B_s - bar B_s Mixing and Its Implication for b to s Transitions in Supersymmetry

We investigate the effect of the current measurement of the neutral $B_s$ meson mass difference, $\Delta M_{B_s}$, on SUGRA models which have non-zero values of the soft breaking terms $(m^2_{LL,RR})_{23}$ and $A^{u,d}_{23}$ at the GUT scale. We use non-zero values of these parameter to explain the $B\to K\pi$ puzzle and find that even after satisfying the experimental result on $\Delta M_{B_s}$ and the branching ratio (BR) of $b\to s\gamma$ we still can explain the puzzle. Further we show that in this parameter space it is possible to accommodate the large BR of $B\to \eta' K$ and the current experimental data for CP asymmetries of $B\to \eta' K^0$ and $B\to \phi K^0$. The predicted value of $\sin (2\beta^{\rm eff})_{\eta' K^0}$ is about $0.52-0.67$.