the nature of the $X_0(2866)$ and related ones that could be observed.

Explicit calculations for tetraquak structures based on the quark interaction have also been performed. In [35], a compact csūd tetraquark is favored for the X 0 (2866) based on a study using effective masses of quarks in heavy mesons or baryon states, and also using universal quark masses and the concept of string junctions which ascribes a mass contribution S to each QCD string junction, where the meson has none, a baryon has one, and a tetraquark has two string junctions. A radially excited tetraquark and orbitally excited tetraquark are also proposed as candidates for the 0 + and 1 − states, and predictions for other states are also done in [36] based on former work done in [37]. On the other hand, an explicit tetraquark study in an extended relativized quark model done in [38] disfavors the compact tetraquark assignment to the X 0 (2866). Finally, it is also interesting to quote the possibility that the observed peaks could correspond to triangle singularities as pointed out in [39]. So far, the support for the D * K * bound state nature of the X 0 (2866) is getting more consensus [9,23,25,[27][28][29]32] and further work and data will help to clarify the panorama in the near future. In this context it is worth exposing our point of view on the issue.
The first thing to point out is that in 2010, in Ref. [40], we made neat predictions for the existence of a bound D * K * state with I = 0, J P = 0 + with mass, 2848 MeV decaying to DK with a width of about 59 MeV. This is in remarkable agreement with the experimental findings for the X 0 (2866), both in the mass and the width, without fitting any parameter to unexisting data at that time. One may wonder how this prediction could be made, which we explain below.
In the first place the use of extensions of the chiral unitary approach in coupled channels [15] in the D sector produced the D * s0 (2317) [41] and the D s1 (2460) [42] as molecular states of mostly DK and D * K respectively. Lattice QCD calculations support this picture [43,44]. The next step would be to investigate the D * K * states and this was done in [40], where, among other states, the D * s2 (2573) state was obtained being also well described, using the fine tuning allowed for the parameters of the theory, once they are fitted to the bulk data of other states. With the input used to obtain the D * s2 (2573), bound states of D * K * nature were also obtained. The width is obtained via box diagrams with DK in the intermediate states mediated by pion exchange. In fact, an exact evaluation of the four meson loops was done. The loop was regularized with a form factor that determines the width of the states. Fine tuning of this form factor was done to get the experimental D * s2 (2573) width. The use of this form factor provided about 59 MeV for the width of the D * K * 0 + bound state and the mass found was 2848 MeV.
The vector-vector interaction is much less studied than the corresponding pseudoscalar-pseudoscalar and pseudoscalar-vector ones. The reason is that for the latter ones there are available standard chiral Lagrangians in [45,46], which are absent for the vector-vector case. Yet, the vector-vector interaction could be well taken into account by means of the local hidden gauge approach [47][48][49][50]. It is a welcome feature that the same approach also leads to the chiral Lagrangians of [45,46] assuming vector meson dominance, as found in [51], so it is a natural extension of the chiral Lagrangians to the vector-vector case. The unitarization in coupled channels using the local hidden gauge approach as a source of the potential leads to the chiral unitary approach for vector-vector interaction and gives rise to several dynamically generated states, molecular states, like the f 2 (1270) and f 0 (1370) in [52] from the ρρ interaction and the f ′ 2 (1525), f 0 (1710), K * 2 (1430) among others from K * K * and related SU(3) channels in [53]. Limitations of the method used in [52,53] for very bound states were discussed in [54,55], but the alternatives proposed were not suited for these energies as discussed in [56,57]. Instead, an improved method was developed in [56] which actually gives very close results to those in [52,53]. The results obtained also explain radiative decays of these resonances [58] and other decays [59].
