  In the heavy quark expansion (HQE) of the total decay rates of $B_s$ and
$D_s^+$ mesons non-perturbative matrix elements of four quark operators are
arising as phase space enhanced contributions. We present the first
determination of $m_s$ effects to the dimension six matrix elements of these
four quark operators via a heavy quark effective theory (HQET) sum rule
analysis. In addition we calculate for the first time eye contractions of the
four quark operators as well as matrix elements of penguin operators. For the
perturbative part we solve the 3-loop contribution to the sum rule and we
evaluate condensate contributions. In this study we work in the strict HQET
limit and our results can also be used to estimate the size of the matrix
element of the Darwin operator via equations of motion.
