Masses and widths of the exotic molecular B (s) (∗) B (s) (∗) states

Masses and widths of the exotic molecular BðÞ ðsÞBðÞ ðsÞstates L. R. Dai ,1,2,* E. Oset,2,†A. Feijoo ,2,‡R. Molina,2,§ L. Roca,3,∥A. Martínez Torres ,4,¶ and K. P. Khemchandani5,** 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 3Departamento de Física, Universidad de Murcia, E-30100 Murcia, Spain 4Universidade de Sao Paulo, Instituto de Fisica, C.P. 05389-970 Sao Paulo, Brazil 5Universidade Federal de Sao Paulo, C.P. 01302-907 Sao Paulo, Brazil (Received 27 January 2022; accepted 28 March 2022; published 21 April 2022) We study the interaction of the doubly bottom systems BB,BB,BsB,B sB,BB,BBs,BB s,BsBs, BsB s,B sB sby means of vector meson exchange with Lagrangians from an extension of the local hidden gauge approach. The full s-wave scattering matrix is obtained implementing unitarity in coupled channels by means of the Bethe-Salpeter equation. We find poles below the channel thresholds for the attractively interacting channels BBin I¼0,B sB−BBsin I¼1 2,BBin I¼0, and B sBin I¼1 2, all of them with JP¼1þ. For these cases the widths are evaluated identifying the dominant source of imaginary part. We find binding energies of the order of 10–20 MeV, and the widths vary much from one system to the other: of the order of 10–100 eV for the BBsystem and B sB−BBs, about 6 MeV for the BBsystem and of the order of 0.5 MeV for the B sBsystem. DOI: 10.1103/PhysRevD.105.074017 I. INTRODUCTION The discovery of the Tcc state by the LHCb Collaboration [1,2] is a turning point for our understanding of meson spectroscopy. While many studies had been done on doubly heavy meson states (see recent review in [3] and references in [4]) the small binding of around 360 keVand small width of about 48 keV were not anticipated, although a small binding energy had been predicted in [5,6]. The discovery has triggered many theoretical works, tuning parameters of the theory to obtain the right mass and in some cases the width also [4,7–21]. The proximity of the Tcc state to the DþD0threshold and the results of the works mentioned above, leave little doubt that one has a molecular state of components DþD0and D0Dþand very close to I¼0 [4,20]. The experimental analysis in [2] shows indeed no signal in the DþD0πþmass distribution which corresponds to an I¼1DDstate. The smallness of the width finds a natural interpretation within the molecular picture [4,7,8,19,20] and is tied to the D→πDdecay width. With this background it is obviously tempting to make accurate predictions for states of BðÞ ðsÞBðÞ ðsÞnature, which might be experimentally observed in the near future. Yet, it is interesting to look into predictions of such states made before the Tcc discovery. The history of possible BðÞBðÞ bound systems is long. In Ref. [22] its possible existence driven by pion exchange was already investigated. Pion exchange supplemented by vector exchange was also considered in [23], and bound states were found. A similar study was conducted in [24] where, using a boson exchange model, bound states were found for the cases BðÞBðÞ with IðJPÞ¼0ð1þÞ;1ð1þÞ, ðBðÞBðÞÞs½JP¼1þ;2þand ðBðÞBðÞÞss½JP¼1þ;2þ. One boson exchange together with arguments of heavy quark symmetry are used in [25,26] to obtain bound states for some of these systems. In the same line, in [27,28] isoscalar bound states of BBnature are found while an isovector appears for BBin [28]. A different perspective is taken in [29–31] using the constituent quark model where also the potential is compared with lattice QCD calculations [30,31]. Again, a bound state is found for BB in the I¼0sector. Other lattice QCD calculations also provide BB potentials that could lead to binding for some configurations [32–35]. The Born-Oppenheimer approximation in the MIT bag model [36], or with lattice