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

A bstract The precise measurement of the width difference ∆Γ s among the mass eigenstates of the B s - $$ {\overline{B}}_s $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msub> </mml:math> 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 $$ \overline{\textrm{MS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>MS</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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.