with the constant where the mean-field link u$_{0}$ = P 1 / 4 is obtained from the expectation value of the plaquette P . In Eqs. (39)-(41) we match A HQET and O HQET L at one-loop, but match O HQET S at tree-level. This is sufficient because the leading-order mixing between the continuum QCD operator O QCD L and the continuum HQET operator O HQET S is already of O ( α$_{s}$ ), and has no tree-level component. The lattice operators have the same form as in the continuum static effective theory The O ( pa ) derivative operator in the equation for the axial current is given by where we have simplified the expression using the equations-of-motion. The domain-wall specific renormalization factor Z MF w that enters the above equations was calculated pertur- batively in Ref. [58]. The superscript “MF” denotes mean-field improvement [59], in which we modify the bare lattice coupling using the value of the mean-field link. Use of this “boosted” coupling as the new expansion parameter improves the convergence of lattice perturbation theory. The mean-field improved coupling α MF has several definitions which differ only at higher-order in perturbation theory than we consider. These differences enter our estimate of the systematic error in Section VI. Our choice for obtaining the mean-field improved coupling from the bare lattice coupling g 2 0 is where N$_{f}$ = 3 is the number of dynamical flavors. We use the plaquette value averaged over different light sea quark mass, P = 0 . 5881, because the difference in the value of P for different light sea quark masses is less than 0 . 05% for the ensembles used in this work. The constants d$_{g}$ = 0 . 1053 and c$_{p}$ = 0 . 1401 were calculated for the Iwasaki gauge action in 1 ( g $^{MF}$)2 = P g 2 0 + d$_{g}$ + c$_{p}$ + N$_{f}$d$_{f}$, (47) A lat ∂, 0 = ∂$_{0}$ ( hγ$_{5}$q ) , (46) O lat S = [ h (1 − γ$_{5}$ ) q ][ h (1 − γ$_{5}$ ) q ] . (45) O lat L = [ hγ$_{µ}$ (1 − γ$_{5}$ ) q ][ hγ$_{µ}$ (1 − γ$_{5}$ ) q ] , (44) A lat 0 = hγ$_{0}$γ$_{5}$q, (43) w MF 0 = 1 − M$_{5}$ + 4(1 − u$_{0}$ ) , (42) 14