QCD *[email protected]n †[email protected].es ‡[email protected].es §[email protected].es ∥[email protected] ¶[email protected] **kanchan.khemchanda[email protected] 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, 074017 (2022) 2470-0010=2022=105(7)=074017(11) 074017-1 Published by the American Physical Society results [37], is used to get the BB interaction. Possible formation of BBs0and BB s1molecules is also investigated by means of kaon exchange [38]. Contact terms and pion exchange are considered in [39], and bound states are obtained in the BBand BBin I¼0;JP¼1þ, with binding energies ranging from 12 to 24 MeV, with large uncertainties. Similar results are obtained using quark model interactions in [40]. The boson exchange model is again used in [41] with the result that no bound state is found for BB,aIðJPÞ¼0ð1þÞbound state is found for BBand bound states in 0ð1þÞ;0ð2þÞand 1ð2þÞare obtained for the BBsystem. An extension of the model to incorporate strange quarks is also done in [42]. Using again a quark model, a compact very bound tetraquark state and a shallow BBmolecular state are also reported in [43]. Further details and discussion of compact tetraquarks predictions can be found in the review of [3]. While there seems to be a common ground in all these models that some exotic double bottom meson states should exist, the predictions are quite different. The recent experimental finding of the Tcc state, with small binding and width, provides an extremely useful information to constrain the freedom in the models and come with more accurate predictions before these states are hopefully found in the near future. On the other hand, none of these works evaluate the width of these states. The aim of the present work is to use the information obtained from the Tcc state and, using tools proved accurate in former studies, make predictions for possible BðÞ ðsÞBðÞ ðsÞstates, evaluating also the decays widths. For this purpose we shall use the extension of the local hidden gauge approach [44–47] to the bottom sector. The interaction is obtained from the exchange of vector mesons, and only the exchange of the light vectors will be considered, since other terms are negligible. The b quarks are then spectators in the interaction, and the rules of heavy quark symmetry are automatically fulfilled. The approach has been often used, but concretely concerning exotic states with two open quarks, the approach was used in [48] to study an exotic D¯ Kbound state that could be identified with the recently discovered Xð3866Þmeson [49] (see update in [50]), and also the DDsystem, where the DDstate with I¼0;JP¼1þand the D sDwith I¼1 2, JP¼1þwere found slightly bound (see update in [21]). The same approach has been used in the description of the Tcc state in [4], where the width was predicted to be small, much smaller than the experimental one claimed in [4] before the analysis of [23], correcting for the experimental resolution, gave a width of the order of 40 keV. The theoretical approach has only 1 degree of freedom, the cutoff used to regulate the meson meson loops. We shall follow the same approach here and, considering the findings of [51,52] which advise the use of the same cutoff in the different heavy sectors to respect heavy quark symmetry; we shall do this to obtain the masses of the possible BðÞ ðsÞBðÞ ðsÞfrom the Tcc states [4]. Furthermore, we shall also evaluate the widths of the states, which should be helpful to identify the nature of these states when they are hopefully discovered in the near future. II. FORMALISM The basic dynamics in the extended local hidden gauge approach is the exchange of vector mesons, as shown in Fig. 1 and a contact term in the case of VV →VV (Vis vertex). There are two basic vertices, the vector-pseudoscalarpseudoscalar ðVPPÞvertex