Study of $B{\overline{B}}^{*}/D{\overline{D}}^{*}$ bound states in a Bethe-Salpeter approach

In this work the $B{\overline{B}}^{*}/D{\overline{D}}^{*}$ system is studied in the Bethe-Salpeter approach with quasipotential approximation. In our calculation both direct and cross diagrams are included in the one-boson-exchange potential. The numerical results indicate the existence of an isoscalar bound state $D{\overline{D}}^{*}$ with $J^{PC}=1^{++}$, which may be related to the $X(3872)$. In the isovector sector, no bound state is produced from the interactions of $D{\overline{D}}^{*}$ and $B{\overline{B}}^{*}$, which suggests the molecular state explanations for $Z_b(10610)$ and $Z_c(3900)$ are excluded.

Figure 1

The BS equation for $|{Z_{P{\overline{P}}^{*}}^{(T)}}^{+}⟩$. The thick and thin lines are for pseudoscalar and vector mesons, respectively. The red (gray) and blue (black) lines are for charged and neutral mesons, respectively. The black lines are for the diagram after isolating the flavor factors to $I^i$. In the last line the $SU(2)$ symmetry is applied.