we run the coefficients at two-loops in three-flavor continuum HQET from m$_{c}$ to the scale µ = 1 /a , where the matching to three-flavor lattice HQET is done. The matching factors and anomalous dimensions needed to compute these coefficients are given in Refs. [30, 48, 49] for the heavy-light current and in Refs. [50–53] for the four-quark operator. For completeness, we present them in Appendix B. Here we simply quote the results for the matching coefficients, which already contain some terms of O ( α 2 $_{s}$): C$_{A}$ ( µ ) = [ α$_{s}$ ( m$_{b}$ ) α$_{s}$ ( m$_{c}$ ) ] − 6 25 · [ α$_{s}$ ( m$_{c}$ ) α (3) s ( µ ) ] − 2 9 · ( 1 − 8 3 α$_{s}$ ( m$_{b}$ ) 4 π ) · ( 1 + J (4) A α$_{s}$ ( m$_{b}$ ) − α$_{s}$ ( m$_{c}$ ) 4 π ) · ( 1 + J (3) A α$_{s}$ ( m$_{c}$ ) − α (3) s ( µ ) 4 π ) + O ( α 2 $_{s}$) , (33) Z$_{2}$ ( m$_{b}$, µ ) = − 8 α$_{s}$ ( m$_{b}$ ) 4 π [ α$_{s}$ ( m$_{b}$ ) α$_{s}$ ( m$_{c}$ ) ] − 4 25 [ α$_{s}$ ( m$_{c}$ ) α (3) s ( µ ) ] − 4 27 + O ( α 2 $_{s}$) , (35) Z$_{1}$ ( m$_{b}$, µ ) = [ α$_{s}$ ( m$_{b}$ ) α$_{s}$ ( m$_{c}$ ) ] − 12 25 · [ α$_{s}$ ( m$_{c}$ ) α (3) s ( µ ) ] − 4 9 · {( 1 − 14 α$_{s}$ ( m$_{b}$ ) 4 π ) · ( 1 + J (4) 11 α$_{s}$ ( m$_{b}$ ) − α$_{s}$ ( m$_{c}$ ) 4 π ) · ( 1 + J (3) 11 α$_{s}$ ( m$_{c}$ ) − α (3) s ( µ ) 4 π ) + 2 α$_{s}$ ( m$_{b}$ ) 4 π {( 1 − [ α$_{s}$ ( m$_{b}$ ) α$_{s}$ ( m$_{c}$ ) ] 8 25 )( 1 + J (3) 11 α$_{s}$ ( m$_{c}$ ) − α (3) s ( µ ) 4 π ) + [ α$_{s}$ ( m$_{b}$ ) α$_{s}$ ( m$_{c}$ ) ] 8 25 $_{·}$  1 − [ α$_{s}$ ( m$_{c}$ ) α (3) s ( µ ) ] 8 27   }} + O ( α 2 $_{s}$) , (34) − 4 − 4 with the parameters J$_{A}$ and J$_{11}$ given by J (3) A = − 0 . 7545 , J (4) A = − 0 . 9098 , (36) J (3) 11 = − 1 . 6980 , J (4) 11 = − 1 . 8637 . (37) Note that the leading-order mixing between O QCD L and O HQET S is of O ( α$_{s}$ ). To determine the coupling constant α$_{s}$ at different scales, which is required to obtain these results, we fix the value of α$_{s}$ at the Z -boson mass to the PDG value α$_{s}$ ( m$_{Z}$ = 91 . 1876 GeV) = 0 . 1176 [6]. Using four-loop running [54, 55], we obtain α$_{s}$ ( m$_{b}$ = 4 . 20 GeV [6]) = 0 . 2228, α$_{s}$ ( m$_{c}$ = 1 . 27 GeV [6]) = 0 . 3819, and α (3) s ( a − 1 = 1 . 729 GeV [34]) = 0 . 3141. Hence we find C$_{A}$ ( a − $^{1}$) = 1 . 0459 , Z$_{1}$ ( m$_{b}$, a − $^{1}$) = 0 . 9100 , Z$_{2}$ ( m$_{b}$, a − $^{1}$) = − 0 . 1502 (38) 12