$X(5568)$ as a $B\bar{K}$ molecule in the Bethe-Salpeter equation approach in the heavy quark limit.

In the heavy quark limit, we study the $X(5568)$ state as a $B\bar{K}$ molecule in the Bethe-Salpeter equation approach. With the kernel containing one-particle-exchange diagrams, we solve the Bethe-Salpeter equation numerically in the covariant instantaneous approximation and find that the $X(5568)$ can exist as a $B\bar{K}$ molecular state with quantum numbers $I (J^P) = 1(0^+)$. In this picture we calculate the strong decay width of $X(5568)\rightarrow B_s^0\pi^+$ and find it to be in the range 19.83 - 22.45 MeV, which is consistent with the experimental data from the D0 Collaboration.