and the vector-vector-vector ðVVVÞvertex, given by the Lagrangians, LVPP ¼−igh½P; ∂μPVμi;ð1Þ LVVV ¼ighðVμ∂νVμ−∂νVμVμÞVνi;ð2Þ with g¼MV 2fðMV¼800 MeV;f¼93 MeVÞwhere P,V are the q¯ qmatrices written in terms of the pseudoscalar or vector meson fields. We consider u,d,s,bquarks and no charm here, (BD and related states are studied in [53]). Then, the pseudoscalar and vector matrices are P¼ 0 B B B B B B B B @ η ffiffi3 pþη0 ffiffi6 pþπ0 ffiffi2 pπþKþBþ π−η ffiffi3 pþη0 ffiffi6 p−π0 ffiffi2 pK0B0 K−¯ K0−η ffiffi3 pþffiffi2 3 qη0B0 s B−¯ B0¯ B0 sηb 1 C C C C C C C C A ; ð3Þ where we have taken the standard η;η0mixing of [54] and V¼ 0 B B B B B B @ ω ffiffi2 pþρ0 ffiffi2 pρþKþ Bþ ρ−ω ffiffi2 p−ρ0 ffiffi2 pK0B0 K−¯ K0ϕB0 s B−¯ B0¯ B0 sϒ 1 C C C C C C A :ð4Þ (a) (b) FIG. 1. Schematic vector exchange between BðÞ mesons. H corresponds to mesons that can be pseudoscalar or vector. L. R. DAI et al. PHYS. REV. D 105, 074017 (2022) 074017-2 Since we work close to the threshold of the B; Bstates, we neglect the three momentum of the external vectors compared to their mass, which allows us to take ϵ0¼0in the polarization vector ϵμof the external vector states, by virtue of the Lorenzconditionof the free massivevector meson field, kμϵμ¼0. Then, in Eq. (2) Vνcannot correspond to an external vector in Fig. 1because ∂νwill be ∂iand produces a three momentum which is taken zero. Then Vνin Eq. (2) corresponds to the Vexchanged vector in Fig. 1and VμVμ gives rise to ϵμϵμ¼−ϵϵ0of the external vectors in the vertex. Equations (1) and (2) are formally identical except for the extra factor ϵϵ0inthe vector-vector interaction. The evaluation of the amplitude stemming from Fig. 1is straightforward, but some caution must be taken. We show below how it proceeds. A. BB system We only consider the interaction in s-wave. The BB is a system of identical particles, with the isospin doublet ðBþ;B 0Þ. Hence, jBB; I ¼0i¼1 2ðBþðpÞB0ð−pÞ−B0ðpÞBþð−pÞÞ; where the extra factor 1 ffiffi2 pin the normalization is taken to work in the unitary normalization, convenient for identical particles [55]. Similarly, with the same normalization, jBB; I ¼1;I3¼1i¼ 1 ffiffiffi 2 pðBþðpÞBþð−pÞÞ: We can see that the I¼0state is antisymmetric under the exchange of the two mesons and must be discarded. Only I¼1exists, and to get the interaction we must evaluate the diagrams of Fig. 2. The interaction stemming from the diagrams of Fig. 2 comes from the exchange of ρ0;ωand gives a potential function, VBþBþ;BþBþ¼1 41 m2 ρþ1 m2 ω½ðp1þp3Þðp2þp4Þ þðp1þp4Þðp2þp3Þ:ð5Þ We must project this in s-wave, and we have ðp1þp3Þðp2þp4Þ¼1 23s−ðM2þm2þM02þm02Þ − 1 sðM2−m2ÞðM02−m02Þ;ð6Þ with ffiffiffis pbeing the rest frame energy of the initial two mesons, and M;m; M0;m 0corresponding to upper, lower (initial), upper, lower (final) masses in general. Once the potential is obtained we construct the scattering matrix via the coupled-channel Bethe-Salpeter equation, T¼½1−VG−1V; ð7Þ where Gis the diagonal loop function for intermediate mesons, that we choose to regularize with the cutoff method [55], integrating over three momenta smaller than a certain qmax. In Eq. (5) we see that the interaction is repulsive, and hence the Tmatrix of Eq. (6) does not produce any bound state. We thus conclude that there are no bound states for the BB system. Using the same formalism we conclude that the interaction in the BsBand BsBschannels is also repulsive. B. BBsystem In this case the particles are not identical. Despite the fact that one can express the states in isospin (I¼0, 1) basis, it is convenient to treat the problem with coupled channels as it was done in [4] for the Tcc state since it was made from DþD0and D0Dþwith different thresholds, and it is closer to the DþD0one. The channels in the present case are BþB0(1), B0Bþ (2) with masses mBþ¼5279.34 MeV;m