The Anatomy of eps'/eps Beyond Leading Logarithms with Improved Hadronic Matrix Elements

Results of a recent calculation of the effective $\dS$ Hamiltonian at the next-to-leading order will be presented\footnote{Work done in collaboration with A. J. Buras and M. E. Lautenbacher.}. These, together with an improved treatment of hadronic matrix elements, are used to evaluate the measure for direct CP-violation, $\epe$, at the next-to-leading order. Taking $\mt=130\,GeV$, $\Lms=300\,MeV$ and calculating $\langle Q_6 \rangle$ and $\langle Q_8 \rangle$ in the $1/N$ approach, 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 $\langle Q_6 \rangle$ by only a factor of two gives $\epe = (20.0 \pm 6.5)\times 10^{-4}$ in agreement with the result of NA31. The dependencies of $\epe$ on $\Lms$, $\mt$, and some $B$-parameters, parameterizing hadronic matrix elements, are briefly discussed.