Decay matrix of B-B̄ mixing: mixing of dimension-seven operators into dimension-six operators under renormalization

The precise measurement of the width difference ΔΓ_s among the mass eigenstates of the B_s-B̄_s system requires the calculation of the corresponding decay matrix to order α_s/m_b. 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 MS̄ 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.