B0¼5279.65 MeV; mB¼5324.70 MeV:ð8Þ We also give the masses of Bsand B sfor later purposes, mBs¼5366.88 MeV;m B s¼5415.4MeV:ð9Þ The elementary interaction is obtained with the diagrams of Fig. 3. One can see that at the quark level one cannot exchange q¯ qin the diagram of Fig. 3(a) because the upper light quark in Bþ is a uquark and in B0is a dquark. In the picture that we have, this translates into a cancellation of ρ0;ω exchange when we take a common mass for the two. The same happens with the diagram of Fig. 3(b). However, (a) (b) FIG. 2. The two diagrams for the BþBþ→BþBþinteraction demanded by the symmetry of the particles. MASSES AND WIDTHS OF THE EXOTIC MOLECULAR …PHYS. REV. D 105, 074017 (2022) 074017-3 in the nondiagonal term of Fig. 3(c) one can exchange a ρþ, for which we obtain the following matrix interaction potential: Vij ¼Cijg2ðp1þp3Þðp2þp4Þϵϵ0;ð10Þ with the matrix Cij given by Cij ¼ 01 m2 ρ 1 m2 ρ 0!:ð11Þ Now the Gmatrix containing the BBloops entering the Bethe-Salpeter equation (7) is G¼GBþB00 0GB0Bþ:ð12Þ If we take the isospin states, jBB; I ¼0i¼ 1 ffiffiffi 2 pðBþB0−B0BþÞ; jBB; I ¼1;I3¼0i¼ 1 ffiffiffi 2 pðBþB0þB0BþÞ;ð13Þ we can see that we would get an attraction with CðI¼ 0Þ¼−1 m2 ρfor I¼0and a repulsion with CðI¼1Þ¼ 1 m2 ρfor I¼1, indicating that we can get a bound state for I¼0but not for I¼1. The spin in the present case is JP¼1þ. Should the binding of the states be small, like in the case of the Tcc, there could be a small violation of isospin, as found in [4], and thus we work in coupled channels. The interaction is formally the same as found for the Tcc in [4], and we follow then the same procedure as there, changing the masses, and using the same cut off around qmax ¼420 MeV to regularize the BBloop functions. Anticipating some results, the pole of the Tmatrix that we will find in the result section associated with the doubly bottom state thus generated is in principle located on the real axis about 20 MeV below the BBthreshold and hence has no width since the only decay channels considered (BþB0and B0Bþ) are closed. The only possible mesonmeson double bottom decay channel with lower threshold could be BB, but it is forbidden for the strong interaction since, to get JP¼1þ, we need L¼1in the BB system, and then parity is violated. Therefore, the only way to obtain a width for the doubly bottom state is from the decay of the Binto Bγ, which has not been measured but has been evaluated theoretically. We shall take from [56–59] the average value of ΓB≃0.40 keV:ð14Þ In order to take into account this effect we include the B→Bγdecay width into the Bpropagators in the loop functions of Eq. (12) (see Fig. 4), GðsÞ¼iZd4q ð2πÞ4 1 q2−m2 Bþiϵ 1 ðP−qÞ2−m2 Bþiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðP−qÞ2 pΓBððP−qÞ2Þ;ð15Þ where for the masses of Band Bwe distinguish between Bþ,B0, and B0,Bþcorrespondingly. Given the off shellness of the Bin the loop, we have to consider an energy dependent B→Bγdecay width, ΓBðs0Þ¼ΓBðm2 BÞm2 B s0pγðs0Þ pγðm2 BÞ3 Θðffiffiffiffi s0 p−mBÞ;ð16Þ where ΓBðm2 BÞ≃0.4keV is the on shell width mentioned above; Θis the step function and pγis the photon decay momentum, pγðsÞ¼λ1=2ðs; m2 B;0Þ=ð2ffiffiffis pÞ, with λstanding for the Källen function. After performing the q0integration in Eq. (15) we get GðsÞ≃Zqmax 0 dq q2 4π2 ωBþωB ωBωB 1 ffiffiffis pþωBþωB ×1 ffiffiffis p−ωB−ωBþiffiffiffi s0 p 2ωBΓBðs0Þ;ð17Þ (a) (b) (c) FIG. 3. Diagrams to calculate the BBinteraction. L. R. DAI et al. PHYS. REV. D 105, 074017 (2022) 074017-4 where ωBðBÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  q2þm2 BðBÞ qand s0¼ð ffiffiffis p−ωBÞ2−  q2. Note that now Eq. (17) provides a small but finite imaginary part for the Tmatrix corresponding to the cut depicted in Fig. 4. C. B sB;BBssystem It is straightforward to extend the previous