ing behavior at inverse lattice spacings below a − 1 ≈ 2 . 5 GeV, the difference between the predicted value for f$_{B}$$_{s}$ at a ≈ 0 . 11 fm given O ( a $^{2}$) scaling and the value of f$_{B}$$_{s}$ that they obtain in the continuum limit using data within the scaling region is only about 25%, which is again close to our power-counting estimate. Thus we expect that naive power-counting should lead to a reasonable estimate of the discretization error in the SU (3)-breaking ratios. For the error estimates in this subsection, we evaluate the strong coupling constant at the lattice scale, α MS s (1 /a ) ∼ 1 / 3. We choose Λ$_{QCD}$ = 500 MeV because the typical QCD scale that enters heavy-light quantities tends to be larger than for light-light quantities, as indicated by fits to moments of inclusive B -decays using the heavy-quark expansion [74]. Fortunately, some of the finite lattice-spacing effects cancel in the ratios f$_{B}$$_{s}$/f$_{B}$$_{d}$ and ξ . This can be seen by the fact that, although the APE and HYP data differ by about 15–20% for the individual decay constants and matrix elements, they agree within statistical errors for the ratios. In SU (3)-breaking quantities, errors must be proportional to the difference in quark masses ( m$_{s}$ − m$_{d}$ ). Dimensional analysis therefore suggests that contributions to the $^{total discretization error are suppressed by the factor (}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$ ∼ 1 / 5, where we use ˜ m$_{s}$ and ˜ m$_{d}$ to denote the renormalized quark masses in the MS scheme [6] (as opposed to the bare lattice quark masses) in this subsection and the next. The observed size of SU (3)-breaking effects in the B -meson decay constants ( f$_{B}$$_{s}$/f$_{B}$$_{d}$ − 1) and in the B − mixing matrix elements ( ξ − 1) are consistent with this expectation. Discretization errors in f$_{B}$$_{s}$/f$_{B}$$_{d}$ and ξ can arise from both the actions and the operators. We estimate each source of error separately, and add them in quadrature to obtain the total discretization error. None of the actions that we are using are O ( a $^{2}$)-improved. Therefore the leading dis- cretization errors from the domain-wall fermion action and Iwasaki gauge action are of O ( a $^{2}$Λ2$_{QCD}$). When combined with the SU (3)-breaking suppression factor, this leads to dis- cretization errors in the ratios f$_{B}$$_{s}$/f$_{B}$$_{d}$ and ξ of O ( a $^{2}$Λ2 QCD × $^{(}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$) ∼ 1 . 7%. The leading heavy-quark discretization errors from the static action are also of O ( a $^{2}$Λ2$_{QCD}$). Hence heavy-quark discretization errors also contribute ∼ 1 . 7% to the total error in the ratios. Be- cause we improve the heavy-light axial current used to compute the decay constant through O ( α$_{s}$ap ), the leading discretization errors from the heavy-light current are of O ( α$_{s}$am ), O ( α 2 $_{s}$a Λ$_{QCD}$), and O ( a $^{2}$Λ2$_{QCD}$).4 When combined with the SU (3)-breaking suppression fac- 4 There are also discretization errors from mixing with operators of other chiralities that are proportional 37