Weak Decays Beyond Leading Logarithms

TL;DR

The paper surveys the status of QCD corrections to weak decays beyond leading logarithms, employing the operator product expansion and renormalization-group methods to construct and evolve a comprehensive set of effective Hamiltonians. It provides a detailed compendium of ΔF=1 and ΔF=2 Hamiltonians, including current-current, QCD penguin, and electroweak penguin operators, with explicit Wilson coefficients, anomalous-dimension matrices, and threshold-matching procedures at NLO. The work analyzes the reduction of renormalization-scale and scheme uncertainties at NLO and demonstrates the resulting impact on CKM parameter extractions and CP-violating observables across K and B meson decays, including rare processes like B → X_s γ and K → π νν̄. It also discusses the special treatment of evanescent operators and the implications for HQET-related calculations, highlighting the importance of precision QCD corrections for testing the Standard Model flavor sector and probing new physics. Overall, the review provides a foundational, technically detailed framework for applying NLO QCD corrections to a broad class of weak decays and their phenomenology.

Abstract

We review the present status of QCD corrections to weak decays beyond the leading logarithmic approximation including particle-antiparticle mixing and rare and CP violating decays. After presenting the basic formalism for these calculations we discuss in detail the effective hamiltonians for all decays for which the next-to-leading corrections are known. Subsequently, we present the phenomenological implications of these calculations. In particular we update the values of various parameters and we incorporate new information on m_t in view of the recent top quark discovery. One of the central issues in our review are the theoretical uncertainties related to renormalization scale ambiguities which are substantially reduced by including next-to-leading order corrections. The impact of this theoretical improvement on the determination of the Cabibbo-Kobayashi-Maskawa matrix is then illustrated in various cases.

Figures (30)

• Figure 1: Unitarity triangle in the complex $(\bar{\varrho},\bar{\eta})$ plane.
• Figure 2: One-loop current-current (a)--(c), penguin (d) and box (e) diagrams in the full theory. For pure QCD corrections as considered in this section and e.g. in \ref{['sec:HeffdF1:66']} the $\gamma$- and $Z$-contributions in diagram (d) and the diagram (e) are absent. Possible left-right or up-down reflected diagrams are not shown.
• Figure 3: One loop current-current (a)--(c) and penguin (d) diagrams contributing to the LO anomalous dimensions and matching conditions in the effective theory. The 4-vertex "$\otimes\,\,\otimes$" denotes the insertion of a 4-fermion operator $Q_i$. For pure QCD corrections as considered in this section and e.g. in \ref{['sec:HeffdF1:66']} the contributions from $\gamma$ in diagrams (d.1) and (d.2) are absent. Again, possible left-right or up-down reflected diagrams are not shown.
• Figure 4: Typical diagrams in the full theory from which the operators (\ref{['eq:Q12']})--(\ref{['eq:Qnnmm']}) originate. The cross in diagram (d) means a mass-insertion. It indicates that magnetic penguins originate from the mass-term on the external line in the usual QCD or QED penguin diagrams.
• Figure 5: Wilson coefficients $y_{7}(m_{\rm c})/\alpha$ and $y_{8}(m_{\rm c})/\alpha$ as functions of $m_{\rm t}$ for $\Lambda_{\overline{\rm MS}}^{(4)}=325\, {\rm MeV}$.