BBformalism to the B sB; BBssystem. We now work with the couple channels BþB0 sand B0 sBþ. The interaction potential is now identical to the one of Eqs. (10) and (11) changing the masses accordingly, and exchanging a Kinstead of a ρ meson, which implies substituting 1 m2 ρby 1 m2 Kin Eq. (11).We can anticipate that the combination 1 ffiffi2 pðB0 sBþ−BþB0 sÞis the one that gets bound, or in other words, that when calculating the couplings of the state that we obtain to B0 sBþand BþB0 swe will obtain results with about the same strength and opposite sign. Now the width of the state comes from the only possible sources of imaginary part, which in this case are the B→Bγand B s→Bsγdecays in the corresponding loops. For the B s→Bsγdecay in the B0 sBþloop we use an analogous expression to Eq. (16) but changing the masses correspondingly and using for the on shell B s→Bsγthe theoretical value ΓB s ≃0.22 keV, obtained from QCD sum rules [59]. On the other hand, the B sBssystem contains only one channel, and the interaction is mediated by ϕexchange, and it is repulsive, thus preventing the existence of any bound states for that system. D. BBsystem The vector-vector interaction in a unitarized form was addressed in [60,61]. It is particularized to the DD systems in [48]. We can sketch how the potentials are obtained. We work here in the isospin basis anticipating that the widths from BB and BBwill be of the order of a few MeV, as found in [21] for the DDsystem. In the unitary normalization suited for identical particles the states are jBB;I ¼0i¼1 2ðBþB0−B0BþÞ; jBB;I ¼1;I3¼0i¼1 2ðBþB0þB0BþÞ; jBB; I ¼1;I3¼1i¼ 1 ffiffiffi 2 pðBþBþÞ:ð18Þ The interaction is obtained in the same way as in the case of BB, except that now we have the extra factors, ϵið1Þϵið3Þϵjð2Þϵjð4Þð19Þ for the diagonal terms [Figs. 3(a) and 3(b)], and ϵið1Þϵið4Þϵjð2Þϵjð3Þð20Þ for the crossed terms [Fig. 3(c)]. The apparent complexity due to the presence of the four polarization vectors is trivially solved by means of the spin projection operators [60],Pð0Þ,Pð1Þ,Pð2Þ, and the combinations of Eqs. (19) and (20) are decomposed into B* B B P q P−q B* FIG. 4. BBloop considering the B→Bγ, which is the source of the imaginary part of the unitarized BBscattering amplitude, and hence, the width of the double bottom generated state. TABLE I. Amplitudes for B¼2,S¼0and I¼0. JAmplitude V-exchange 0BB→BB0 1BB→BB1 4g2ð1 m2 ω −3 m2 ρÞfðp1þp4Þ:ðp2þp3Þþðp1þp3Þ:ðp2þp4Þg 2BB→BB0 TABLE II. Amplitudes for B¼2,S¼0and I¼1. JAmplitude V-exchange 0BB→BB1 4g2ð1 m2 ωþ1 m2 ρÞfðp1þp4Þ:ðp2þp3Þþðp1þp3Þ:ðp2þp4Þg 1BB→BB0 2BB→BB1 4g2ð1 m2 ωþ1 m2 ρÞfðp1þp4Þ:ðp2þp3Þþðp1þp3Þ:ðp2þp4Þg MASSES AND WIDTHS OF THE EXOTIC MOLECULAR …PHYS. REV. D 105, 074017 (2022) 074017-5 ϵiϵiϵjϵj¼3Pð0Þ; ϵiϵjϵiϵj¼Pð0ÞþPð1ÞþPð2Þ; ϵiϵjϵjϵi¼Pð0Þ−Pð1ÞþPð2Þ;ð21Þ where we have assumed that the polarization vectors appear in the order of the particles 1–4 as in Fig. 2(a). One can then obtain the interaction in all spin channels J¼0, 1, 2 (recall we have L¼0,Jcomes from spin combinations). The symmetry rules are automatically fulfilled and BBin I¼ 0(antisymmetric) can only appear in J¼1(antisymmetric) and in I¼1(symmetric) can only appear in J¼0,2 (symmetric). The results obtained for BB,B sB,B sB s, are identical to those obtained for the DD,D sD,D sD s in Tables XVI–XIX of [48] which we reproduce in Tables I–IV, omitting the contact term and the exchange of J=ψ(here ϒ) which are negligible. We can see that the BBin I¼0and JP¼1þis attractive, BBin I¼1is repulsive in the two J¼0,2 allowed channels. The B sBchannel is attractive in JP¼1þ, and the B sB sis repulsive in the two J¼0,2 allowed channels. We thus expect only bound states for BB;I ¼0;JP¼1þand B sB;I ¼1 2;JP¼1þ. III. EVALUATION OF THE WIDTH The evaluation of the width of the states follows exactly the same steps as the one