The coefficients in the evolution factor are with and the one- and two-loop anomalous dimensions The one-loop QCD to HQET matching factors for the Δ B = 2 four quark operator are calculated in Refs. [50, 53]: The operators O$_{L}$ and O$_{S}$ mix under RG evolution (see Eq. (32)), such that the evolution factors for the matching coefficients can be written as a 2 × 2 matrix: Note that the matrix element U ( N$_{f}$ ) 12 is zero to all-orders in perturbation theory due to the fact that heavy-quark spin symmetry prohibits mixing from O$_{S}$ into O$_{L}$ . Because ˜ Z$_{2}$ in Eq. (B11) has no tree-level contribution, the two-loop expression for U ( N$_{f}$ ) 11 and one-loop expressions for U ( N$_{f}$ ) 21 and U ( N$_{f}$ ) 22 are sufficient to determine the O ( α$_{s}$ ) matching coefficients. 48 U ( N$_{f}$ ) L =   U ( N$_{f}$ ) 11 0 U ( N$_{f}$ ) 21 U ( N$_{f}$ ) 22   . (B12) Z$_{1}$ ( m$_{b}$, m$_{b}$ ) = 1 − 8 N 2 c + 9 N$_{c}$ − 15 2 N$_{c}$ α$_{s}$ ( m$_{b}$ ) 4 π + O ( α 2 $_{s}$) , (B10) ˜ Z$_{1}$ ( m$_{b}$, m$_{b}$ ) = 1 − c 2 N$_{c}$ 4 π + O ( α 2 $_{s}$) , (B10) Z$_{2}$ ( m$_{b}$, m$_{b}$ ) = − 2( N$_{c}$ + 1) α$_{s}$ ( m$_{b}$ ) 4 π + O ( α 2 $_{s}$) . (B11) γ (1) A = − 16 C$_{F}$ ( 49 96 C$_{A}$ − 5 32 C$_{F}$ − 5 24 T$_{F}$N$_{f}$ − ( 1 4 C$_{A}$ − C$_{F}$ ) π 2 6 ) . (B9) γ (0) A = − 3 C$_{F}$, (B8) β$_{1}$ = 34 3 C 2 A − 4 C$_{F}$T$_{F}$N$_{f}$ − 20 3 C$_{A}$T$_{F}$N$_{f}$, (B7) β$_{0}$ = 11 3 C$_{A}$ − 4 3 T$_{F}$N$_{f}$, (B6) J ( N$_{f}$ ) A = γ (1) A / 2 β$_{0}$ − γ (0) A β$_{1}$/ 2 β 2 $_{0}$, (B5) d ( N$_{f}$ ) A = γ (0) A / 2 β$_{0}$, (B4)