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

TL;DR

This paper calculates the mixing of dimension-seven operators into dimension-six operators in B_s-ar{B}_s mixing, addressing renormalization issues and ensuring consistency with power counting and large N_c limits.

Contribution

It develops a methodology for renormalizing dimension-7 operators in B-meson mixing calculations, including counterterm calculations and consistency checks.

Findings

Infrared-finite terms can be absorbed into finite counterterms.

Calculated all counterterms for dimension-7 operator mixing.

Verified results with large N_c limit hadronic matrix elements.

Abstract

The precise measurement of the width difference \Delta\Gamma_s among the mass eigenstates of the B_s-\bar{B}_s system requires the calculation of the corresponding decay matrix to order \alpha_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{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.