Looking for the exotic X0 (2866) and its JP=1+ partner in the B ¯ 0 →d (∗)+K-K (∗)0 reactions

Looking for the exotic X0ð2866Þand its JP=1+partner in the ¯ B0→DðÞ+K−KðÞ0reactions L. R. Dai ,1,2,* R. Molina,2,†and E. Oset2,‡ 1School of Science, Huzhou University, Huzhou 313000, Zhejiang, China 2Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.22085, 46071 Valencia, Spain (Received 16 March 2022; accepted 2 May 2022; published 18 May 2022) We propose two reactions, ¯ B0→K0DþK−and ¯ B0→K0DþK−, which have been already measured at Belle, to look into the JP¼0þ,X0ð2866Þstate and a 1þpartner of molecular D¯ Knature by looking at the DþK−and DþK−invariant mass distributions, respectively. Very clear peaks over the background are predicted and the branching ratios for the production of these states are evaluated to facilitate the task of determining the needed statistics for their observation. We conclude that with the upgrade of Belle II clear peaks should be seen in both reactions for the two resonances discussed. DOI: 10.1103/PhysRevD.105.096022 I. INTRODUCTION In the paper [1] the Belle collaboration reported on the ¯ B0→DðÞþK−KðÞ0decays, giving a list of eight reactions for which the branching ratios were provided. In some of the reactions, (i) ¯ B0→DþK−K0, (ii) ¯ B0→DþK−K0, (iii) ¯ B0→DþK−K0,(iv) ¯ B0→DþK−K0, one finds pairs [DþK−in (i) and (ii), DþK−in (iii) and (iv)] that contain open charm and strangeness with cand squarks. Should these pairs result from the decay of a physical state, it would be genuinely exotic since it cannot come from a q¯ q conventional meson. The chosen pairs could correspond to states with isospin I¼0, while the other four cases of [1] would correspond to DðÞþ ¯ Kstates with isospin I¼1. The limited statistics prevented the authors from getting DðÞþK−mass distributions, while the accumulation of K−K0events from four reactions allowed them to get a K−K0mass distribution that evidenced the B→DðÞþa1 ð1260Þdecay with a1ð1260Þ→K−K0. Yet, the abundant literature on tetraquark states from the very beginning of the quark model [2–12] (see Refs. [13–18] for reviews on more recent works) would have made it advisable to look at the DðÞþ ¯ Kmass distributions in search of possible peaks corresponding to exotic states. Recently, the answer to this question was provided by the LHCb collaboration [19,20] with the finding of the X0ð2866Þand X1ð2900Þstates in the Bþ→DþD−Kþ decay by looking at the D−Kþinvariant mass distribution. In the charge conjugate reaction B−→DþD−K−one would find the peaks in the DþK−invariant mass distribution. Interestingly, the existence of a I¼0;JP¼0þ molecular state of D¯ Knature, decaying to D¯ K, had been predicted in [21], with a mass of 2848 MeV and a width between 23–59 MeV, which is very close to the data of the X0ð2866Þwith mass 2866 7MeV and width 57.212.9MeV. An update of that work in regard to the LHCb results is presented in [22]. The findings of Refs. [19,20] prompted many works offering an explanation for the X0ð2866Þas a tetraquark state [23–26] or a molecular D¯ Kstate [27–32]. The sum rules studies [33–37] have also contributed its share to the discussion, some of them proposing a molecular structure [35–37]. Other studies suggest a structure coming from a triangle singularity [38] or cusps and analytical properties of triangle diagrams [39,40]. A triangle mechanism is also suggested in [41], and in [42] a detailed quark model calculation is shown to disfavor the compact tetraquark picture. In Ref. [21], apart from the JP¼0þstate, two other states with JP¼1þ;2þalso in I¼0were found. In the update of [22], where the free parameters of the model were adjusted to experiment [19] for the X0ð2866Þ, the masses, widths, and couplings to the D¯ Kchannel were evaluated, which are