of the DDstates done in [21], simply changing Dby B, and the results are identical, simply changing the masses. The decay of the BB channels to BB is not allowed because all the BBstates have parity positive and spin S¼1. One needs L¼1for BB to match the angular momentum but then parity is violated. Thus, the only allowed decay channel is the BB which involves an anomalous coupling. The VV decay into VP was addressed in [50] in the evaluation of the width of the D¯ KX0ð3866Þstate. To give a width to the state the box diagrams including all possible BBintermediate states are evaluated, and the imaginary part is obtained and added to the real potential evaluated in the former subsections. Then the Bethe-Salpeter equation is solved with the complex potential. We plot jTj2for the bound state, from where we obtain the mass and the width. Translating from [21] to our case we obtain (see Figs. 2–4,6 of [21] and replace D;D;D  s;D sby B;B;B  s;B sto obtain the diagrams involved in the evaluation of the widths), (a) BB,I¼0JP¼1þ V¼− g2 m2 ρðp1þp3Þ:ðp2þp4Þ; ImVbox ¼− 6 8π 1 ffiffiffis pq5G0 22 ðffiffiffi 2 pgÞ2E2 B ×1 ðp0 2−EBðqÞÞ2−q2−m2 π2 F4ðqÞmB mK2 ; ð22Þ with q¼λ1=2ðs; m2 B;m 2 BÞ 2ffiffiffis p;p0 1¼p0 2¼EB; EB¼ffiffiffis p 2;q0¼sþm2 B−m2 B 2ffiffiffis p; and G0¼3g0 4π2f;g0¼− GVmρ ffiffiffi 2 pf2;GV¼55 MeV; f¼93 MeV; FðqÞ¼e½ðp0 1−q0Þ2−q2=Λ2:ð23Þ TABLE III. Amplitudes for B¼2,S¼1and I¼1 2. JAmplitude V-exchange 0B sB→B sBg2ðp1þp4Þ:ðp2þp3Þ m2 K 1B sB→B sB−g2ðp1þp4Þ:ðp2þp3Þ m2 K 2B sB→B sBg2ðp1þp4Þ:ðp2þp3Þ m2 K TABLE IV. Amplitudes for B¼2,S¼2and I¼0. JAmplitude V-exchange 0B sB s→B sB sg2 2 1 m2 ϕfðp1þp4Þ:ðp2þp3Þþðp1þp3Þ:ðp2þp4Þg 1B sB s→B sB s0 2B sB s→B sB sg2 2 1 m2 ϕfðp1þp4Þ:ðp2þp3Þþðp1þp3Þ:ðp2þp4Þg L. R. DAI et al. PHYS. REV. D 105, 074017 (2022) 074017-6 (b) B sB,I¼1 2JP¼1þ V¼− g2 m2 Kðp1þp4Þ:ðp2þp3Þ; ImVbox ¼− 1 8π 1 ffiffiffis pq51 3ð2gÞ2G0 ffiffiffi 2 p2 ðE2 B sþE2 BÞ ×1 ðp0 2−EBsðqÞÞ2−q2−m2 K2 F4ðqÞmB mK2 ; ð24Þ where p0 1¼EB s;p0 2¼EB; q¼λ1=2ðs; m2 B;m 2 BsÞ 2ffiffiffis p;q0¼sþm2 B−m2 Bs 2ffiffiffis p; and FðqÞgiven by Eq. (23). In the equations for q¼ λ1=2ðs; m2 1;m 2 2Þ=ð2ffiffiffis pÞaΘðs−ðm1þm2Þ2Þis implied. We take now a potential, V0¼VþiImVbox; with Λ≃1300 MeV as in [48] and solve the BetheSalpeter equation of Eq. (7). IV. RESULTS A. BBstates In the first place we show the results that we obtain for the I¼0;JP¼1þBBsystem. As we pointed out, the only source of imaginary part comes from the B→Bγ decay, with a very small width, as shown in Eq. (14).We thus should expect bound states with a very narrow width. Indeed, in Fig. 5we plot the modulus squared of the BþB0→BþB0amplitude. The results are shown for three different values of qmax ranging from 400 MeV to 450 MeV, in line with the 420 MeVused in [4] to obtain the binding of the Tcc state. The plots peak at positions 10587, 10583 and 10577 MeV for qmax ¼400, 420 and 450 MeV respectively, which give an idea of the uncertainty in the mass of the generated double bottom state. The vertical lines represent the thresholds of the BþB0and B0Bþ channels. It is worth noting that we get bindings bigger than those for the Tcc case [4], of the order of 20 MeV with respect to the BBthreshold. This is in contrast with the 360 keV binding found in [2] for the Tcc. It could be surprising at a first sight the fact that, using the same cutoff, the binding obtained is bigger than for the Tcc case. We would like to note that this finding is common to observations done in quark model studies of tetraquarks, indicating a stronger attraction as the mass of the heavy quark increases [62]. Similar conclusions are reached in [28,36,63]. We can also evaluate the couplings of the generated states to the different channels. In the real axis and close to the pole position we can define the couplings gito the ith channels as tij ≃ gigj