Possible bound states with hidden bottom from $\bar{K}^{(*)}B^{(*)}\bar{B}^{(*)}$ systems

We study the three-body systems of $\bar{K}^{(*)}B^{(*)}\bar{B}^{(*)}$ by solving the Faddeev equations in the fixed-center approximation, where the light particle $\bar{K}^{(*)}$ interacts with the heavy bound states of $B\bar{B}$ ($B^*\bar{B}^*$) forming the clusters. In terms of the very attractive $\bar{K}^*B$ and $\bar{K}^*B^*$ subsystems, which are constrained by the observed $B_{s1}(5830)$ and $B_{s2}^*(5840)$ states in experiment, we find two deep bound states, containing the hidden-bottom components, with masses $11002\pm 63$ MeV and $11078\pm 57$ MeV in the $\bar{K}^*B\bar{B}$ and $\bar{K}^*B^*\bar{B}^*$ systems, respectively. The two corresponding states with higher masses of the above systems are also predicted. In addition, using the constrained two-body amplitudes of $\bar{K}B^{(*)}$ and $\bar{K}\bar{B}^{(*)}$ via the hidden gauge symmetry in the heavy-quark sector, we also find two three-body $\bar{K}B\bar{B}$ and $\bar{K}B^{*}\bar{B}^*$ bound states.

In the present work, we extend the study of Ref. [27] to the bottom sector to investigate the possible bound states from theK ( * ) B ( * )B( * ) systems with isospin I = 1/2 using the fixed-center approximation of the Faddeev equations. We take BB, B * B * as clusters (denoted as R in the following) andK ( * ) as a third particle to scatter, which satisfies the general criteria of the FCA to the Faddeev equations: that the mass of the third particle P 3 should be smaller than a stable cluster (such as a bound state) composed of the two other particles P 1 and P 2 . In Ref. [61], a single bound state of BB system with J P = 0 + and the three-degenerate states of B * B * system with J P = 0 + , 1 + , 2 + was found in isospin 0 using the coupled-channel chiral unitary approach. In principle, the spin state of B * B * cluster can be chosen as J R = 0, 1, or 2. Similar to Ref. [24], we are interested in the largest total spin of a molecular state from theK ( * ) B * B * systems, therefore, the J R = 2 cluster of B * B * is preferred. The corresponding wave function is simple with the spins of B * andB * aligned, which will relatively simplify the practical calculation. Besides, for the two-body subsystem, such asK * B * , which can also produce the three spin bound states with 0 + , 1 + , and 2 + , the spin-aligned 2 + state is more bound, by around 60 MeV, than the other spin states. Thus, the largest spin state ofK ( * ) B * B * would produce a larger binding than the other total spin systems. Therefore, we will investigate the J = 0KBB, J = 1K * BB, J = 2KB * B * and J = 3K * B * B * systems with isospin I = 1/2 and study the possibility to produce the bound states.
In the following Sec. II, we will first present the details of the formalism employed in the FCA framework. The input of two-body amplitudes are also calculated in the chiral unitary approach.
The predicted bound states fromK ( * ) B ( * )B( * ) are shown in Sec. III with the corresponding discussion. Finally, a summary is given in Sec. IV.

II. THEORETICAL FRAMEWORK
A. Fixed-center approximation to Faddeev equations Under the FCA, the total elastic scattering T -matrix of three-body systems can be simplified as the sum of two Faddeev partitions, where T 1 and T 2 describe the iterated interactions of theK ( * ) scattering off the cluster R with a first collision on B (B * ) andB (B * ), respectively. These interactions are illustrated in Fig. 1 and can be expressed as the two coupled equations, basis can be written as One introduces two scattering T -matrices of the left/right collision, where the two-body amplitudes, t I=0,1 K ( * ) B ( * ) and t I=0,1 K ( * )B( * ) , are the input of the FCA equations.

