Abstract
The hidden charm $X(3872)$ resonance is usually thought to be a $D^{*0} \bar{D}^0$ meson-antimeson molecule with quantum numbers $J^{PC} = 1^{++}$. If this is the case, there is the possibility that there might be three body bound states with two charmed mesons and a charmed antimeson. Here we argue that the theoretical existence of this type of three body molecules is expected from heavy quark spin symmetry. If applied to the two body sector, this symmetry implies that the interaction of the $D^{*0} \bar{D}^{*0}$ meson-antimeson pair in the $J^{PC} = 2^{++}$ channel is the same as in the $J^{PC} = 1^{++}$ $D^{*0} \bar{D}^0$ case. From this we can infer that the $J^P = 3^{-}$ $D^{*0} D^{*0} \bar{D}^{*0}$ molecule will be able to display the Efimov effect if the scattering length of the $2^{++}$ channel is close enough to the unitary limit. Heavy quark spin symmetry also indicates that the $J^P = 2^{-}$ $D^{*0} D^{*0} \bar{D}^0$ molecule is analogous to the $J^P = 3^{-}$ $D^{*0} D^{*0} \bar{D}^{*0}$ one. That is, it can also have a geometric spectrum. If we consider these triply heavy trimers in the isospin symmetric limit, the Efimov effect disappears and we can in principle predict the fundamental state of the $2^-$ $D^*D^*\bar{D}$ and $3^-$ $D^* D^* \bar{D}^*$ systems. The same applies to the $B^*B^* \bar{B}^*$ system: if the $Z_b(10650)$ is an isovector $B^* \bar{B}^*$ molecule then the $0^-$ isodoublet and the $1^-$, $2^-$ isoquartet $B^* B^* \bar{B}^*$ trimers might bind, but do not display Efimov physics. Finally from heavy flavour symmetry it can be argued that scattering in the $B D$ two-body system might be resonant. This would in turn imply the possibility of Efimov physics in the $B B D$ three body system.
Highlights
The three boson system in the unitary limit, i.e., when the two-body scattering length goes to infinity, shows a geometric spectrum in which the ratio of the energy of the nth and (n þ 1)th excited trimer is En=Enþ1 ≃ 521
From heavy quark spin symmetry (HQSS) we expect the spectrum of hadron systems containing a mixture of heavy (Q 1⁄4 c, b) and light (q 1⁄4 u, d, s) degrees of freedom to be independent of the spin of the heavy quarks [35,36,37,38,39]
We find that aðpÞ follows the integral equation
Summary
The three boson system in the unitary limit, i.e., when the two-body scattering length goes to infinity, shows a geometric spectrum in which the ratio of the energy of the nth and (n þ 1)th excited trimer is En=Enþ1 ≃ 521. The Efimov effect extends to other three-body systems in the unitary limit: relevant to the present investigation is the case of two identical noninteracting boson of species A and a third particle of species B that interacts resonantly with the other two, i.e., the AB scattering length diverges. This system displays a geometric spectrum too [1]. In the isospin symmetric limit the Xð3872Þ can be considered to be an isoscalar 1þþ DÃDmolecule with a binding energy of about 4 MeV In this limit there is no Efimov effect, independently of the location of the bound state. We will present the conclusions of this work at the end
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