Confidence on one approach grows when apart from explaining known features of many states, new states are predicted that are later on found experimentally. In this sense it is worth noting that in the application of the method to study the ρ(ω)D * interaction in [60] three states were found corresponding to 0 + , 1 + , 2 + , the D 0 (2600), D * (2640), and D * 2 (2460). The D 0 (2600) was a prediction at that time and soon it was found in [61]. Since HQSS is invoked in constructing potentials for the D * K * system, it is worth mentioning that the local hidden gauge relies upon the exchange of vector mesons and a contact term. The dominant terms of the interaction stem from the exchange of light vectors, and this respects HQSS in an obvious way, because the heavy quarks are spectators in this exchange (see technical details in [62]). The exchange of heavy vectors and the contact terms are subleading in the HQSS counting and thus are not subject to the strict HQSS rules. Then it is interesting to note that the local hidden gauge interaction when the exchange of heavy vectors and the local terms are eliminated is the same for J = 0 + , 1 + , 2 + (see Table XI of [40]), and hence in this limit we would obtain three degenerate states, as is the case of [27]. When the subleading terms are kept, the degeneracy is broken and in [40] we found three states with masses 2848 MeV for 0 + , 2839 MeV for 1 + and 2733 for 2 + . With the advent of the new LHCb experiment [22], we can do minor changes in the parameters of the theory to perfectly fit the mass and width of the X 0 (2866) and then refine the predictions for the 1 + and 2 + states. These predictions, backed by the success of the method used for the vector-vector interaction in related problems should be a stimulus to look into the experiment with improved statistics to see new peaks for the states that the theory predicts. Success in this enterprise would provide a strong backing to the molecular picture of the X 0 (2866) state. As to the X 1 (2900), with 1 − , our approach based on the s-wave interaction of vector mesons, clearly cannot provide this state. This is the same conclusion reached in [23,25,[27][28][29].

II. FORMALISM
We follow the steps of [40] to describe the V − V interaction for the D * K * case (for the conjugate state,D * K * , the interaction would be identical). At the same time we discuss issues related to recent work on the subject. The V − V interaction at tree level is given in terms of two Lagrangians of the local hidden gauge approach extended to the charm sector The first Lagrangian in Eq. (1) is a contact term involving four vectors. The second one is a three vector vertex which gives rise to a vector exchange diagram. The mechanisms are depicted in Fig. 1. The mechanism of Fig. 1 (b) corresponds to light vector meson exchange. In this mechanism the c quarks are spectators and therefore, the interaction does not depend on the c quark. As a consequence of that, the rules of HQSS are automatically fulfilled. Actually, the s quark inK * is also a spectator in this case and hence, one is involving only u, d quarks in the process. Even if formally we can evaluate the term using the SU(4) structure of Eqs. (1) and (2), only the SU(2) subgroup of it is effectively used. In fact, the vertices D * D * ρ(ω) can be evaluated directly using the flavor wave functions of the ρ and ω without invoking any SU(4) structure [63]. The diagram of Fig. 1 (c) is suppressed because of the mass of the heavy vector exchanged and is subleading in the HQSS counting, and so is the contact term of Fig. 1 (a) as shown in [64].
The amplitudes corresponding to these three mechanisms are evaluated in terms of their polarizations and then they are projected over J = 0, 1, 2 using the projector operators [52] As a consequence, we obtain the tree level terms, V , shown in Table I.  In Table I we can see explicitly the contribution of the contact term, which is different for each value of J, and the contribution of the D * s exchange, which is the same for J = 0, 2, but has opposite sign for J = 1. We should note that in the absence of the subleading terms, contact and D * s exchange, the interaction is the same for all the three cases and thus we should expect a degenerate spectrum. This is indeed what happens in [27]. The subleading terms are smaller than the dominant term from ρ, ω exchange, but not negligible, and they lead to a breakup of the degeneracy of the three J states.
The full amplitude is obtained by solving the Bethe-Salpeter equation with this potential, with G the loop function of the intermediate D * andK * . The width of theK * is taken into account convolving the D * K * loop function with theK * spectral function [40], and G is regularized in dimensional regularization by means of a subtraction constant as where s is the squared c.m. energy, p is the on-shell three-momentum of the two mesons, and M 1 , M 2 the masses of the two mesons. Eq. (4) with the single channel D * K * , and using the value of α necessary to reproduce the D s2 (2573), gives rise to a bound D * K * state with 0 + around 2848 MeV.
In the absence of the K * width the state has zero width. The convolution with theK * spectral function gives rise to a tiny width, which is not the one observed. The experiment sees the state in theDK (DK in our case) decay. This decay was also studied in [40] by means of the box diagram that leads to this decay, as shown in Fig. 2.
