$T_{bbb}$: a three $B-$meson bound state

By solving exactly the Faddeev equations for the bound-state problem of three mesons, we demonstrate that current theoretical predictions pointing to the existence of a deeply-bound doubly bottom axial vector tetraquark lead to the existence of a unique bound state of three $B$ mesons. We find that the $BB^*B^*-B^*B^*B^*$ state with quantum numbers $(I)J^P=(1/2)2^-$, $T_{bbb}$, is about 90\,MeV below any possible three $B$-meson threshold for the reported binding of the doubly bottom axial vector tetraquark, $T_{bb}$.

relation is true for the bb binding energy in a tetraquark, it concludes that the axial vector bbūd state is stable. The Heavy-Quark Symmetry analysis of Ref. [3] predicts the existence of narrow doubly heavy tetraquarks. Using as input for the doubly bottom baryons, not yet experimentally measured, the diquark-model calculations of Ref.
The possible existence of deuteron-like hadronic molecular states made of vector-vector or pseudoscalar-vector two-meson systems was proposed in Ref. [12] in an exploratory study suggesting the deusons, two-meson states bound by the one-pion exchange potential. This scenario of meson-meson stable states bound by some interacting potential, has later on been frequently used to draw conclusions about the existence of hadronic molecules [13][14][15] (see Refs. [16] for a recent compendium). The constituent quark and the meson-meson approaches to hadronic molecules must be equivalent [17], although, as will be discussed below, to get the results of the constituent quark approach would, in general, require a coupled-channel meson-meson study [18].
It is also worth to emphasize that when a two-body interaction is attractive, if the twobody system is merged with nuclear matter and the Pauli principle does not impose severe restrictions, the attraction may be reinforced. We find the simplest example of the effect of additional particles in the two-nucleon system. The deuteron, (i)j p = (0)1 + , is bound by 2.225 MeV, while the triton, (I)J P = (1/2)1/2 + , is bound by 8.480 MeV, and the α particle, (I)J P = (0)0 + , is bound by 28.295 MeV. The binding per nucleon B/A increases as 1 : 3 : 7. Thus, a challenging question is if the existence if a deeply bound two B-meson system 1 could give rise to bound states of a larger number of particles. As it was shown in Ref. [19] the answer is by no means trivial, because when the internal two-body thresholds of a three-body system are far away, they conspire against the stability of the three-body system.

II. COLOR DYNAMICS.
As it has been stated above, results based on meson-meson scattering or a constituent quark picture should be equivalent, provided that, in general, a coupled-channel mesonmeson approach would be necessary to reproduce the constituent quark picture [17,18]. To be a little more specific, let us note that four-quark systems present a richer color structure than standard baryons or mesons. Although the color wave function for standard mesons and baryons leads to a single vector, working with four-quark states there are different vectors driving to a singlet color state out of colorless meson-meson (11) or colored two-body (88, 33, or 66) components. Thus, dealing with four-quark states an important question is whether one is in front of a colorless meson-meson molecule or a compact state (i.e., a system with two-body colored components). Note, however, that any hidden color vector can be expanded as an infinite sum of colorless singlet-singlet states [17]. This has been explicitly done for compact QQqq states in Ref. [18].
In the heavy-quark limit, the lowest lying tetraquark configuration resembles the helium atom [3], a factorized system with separate dynamics for the compact color3 QQ nucleus and for the light quarks bound to the stationary color 3 state, to construct a QQqq color singlet.
The validity of this argument has been mathematically proved and numerically checked in Ref. [18], see the probabilities shown in Table II for the axial vector bbūd tetraquark.
It has been recently revised in Ref. [6], showing in Fig can be expanded as the mixture of several physical meson-meson channels [17], BB * and B * B * for the axial vector bbūd tetraquark (see Table II of Ref. [18]) and, thus, they can be also studied as an involved coupled-channel problem of physical meson-meson states [20,21].
Our aim in this work is to solve exactly the Faddeev equations for the three-meson bound state problem using as input the two-body t−matrices of Refs. [5,[18][19][20], driving to the axial vector bbūd bound state, T bb , as an involved coupled-channel system made of pseudoscalar-vector and vector-vector two B-meson components. We show that for any of the recently reported values of the T bb binding energy [1-10], the three-body system Out of the possible spin-isospin three-body channels (I)J P made of B and B * mesons, we select those where, firstly, two-body subsystems containing two B-mesons are not allowed, because the BB interaction does not show an attractive character; and, secondly, they contain the axial vector (i)j p = (0)1 + doubly bottom tetraquark, T bb . The three-body channel (I)J P = (1/2)2 − is the only one bringing together all these conditions to maximize the possible binding of the three-body system 2 . We indicate in Table I the two-body channels  contributing to this state that we examine in the following.
The Lippmann-Schwinger equation for the bound-state three-body problem is where V i is the potential between particles j and k and G 0 is the propagator of three free particles. The Faddeev decomposition of Eq. (1), leads to the set of coupled equations, The Faddeev decomposition guarantees the uniqueness of the solution [22]. Eqs. (3) can be rewritten in the Faddeev form with where t i are the two-body t−matrices that already contain the coupling among all two-body channels contributing to a given three-body state, see Table I. The two sets of equations (3) and (4)  all identical [19]. The additional terms in Fig. 1 are, of course, those responsible for the coupling between the BB * B * and B * B * B * components of the system.

IV. RESULTS.
We show in Fig. 2  and antibaryon (B 2 ) is forbidden if the transition amplitude B 1B2 |T |Ψ T bbb vanishes. In principle T is the transition matrix (or S matrix) which is roughly e iH , but since |Ψ T bbb is a true eigenstate of H, the transition amplitude vanishes if the overlap B 1B2 |Ψ T bbb vanishes itself [24]. Since there are no experimental data for the Ω bbb mass and there is a wide variety of theoretical estimations (see Table 1 of Ref. [25]) it has to be calculated within the same scheme. For the binding energy of the T bb axial vector tetraquark obtained in Ref.
[1], the Ω bbb has a mass of 14.84 GeV. Thus, the Ω bbb −p threshold would lie at 15.78 GeV, above the T bbb state. Let us note that even if the Ω bbb −p threshold would lie below the three B−meson energy, the T bbb state will show up as a narrow resonance as recently discussed in Ref. [26], due to the negligible interaction between the Ω bbb and thep. The dynamics of this type of states would come controlled by the attraction in the three-body system and the channel made of almost non-interacting hadrons is mainly a tool for the detection. This is exactly  It is appealing that the stability of such hexaquark state with respect the lowest tetraquark-meson threshold was already anticipated in the exploratory study of Ref. [28] within a quark string model. Let us finally note that our discussion above could be extended to the charm sector, where the two-body bound state would lie close to threshold [21,29].
However, as we have noted above, going from the bottom to the charm sector there is a factor 3 in the mass difference between pseudoscalar and vector mesons, what makes the coupled-channel effect much less important in the charm case than in the bottom one. Thus, one does not expect binding in the three-meson charm sector.