O ( a $^{2}$Λ2 QCD × $^{(}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$) ∼ 1 . 7%. Adding the contributions from light-quark and gluon discretization errors, heavy-quark discretization errors, and discretization errors in the heavy-light current or four-fermion operator in quadrature, we estimate the error in f$_{B}$$_{s}$/f$_{B}$$_{d}$ to be ∼ 3.2% and the error in ξ to be ∼ 3.7%. tor, this leads to discretization errors in the ratio f$_{B}$$_{s}$/f$_{B}$$_{d}$ of O ( α$_{s}$ × ( am$_{s}$ − am$_{d}$ )) ∼ 1 . 2% plus O ( α 2 $_{s}$a Λ$_{QCD}$ × $^{(}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$) ∼ 0 . 6% plus O ( a $^{2}$Λ2 QCD × $^{(}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$) ∼ 1 . 7%. Although we do not improve the heavy-light four fermion operator used to compute the B -mixing matrix element, the operator does not have any tree-level O ( a ) errors [38]. Thus the leading discretization errors in the ratio ξ from the four-fermion operator are of O ( α$_{s}$ × ( am$_{s}$ − am$_{d}$ )) ∼ 1 . 2% plus O ( α$_{s}$a Λ$_{QCD}$ × $^{(}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$) ∼ 1 . 9% plus O ( a $^{2}$Λ2 QCD × m$_{s}$ − m$_{d}$ ) / Λ$_{QCD}$) ∼ 1 . 7%. We compute the renormalization factors needed to match the lattice axial current and four-fermion operator to the continuum using one-loop lattice perturbation theory. This leaves a residual error due to the omission of higher-order terms. Based on power-counting, we estimate the truncation error in the coefficients to be of O ( α 2 $_{s}$), which is the size of the first neglected term in the series. As we noted earlier in Sec. III C, however, the matching coefficient Z$_{Φ}$ cancels in the ratio of decay constants f$_{B}$$_{s}$/f$_{B}$$_{d}$ ; thus its contribution to the error in f$_{B}$$_{s}$/f$_{B}$$_{d}$ is zero. Although such an exact cancellation does not occur for the ratio of mixing matrix elements ξ , the error in ξ due to the uncertainty in the ratio of matching coefficients Z$_{SP}$/Z$_{V A}$ is suppressed by the SU $^{(3)-breaking factor (}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$. This is because, in the SU (3) limit, the four-fermion operator matrix elements would be equal in the numerator and denominator, so the error in ξ from the renormalization factor uncertainty would be zero. We therefore expect the error in f$_{B}$$_{s}$/f$_{B}$$_{d}$ to be 0% and the error in ξ to be of O ( α 2 s × $^{(}$˜ m$_{s}$ − ˜ m$_{d}$ ) / Λ$_{QCD}$) ∼ 2 . 2%. This error will decrease with the inclusion of data at a finer lattice spacing because the smaller coupling constant will improve the convergence of the series. to am$_{res}$ . These effects, however, are expected to be sub-percent level in the matrix elements [75], and therefore negligible in the SU (3)-breaking ratios. D. Heavy-light current and four-fermion operator renormalization 38