shown in Table I. For reasons of parity and angular momentum conservation, the 0þstate only decays to D¯ Kwhile the 1þstate decays to D¯ K. *[email protected]n †[email protected].es ‡[email protected].es Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 105, 096022 (2022) 2470-0010=2022=105(9)=096022(7) 096022-1 Published by the American Physical Society The purpose of the present work is to investigate whether by looking at the ¯ B0→DðÞþK−KðÞ0reactions one can observe clear peaks in the DðÞþK−spectrum. The reaction is similar to the B−→DþD−K−one studied in [19,20]. The KðÞ0in the Belle reactions would play the role of the D−in the LHCb one. The study is stimulated by the success found in [43], fairly reproducing the D¯ Kpeak versus the background of [19] in the study of the B−→DþD−K− reaction. This success was used in [43] to suggest the ¯ B0→ DþD0K−decay in order to investigate the 1þstate of Table Iby looking at the DþK−mass distribution. It was found that the 1þstate generated a peak in the distribution with a strength about seven times bigger than the background at the peak of the 1þcontribution. Based on these findings, we propose here to study the ¯ B0→DþK−K0 and ¯ B0→DþK−K0reactions. The reason to choose these two reactions from the eight reactions of Belle [1] is that, both in the signal for the exotic states as in the background, the amplitudes can proceed in the swave, which is assumed to be dominant as usual, and one can correlate the background and the signal for the production of the 0þand 1þ states. II. FORMALISM AND RESULTS A. Production of the 1+state in ¯ B0→D+K−K0→ D+K−K0 The D¯ K1þstate can only decay in D¯ K. Thus, we choose the ¯ B0→DþK−K0reaction and look at the DþK−mass distribution. The signal, however, will come from the ¯ B0→DþK−K0reaction, after which the final state interaction of DþK−will give the 1þstate (R1) and posterior decay into DþK−. The primary step proceeds via external emission as depicted in Fig. 1. The ¯ ud component after the W−vertex is hadronized with an s¯ scomponent to give rise to K−K0and the c¯ dgives the Dþ. One has three vectors and one can write the s-wave component of the transition matrix matching the angular momentum of the ¯ B0as ˜ t¼Cϵð1Þ·ðϵð2Þ×ϵð2ÞÞ¼Cϵijkϵð1Þ iϵð2Þ jϵð3Þ k;ð1Þ where the indices 1,2,3 apply to the K0,D0, and K−, respectively. We observe how the spins of the particles 2 and 3 combine to J¼1. The next step is to consider the DþK−interaction. With our phase convention ðDþ;−D0Þ,ð¯ D0;D −Þ,ðKþ;K0Þ,ð¯ K0;−K−Þ, the I¼0D¯ Kstate is written as jD¯ K;I¼0i¼− 1 ffiffiffi 2 pðDþK−þD0¯ K0Þ:ð2Þ The final state interaction of DþK−to produce the R1 state is taken into account as shown diagrammatically in Fig. 2. We also need the vertex R1D¯ K, which incorporates the spin projection generator Di ˜ gi¼giVðiÞ;ð3Þ with VðiÞgiven by [44] Vð0Þ¼1 3ϵð2Þ lϵð3Þ lδij; Vð1Þ¼1 2ðϵð2Þ iϵð3Þ j−ϵð2Þ jϵð3Þ iÞ; Vð2Þ¼1 2ðϵð2Þ iϵð3Þ jþϵð2Þ jϵð3Þ iÞ− 1 3ϵð2Þ lϵð3Þ lδij:ð4Þ Considering the ˜ tmatrix of Eqs. (1) and (3) for Vð1Þ, Vð1Þ¼1 2ðϵð2Þ i0ϵð3Þ j0−ϵð2Þ j0ϵð3Þ i0Þ, and that in the loop function we sum over the spin polarization Ppol ϵðrÞ iϵðrÞ j¼δij (r¼2, 3), we obtain t¼Cϵð1Þϵii0j0GD¯ KðMinvðD¯ KÞÞ−1 ffiffiffi 2 pgR;D¯ K; FIG. 1. Diagrammatic decay of ¯ B0→DþK−K0at the quark level. TABLE I. Properties of the D¯ Kstates from Ref. [22] accounting for D¯ Kand D¯ Kdecays. I½JPM[MeV] Γ[MeV] Coupled channels gR;D¯ K[MeV] State 0½2þ2775 38 D¯ K16536 ? 0½1þ2861 20 D¯ K12056 ? 