SU(3) Flavor Breaking in Hadronic Matrix Elements for $B-\bar{B}$ Oscillations

We present an analysis, using quenched configurations at $6\mathrm{/}{g}^2=$5.7, 5.85, 6.0, and 6.3 of the matrix element $\MP\equiv\langle \bar P_{hl}|\bar h \gamma_\mu (1-\gamma_5)l \bar h \gamma_\mu(1-\gamma_5)l|P_{hl}\rangle$ for heavy-light pseudoscalar mesons. The results are extrapolated to the physical $B$ meson states, $\text{Bd}$ and $\text{Bs}$. We directly compute the ratio $\text{MS}\mathrm{/}\text{MB}$, and obtain the preliminary result $\text{MS}\mathrm{/}\text{MB}=1.54(13)(32)$. A precise value of this SU(3) breaking ratio is important for determining $V_{td}$ once the mixing parameter $x_s$ for $\text{Bs}-\overline{\text{Bs}}$ is measured experimentally. We also determine values for the corresponding B parameters, $B_{bs}(2 \rm{GeV})=B_{bd}(2 \rm{GeV})=1.02(13)$, which we cannot distinguish in the present analysis.