is as follows: We use the HYP smearing parameters ( α$_{1}$, α$_{2}$, α$_{3}$ ) = (1 . 0 , 1 . 0 , 0 . 5), sometimes referred to as HYP2. These were shown to approximately minimize the noise-to-signal ratio in Ref. [25]. In Eqs. (19)–(23), Proj$_{SU(3)}$ ( V ) indicates the projection of the link V onto an SU (3) matrix. This projection reduces the statistical noise and thus enhances the smearing effect without increasing the level of smearing. In some cases the SU (3) projection also suppresses quantum corrections to lattice operators in perturbation theory [45]. The projection of the smeared link onto SU (3) is not unique, and we use two different schemes: for the case of APE smearing, Eq. (19), we project by the unit circle method based on polar decomposition [46], while for the case of HYP smearing, Eqs. (21)–(23), we obtain the projected matrix by an iterative procedure seeking the SU (3) matrix U$_{max}$ that maximizes Re Tr ( U$_{max}$V $^{†}$), where V is the HYP smeared link matrix [47]. We describe the details of the two schemes and show their equivalence in the weak coupling limit in Appendix A. In order to renormalize the heavy-light axial current and Δ B = 2 four-fermion operator, we adopt a two-step matching procedure. In the first step, we match the QCD operators renormalized in the MS scheme using naive dimensional regularization (NDR) at a scale µ$_{b}$ to continuum static effective theory operators renormalized at a scale µ . This step is described in Sec. III A. In this paper, we choose µ$_{b}$ to be the b -quark mass m$_{b}$ and µ to be the inverse lattice spacing a − $^{1}$. In the second step, we match the continuum static effective theory operators to the lattice ones. This step is described in Sec. III B. We combine the 10 III. PERTURBATIVE MATCHING OF HEAVY-LIGHT CURRENT AND FOUR- FERMION OPERATORS V$_{0}$ ( x ) = Proj$_{SU(3)}$ [ (1 − α$_{1}$ ) U$_{0}$ ( x ) + α$_{1}$ 6 ± 3 ∑ ν = ± 1 ˜ V$_{ν}$$_{;0}$ ( x ) ˜ V$_{0;}$$_{ν}$ ( x + ̂ ν ) ˜ V † ν $_{;0}$( x + ̂ 0) ] , (21) ˜ V$_{µ}$$_{;}$$_{ν}$ ( x ) = Proj$_{SU(3)}$   (1 − α$_{2}$ ) U$_{µ}$ ( x ) + α$_{2}$ 4 ± 3 ∑ ρ = ± 0 ρ = µ,ν V$_{ρ}$$_{;}$$_{νµ}$ ( x ) V$_{µ}$$_{;}$$_{ρν}$ ( x + ̂ ρ ) V † ρ ; $_{νµ}$( x + ̂ µ )   , (22) V$_{µ}$$_{;}$$_{νρ}$ ( x ) = Proj$_{SU(3)}$    (1 − α$_{3}$ ) U$_{µ}$ ( x ) + α$_{3}$ 2 ± 3 ∑ η = ± 0 η = µ,ν,ρ U$_{η}$ ( x ) U$_{µ}$ ( x + ̂ η ) U † $_{η}$( x + ̂ µ )    . (23)