0½0þ2866 57 D¯ K11276 X0ð2866Þ L. R. DAI, R. MOLINA, and E. OSET PHYS. REV. D 105, 096022 (2022) 096022-2 where gR1;D¯ Kis the coupling of the resonance R1to the (I¼0)D¯ Kstate and GD¯ Kis the loop function of the D;¯ Kintegrating the product of the propagators of the two particles. We use dimensional regularization for this loop with α¼−1.474 for a chosen μ¼1500 MeV as was needed in [22] to obtain the right mass of the X0ð2866Þstate. The sum over the final vector polarization in Pjtj2is implemented by summing Pjtj2over the indices i0;j 0for the implicit VV components of R1and over the index ito sum over the K0polarization. We find X pol jtj2¼3C2jgR1;D¯ Kj2jGD¯ KðMinvðD¯ KÞÞj2: The next step is to consider the decay of R1into DþK− as depicted in Fig. 2(b). This leads to a t0matrix containing the coupling of R1to DþK−. This is accomplished by an effective coupling gR1;D¯ Kto the D¯ K;I ¼0state, such that the coupling to DþK−is −1 ffiffi 2 pgR1;D¯ K. To get the gR1;D¯ K coupling we use the R1decay width via ΓR1¼1 8π 1 M2 R1jgR1;D¯ Kj2q¯ K; q¯ K¼λ1=2ðM2 R1;m 2 D;m 2 ¯ KÞ 2MR1 ;ð5Þ taking the value of ΓR1from Table I. Hence X pol jt0j2¼6 4C2jgR1;D¯ Kj2jGD¯ KðMinvÞj2 ×jgR1;D¯ Kj2    1 M2 invðR1Þ−M2 R1þiMR1ΓR1     2 :ð6Þ The invariant mass distribution is then given by dΓ dMinvðDþK−Þ¼1 ð2πÞ3 1 4M2 ¯ B0 p¯ K0 ˜ pK−Xjt0j2;ð7Þ where p¯ K0¼λ1=2ðM2 ¯ B0;m 2 ¯ K0;M2 invðDþK−Þ 2M¯ B0 ; ˜ pK−¼λ1=2ðM2 invðDþK−Þ;m 2 D;m 2 ¯ KÞ 2MinvðDþK−Þ:ð8Þ We would like to compare this mass distribution with the one of the background for the same reaction, ¯ B0→K0DþK−. The process proceeds with the same topology as in Fig. 1, changing K−by K−. As shown in [45] the difference between pseudoscalar and vector production can be taken into account by means of Racah coefficients of the same order of magnitude, so approximately we can put for the ¯ B0→K0DþK−background t¼CϵðKÞϵðDÞ with the same constant Cas in Eq. (1), such that now the background distribution is given by dΓbac dMinvðDþK−Þ¼1 ð2πÞ3 1 4M2 ¯ B0 pK0 ˜ p¯ K3C2:ð9Þ The assumption of taking the same constant Cis supported by the results of [43], reproducing fairly well the signal versus the background of the LHCb experiment [19]. The results can be seen in Fig. 3. We can see a peak clearly sticking out of the background, as was also found in [43] with a different reaction. It is clear that even if there were uncertainties of a factor of two or three, the signal should be clearly seen. In order to test the feasibility of the experiment, we integrate the mass distributions over the whole invariant mass range, for both, the resonance peak and the background. We find Γpeak Γback ¼0.125:ð10Þ Next, we see from Fig. 2(d) of Ref. [1] that for ¯ B0→ K0DþK−there are about 45 events reported in [1] at the ¯ B0peak. This means we can expect about six events in the (a) (b) FIG. 2. (a) Rescattering of DþK−to give the resonance R1; (b) further decay of R1into DþK−. LOOKING FOR THE EXOTIC X0ð2866ÞAND ITS …PHYS. REV. D 105, 096022 (2022) 096022-3 peak with the present setup but not enough to see a clean structure. Yet, with the Belle II prospects where there will be about 30 times more events than so far collected in BABAR and Belle [46], one could have 170 events, which is much more than sufficient to see clearly the peak, given the clear signal of ¯ B0mesons seen with 45 events in [1].Even accepting a rate three times smaller than estimated, there would be enough statistics to see clearly the peak. B. Production of the 0+state in ¯ B0→K0D+K− → K0D+K− Proceeding like in the former subsection, we would now compare the signal for the 0þstate from the ¯ B0→ K0DþK−with DþK−interaction leading to the 0þstate (R0) and its decay into DþK−, and the background from the ¯ B0→K0DþK−reaction. We can proceed as before and for the ¯ B0→K0DþK−we assume a transition matrix t¼C0ϵðDÞϵðKÞð11Þ and similarly for the ¯ B0→K0DþK− t¼C0ð12Þ with the same C0, for both reactions, as we have done before. We shall come back to this assumption. Following the same steps as before, we obtain dΓ0 dMinvðDþK−Þ¼1 ð2πÞ3 1 4M2 ¯ B0 pK0 ˜ pK−Xjt0j2;ð13Þ where Xjt0j2¼3 4C02jGD¯ KðMinvðDþK−ÞÞj2jgR0;D¯ Kj2 ×    1 M2 invðDþK−Þ−M2 R0þiMR0ΓR0     2 jgR0;D ¯ Kj2; ð14Þ with pK0¼λ1=2ðM2 ¯ B0;m 2 K0;M 2 invðDþK−Þ 2M¯ B0 ; ˜ pK−¼λ1=2ðM2 invðDþK−Þ;m 2 Dþ;m 2 K−Þ 2MinvðDþK−Þ;ð15Þ with gR0;D¯ Kgiven in Table I, and the effective jgR0;D ¯ Kj2 coupling obtained from ΓR0¼1 8π 1 M2 R0jgR0;D ¯ Kj2q¯ K; q¯ K¼λ1=2ðM2 R0;m 2 Dþ;m 2 ¯ KÞ 2MR0 :ð16Þ For the background we find dΓ0 bac dMinvðDþK−Þ¼1 ð2πÞ3 1 4M2 ¯ B0 pK0 ˜ pK−C02:ð17Þ The results for these two distributions are shown in Fig. 4. We also see a signal that sticks out of the background clearly, as was shown in [43] for the B−→ DþD−K−reaction in the production of the X0ð2866Þ. As we have done before, we integrate over the range of the invariant mass the signal and background in Fig. 4and we find FIG. 3. dΓ dMinv for R1production and dΓbac dMinv for background in the ¯ B0→K0DþK−reaction in global arbitrary units. Minv is the DþK−invariant mass. FIG. 4. dΓ0 dMinv for R0production and dΓ0 bac dMinv for background in the ¯ B0→K0DþK−reaction in global arbitrary units. Minv is the DþK−invariant mass. L. R. DAI, R. MOLINA, and E. OSET PHYS. REV. D 105, 096022 (2022) 096022-4 Γ0 peak Γ0 bac ¼0.124:ð18Þ In Ref. [1] [see Fig. 2(f) of [1] ] one finds about 30 events for ¯ B0→DþK−K0 saround the ¯ B0peak. This means that one could expect around four events in the peak of the resonance with present statistics, which is clearly insufficient to determine the peak. With 30 times more statistics from the Belle II upgrade there would be about 110 events, more than sufficient to see clearly the peak. As this point we would like to make some discussion. The ¯ B0→K0DþK−can proceed with the topology of Fig. 1changing K0by K0where the K0and K−are produced by hadronization of the ¯ ud component. Yet, the production of K0K−from the same vertex is suppressed, as discussed in [47]. Indeed, the vertex WPP is given by Wμh½P; ∂μPi in chiral theory [48,49], the swave going as the difference in the energies of the two pseudoscalars for the WPP vertex, which vanishes in the Wrest frame if the particles have the same mass. The argument does not hold if one produces a vector and a pseudoscalar, as it was the case for the signal of the 0þstate. The argument given above is corroborated by the branching ratio of the ¯ B0→DþK−K0, which is about one order of magnitude smaller than the one of ¯ B0→DþK−K0(see Table II of Ref. [1]). Certainly we could now have contributions from higher partial waves, but the argumentation given above, with the support of the small ¯ B0→DþK−K0branching ratio, would tell us that we can expect in practice a peak showing even stronger with respect to the background than what is shown in Fig. 4. In order to make this argument more quantitative, we take from Table II of Ref. [1] the following branching ratios: Bð¯ B0→K−DþK0Þ¼ð1.29 0.22 0.25Þ10−3;ð19Þ Bð¯ B0→K0DþK−Þ¼ð0.16 0.08 0.03Þ10−3:ð20Þ By analogy to Eq. (11) we would assume now for the ¯ B0→ K0DþK−amplitude, t¼ ˜ C0ϵðDÞ·ϵðKÞ;ð21Þ and for ¯ B0→K0DþK−the amplitude of Eq. (12) with a different coupling, t¼ ˜ C00:ð22Þ The mass distribution for the case of Eq. (21) is given as dΓ dMinvðDþK0Þ¼1 ð2πÞ3 1 4M2 B0 pK− ˜ pK03 ˜ C02;ð23Þ with pK−being the K−momentum in the ¯ B0rest frame and ˜ pK0the momentum of the K0in the DþK0rest frame. The mass distribution for the case of Eq. (22) is given by dΓ dMinvðDþK0Þ¼1 ð2πÞ3 1 4M2 B0 pK− ˜ pK0 ˜ C002;ð24Þ with the same meaning for the pK−and ˜ pK0as before. By integrating Eqs. (23) and (24) and writing B¼Γ=Γtot, we find using Eqs. (19) and (20), ˜ C02 Γtot ¼5.86 ×10−3MeV−1; ˜ C002 Γtot ¼1.42 ×10−3MeV−1;ð25Þ which leads to ˜ C0= ˜ C00 ≃2. By assuming ˜ C0¼ ˜ C00 as we have done in Eqs. (11) and (12) we would be underestimating the signal for the resonance in about a factor of 4. This makes more