The Spin-orbit force, recoil corrections and possible $B \bar{B}^{*}$ and $D \bar{D}^{*}$ molecular states

In the framework of the one boson exchange model, we have calculated the effective potentials between two heavy mesons $B \bar{B}^{*}$ and $D \bar{D}^{*}$ from the t- and u-channel $π$, $η$, $ρ$, $ω$ and $σ$ meson exchange with four kinds of quantum number: $I=0$, $J^{PC}=1^{++}$; $I=0$, $J^{PC}=1^{+-}$; $I=1$, $J^{PC}=1^{++}$; $I=1$, $J^{PC}=1^{+-}$. We keep the recoil corrections to the $B \bar{B}^{*}$ and $D \bar{D}^{*}$ system up to $O(\frac{1}{M^2})$. The spin orbit force appears at $O(\frac{1}{M})$, which turns out to be important for the very loosely bound molecular states. Our numerical results show that the momentum-related corrections are unfavorable to the formation of the molecular states in the $I=0$, $J^{PC}=1^{++}$ and $I=1$, $J^{PC}=1^{+-}$ channels in the $D \bar{D}^{*}$ systems.

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