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We compute perturbative matching coefficients to the Heavy Quark EffectiveTheory representation for the QCD effective local $\Delta B=2$ Hamiltonian thatdetermines the mass difference in $B^0-\bar{B}^0$ system of states. We reporton the results at NNLO in the strong coupling constant for matchingcoefficients of two physical operators in HQET. Our results provide the firmconfirmation that the recent NLO sum rules analysis of the bag parameter $B_q$is stable with regard of inclusion of higher order radiative corrections. As aby-product of our calculation we give a fully analytical solution for the oneloop QCD-to-HQET matching problem: we present the explicit formulas for therenormalization of four quark operators of the full bases in both QCD and HQETand the expressions for matching coefficients in a closed form.

$B^0$-$\bar{B}^0$ mixing: matching to HQET at NNLO