The amplitude given by the diagram of Fig. 2 is added to the potential discussed above and iterated with the Bethe Salpeter equation to obtain bound states which now decay into DK. It is interesting to see that, since D andK have spin zero, we need L = 0, 1, 2 for these intermediate states to match the J = 0, 1, 2 of the states obtained. Since all V − V states obtained have positive parity in the s-wave that we study, only J = 0, 2 decay into DK, and the state with J = 1 cannot. This means that the J = 1 state that we predict cannot be seen in the experiment of [22], but J = 0, 2 could be. The J = 1 state can be seen in D * K decay, which requires an anomalous vertex and should be suppressed. Indeed, this is what happens for the D * (2640) state that we obtained with J = 1 in [60], with a small width compared to the D 0 (2600) and D * 2 (2460), which have a large width. In the PDG [65], the widths of D 0 (2600), The evaluation of the diagram in Fig. 2 requires a regularizing form factor. In [40] we took a form factor from [66] with q 0 = (s + m 2 D − m 2 K )/2 √ s and Λ of the order of 1 − 1.2 GeV. The amplitude of Fig. 2 gave negligible contribution to the real part of the energy of the bound states, but provides the width for DK decay. Choosing α = −1.6 (with µ = 1500 MeV) and Λ = 1200 MeV, we obtained in [40] the results of Tables II and III. The masses and widths are evaluated normally from the poles in the second Riemann sheet. When the convolution in the G-function is done it is common to obtain them from the |T | 2 plot instead.
In view of the new experimental data of [22] we do a fine tuning of the α and Λ parameters to obtain the experimental numbers. In Figs. 3, 4 and 5, we show |T | 2 for the D * K * states with L = 0 + , 1 + , 2 + . By inspecting these figures we obtain M (0 + ) = 2866 MeV, Γ = 57 MeV with α = −1.474, Λ = 1300. The new results for the three D * K * states are shown in Table IV. As we can see, the parameters are very close to those used in [40] since only a small difference in the mass and width has to be accommodated. As a consequence, the results for the 1 + , 2 + states, which are predictions of our theoretical framework, are very similar to those obtained in [40] (see Table III).
As discussed above, the J P = 1 + state cannot be seen in the DK spectrum (or D − K + in the experiment [22]) but the 2 + state could. If one looks into the D − K + spectrum of [22], there is not enough statistics in that region to make any claim so far. By simple analogy to the D s2 (2572) state from D * K * , the 2 + state from D * K * should also Λ=1200 MeV exist, and this simple observation, in addition to our predictions, should give incentives to look for this additional state with more statistics. The search for the 1 + state in the D * K spectrum would be a complement to this search to find the whole family of exotic D * K * molecular states.

III. CONCLUSIONS
The observation in [22] of two mesons in the D − K + spectrum, with csūd content (for D + K − ), provides the first example of an open heavy flavor exotic state. The discovery has immediately triggered a response from the theoretical community, with different suggestions for the interpretation of these states. One of the most promising interpretations is that the X 0 (2866) is a molecular state ofD * K * nature, while the X 1 (2900) is not easy to accommodate in that picture. In this paper we have recalled that a prediction of a bound D * K * state with 0 + was already been done in [40], being in remarkable agreement with the mass and width of the X 0 (2866). We have taken the opportunity to discuss that idea to the light of developments done after the theoretical paper [40] in connection with the data of [22]   and recent theoretical papers done after the experimental discovery. One of the issues is heavy quark spin symmetry, that has become fashionable in the study of heavy quark systems. We have shown that the approach followed in [40] respects HQSS. Indeed, the formalism followed to deal with the vector-vector interaction is the local hidden gauge approach, where one has a contact term and other terms stemming from the exchange of vector mesons. The dominant terms come from the exchange of light vector mesons, and since in this case the heavy quarks are spectators in the process, the amplitudes do not depend upon them and HQSS is automatically fulfilled. Yet, even if subdominant in the heavy quark counting, the contact term and the exchange of heavy vectors are not negligible and have as a consequence the splitting of the J = 0, 1, 2 states generated in s-wave, which are degenerate in the strict heavy quark limit. We have taken advantage of the measurements in [22] to fine tune the two parameters of the model such as to adjust exactly the mass and the width of the 0 + state to the X 0 (2866), and then made predictions for the 1 + and 2 + states. The 1 + state cannot be seen in the DK spectrum because of parity reasons and we expect a small width. We suggest to look at it in the D * K spectrum. As to the 2 + state, it can be seen in the DK spectrum, but it falls in a region of energies where the experiment has small statistics. Our results and the existence of the closely related D * s2 (2573) state, which stems from the D * K * interaction, should provide an incentive to look for these predicted states. Their observation would give a strong support to the molecular picture for the X 0 (2866) state and related ones that could come from the experiment.