B. Two-body amplitudes of subsystems
In order to obtain the two-body amplitudes in Eq. (7), we follow the calculation details in Ref. [62] and employ the lowest order Lagragians with the local hidden gauge symmetry in the SU(4) sector, 1 where f π denotes the pion decay constant f π = 93 MeV and the coupling g is determined through the SU(4) symmetry [62]. The fields of pseudoscalar mesons (P ) and vector mesons (V ) are 1 Note that the extension of the local hidden gauge approach from the light-quark sector [63,64] to the heavy-quark sector is possible if the heavy quarks of hadrons are just spectators and the major contributions of the interaction is from the exchange of light vector mesons (ρ, ω, φ). In this case, one can employ the SU(4) symmetry formally in the Lagrangians (e.g. Eq. [8)] and the actual SU(3) subgroup is used in the evaluation of the vertices. For more discussions and the proof, one can refer to Sec. II of Ref. [65] and Sec. II and the Appendix of Ref. [66]. collected in the 4 × 4 matrices, After considering the contact interactions and one-boson exchange contributions within the coupled-channel approach, the s-wave potentials, vK( * ) B ( * ) and vK( * )B( * ), are projected, as shown in Ref. [62]. To keep the self-consistency of the current work, we summarize all the needed twobody potentials in the Appendix.
In the chiral unitary approach, the interaction kernels vK( * ) B ( * ) or vK( * )B( * ) can be resumed in the Bethe-Salpeter equation, where G is a diagonal matrix with the element being a two-meson loop function for the ith particle channel, with the total four-momentum of two-meson system p µ = ( √ s i , 0) and √ s i the center-of-mass (c.m.) energy. Using the cutoff regularization, the loop function changes as where k max denotes as the momentum cutoff and ω i = To evaluate the amplitudes t I=0,1 KB ( * ) and t I=0,1 K * B( * ) , the momentum cutoff is chosen to be k max = 1070 MeV as given in Ref. [62]. As demonstrated in Ref. [43], in the above calculation of two-body scattering, we use the normalization of Mandl and Shaw [67], which introduces different weight factors for the particle fields. Therefore, one has to consider these factors in our two-body amplitudes Here we have taken the approximations , which is suitable for heavy-bottom particles as demonstrated in Ref. [24].

C. Total amplitude of three-body system
Finally, we obtain the total amplitude TK( * ) R ofK ( * ) B ( * )B( * ) by solving the FCA Eqs. (1)(2)(3), which is apparently a function of the total invariant mass of the three-body system. The arguments in the two-body amplitudes, tK( * ) B ( * ) and tK( * )B( * ), are s 1 and s 2 , which are (commonly) determined through Besides, in Ref. [22], another set of transformation for the s i in terms of s are proposed with the consideration of the recoil. Here the total three-momentum of the two-particle system, , is estimated in terms of the binding energy of B ( * ) andB ( * ) in the cluster R, In the following, we will take this choice to evaluate the uncertainties of our prediction.
The propagator ofK ( * ) inside the cluster, G 0 in Eq. (14), can be expressed as where q 0 denotes the energy carried byK ( * ) in the cluster rest frame, and F R (q 2 ) is the form factor of B ( * )B( * ) cluster, which is introduced to consider the molecular dynamics by using the Fourier transformation of the s-wave cluster R [43] with the normalization factor N = F R (0) and It is worth noting that the form factor F R (q 2 ) has implicitly taken into account the interaction of B ( * ) andB ( * ) which leads to the binding of B ( * )B( * ) system. Hence, the upper integration limit Λ R should take the same value of the momentum cutoff as the one used to regularize the B ( * )B( * ) loop to get the bound state R.