quantitative the discussion made above, indicating that we should expect a fairly larger signal over the background than shown in Fig. 4. In the LHCb case [19] a different reaction was used, the Bþ→DþD−Kþ, or analogously B−→D−DþK−.Evenif the reaction seems the same except for small changes as ¯ B0→K0DþK−, replacing the D−with K0, the reactions are topologically different since the LHCb one, as well as the associated B−→D−DþK−reaction, proceeds via internal emission, and the argument discussed above is peculiar to the Wμh½P; ∂μPivertex of external emission. In the LHCb reaction the formalism used here for the signal and background gave rise to a distribution in fair agreement with experiment. There is no analog reaction to the B−→ D−DþK−that proceeds with the internal emission of the type B→DK ¯ K. The reaction that we have chosen to observe the 0þ,X0ð2866Þstate, ¯ B0→K0DþK−, stands as a good one, where the signal over background is expected to be even bigger than shown in Fig. 4. III. CONCLUSIONS We have chosen two reactions, already performed by the Belle collaboration [1], to observe the 0þ;1þstates obtained from the D¯ Kinteraction, where the 0þstate is associated to the X0ð2866Þstate. From the eight reactions of the type ¯ B→DðÞK−K0of Ref. [1] we have selected two, the ¯ B0→K0DþK−and ¯ B0→K0DþK−, in order to observe the 1þand 0þstates, respectively. In the first case the signal of the 1þstate stems from the original ¯ B0→ K0DþK−reaction, followed by DþK−interaction to give the R1resonance, which decays posteriorly to DþK−. In the second case, the signal for the 0þstate comes from the ¯ B0→K0DþK−with the posterior DþK−interaction producing the 0þstate, which decays lately into DþK−.We LOOKING FOR THE EXOTIC X0ð2866ÞAND ITS …PHYS. REV. D 105, 096022 (2022) 096022-5 could relate the mass distributions of the signal and the background, finding very clear peaks for the 1þand 0þ states. However, we have argued that in the case of the 0þ state we expect the signal to be even more pronounced with respect to the background than what is calculated here because of the suppressed ¯ B0→DþK−K0decay versus ¯ B0→DþK−K0decay at the tree level. These reactions, already measured at Belle [1], would need somewhat more statistics to show the DðÞþK−peaks clearly. Based upon the number of events presently observed at Belle [1], we have estimated a few events for the peak of the resonances, but with a factor of 30 increase in the number of events expected in Belle II, the number of events in the peak would be fairly larger than 100, which is much more than sufficient to see the strength and shape of the peaks, corroborating the existence of the X0ð2866Þand observing its 1þpartner predicted around 2861 MeV with around 20 MeV width. ACKNOWLEDGMENTS This work is partly supported by the National Natural Science Foundation of China under Grants No. 11975009, No. 12175066, and No. 12147219. R. M. acknowledges support from the program “Contratación de investigadores de Excelencia de la Generalitat valenciana”(GVA) with Ref. No. CIDEGENT/2019/015 and from the Spanish national Grants No. PID2019–106080 GB-C21 and No. PID2020–112777 GB-I00. This work is also partly supported by the Spanish Ministerio de Economia y Competitividad (MINECO) and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P B and No. PID2020–112777 GB-I00, and by Generalitat Valenciana under Contract No. PROMETEO/2020/023. This project has received funding from the European Union Horizon 2020 research and innovation program under the program H2020-INFRAIA-2018-1, Grant Agreement No. 824093 of the STRONG-2020 project. [1] A. Drutskoy et al. (Belle Collaboration), Phys. Lett. B 542, 171 (2002). [2] M. Gell-Mann, Phys. Lett. 8, 214 (1964). 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