Decay matrix of B-$\overline{B}$ mixing: mixing of dimension-seven operators into dimension-six operators under renormalization

The precise measurement of the width difference ∆Γ$_s$ among the mass eigenstates of the B$_s$- $\overline{B}$$_s$ system requires the calculation of the corresponding decay matrix to order αs/mb. QCD corrections to power-suppressed terms in the Heavy Quark Expansion involve the renormalization of dimension-7 four-quark operators for which no general methodology is available yet. In the $\overline{MS}$ scheme one-loop corrections to matrix elements of dimension-7 operators violate the power counting, but we find the responsible terms to be infrared-finite and show that they can be absorbed into finite counterterms proportional to dimension-6 operators. We calculate all these counterterms and subsequently verify the consistency of our results with hadronic matrix elements calculated in the limit of a large number N$_c$ of colours. The condition of correct power counting implies constraints on the possible definitions of evanescent operators.