$B_s\pi$--$B\bar{K}$ interactions in finite volume and the $X(5568)$

The recent observation of $X(5568)$ by the D0 Collaboration has aroused a lot of interest both theoretically and experimentally. In the present work, we first point out that $X(5568)$ and $D_{s0}^*(2317)$ cannot simultaneously be of molecular nature, from the perspective of heavy-quark symmetry and chiral symmetry, based on a previous study of the lattice QCD scattering lengths of $DK$ and its coupled channels. Then we compute the discrete energy levels of the $B_s\pi$ and $B\bar{K}$ system in finite volume using unitary chiral perturbation theory. The comparison with the latest lattice QCD simulation, which disfavors the existence of $X(5568)$, supports our picture where the $B_s\pi$ and $B\bar{K}$ interactions are weak and $X(5568)$ cannot be a $B_s\pi$ and $B\bar{K}$ molecular state. In addition, we show that the extended Weinberg compositeness condition also indicates that $X(5568)$ cannot be a molecular state made from $B_s\pi$ and $B\bar{K}$ interactions.