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

TL;DR

This paper addresses the challenge of predicting the width difference $\Delta\Gamma_s$ in the $B_s$ system by computing the decay matrix at order $\alpha_s/m_b$ within the Heavy Quark Expansion. The authors show that IR-finite, power-counting-violating terms arising from one-loop corrections to dimension-7 four-quark operators can be absorbed into finite counterterms proportional to dimension-6 operators, and they derive the renormalization constants and counterterms in the $\overline{\rm MS}$ scheme. They define evanescent operators and demonstrate that their finite pieces must be chosen to respect Fierz symmetry, particularly for the color-flipped operators, to maintain correct power counting. A large-$N_c$ analysis provides nontrivial cross-checks, confirming that the renormalized matrix elements scale as $\mathcal{O}(1/m_b)$ at NLO and guiding the definition of the evanescent operators. This work lays essential groundwork for including NLO $\alpha_s$ corrections to the $1/m_b$ terms in $\Gamma_{21}^s$, improving the theoretical control over $\Delta\Gamma_s$ and informing lattice-continuum matching, with further work needed to complete the full $1/m_b$ and NNLO contributions.

Abstract

The precise measurement of the width difference ΔΓ_s among the mass eigenstates of the B_s-\bar{B}_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 \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.

Figures (2)

• Figure 1: Infrared-finite diagrams originating from the couplings of the operators $\widetilde{R}_2$ and $\widetilde{R}_3$ with a gluon field and a derivative acting on a $b$ field. The $b$ line with derivative is marked with a dot.
• Figure 2: Diagrams $E_1$, $E_1'$, $E_2$ and $E_2'$ which contribute to the finite renormalization of operators $R_2$, $R_3$, $\widetilde{R}_2$ and $\widetilde{R}_3$. The derivatives, acting on the $b$ and $s$ fields, are denoted as dots on the corresponding lines.