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

We compute perturbative matching coefficients to the Heavy Quark Effective Theory representation for the QCD effective local $\Delta B=2$ Hamiltonian that determines the mass difference in $B^0-\bar{B}^0$ system of states. We report on the results at NNLO in the strong coupling constant for matching coefficients of two physical operators in HQET. Our results provide the firm confirmation 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 a by-product of our calculation we give a fully analytical solution for the one loop QCD-to-HQET matching problem: we present the explicit formulas for the renormalization of four quark operators of the full bases in both QCD and HQET and the expressions for matching coefficients in a closed form.