The Anatomy of ${\epsilon }^{\prime }/\epsilon$ Beyond Leading Logarithms with Improved Hadronic Matrix Elements

Abstract

We use the recently calculated two--loop anomalous dimensions of current-current operators, QCD and electroweak penguin operators to construct the effective Hamiltonian for $\Delta S=1$ transitions beyond the leading logarithmic approximation. We solve the renormalization group equations and give the numerical values of Wilson coeff. functions. We propose a new semi-phenomenological approach to hadronic matrix elements which incorporates the data for $CP$-conserving $K\to \pi \pi$ amplitudes and allows to determine the matrix elements of all $(V-A)\otimes (V-A)$ operators in any renormalization scheme and do a renormalization group analysis of all hadronic matrix elements $⟨Q_i(\mu )⟩$. We compare critically our treatment of these matrix elements with those given in the literature. We find in the NDR scheme $\epe = (6.7 \pm 2.6)\times 10^{-4}$ in agreement with the experimental findings of E731. We point out however that the increase of $⟨Q_6⟩$ by only a factor of two gives $\epe = (20.0 \pm 6.5)\times 10^{-4}$ in agreement with the result of NA31. The dependence of $\epe$ on ${\Lambda }_{\overline{MS}}$, $m_t$ and $⟨Q_{6,8}⟩$ is presented.

91 pages (A4) with 20 PostScript figures (distributed separately); TUM-T31-35/93, MPI-Ph/93-11 and CERN-TH 6821/93

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• Two--Loop Anomalous Dimension Matrix for $\Delta S=1$ Weak Non-Leptonic Decays II: $\mathcal{O}({\alpha }_e{\alpha }_s)$
• ${\epsilon }^{\prime }/\epsilon$ at the next-to-leading order in QCD and QED

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