Abstract
Heavy-quark symmetry as applied to heavy hadron systems implies that their interactions are independent of their heavy-quark spin (heavy-quark spin symmetry) and heavy flavour contents (heavy flavour symmetry). In the molecular hypothesis the $X(3872)$ resonance is a $1^{++}$ $D^*\bar{D}$ bound state. If this is the case, the application of heavy-quark symmetry to a molecular $X(3872)$ suggests the existence of a series of partner states, the most obvious of which is a possible $2^{++}$ $D^*\bar{D}^*$ bound state for which the two-body potential is identical to that of the $1^{++}$ $D^*\bar{D}$ system, the reason being that these two heavy hadron-antihadron states have identical light-spin content. As already discussed in the literature this leads to the prediction of a partner state at $4012\,{\rm MeV}$, at least in the absence of other dynamical effects which might affect the location of this molecule. However the prediction of further heavy-quark symmetry partners cannot be done solely on the basis of symmetry and requires additional information. We propose to use the one boson exchange model to fill this gap, in which case we will be able to predict or discard the existence of other partner states. Besides the isoscalar $2^{++}$ $D^*\bar{D}^*$ bound state, we correctly reproduce the location and quantum numbers of the isovector hidden-bottom $Z_b(10610)$ and $Z_b(10650)$ molecular candidates. We also predict the hidden-bottom $1^{++}$ $B^*\bar{B}^*$ and $2^{++}$ $B^*\bar{B}^*$ partners of the $X(3872)$, in agreement with previous theoretical speculations, plus a series of other states. The isoscalar, doubly charmed $1^+$ $D D^*$ and $D^* D^*$ molecules and their doubly bottomed counterparts are likely to bind, providing a few instances of explicitly exotic systems.
Highlights
Hadronic molecules were conjectured four decades ago from a direct analogy to the deuteron and the nuclear forces that bind it [1,2]
II we review the application of heavy-quark spin symmetry to the heavy meson-antimeson system
Gσ is derived from the quark model, for which there are no reliable error estimations, though 30% looks like a sensible figure; gω has been derived from SU(3)flavor symmetry and the OZI rule, where at least a 20% error is to be expected
Summary
Hadronic molecules were conjectured four decades ago from a direct analogy to the deuteron and the nuclear forces that bind it [1,2]. The one-boson-exchange (OBE) model [36,37], besides having played a central role in the seminal speculations about the existence of hadron molecules, is able to provide this missing information about the hadron-hadron interactions In this model the potential between two hadrons is the consequence of the exchange of a series of light mesons (π, σ, ρ, and ω) that provide the necessary dynamics for binding. After determining the cutoff Λ with this condition, we can explore how the OBE model applies to the particular cases of the heavy meson-meson and heavy meson-antimeson systems and what predictions are to be expected.
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