$B^0\text{−}{\overline{B}}^0$ mixing: Matching to HQET at NNLO

We compute perturbative matching coefficients to the heavy quark effective theory (HQET) representation for the QCD effective local $\mathrm{\Delta }B=2$ Hamiltonian that determines the mass difference in the $B^0\text{−}{\overline{B}}^0$ system of states. We report on the results at next-to-next-to-leading order in the strong coupling constant for matching coefficients of two physical operators in HQET. Our results provide firm confirmation that the recent next-to-leading order sum rules analysis of the bag parameter $B_q$ is stable with regard of inclusion of higher order radiative corrections. As a byproduct 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.