s−sR ;i¼1;2for BþB0;B 0Bþ;ð25Þ with sR≡M2 Rthe square of the energy of the bound state. Therefore, gigj¼lim s→sRðs−sRÞtijðsÞ;ð26Þ which is nothing but the residue at the pole. We find, for qmax ¼420 MeV, g1¼35954 MeV;g 2¼−35798 MeV;ð27Þ where g1,g2, have opposite sign as we anticipated. According to Eq. (13) this indicates a very neat I¼0 state, as we anticipated that only the I¼0component could lead to a bound state. The larger distance to the thresholds of the BþB0, B0Bþstates has as a consequence a smaller isospin breaking than the one found in the Tcc state, as can be seen by the proximity of g2to −g1. The width of the states can be obtained directly from the width of the peak zooming in the plots in Fig. 5or alternatively using that, at the peak, T11 ¼g2 1 s−sRþiMRΓR ⇒ΓR¼− g2 1ImfT11g MRjT11j2:ð28Þ Either way gives the width of the doubly bottom generated state: 25, 14 and 4 eV for qmax ¼400, 420 and 450 MeV respectively. These quantities are indeed extremely small, in line with the estimated 0.4 keV of the B→Bγdecay FIG. 5. Squared amplitude jTBþB0→BþB0j2. The vertical lines indicate the B0Bþand BþB0thresholds at 10604.04 MeV and 10604.35 MeV, respectively. MASSES AND WIDTHS OF THE EXOTIC MOLECULAR …PHYS. REV. D 105, 074017 (2022) 074017-7 width and have a large uncertainty. The smaller values of the width compared to the 400 eVof Eq. (14) stem from the use of the energy dependence of Eq. (16). If one uses a constant width for B→Bγ, the widths obtained for the states are more in line with that latter number.1This smallness could then make difficult to determine the width of this doubly bottom state experimentally. Yet the mode to observe it would be looking at the BBγinvariant mass distribution. B. B sB;BBsstates In Fig. 6we show the position of the peaks for the B sB; BBssystem as described in Sec. II C. The positions are obtained at 10683,10681 and 10677 MeV for qmax ¼400, 420 and 450 MeV respectively, which are about 10–15 MeV below the thresholds (10691.6 MeV for BþB0 sand 10694.74 MeV for B0 sBþ). The couplings of the generated state to BþB0 sð1Þand B0 sBþð2Þ, for qmax ¼420 MeV, are g1¼25240 MeV;g 2¼−26845 MeV;ð29Þ and, as anticipated in Sec. II C, the couplings have opposite sign. The widths obtained for the doubly bottom state are 60, 45 and 25 eV for qmax ¼400, 420 and 450 MeV respectively. The results are qualitatively analogous to those found for the BBstates and then similar conclusions as in Sec. IVA can be deduced. C. BBstates In this subsection we show the results obtained for the BBsystem. Once again we obtain bound states for the same range of the qmax values in the line of those used for the Tcc. As shown in Fig. 7, we get bindings of the order of 20 MeV with respect to the BBthreshold. Changing qmax from 400 MeV to 450 MeV causes an increase of the binding by about 9 MeV. These bindings, although small, are considerably bigger than those found for the analogous DDsystem in [21], of the order of 1–2 MeV. The width of the state is of the order of 8 MeV, and the state becomes narrower as it approaches threshold, something already observed in [21], resulting from the general rule that the couplings of a bound state to its components go to zero as the binding goes to zero [64], which is generalized to coupled channels in [65,66]. The width is modulated by the form factor of Eq. (23), but its dependence on the Λparameter is smooth as one can see in Fig. 8. FIG. 7. Squared amplitude jTBB→BBj2with Λ¼1200 MeV. The vertical line indicates the BBthreshold at 10649.4 MeV. FIG. 6. Squared amplitude jTBþB0 s→BþB0 sj2. The vertical lines indicate the thresholds of 10691.74 MeV for BþB0 sand 10694.6 MeV for B0 sBþ. FIG. 8. Squared amplitude jTBB→BBj2with qmax ¼420 MeV. 1We take advantage to mention that the present formalism is different, but related to the