For the HYP-smeared wall-source data, however, we must first extract the bag parameter B$_{B}$$_{q}$ then we obtain the matrix element via The derivation Eq. (72) is more complex than that of either Eq. (63) or Eq. (73) because the box sources in the amplitudes appearing in both the numerator and denominator are not translationally invariant and will create B -meson states carrying a variety of spatial momenta. As explained in greater detail in App. C, these states are degenerate with the lowest energy B -meson state and so cannot be suppressed by simply going to large Euclidean time separations. Instead, Eq. (72) is derived in App. C using the local conservation of heavy- quark number, a property special to the static approximation. Finally, for both the APE and HYP-smeared data, we compute the renormalized matrix element through O ( α$_{s}$ ), using the perturbative matching factors given in Table III. We compute the ratio of mixing matrix elements in the same manner as we compute the ratio of decay constants in the previous subsection. For example, Fig. 3 shows the determination of the numerator M ren B$_{s}$ and the denominator M ren B$_{l}$ on the am$_{l}$ = 0 . 02 ensemble for the APE (left-hand plot) and HYP data sets (right-hand plot). We compute the values of the plateaux using time-slices 8–12 because we do not observe excited-state contamination in this region. Table VI shows the results for the SU (3)-breaking ratios √ M ren B$_{s}$ /M ren B$_{l}$ on the three sea quark ensembles; the corresponding plateau plots are shown in Fig. 4. We extrapolate our results for the ratio of the decay constants and for the ratio of mixing matrix elements to the physical point using a phenomenologically-motivated function based on next-to-leading order partially quenched SU (2) heavy-light meson chiral perturbation theory. In SU (2) χ PT, the pesudoscalar mesons containing strange quarks (i.e. kaons and 23 V. CHIRAL EXTRAPOLATION M ren B$_{q}$ = Z$_{V A}$M lat V V + AA + Z$_{SP}$M lat SS + $_{PP}$, (75) M$_{O}$$_{i}$ = 8 3 B$_{O}$$_{i}$ (Φ ren $_{B$_{q}$}$) $^{2}$. (74) B$_{O}$$_{i}$ = lim t$_{f}$ ≫ t ≫ t$_{0}$ 3 8 L 3 C W $_{O$_{i}$}$( t$_{f}$,t,t$_{0}$ ) C $^{LW}$( t,t$_{f}$ ) C $^{LW}$( t,t$_{0}$ ) , (73)