III. RESULTS AND DISCUSSION
In our numerical evaluation, the meson masses are taken from Ref.  [61]. Generally speaking, the cutoff is a free and important parameter in the analysis of two-body bound states. One has to rely on some experimental information to determine/constrain it. In Ref. [61], the value of the cutoff is chosen as Λ R = 415 MeV, which is fixed to produce a bound state X(3700) of the DD system [15]. andK * B * are well constrained by the observed bound states, B s1 (5830) and B * s2 (5840), therefore, in the following, we will first study theK * B ( * )B( * ) systems and then briefly mention the results ofKB ( * )B( * ) systems.
Using the momentum cutoff Λ R = 415 MeV, in Fig. 2, we present the shape of total amplitude ofK * BB system as a function of the three-body total energy √ s. To further analyze the uncertainties, as in Refs. [24,53], the two schemes are used to share the three-body total energy into the two subsystems. We denote the relationship between s 1, 2 and s given in Eq. (15) Fig. 3. One can see that there are two peaks in |T Num | 2 and |T Den | 2 , which are produced from the bound state of two-body amplitudes t I=0 KB and t I=1 K * B and locates at the same √ s as the pole position of two-body peaks when transforming the √ s 1, 2 to the total energy √ s using the method A, as shown in the lower panel of Fig. 3. The total amplitude ofK * BB is obtained where the superscripts Re and Im denote the real and imaginary parts of T Num and T Den . Although the imaginary parts T Im Num and T Im Den are small, one cannot ignore the imaginary part of TK * BB which also relates to the real parts of T Re Num and T Re Den as shown in Eq. (21). This nonzero imaginary contribution will slightly change the pole position of the modulus square ofK * BB system, such as the position of the lower pole varying from 11039.8 MeV to 11039 MeV and the position of the upper pole changing from 11310 MeV to 11307 MeV. The deep bound state is produced by theK * B interaction with isospin I = 0 and the bound state with higher mass is originated from theK * B interaction with isospin I = 1, as shown in the lower panel of Fig. 3. Thus, the physical picture is that the two poles correspond to theK * sticking closer to B orB.
For the three-body systemK * B * B * with spins aligned (J = 3), a similar shape of three-body amplitude with the different cutoffs is obtained. The results for position of peaks are summarized in Table I  , we can conclude that theK * B ( * )B( * ) three-body system is more bound than either pair, as a consequence of the combination of two subsystems. It is worth noting that a similar superbound state was also predicted in the bottom sector, ρB * B * system [24], using the same theoretical framework.
On the other hand, in Table II, the masses and binding energies of the heavier bound states, e.g. shown in Fig. 2, are summarized for theK * BB andK * B * B * systems with the momentum cutoff Λ R = 415 and 830 MeV, respectively. A similar mass difference between method A and B as Table I is  Furthermore, we also study theKB ( * )B( * ) three-body systems using the FCA of the Faddeev equations. The total amplitude of the three-body interaction is determined via Eq. (14). The corresponding inputs of the two-body amplitudes, t I=0, 1 KB ( * ) , t I=0, 1 KB ( * ) , are evaluated using the effective Lagrangians Eq. (8) with the local hidden gauge approach. As given in Ref. [62], the two deep bound states, the 0(0 + ) state with mass around 5460 MeV ofKB system and the 0(1 + ) state with mass around 5665 MeV ofKB * , are predicted. These very attractive interactions will guarantee that bound states in the three-body systems are produced. In Table III, we have tabulated our findings in theKB ( * )B( * ) systems with two cutoffs, as employed in the study ofK * B ( * )B( * ) systems. After averaging, one finds two bound states with mass 10659 ± 69 MeV for theKBB system and with mass 10914 ± 62 MeV for theKB * B * system. We also found that the difference of binding energy betweenKBB andKB * B * systems is around 150 MeV, which originates from the difference (∼ 140 MeV) of the binding energies of the two-body subsystemsKB andKB * .
Finally, we want to mention that, although theKBB andKB * B * systems satisfy all the criteria for a reliable application of the FCA and produce two bound states, one should take a critical look at the masses and the binding energies obtained in Table III. Because the two-body amplitudes of K B andKB * determined in Ref. [62] are different with the ones from Refs. [69][70][71], where the B * s0 (5725) and B s1 (5778) mesons were predicted, respectively. More efforts are needed to obtain the final conclusion of theKB andKB * interactions in order to well determine the masses of the KB ( * )B( * ) bound states.

IV. SUMMARY
We have performed a three-body study of theK three-body systems. We expect that the current study and Ref. [27] will arouse interest to the study of the hadronic states with hidden charm/bottom in the strange sector.