Neutral $B$-meson mixing from three-flavor lattice quantum chromodynamics: Determination of the $SU(3)$-breaking ratio $\xi$

We study $SU(3)$-breaking effects in the neutral $B_{d}-\overline{B}_{d}$ and $B_{s}-\overline{B}_{s}$ systems with unquenched $N_{f}=2+1$ lattice quantum chromodynamics (QCD). We calculate the relevant matrix elements on the MILC collaboration’s gauge configurations with asqtad-improved staggered sea quarks. For the valence light-quarks ($u, d$, and $s$) we use the asqtad action, while for b quarks we use the Fermilab action. We obtain $\xi=f_{B_{s}}\sqrt{B_{B_{s}}}/f_{B_{d}}\sqrt{B_{B_{d}}}=1.268 \pm 0.063$. We also present results for the ratio of bag parameters $B_{B_{s}}/B_{B_{d}}$ and the ratio of Cabibbo-Kobayashi-Maskawa matrix elements $|V_{td}|/|V_{ts}|$. Although we focus on the calculation of $\xi$, the strategy and techniques described here will be employed in future extended studies of the $B$ mixing parameters $\vartriangle M_{d,s}$ and $\vartriangle\Gamma_{d,s}$ in the standard model and beyond