TABLE III: Perturbative matching factors for the decay constants and mixing matrix elements evaluated for APE smeared and HYP smeared static-quark gauge links for the choices of the strong coupling constant α MF s . smearing Z$_{Φ}$ Z$_{V A}$ Z$_{SP}$ APE 0.9507 0.7485 -0.1448 HYP 0.9813 0.8108 -0.1448 as and present their values for our choice of simulation parameters in Table III. These will be used in the following section to extract the physical decay constants and Δ B = 2 four- fermion matrix elements via the relations In practice, the renormalization factor Z$_{Φ}$ cancels in the ratio f$_{B}$$_{s}$/f$_{B}$$_{d}$ , and only the quantity Z$_{SP}$/Z$_{V A}$ enters the ratio ξ . Therefore we do not need Z$_{Φ}$ (or Z$_{V A}$ and Z$_{SP}$ by themselves) for our current analysis of the SU (3)-breaking ratios. We present all three matching coefficients for completeness, however, because they will be necessary for calcu- lating the individual decay constants and four-fermion operator-mixing matrix elements in future work. In this section we calculate the ratios of the B -meson decay constants and mixing matrix elements at unphysical values of the light and strange quark masses. On each sea quark ensemble, we compute the necessary 2-point and 3-point correlation functions at two values of the valence quark mass: the unitary point m$_{x}$ = m$_{l}$ and a point tuned to the physical 16 IV. LATTICE CALCULATION OF SU (3) BREAKING RATIOS Z$_{Φ}$ ( a − $^{1}$) = C$_{A}$ ( a − $^{1}$) · Z MF A ( a − $^{1}$, a − $^{1}$) , (52) Z$_{V A}$ ( µ$_{b}$, a − $^{1}$) = Z$_{1}$ ( µ$_{b}$, a − $^{1}$) · Z MF L ( a − $^{1}$, a − $^{1}$) , (53) Z$_{SP}$ ( µ$_{b}$, a − $^{1}$) = Z$_{2}$ ( µ$_{b}$, a − $^{1}$) · Z MF S ( a − $^{1}$, a − $^{1}$) , (54) A QCD 0 = Z$_{Φ}$ ( a − $^{1}$) ( A lat 0 ( a − $^{1}$) + c MF A aA lat ∂, $_{0}$( a − $^{1}$) ) , (55) − lat − − lat − O QCD L ( µ$_{b}$ ) = Z$_{V A}$ ( µ$_{b}$, a − $^{1}$) O lat L ( a − $^{1}$) + Z$_{SP}$ ( µ$_{b}$, a − $^{1}$) O lat S ( a − $^{1}$) . (56)