one used in [4], where a convolution of the Gfunction was made. If we use the present method we obtain a width of 39 keV for the Tcc state using the mass of the LHCb analysis of Ref. [2]. L. R. DAI et al. PHYS. REV. D 105, 074017 (2022) 074017-8 D. B sBstates In this subsection we show the results for the I¼1 2,JP¼ 1þB sBsystem. In Fig. 9we show the results of jTj2for the B sBamplitude. Depending on the choice of qmax we obtain again peaks corresponding to bound states of that system, more bound as qmax increases. The binding is of the order of 12 MeV and changing qmax from 400 MeV to 450 MeV increases the binding by about 6 MeV. The width is of the order of 0.5 MeV. The smaller width of the state, similar to the case of the D sDversus DDfound in [21], is due to the fact that in the decay diagrams of B sB→B sB or BsBone is exchanging kaons rather than pions (see detailed related figures replacing D;D;D  s;D sby B;B;B  s;B sin Figs. 2–4,6 of [21]). Once again, in Fig. 10 we show how the width changes with a change of the parameter Λ, and we observe that the changes are minor for a reasonable change of Λ. We summarize our results in Table Vtaking qmax ¼ 420 MeV and Λ¼1200 MeV. V. CONCLUSIONS We have studied the interaction of the BB,BB,BsB, B sB,BB,BBs,BB s,BsBs,BsB s,B sB ssystems with an extension of the local hidden gauge approach, where one exchanges vector mesons between the bottom mesons. Only the exchange of the light vectors is taken into account, the exchange of the heavy ones being irrelevant. This picture, having the heavy quarks as spectators, automatically fulfills the rules of heavy quark symmetry. The picture shows that we only have four systems bound, the BBin I¼0,B sB−BBsin I¼1 2,BBin I¼0and B sBin I¼1=2, all of them with JP¼1þ. We have also considered the decay channels of theses systems: the BBγfor the BBsystem, BsBγfor the B sB−BBssystem, BBfor the BBsystem, and B sBor BBsfor the B sBsystem. The binding energy of these states is tied to the regulator of the loops in the intermediate states in the Bethe-Salpeter equation, but for that we use a cutoff in the range of the one needed to obtain the binding energy of the Tcc state. With this input we can make predictions and find bound states in the four cases varying from 10–20 MeV binding. The widths vary much, from the order of 10–50 eV for the BBand B sB−BBssystems to about 8 MeV in the case of the BBsystem, or 0.5 MeV for the B sBsystem. The accuracy of former predictions using the present framework make us confident on the predictions made here and should encourage the experimental search for these states with LHCb or other facilities. ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China under Grants No. 11975009, No. 12175066, No. 12147219. This work is also supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under Contracts No. FIS201784038-C2-1-P B and No. FIS2017-84038-C2-2-P B and by Generalitat Valenciana under Contract No. PROMETEO/ 2020/023. This project has received funding from the European Unions Horizon 2020 research and innovation programme under Grant Agreement No. 824093 for the STRONG-2020 project. R. M. acknowledges support from the Contratación de investigadores de Excelencia de la Generalitat valenciana (GVA) program with Ref. No. CIDEGENT/2019/015 and from the spanish national TABLE V. States of JP¼1þobtained from different configurations. The binding Bis referred to the closest threshold. States M(MeV) B(MeV) Γ BBðI¼0Þ10583 21 14 eV B sB−BBsðI¼1 2Þ10681 11 45 eV BBðI¼0Þ10630 19 8 MeV B sBðI¼1 2Þ10728 12 0.5 MeV FIG. 9. Squared amplitude jTB sB→B sBj2with Λ¼1200 MeV. The vertical line indicates the B sBthreshold at 10740.1 MeV. FIG. 10. Squared amplitude jTB sB→B sBj2with qmax ¼ 420 MeV. MASSES AND WIDTHS OF THE EXOTIC MOLECULAR …PHYS. REV. D 105, 074017 (2022) 074017-9